Angel Dishliev Katya Dishlieva Svetoslav Nenov

Transcription

1 SPECIFIC ASYMPTOTIC PROPERTIES OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS. METHODS AND APPLICATIONS Angel Dishliev Katya Dishlieva Svetoslav Nenov PA Academic Publications

2 Specific Asymptotic Properties of the Solutions of Impulsive Differential Equations. Methods and Applications Authors: Angel Dishliev 1, Katya Dishlieva 2, Svetoslav Nenov 3 1,3 Address: Department of Mathematics University of Chemical Technology and Metallurgy 8, Kliment Ohridsky, Sofia, 1756, BULGARIA 1 dishliev@uctm.edu 3 svety@math.uctm.edu 2 Address: Faculty of Applied Mathematics and Informatics Technical University of Sofia 8, Kliment Ohridsky, Sofia, 1756, BULGARIA 2 kgd@tu-sofia.bg Reviewers: Professor J. Henderson, Baylor University Professor Paul Eloe, University of Dayton ISBN: Pages: 309 Typesetting system: L A TEX Figures: PSTricks c Academic Publications, Ltd., 2012 All rights reserved. Thisworkmaynotbetranslatedorcopiedinwholeorin part without the written permission of the publisher Academic Publications, Ltd., except for scientific or scholarly analysis. c Academic Publications, Ltd.,

3 With dedication to our Teacher Professor D. Bainov

4

5 Introduction Contents Chapter 1. Continuous Dependence and Stability of the Solutions of Impulsive Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments 1 1. Continuous Dependence of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments 3 2. Stability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments Application: Continuous Dependence and Stability of the Solutions of Pharmacokinetic Model with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments 23 Chapter 2. Continuous Dependence and Diferentiability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects Continuous Dependence of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects Differentiability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects Application: Continuous Dependence and Differentiability of the Solutions of Logistic Model with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects 49 Chapter 3. Continuous Dependence and Uniform Stability of the Solutions of the Differential Equations v viii

6 with Variable Moments of Impulses on the Impulsive Hypersurfaces and Impulsive Effects Sufficient Conditions for the Absence of the Phenomenon Beating Continuous Dependence of the Solutions of the Differential Equations with Variable Impulsive Moments on the Impulsive Hypersurfaces Uniform Stability of the Solutions of the Differential Equations with Variable Impulsive Moments on the Initial Condition and Impulsive Perturbations 99 Chapter 4. Continuous Dependence of the Solutions of Differential Equations with Variable Moments of Impulses on the Initial Condition and Barrier Curves Continuous Dependence of the Solutions of Differential Equations with Not Fixed Moments of Impulses on the Initial Condition and Barrier Curves Application: Continuous Dependence of the Solutions of the Gompertz Model with Non Fixed Moments of Impulses on the Initial Condition and Barrier Curves 129 Chapter 5. Orbital Hausdorff Continuous Dependence of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects Orbital Hausdorff Continuous Dependence of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects Application: Orbital Hausdorff Continuous Dependence of the Solutions of Lotka-Volterra Model with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects 147 Chapter 6. Orbital Hausdorff Stability of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition Orbital Hausdorff Stability of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition 159

7 2. Orbital Hausdorff Stability of the Solutions of Lotka-Volterra Model without Impulses on the Initial Condition Orbital Hausdorff Stability of the Solutions of Model of Harmonic Oscillator 189 Chapter 7. Optimization Problems in Population Dynamics Minimization of the Time Required for Reproduction of an Isolated Population Application: A Model of Optimal Regime of Outer Effects Impulsive Controllability and Optimization Problems. Lagrange s Method Application: Impulsive Controllability and Optimization Problems in Population Dynamics 219 Chapter 8. Continuous Dependence of the Solutions of Differential Equations with Variable Structure and Non Fixed Moments of Impulses with Respect to the Switching Functions Continuous Dependence of the Solutions of the Differential Equations with Variable Structure and Non Fixed Moments of Impulses with Respect to the Switching Functions Modelling by the Differential Equations with Variable Structure and Non Fixed Moments of Impulses 247 Bibliography 261

8 viii Introduction The questions under consideration in this monograph refer to the qualitative theory of certain types impulsive differential equations and their applications in different mathematical models. This class of equations is a suitable mathematical apparatus for modeling the processes and phenomena, which are subjected to the short term external effects during their development. The duration of these effects is negligible, compared with the total duration of the processes or phenomena, which the impulsive equations describe. Therefore, it can be assumed that the effects are instantaneous and they take form of impulses. The study of intermittently developing dynamical states is also subject to other sciences, such as mechanics, pharmacokinetics, population dynamics, economics, control theory, etc. Mathematical apparatus in the form of modeling impulsive differential equations in these cases as a rule is required. Impulsive phenomena in the evolution processes may be found in the following cases: Operation of a damper, subjected to the percussive effects, see Chernousko et al. [94], Simeonov [303]; Change of the valve shutter speed in its transition from open to closed state, see Bainov et al. [61], Matov [240]; Fluctuations of pendulum systems in the case of external impulsive effects, see Kalitin [194, 195, 196]; Percussive model of a clock mechanism, see Bautin [75], Krilov and Bogolyubov [205]; Percussive systems with vibrations, see Babitskii and Krupenin [29], Kobrinskii and Kobrinskii [202]; Relaxational oscillations of the electromechanical systems, see Andronov et al. [23]; Electronic schemes, see Popov [283], Zavalishchin and Sesekin [353]; Remittent oscillator subjected to the impulsive effects, see Simeonov [303]; Dynamic of system with automatic regulation, see Popov [283], Zipkin [376]; The passage of the solid body from a given fluid density to another fluid density, see Simeonov [303]; Control of the satellite orbit, using the radial acceleration, see Zavalishchin and Sesekin [353], Zavalishchin et al. [355]; Change of the speed of a chemical reaction in the addition or removal of a catalyst, see Simeonov [303];

9 INTRODUCTION ix Disturbances in cellular neural networks, see Ahmad et al. [7], Ahmad and Stamova [13], Chen and Shen [91], Chua and Yang [95, 96], Slavova [307], Stamov and Stamova [311], Yang and Xu [346]; Impulsive perturbations in the speed of development of globular bacteria, obeying the Schmalhausen s law, see Dishliev and Dishlieva [120]; Impulsive external intervention and optimization problems in the dynamics of isolated populations, see Angelova and Dishliev [24], Bainov and Dishliev [41, 42], Ballinger and Liu [74], Dong et al. [131], Gopalsamy and Zhang [164], Liu [226], Stamova and Stamov [318], Zhang et al. [367]; Death in the populations as a result of impulsive effects, see Meng et al. [247]; Impulsive external interference and the optimization problems in population dynamics of the predator-prey types, see Angelova et al. [26, 27], Baek [30], Jiao et al. [190], Juang and Lu [192, 193], Nie et al. [263, 264], Yan et al. [344], Zeng et al. [358]; Ecology, see Xiao et al. [342]; Pharmacokinetics, see D onofrio [132], Gao et al. [154, 155], Saker and Alzabut [290]; Epidemiology, see Fuhrman et al. [153], Meng et al. [244], Stone et al. [321], Zhang et al. [363]; Shock changes oftheprices intheclosedmarkets, see Stamova and Emmenegger [317]; etc. Mathematical theory of the systems of differential equations which change at a leap develops in two main directions. One of these directions is introduced in the papers of Zavalishchin et al. [355], Halanay and Veksler [168], Pandit and Deo [274], etc. The basic mathematical tools for describing the impulsive effects are generalize functions type Dirac delta function). In the problems, studied in this direction of the impulsive differential equations theory, the moments of impulsive effect are fixed in advance. Note that, this approach, used for description the leaps processes, is not suitable for the modeling systems, where the impulsive moments are determined dynamically, depending on the values of the solution at every moment of its domain. The start of the second direction in the research of leaps processes was set by V. Milman and A. Myshkis in the sixties of the 20-th century, see Milman and Myshkis [252] and Milman and Myshkis [253]. Some

10 x general considerations about the necessity of studying the systems with impulsive effect were given in their papers. The first results for the solutions stability were obtained. Immediately after the initial studies of V. Milman and A. Myshkis, a number of Ukrainian mathematicians continue the development of this mathematical theory. We will point: Á. Samoilenko, N. Perestyuk, V. Plotnikov, S. Borysenko, A. Ignatiev, N. Skripnik, Y. Rogovchenko, M. Akhmetov, O. Chernikova and P. Zabrejko see respectively: Akhmetov et al. [20], Bainov et al. [56], Borysenko [84, 85], Perestyuk and Chernikova [277], Plotnikov et al. [279], Plotnikov and Kitanov [280, 281], Plotnikov and Plotnikova [282], Rogovchenko [287], Samoilenko [291], Samoilenko and Perestyuk [292, 293], Zabrejko et al. [351]). In the early eighties and then, several mathematical groups were created, continuing the scientific research on the fundamental and qualitative theory of impulsive differential equations and their applications. We will indicate the groups of M. Pinto see Fenner and Pinto [145], Gonzalez and Pinto [162], Naulin and Pinto [260]), J. Nieto see Franco and Nieto [147], Li and Nieto [214], Nieto [265], Nieto and Rodríguez-López [268]), V. Lakshmikantham see Lakshmikantham and Liu [212], Lakshmikantham and Vasundhara [213], Zhao and Lakshmikantham [369]), P. Eloe see Benchohra and Eloe [77], Doddaballapur et al. [129], Eloe and Henderson [137], Eloe et al. [138], Eloe and Sokol [140], Eloe and Usman [141]), J. Henderson see Benchohra et al. [78, 79, 80, 81], Eloe and Henderson [136], Eloe et al. [139], Henderson and Thompson [170]) and D. Bainov see Bainov et al. [36, 37, 38, 39, 40, 44, 46, 47, 48, 50, 51, 53, 55, 57, 58, 59, 60, 62], Bainov and Milusheva [63, 64], Bainov and Minchev [65], Bainov and Nenov [66], Bainov et al. [67], Hristova and Bainov [175], Kulev and Bainov [206, 207, 208], Milev and Bainov [249, 250], Milev et al. [251], Nenov and Bainov [262], Simeonov and Bainov [305, 306]). Many monographs are devoted to the impulsive differential equations. We will mention: V. Lakshmikantham, D. Bainov and P. Simeonov [211]; D. Bainov and V. Covachev [33]; D. Bainov, S. Kostadinov and N. Minh [54]; D. Bainov andp. Simeonov [68 72]; I. Stamova [316]; Á. Samoilenko and N. Perestyuk [295, 297]; N. Perestyuk, V. Plotnikov, Á. Samoilenko andn.scripnik[278]; S.Borysenko, V.KosolapovandÁ.Obolenskii [86]; Á. Halanay and D. Veksler [168]; S. Zavalishchin, A. Sesekin [354]; S. Pandit and S. Deo [274] and M. Benchohra, J. Henderson and S. Ntouyas [79].

11 INTRODUCTION xi In general, the impulsive equations consist of three partssee Anokhin et al. [28], Nieto and O Regan [267], Nieto and Rodríguez-López [270], Perestyuk and Chernikova [276], Xian et al. [341], etc.): Differential equation which describes the differentiable part of the solution and usually has the form dx = f t,x), 0.1) dt where function f is continuous in the domain of the variables t and x; Condition for consistently determination the moments of impulsive effects on the solution. In the simplest case, the impulsive moments are fixed in advance. In more complicated and more general cases, the impulsive moments are consecutive solutions of equationor of equations) of the form gt,xt)) = 0, 0.2) where the function g is defined and continuous in the extended phase or only in the phase) space of the differential equations, and x = xt) is a solution of the same; Impulsive function, which defines the magnitude and direction of the impulsive effect. This function usually has the form: xt imp +0) = Ixt imp )), 0.3) where the function I is defined and continuous in the phase space of the considered impulsive system and t imp is a solution of the previous equation 0.2). The system 0.1), 0.2), 0.3) is called impulsive system. There are different types of impulsive differential equations depending on the way of determining the moments of impulsive effects. We will cite: Fixed moments of impulsive effects see Agarwal et al. [1], Agarwal and O Regan [4, 5], Benchohra et al. [78], Fu and Li [151], etc.); Impulsive moments, which coincide with the moments, at which the integral curve meets the predefined sets, located in the extended phase space. The most frequently, these sets are not intersecting hypersurfaces see Bainov and Dishliev [43], Bainov et al. [45], Benchohra et al. [79, 80], Benchohra and Ouahab [82], Frigon and O Regan [149, 150], etc.); The impulsive moments, which coincide with the moments, at which the trajectory meets the predefined sets, located in the phase space, see Angelova et al. [27], Bajo [73], etc.;

12 xii The impulsive moments, which coincide with the moments, at which the solution minimizes a given functional, see Bainov and Dishliev [41]; The impulsive moments, which are occasional in their nature and they satisfy a certain law of distribution, etc. The solutions of the corresponding initial problem for impulsive differential equations are piecewise continuous functions with first type points of discontinuity, at which the solutions are continuous on the left. The specificities, related to the study of impulsive differential equations and difficulties, arising in their investigation are: Discontinuity of the solution: There are first-type points of discontinuity, i.e., the leaps are limited. Usually, it is assumed that the solution is continuous on left at the points of impulsive effect see the monographs of Lakshmikantham et al. [211], Samoilenko and Perestyuk [295]); The effect of beating existencesee Dishliev and Bainov[110, 111], Samoilenko et al. [296]): In this case, the integral curve or trajectory of the equation meets repeatedlyeven infinitely many) impulsive set. Then it is possible a specific situation, in which the impulsive moments have a compression point and therefore, the solution is not continuable to the right from this point, i.e. the solution dying. Therefore, in the situation, described above, investigation of different aspects of the qualitative theory of such type of equations continuous dependence, periodicity, stability etc.) is impossible by using the classical results; Loss of the autonomous property: Note that, even the cases in which the right-hand sides of the equations with impulses do not depend on time, the impulsive moments are obtained as solutions of equations involving a solution, which of course) is a function of time. Consequently, these moments depend on the time, including the initial moment. This means that, the solution of problem with impulses, which depends on the impulsive moments, is a composite function of the initial point. Thus, we conclude that, it is not an autonomous; Fusion of the solutions: Usually, it takes place after the impulsive effect; Impulsive moments change after changing the initial condition: Different solutions of one and the same impulsive equation with initial conditions which do not coincide) have different impulsive moments, including the possibility, one of these solutions to be without impulses. After

13 INTRODUCTION xiii perturbations in the initial conditions, it is possible, to obtain some differences in the sizes and directions of the basic and perturbed problems; Changing the impulsive moments at the perturbations of the system. The interferences of the system may consist in: Modification of the right side, Changing the equation parameters, Changing the impulsive functions, etc. The solution of the basic problem and the corresponding solution of the perturbed problem with the same initial condition) in the general case have different impulsive moments and impulsive effects different in size and direction; Accumulation of errors: The perturbations and inaccuracies at the impulsive effects accumulate in time and they have significant influence in determining the solution behavior. For example, for unlimited number of impulses these perturbations may have insuperable character, leading to the solutions formation which differ unlimited from the studied nonperturbed solution, etc. Further, under the term classical qualitative theory and classical results are implied all basic statements, which are included in the qualitative theory of differential equations without impulses. We pay attention to the fact that most famous classical results, concerning differential equations without impulses see the monographs Bellman [76], Gopalsamy [163], Rush et al. [288], Yoshizawa [349]), are reformulated and summarized some of the mentioned above classes of impulsive differential equations. The classical qualitative theory recast almost completely in the class of equations with pre fixed moments of impulsive effects, for example. We mention that, some results of the equations with fixed moments of impulsive effect are trivial generalization of the famous classical theorems. Therefore, they are not reflected in the scientific literature. On the other hand, for some special classes of impulsive differential equations the corresponding to the classical qualitative theory is still at an initial stage. Here, as an example we will point the equations with the impulsive moments which have occasional nature. These equations are suitable for modeling a number of engineering processes, such as power and voltage of the ionic current at electrolysis. It is observed an accumulation of insulating film on the electrodes in this technology. Empirically, it is found out that the current breaks film randomly, which means that, the

14 xiv current power rapidly changes. This phenomenon is described mathematically by the impulsive leap of the solution of modeling differential equation, respectively differential inclusion. Various aspects of the theory of impulsive differential equations are discussed in the articles Bajo [73], Duanet al. [133], Feng andchen [144],Graef et al. [165], LiuandGe [232], Luo [236], Luo et al. [237], Luo and Shen [238], Rao and Sivasundaram [285], Rao et al. [286], Shen and Zou [299], Wen et al. [335], Yan [343, 343], Zhang and Feng [360], Zhang et al. [361, 362, 366]. The investigation of impulsive differential equations has a number of specific features, sa why the study of qualitative characteristics of their solutions is difficult. We will recall the results, mentioned above and dedicated the phenomenon of beating. They refer to the equations, in which the impulsive moments coincide with the moments, in which the integral curve meets predefined, not intersecting hypersurfaces. We will complete the list of the specific properties of the studied class impulsive equations, indicating some articles, devoted to the uniform stability of the solutions on the changes in the size of impulsive effects see Dishliev and Bainov [119]) and impulsive hypersurfaces see Bainov and Dishliev [43]). Such results are meaningless in the equations without impulses. We note that the impulsive moments depend on the initial condition of the solution for these equations. In general, these moments are different for each solution. In other words, the impulsive moments substantially depend on the solution. That means, the proximity between two different solutions could not be expected in the sufficiently small surrounding of the impulsive moments. Imagine a situation, in which one impulsive leap is realized for a solution, while for the other solution such a leap is coming. We will explicitly emphasize that the main difficulties in investigation the asymptotic properties of the solutions of differential equations with impulses, in cases where perturbations are in the impulsive sets in particular impulsive moments) or in the size of impulsive functions, due to the fact that these effects are repeated in general infinitely many). It is possible also, the moments when the impulsive effects occur, to be unlimited in time. It means that, the external perturbations are not vanish and do not fade for an arbitrary long distance from the initial moment. The research in this work is devoted to the specific and characteristic only for the impulsive equations continuous dependencies differentiabilities and stabilities of the solutions. The perturbations are mainly in the impulsive sets particularly the impulsive moments) or in the size of the impulsive functions size of the impulsive leaps ). As a rule,

15 INTRODUCTION xv each of the results is applied to well known mathematical model of population dynamics, pharmacokinetics, hydrodynamics, etc. The obtained asymptotic properties of the solutions are interpreted in the terms of the corresponding model. We use two classical theorems, formulated below: Theorem 0.1. Let the following conditions hold: 1) Function f C[[t 0,T] D,R n ], where D is nonempty area D R n. 2) Function f is Lipschitz continuous in the second argument for [t 0,T] D. 3) Function xt;t 0,x 0 ) is a solution of the initial problem dx dt = f t,x), xt 0) = x 0, defined in the interval [t 0,T], where x 0 D. Then there exists δ =const> 0 such that, for every point x 0 D, for which the inequality x 0 x 0 < δ is valid, the following statements are satisfied: 1) There exists a unique solution xt;t 0,x 0 ) of the problem dx dt = f t,x), xt 0) = x 0, defined in the interval [t 0,T]. 2) For any t [t 0,T], it is fulfilled i.e. lim xt;t 0,x x 0 x 0 ) = xt;t 0,x 0 ), 0 ε > 0) δ = δ ε), 0 < δ < δ) : x 0 D, x 0 x 0 < δ ) xt;t 0,x 0 ) xt;t 0,x 0 ) < ε, t 0 t T. The theorem above is a particular case of Theorem 7.1 in Coddington and Levinson [99]. Theorem 0.2. Let the following conditions hold: 1) Functions u,v C[[t 0,T],R + ]. 2) The constant C is nonnegative.

16 xvi 3) The next inequality is satisfied Then vt) C + t t 0 vτ).uτ)dτ, t 0 t T. t vt) Cexp uτ)dτ, t 0 t T. t 0 The theorem above coincides with Theorem 1.1, Section 1, Chapter 3 in Hartman [169]. The last inequality is called Gronwall inequality. The main results, obtained in the chapters of the monograph are: Chapter 1. The main object of research is nonautonomous nonlinear systems of ordinary differential equations with fixed moments of impulsive effects. Sufficient conditions are found, under which the solutions are continuously dependent and stable on the initial condition and the impulsive moments. The results are applied to the mathematical model of pharmacokinetics. Chapter 2. The basic object of study is nonautonomous nonlinear systems of ordinary differential equations with fixed moments of impulsive effects. Sufficient conditions are found, under which the solutions are continuously dependent on the initial condition and the impulsive effects in the first section of the chapter. The problem with the differentiability of the solutions on the initial condition and impulsive effects is examined in the next section. Chapter 3. A relatively popular class of nonlinear nonautonomous systems of differential equations with non fixed moments of impulsive effects is investigated in this chapter. The impulses take place at the moments, at which the integral curve of the initial problem meets a predefined hyper surfaces. In Section 3.1 sufficient conditions for the absence of the phenomenon of beating are given. In the second section of the chapter sufficient conditions are found, under which the solutions of the studied systems are continuously dependent on the impulsive hypersurfaces. In the last section sufficient conditions are derived, under which the zero solution is uniformly stable on the initial condition and the impulsive perturbations. Chapter 4. The object of investigation is nonautonomous nonlinear system of differential equations with non fixed moments of impulsive effects. The impulsive moments coincide with the moments, in which the integral curve meets any of the so-called barrier curves. In Section 4.1

17 INTRODUCTION xvii sufficient conditions are found, under which the solution of the system considered is continuously dependent with respect to the initial condition and barrier curves. In the next section the obtained results are applied to the general impulsive Gompertz model of population dynamics. Chapter 5. The basic object of research is impulsive nonlinear autonomous system of differential equations. The impulsive moments are non fixed and they take place when the trajectory of the initial problem intersects the so-called impulsive set, situated in the phase space. In Section5.1forthistypeofproblemswestudytheconceptoforbitalHausdorff continuous dependence on the initial condition and the impulsive effects. Sufficient conditions are found, under which the solutions have this property. The results, obtained in the previous section are applied to the impulsive Lotka-Volterra mathematical model, describing the evolutionary dynamics of the predator-prey type in the second section of the chapter. Chapter 6. The initial problem of the previous chapter is a main problem of investigation here. The impulsive set is a part of hyperplane, situated in the phase space of the system of differential equations. In the first section of the chapter sufficient conditions are found, under which if the solution of the corresponding problem without impulses is orbital gravitating, then the research problem with impulses is orbital Hausdorff stable on the initial condition. In the second section of the chapter the classical without impulses) Lotka-Volterra predator-prey model is considered. The sufficient conditions are derived for the orbital gravitation and the Hausdorff stability in respect to the initial point. Chapter 7. In the first and second section of the chapter the development of an isolated population, subjected to the Verhulst Law is investigated. The moments and quantities of the impulsive removals of the population biomass are determined so that, time for reproduction of the taken away biomass is minimal. The optimal reproduction scheme of the taken away quantity of biomass is described. One class of the optimization problems is studied in the next two sections. The optimization is done through the impulsive control. The necessary and sufficient conditions for existence of the optimal impulsive control of the initial problems of the dynamical systems are found. The results are applied to the impulsive Verhulst and Gompertz models, describing the dynamics of the isolated populations. Chapter 8. In this chapter the initial problem for the systems of nonlinear ordinary differential equations with variable structure and impulses is studied. The change of right-hand side of the system and the

18 xviii impulsive perturbations of the solution is made at the moments, when the so-called switching functions become zero. In the first section of the chapter sufficient conditions for continuously dependence of the solution in respect to the initial condition and the switching functions are found. In the second section, using the systems of differential equations with variable structure and impulses, the dynamics of the protected shutter valve is described.

19 Chapter 1 Continuous Dependence and Stability of the Solutions of Impulsive Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments An important specific feature that distinguishes the impulsive equations from the equations without impulses is the fact that their solutions are subjected to the multiple external influences effects), which are discrete in time. Therefore, their solutions are discontinuous functions. The main object of investigation in this chapter is a fairly popular class of nonlinear systems nonautonomous ordinary differential equations with fixed moments of impulsive effects. The main specific feature of the equations considered is the predetermination of the impulsive moments. The impulses take place consistently, when current time reaches the consecutive impulsive moment. This class of equations is well studied because of its relatively simple structure of the impulsive effects, and also due to their wide application in practice see Agarwal and Karacoç [2], Ahmad and Sivasundaram[10], Bai et al.[31], Braverman and Braverman [88], Dishliev and Dishlieva [121], Doddaballapur et al. [129], Duan et al. [133], Eloe and Henderson [136, 137], Eloe et al. [138, 139], Eloe and Sokol [140], Eloe and Usman [141], Feng and Chen [144], Fu and Zhou [152], Graef et al. [165], Guo et al. [167], Ignatyev and Ignatyev [184], Li [215, 218], Li et al. [219], Liu andchen [229], Liu et al. [231], Liu and Ge [232], Lu and Wang [233, 234], Lu et al. [235], Luo [236], Luo et al. [237], Luo and Shen [238], Mohamad et al. [255], Özbekler and Zafer [271], Rao and Tsakos [284], Shen and Li [298], Shen and Zou [299], Shuai et al. [302], Smith and Wahl [308], Stamov and Stamova [311], Stamova [312, 316], Stamova and Emmenegger [317], Stamova and Stamov [318], Sun and Chen [323], Tang et al. [324], Tang and Chen [325], Thomaseth [326], Wang et al. [328, 329, 330, 331, 332, 334], Wen 1

20 2 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES et al. [335], Xia [340], Xiao et al. [342], Yan [343], Yan et al. [344], Yang and Cao [345], Yang and Xu [346, 347], Zavalishchin and Sesekin [354], Zeng et al. [358], Zeng [359], Zhang and Feng [360], Zhang et al. [361, 362, 363], Zhang and Fan [365], Zhang et al. [366, 367], Zhao et al. [372, 374] etc). Many studies have been devoted to the solutions stability of different classes impulsive differential equations with respect to the initial condition. Besides the monographs in the introduction, we cite in additional: Ahmad and Sivasundaram [8, 11], Ahmad and Stamova [13], Anokhin et al. [28], Bainov et al. [35, 50, 59, 60], Gladilina and Icnatyev [160], Gopalsamy [163], Hristova [174], Luo and Shen [239], Medina and Pinto [241], Rao and Tsakos [284], Yang and Cao [345], Yang et al. [348], Ahmad and Sivasundaram [11], Duan et al. [133], Ignatyev [183], Ignatyev and Ignatyev [184, 185], Kosseva et al. [204], Kulev and Bainov [206, 207, 208], Lakshmikantham and Liu[212], Li et al. [216], Liu [226], Liu and Ge [232], Milev and Bainov [249], Milev et al. [251], Perestyuk and Chernikova [277], Rao and Sivasundaram [285], Rao et al. [286], Samoilenko and Perestyuk [294], Stamova [312, 313, 315, 316], Stamova and Stamov [319], Zang and Sun [352]. The concepts continuous dependence and stability in respect of the initial condition and impulsive moments of the equations, mentioned above are studied in first and second sections of this chapter. The basic problem and so-called perturbed problem are considered. Perturbations are repeated and expressed in every possible change of impulsive moment. The changes of impulsive moments in the general case are infinitely many time. The main goal is to find some conditions, under which the solutions of both problems basic and perturbed) are close to each other in a certain sense. We will explicitly emphasize that these concepts are specific to the solutions of impulsive differential equations, which are introduced and studied first by Dishliev and Dishlieva [121]. In Section 1.1 and Section 1.2 sufficient conditions are found under which the solutions of these systems have the asymptotic properties, mentioned above. In Section 1.3, the obtained results are applied to the impulsive mathematical model of the pharmacokinetic, borrowed from, see Mihailova and Staneva-Stoytcheva [248]. The investigation presented of this model is new. Various aspects of the dynamic impulsive mathematical models from the epidemiology and pharmacokinetics are studied in Bailey and Shafer [32], D onofrio [132], Gao et al. [154, 155], Liu and Ye [224], Thomaseth [326], Wang et al. [330], Zhang et al. [363].

21 1. CONTINUOUS DEPENDENCE 3 1. Continuous Dependence of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments The main object of investigation in this section is the following initial problem for impulsive ordinary differential equations: dx dt = f t,x), t t i, 1.1) xt i +0) = xt i )+I i xt i )), i = 1,2,..., 1.2) xt 0 ) = x 0, 1.3) where: f : R + D R n ; n N; D is a domain, D R n ; t i R + ; 0 t 0 < t 1 < t 2 <...; I i : D R n ; Id+I i ) : D D and x 0 D. The identity in R n is denoted by Id. The solution of problem considered is a piecewise function. It is satisfied: 1) For t 0 t t 1, the solution of problem 1.1), 1.2), 1.3) coincides with the solution of problem without impulses) 1.1), 1.3); 2) For t i < t t i+1, i = 1,2,..., the solution of problem 1.1), 1.2), 1.3) is identical to the solution of system 1.1) with initial condition xt i +0) = Id+I i )xt i )). With the problem 1.1), 1.2), 1.3), we discuss the corresponding perturbed problem dx dt = f t,x ), t t i, 1.4) x t i +0) = x t i )+I ix t i )), i = 1,2,..., 1.5) x t 0 ) = x 0, 1.6) where 0 t 0 < t 1 < t 2 <... and x 0 D. Let xt;t 0,x 0 )andx t;t 0,x 0) denotethesolutions ofproblems1.1), 1.2), 1.3) and 1.4), 1.5), 1.6), respectively see Figure 1.1). Further we use the notation: a,b], if a < b; a,b = b,a], if a > b;, if a = b.

22 4 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES x 2 0 x 0 x x 1 x t;t 0,x 0 ) xt;t 0,x 0 ) 0 = t 0 = t 0 t 1 t 1 t 2 t 2 Figure 1.1 t Definition 1.1. We say that the solution of problem 1.1), 1.2), 1.3) depends continuously on the initial point t 0,x 0 ) and impulsive moments t 1,t 2,..., if: ε > 0) T > t 0 ) δ = δε,t) > 0) : t 0,x 0 ) [0,T] D, t 0 t 0 < δ, x 0 x 0 < δ) t 1,t 2, R +, t 0 < t 1 < t 2 <..., t 1 t 1 < δ, t 2 t 2 < δ,...) x t;t 0,x 0) xt;t 0,x 0 ) < ε, t [t 0,T] t i,t i. i=0,1,... The main goal in this section is to find sufficient conditions for continuous dependence of the solution of initial problem 1.1), 1.2), 1.3) on the initial point and impulsive moments. We use the following conditions: H1.1. f C[R + D,R n ]. H1.2. There exist a constant C f > 0, such that t,x) R + D ) f t,x) C f. H1.3. There exist a constant L > 0, such that t,x ),t,x) R + D ) ft,x ) ft,x) L x x. H1.4. lim i t i =. H1.5. t 0,x 0 ) R+ D) the problem without impulses 1.4), 1.6) has a unique solution, defined in interval [t 0, ). H1.6. The functions I i C[D,R n ], i = 1,2,...

23 1. CONTINUOUS DEPENDENCE 5 Theorem 1.1. Let the conditions H1.4 and H1.5 be satisfied. Then t 0,x 0 ) R+ D), the solution of problem with impulses 1.4), 1.5), 1.6) exists and it is unique on the interval [t 0, ). The proof of the theorem is trivial. In particular, the assertion of Theorem 1.1 concerns the solution of problem 1.1), 1.2), 1.3). We introduce the notations and therefore, t min i = min{t i,t i}, t max i = max{t i,t i} t i,t i = ] t min i,t max i, i = 0,1,... Theorem 1.2. Let the conditions H1.1-H1.6 be satisfied. Then the solution of problem 1.1), 1.2), 1.3) depends continuously on the initial point and the impulsive moments t 1,t 2,... Proof. According to condition H1.4, there exists a number k N, such that t k < T t k+1. Further, we assume that x 0 x 0 < δ, t i t i < δ, i = 0,1,... Then, if δ is sufficiently small positive constant the following inequalities are valid: Moreover, we have [t 0,T]\ T < t min k+2, tmax i < t min i+1, i = 0,1,...,k+1. i=0,1,... t i,t i =[t 0,T]\ = where T min = min{t,t min k+1 that T min = t min k+1 the point t min k+1 i=0,1,... i=0,1,...,k 1 ] t min i,t max i t max i,t min i+1 ] ) t max k,t min], }. For convenience from now on we assume. Note that, if the inequality T < tmin k+1 is satisfied, then should be replaced by T in the remaining part of the proof of Theorem 1.2. Under the given assumption it follows: [t 0,T]\ t i,t i = i=0,1,... i=0,1,...,k t max i,ti+1] min.

24 6 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES First, we will obtain an estimation for the difference for t ] t max 0,t min 1. Let t min 0 = t 0 0 = t 0 is similar. We have t max x t;t 0,x 0 ) xt;t 0,x 0 ) and tmax 0 = t 0. The proof of the case t min x t max 0 ;t 0,x 0) xt max 0 ;t 0,x 0 ) t max 0 = x t ) min 0 ;t 0,x 0 + t min 0 f τ,x τ;t 0,x 0 ))dτ xtmax 0 = t 0 and 0 ;t 0,x 0 ) ) x t min 0 ;t 0,x 0 xt max 0 ;t 0,x 0 ) tmax 0 + f τ,x τ;t 0,x 0)) dτ x 0 x 0 + t max 0 C f dτ t min 0 ) = x 0 x 0 +C f t max 0 t min 0 1+C f )δ. 1.7) For t t max 0,t min 1 with the solution of the initial problem t min 0 ], the solution of problem 1.1), 1.2), 1.3) coincides dx = f t,x), xtmax 0 ) = x 0, dt or equivalently with the solution of the integral equation xt) = x 0 + t f τ,xτ))dτ. Consequently, xt;t 0,x 0 ) = x 0 + t max 0 t f τ,xτ;t 0,x 0 ))dτ. 1.8) t max 0 On the interval, considered above, the solution of problem1.4), 1.5), 1.6) coincides with the solution of the problem dx dt = ft,x ), x t max 0 ) = x t max 0,t 0,x 0 ),

25 1. CONTINUOUS DEPENDENCE 7 whence it follows t x t;t 0,x 0 ) = x t max 0 ;t 0,x 0 )+ t max 0 f τ,x τ;t 0,x 0 ))dτ. 1.9) From 1.8) and 1.9) for t ] t max 0,t min 1, we get x t;t 0,x 0 ) xt;t 0,x 0 ) t x t max 0 ;t 0,x 0 )+ f τ,x τ;t 0,x 0 ))dτ x 0 t max 0 t f τ,xτ;t 0,x 0 ))dτ t max 0 x t max 0 ;t 0,x 0 ) x 0 t + f τ,x τ;t 0,x 0)) f τ,xτ;t 0,x 0 )))dτ t max 0 x t max 0 ;t 0,x 0 ) xtmax 0 ;t 0,x 0 ) +L t t max 0 Using the estimate 1.7) we find x τ;t 0,x 0) xτ;t 0,x 0 ) dτ. x t;t 0,x 0 ) xt;t 0,x 0 ) 1+C f )δ +L t x τ;t 0,x 0 ) xτ;t 0,x 0 ) dτ. 1.10) t max 0 From 1.10), using Gronwall s inequality see Hartman [169] or Theorem 2 from introduction), we get the estimate x t;t 0,x 0) xt;t 0,x 0 ) δ1+c f )e Lt tmax 0 ) δ1+c f )e Ltmin 1 t max 0 ) δ1+c f )e LT, t t max 0,t min 1 ].

26 8 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES Let δ 0 be an arbitrary positive number. From the estimate above, it follows that if δ is a sufficiently small constant, then we have x t;t 0,x 0) xt;t 0,x 0 ) < δ 0, t ] t max 0,t min ) We obtain an estimate for the difference x t;t 0,x 0 ) xt;t 0,x 0 ), for t ] t max 1,t min 2. Assume that t min 1 = t 1 and t max 1 = t 1. The case t min 1 = t 1 and t max 1 = t 1 can be considered similarly. As t min 1 is the first impulsive moment of problem 1.1), 1.2), 1.3), we find that x t min 1 +0;t 0,x 0 ) =xt1 +0;t 0,x 0 ) =xt 1 ;t 0,x 0 )+Ixt 1 ;t 0,x 0 )) =x ) )) t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x 0. If t ] t min 1,t min 2, then the solution coincides with the solution of the initial problem: dx dt = ft,x), x ) ) )) t min 1 = x t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x 0. Therefore, it is identical with the solution of the integral equation xt) = x t ) )) t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x 0 + t min 1 f τ,xτ))dτ. In particular, if t = t max 1 +0, we get xt max 1 +0;t 0,x 0 ) =x ) )) t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x t max t min 1 f τ,xτ;t 0,x 0 ))dτ. 1.12) For t ] t min 1,t max 1 = t1,t 1 ], the solution of problem 1.4), 1.5), 1.6) does not undergo impulsive perturbation and satisfies the equality x t;t 0,x 0) = x t ) t min 1 +0;t 0,x 0 + from which, we find t min 1 f τ,x τ;t 0,x 0))dτ, t max 1 x t max 1 ;t 0,x 0) =x ) t min 1 +0;t 0,x 0 + f τ,x τ;t 0,x 0))dτ t min 1

27 1. CONTINUOUS DEPENDENCE 9 t max 1 =x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 ))dτ. t min 1 From the impulsive equality 1.5) for i = 1, we obtain x t max 1 +0;t 0,x 0 ) =x t 1 +0;t 0,x 0 ) t max 1 =x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0))dτ t min 1 t min 1 t max 1 +I 1 x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 ))dτ. 1.13) From 1.12) and 1.13) it follows x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) x t min 1 ;t 0,x 0 t max 1 + t min 1 ) x t min 1 ;t 0,x 0 ) tmax 1 f τ,x τ;t 0,x 0 )) dτ + f τ,xτ;t 0,x 0 )) dτ + I 1 x t min 1 ;t 0,x 0 t max 1 ) + t min 1 t min 1 f τ,x τ;t 0,x 0))dτ I 1 x t min 1 ;t 0,x 0 )). 1.14) Using 1.11), we estimate the first addend on the right-hand side of 1.14) at t = t min 1 : x t min 1 ;t 0,x 0) xt min 1 ;t 0,x 0 ) δ ) For the second addend on the right-hand side of 1.14) using condition H1.2, we obtain t max 1 f τ,x τ;t 0,x 0 )) dτ C f t min 1 ) t max 1 t min 1 Cf δ. 1.16)

28 10 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES In similar way, we receive t max 1 f τ,xτ;t 0,x 0 )) dτ C f δ. 1.17) t min 1 It is true that x t min 1 ;t 0,x 0 t max 1 ) + t min 1 f τ,x τ;t 0,x 0))dτ x ) t min 1 ;t 0,x 0 ) ) tmax 1 x t min 1 ;t 0,x 0 x t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 )) dτ δ 0 +C f δ. 1.18) Let δ1 min be an arbitrary positive number. From 1.18) and using the continuity of the function I 1 it follows that there exist sufficiently small positive values of δ and δ 0 such that I 1 x t min 1 ;t 0,x 0 t max 1 ) + t min 1 t min 1 f τ,x τ;t 0,x 0))dτ I 1 x t min 1 ;t 0,x 0 )) < 1 2 δmin ) From 1.14), 1.15), 1.16), 1.17) and 1.19) we obtain x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) < δ 0 +2C f δ δmin 1. Once again, if δ and δ 0 are sufficiently small, from the inequality above it follows the next estimate x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) δ1 min. 1.20) For t ] t min 1,t max 2 the solution of problem 1.1), 1.2), 1.3) does not undergo any impulsive perturbation and therefore, it satisfies xt;t 0,x 0 ) = xt max 1 +0;t 0,x 0 )+ t t max 1 f τ,xτ;t 0,x 0 ))dτ.

29 1. CONTINUOUS DEPENDENCE 11 Inthesimilar way, fort ] t min 1,t min 2, the solutionofproblem1.1), 1.2), 1.3) satisfies the integral equation t x t;t 0,x 0 ) = x t max 1 +0;t 0,x 0 )+ Thus, taking into account 1.20), we find t max 1 f τ,x τ;t 0,x 0 ))dτ. x t;t 0,x 0) xt;t 0,x 0 ) t x t max 1 +0;t 0,x 0 )+ f τ,x τ;t 0,x 0 ))dτ t max 1 t xt max 1 +0;t 0,x 0 ) f τ,xτ;t 0,x 0 ))dτ t max 1 x t max 1 +0;t 0,x 0) xt max 1 +0;t 0,x 0 ) t + f τ,x τ;t 0,x 0 )) f τ,xτ;t 0,x 0 )))dτ t max 1 <δ min 1 +L t t max 1 Now, we apply Gronwall s inequality: x τ;t 0,x 0 ) xτ;t 0,x 0 ) dτ. x t;t 0,x 0 ) xt;t 0,x 0 ) δ min 1 1+C f )e Lt tmax 0 ) δ1 min 1+C f )e LT, t t max 1,t min 2 Let δ 1 be an arbitrary positive constant. It follows from the estimate above that if δ1 min is sufficiently small constant i.e. if δ and δ 0 are small enough), then x t;t 0,x 0) xt;t 0,x 0 ) < δ 1,t ] t max 1,t min ) In the same way as 1.21), we may obtain the following implication δ i > 0) δ > 0, δ 0 > 0, δ 1 > 0,..., δ i 1 > 0) : x t;t 0,x 0) xt;t 0,x 0 ) < δ i, t t max i,t min i+1], i = 0,1,...,k. 1.22) ].

30 12 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES Let ε be an arbitrary positive number and δ k = ε. Taking into account 1.22), we deduce that, there exist positive constants δ,δ 1,..., δ k 1 such that x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t t max k,tk+1] min. 1.23) Without loss of generality, we may assume that δ k 1 ε. Again, using 1.22), we reach the conclusion: There exist positive constants δ,δ 1,...,δ k 2, for which the following estimate ], 1.24) x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t t max k 1,tmin k holds true. Without loss of generality, we may also impose an additional requirement δ k 2 ε. Finally, from 1.11), or equivalently from 1.22), it follows that, if i = 0, then there exists a constant δ > 0 such that x t;t 0,x 0) xt;t 0,x 0 ) < ε, t t max 0,t min 1 From 1.23), 1.24) and 1.25) we conclude that ]. 1.25) ε > 0) δ > 0) : x 0 x 0 < δ, t i t i < δ, i = 0,1,...,k) x t;t 0,x 0) xt;t 0,x 0 ) < ε, t t max i,ti+1] min, i = 0,1,...,k. The theorem is proved.

31 2. STABILITY Stability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments Definition 1.2. We say that the solution of problem 1.1), 1.2), 1.3) is stable with respect to the initial point t 0,x 0 ) and the impulsive moments t 1,t 2,..., if: ε > 0) δ = δε) > 0) : t 0,x 0) R + D, x 0 x 0 < δ ) ) t 1,t 2, R+, t 0 < t 1 < t 2 <..., t i t i < δ i=0,1,... x t;t 0,x 0) xt;t 0,x 0 ) < ε, t [t 0, )\ i=0,1,... t i,t i. The main aim of this section is to find some sufficient conditions for stability with respect to the initial point and impulsive moments of the solution of initial problem 1.1), 1.2), 1.3). LetXt;t 0,x 0 )bethesolutionoftheinitialproblemwithoutimpulses 1.1), 1.3). Definition 1.3. We say that system 1.1) is gravitating with constant κ, if: t 0 0) x 0,x 0 D) t t 0) Xt;t 0,x 0 ) Xt;t 0,x 0 ) < κ x 0 x 0. Remark 1.1. Let the eigenvalues of real n n matrix A be different and they do not have positive real parts. Then the system dx dt = Ax is gravitating with constant 1. Further, we use the following conditions: H1.7. There are constants L i > 0, i = 1,2,..., such that: H L i ) < ; i=1,2,... H x,x D) I i x ) I i x) L i x x, where i = 1,2,...

32 14 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES H1.8. There exists a constant C I > 0, such that x,x D) x +I i x )) x+i i x)) C I x x, where i = 1,2,... Theorem 1.3. Let the following conditions be satisfied: 1) Conditions H1.1-H1.5 and H1.7 are valid. 2) System 1.1) is gravitating with constant 1. 3) Inequality C f 1 holds true see condition H1.2). Then the solution of the problem with impulses 1.1), 1.2), 1.3) is stable with respect to the initial point t 0,x 0 ) and impulsive moments t 1,t 2,... Proof. We assume that, there exist positive constants δ x, δ 0, δ 1,... and δ, such that: x 0 x 0 < δ x, t i t i < δ i, i = 0,1,...; δ x δ, δ i = δ. From condition H1.4 we have i=0,1,... t = const. > 0) : i = 1,2,...) t i t i 1 > t. From the inequalities above, it follows that if δ is small enough positive constant for example δ < t ), then the following inequalities are valid: Also, we have [t 0, )\ t max i i=0,1,... < t min i+1, i = 0,1,... t i,t i =[t 0, )\ = i=0,1,... Similarly to 1.7), we get the estimate i=0,1,... t max i,ti+1] min. ] t min i,t max i x t max 0 ;t 0,x 0 ) xtmax 0 ;t 0,x 0 ) δ x +C f δ 0 1+C f )δ 0. Taking into account, that system 1.1) is gravitating with constant 1, we obtain: x t;t 0,x 0 ) xt;t 0,x 0 ) 1+C f )δ 0, t ] t max 0,t min ) Let ε be an arbitrary positive constant. From the previous estimate, itfollowsthat, ifδ 0 issufficiently small respectively, ifδ issmallenough),

33 2. STABILITY 15 then x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t ] t max 0,t min ) Let t ] t max 1,t min 2. We have to obtain an estimate for the difference x t; t 0,x 0) xt;t 0,x 0 ). We assume that, t min 1 = t 1 and t max 1 = t 1. The cases t min 1 = t 1 and t max 1 = t 1 are considered in similar way. As t min 1 is the first impulsive moment of problem 1.4), 1.5), 1.6), then x t min 1 +0;t 0 0),x =x t 1 +0;t 0,x 0 ) =x t 1 ;t 0,x 0 )+I 1x t 1 ;t 0,x 0 )) =x ) t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0 0)),x. Fort ] t min 1,t max 1, thesolutionofproblem1.4), 1.5), 1.6)doesnotundergo impulsive perturbation and it coincides with the solution of initial problem dx dt = ft,x ), x ) t ) min 1 = x t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x 0 )), i.e., the following equation holds true x t;t 0,x 0) = x ) t min 1 ;t 0,x )) 0 +I1 x t min 1 ;t 0,x 0 If t = t max 1, we find x t max 1 ;t 0,x 0 ) = x t min 1 ;t 0,x t t min 1 ) )) +I1 x t min 1 ;t 0,x 0 t max 1 t min 1 f τ,x τ;t 0,x 0 ))dτ. f τ,x τ;t 0,x 0 ))dτ. 1.28) ], the solution of problem 1.1), 1.2), On the same interval t min 1,t max 1 1.3) coincides with the solution of the initial problem dx dt = ft,x), x ) ) t min 1 = x t min 1 ;t 0,x 0, or equivalently, with the solution of the equation xt;t 0,x 0 ) = x t ) t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ, t min 1

34 16 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES for t = t max 1 we derive xt max 1 ;t 0,x 0 ) = x ) tmax 1 t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ. Since t max 1 = t 1, we have: t min 1 xt max 1 +0;t 0,x 0 ) =xt max 1 ;t 0,x 0 )+I 1 xt max 1 ;t 0,x 0 )) =x ) tmax 1 t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ t min 1 +I 1 x ) tmax 1 t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ. 1.29) t min 1 Therefore, from 1.28) and 1.29), we find: x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) = x t max 1 ;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) = x t min 1 ;t 0,x 0 ) +I1 x t min 1 ;t 0,x 0 t max 1 )) + x ) tmax 1 t min 1 ;t 0,x 0 f τ,xτ;t 0,x 0 ))dτ I 1 t min 1 t min 1 x ) tmax 1 t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ t min 1 ) ) x t min 1 ;t 0,x 0 x t min 1 ;t 0,x t max 1 t min 1 f τ,xτ;t 0,x 0 ))dτ t max 1 t min 1 f τ,x τ;t 0,x 0 ))dτ f τ,x τ;t 0,x 0 ))dτ

35 + I 1 x t min 1 ;t 0,x 0 )) I1 x t min 2. STABILITY 17 ) tmax 1 1 ;t 0,x 0 + t min 1 f τ,xτ;t 0,x 0 ))dτ. From 1.26), both conditions H1.2 and H1.7 and using the inequality C f 1, it follows: x t max 1 +0;t 0,x 0) xt max 1 +0;t 0,x 0 ) 1+C f )δ 0 +2C f δ 1 +L 1 x t ) min 1 ;t ) 0,x 0 x t min 1 ;t 0,x 0 t max 1 + f τ,xτ;t 0,x 0 ))dτ t min 1 1+C f )δ 0 +2C f δ 1 +L 1 1+C f )δ 0 +C f δ 1 ) =1+L 1 )1+C f )δ 0 +2+L 1 )C f δ 1 1+L 1 )1+C f )δ 0 +δ 1 ). 1.30) Again, taking into account that system 1.1) is gravitating, we obtain the inequality x t;t 0,x 0 ) xt;t 0,x 0 ) x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) 1+L 1 )1+C f )δ 0 +δ 1 ) 1+C f ) 1+L j ) δ1+c f ) j=1,2,... j=1,2,... j=0,1,... δ j 1+L j ), 1.31) where t ] t max 1,t min 2. For the sufficiently small values of δ from the inequality above, we get x t;t 0,x 0) xt;t 0,x 0 ) < ε, t t max 1,t min 2 In the similar way, for i = 2,3,..., we find ]. 1.32) x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) 1+L 1 )...1+L i )1+C f )δ 0 +δ 1 + +δ i ); x t;t 0,x 0) xt;t 0,x 0 ) δ1+c f ) j=1,2,... 1+L j ), t t max i,ti+1] min ;

36 18 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES x t;t 0,x 0) xt;t 0,x 0 ) < ε, t t max i, ti+1] min. Using 1.27), 1.32) and the last inequality it follows that x t;t 0,x 0) xt;t 0,x 0 ) < ε, t t max i,ti+1] min. for sufficiently small values of δ. The theorem is proved. i=0,1,... Theorem 1.4. Let the following conditions be satisfied: 1) Conditions H1.1-H1.5 and H1.8 hold true. 2) The system 1.1) is gravitating with constant κ. 3) The inequalities C I C f 1 and C I κ 1 are valid. Then, the solution of problem with impulses 1.1), 1.2), 1.3) is stable with respect to the initial point t 0,x 0 ) and impulsive moments t 1,t 2,... Proof. We obtain an estimate x t max 0 ;t 0,x 0 ) xtmax 0 ;t 0,x 0 ) 1+C f )δ 0, similarly to the proof of Theorem 1.3. Using the inequality above and condition 2 of Theorem 1.4, we get x t;t 0,x 0 ) xt;t 0,x 0 ) κ1+c f )δ 0, t ] t max 0,t min 1. Let the equalities t min 1 = t 1 and t max 1 = t 1 be valid. Then x t min 1 +0;t 0,x 0 ) =xt1 +0;t 0,x 0 ) =xt 1 ;t 0,x 0 )+I 1 xt 1 ;t 0,x 0 )) Similarly to 1.28), we have =x t min 1 ;t 0,x 0 ) +I1 x t min 1 ;t 0,x 0 )). xt max 1 +0;t 0,x 0 ) = xt max 1 ;t 0,x 0 ) = x ) )) tmax 1 t min 1 ;t 0,x 0 +I1 x t min 1 ;t 0,x 0 + f τ,xτ;t 0,x 0 ))dτ. Analogously to 1.29), we find t max 1 x t max 1 ;t 0,x 0) = x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0))dτ. t min 1 t min 1

37 2. STABILITY 19 Since t max 1 = t 1, then from the equality above, it follows x t max 1 +0;t 0,x 0 ) =x t max 1 ;t 0,x 0 )+I 1x t max 1 ;t 0,x 0 )) t max 1 =x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 ))dτ t min 1 t max 1 +I 1 x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 ))dτ. t min 1 Obviously Id+I 1 ) is Lipschitz continuous function with constant C I see condition H1.8). Therefore x t max 1 +0;t 0,x 0 ) x ) t min 1 ;t 0,x 0 t max 1 = x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0 ))dτ t min 1 t max 1 +I 1 x ) t min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0))dτ t min 1 x ) )) tmax 1 t min 1 ;t 0,x 0 I1 x t min 1 ;t 0,x 0 f τ,xτ;t 0,x 0 ))dτ x t min 1 ;t 0,x 0 t max 1 ) + +I 1 x t min 1 ;t 0,x 0 t min 1 t max 1 ) + t min 1 f τ,x τ;t 0,x 0))dτ t min 1 f τ,x τ;t 0,x 0 ))dτ [ x t min 1 ;t 0,x 0 ) +I1 x t min 1 ;t 0,x 0 ))]

38 20 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES t max 1 + f τ,xτ;t 0,x 0 ))dτ t min 1 t max 1 C I x t ) min 1 ;t 0,x 0 + f τ,x τ;t 0,x 0))dτ x ) t min 1 ;t 0,x 0 t min 1 +C f δ 1 C ) ) I x t min 1 ;t 0,x 0 x t min 1 ;t 0,x 0 t max 1 +C I f τ,x τ;t 0,x 0 ))dτ +C f δ 1 t min 1 C I κ1+c f )δ 0 +C I C f δ 1 +C f δ 1 1+C f )δ 0 +δ 1 ). Again, taking into account that system 1.1) is gravitating, we obtain the inequality x t;t 0,x 0 ) xt;t 0,x 0 ) x t max 1 +0;t 0,x 0 ) xtmax 1 +0;t 0,x 0 ) κ1+c f )δ 0 +δ 1 ) κ1+c f ) <κ1+c f )δ, t ] t max 1,t min 2. j=0,1,... We obtain see the last inequality) x t;t 0,x 0) xt;t 0,x 0 ) < ε, t ] t max 1,t min 2 for sufficiently small values of δ. In the similar way, for i = 2,3,..., we find x t max i +0;t 0,x 0) xt max i +0;t 0,x 0 ) 1+C f )δ 0 +δ 1 + +δ i ); x t;t 0,x 0 ) xt;t 0,x 0 ) κ1+c f )δ 0 +δ 1 + +δ i ) <κ1+c f )δ, t t max i,ti+1] min ; x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t t max i,ti+1] min. Finally, the following inequality is satisfied x t;t 0,x 0) xt;t 0,x 0 ) < ε, t i=0,1,... t max i,ti+1] min, δ j

39 2. STABILITY 21 for sufficiently small values of δ. The theorem is proved.

40 22 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES

41 3. APPLICATION: PHARMACOKINETIC MODEL Application: Continuous Dependence and Stability of the Solutions of Pharmacokinetic Model with Fixed Moments of Impulses on the Initial Condition and Impulsive Moments The treatment of many medical conditions is achieved by the maintaining a therapeutic drug concentration in the blood plasma) of the patient. This concentration can be achieved in two basic ways: By continuous administration of the drug; By intermittentimpulsive) administration of the drug at certain time intervals. In terms of effective treatment, continued application of the medicines is preferable. Unfortunately, this method is difficult in its practical realization. More precisely, in the common case, it is impossible for one patient to take continuously one or more drugs during the treatment period this period may have different duration - several weeks or months). Therefore, the maintenance of therapeutic drug concentration in the blood via discrete impulsive applications of the drug is more common. It is natural to assume that, the drug amount is bounded from below, i.e., there exists a minimum quantity of drug that could be taken as a single dose at once. By using this type of treatment, a physician canmanipulate by two pharmacokinetic parameters: the size of single dose of drug D i and the length of dosing interval T i+1, i = 1,2,... More precisely, D i is the dose of i-th application of the drug, and T i+1 is the time between moment of i-th and i+1)-application of the drug, i = 1,2,... In the cases, when the dosing intervals are shorter than required time for complete elimination of the drug in the body, the drug begins to cumulate. This accumulation is useful for the patient treatment, if it is kept in intervals, determined by the minimum and maximum plasma levels, called therapeutic range therapeutic window) of the drug concentration. Pharmacokinetic model for the therapeutic treatment consists in the appropriate choice of dose scheme which guarantees the maintenance of drug concentration within the therapeutic window. We introduce the following restrictions and notations: 1) The body is presented as a compartment with volume V d in which the active drug is distributed. It is possible the concentration of the drug to vary in different parts of the body. For convenience, later we will assume that the ratio between the drug levels in any part of the body is a constant during the treatment

42 24 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES period. This means that, any change of the drug concentration in plasma reflects in a strictly defined relevant quantitative change of the drug concentration in tissue; 2) The treatment initial moment is denoted by t 0 ; 3) The duration between the medication initial moment t 0 and first moment t 1 in which, the drug with volume D 1 is imported into the body, is denoted by T 1, i.e. it is satisfied t 1 = t 0 +T 1 ; 4) DoseD i of themedicine isimported directly into a compartment at the moment t i = t i 1 +T i = t 0 + T j, j=1,2,...,i where i = 1,2,...; 5) Elimination of the drug is implemented with a speed, proportional to its current amount in the body, i.e. it is considered as a process of first order, which is characterized by speed constant K. The latter constant is the sum of the constant of metabolism and the constant of extraction of the unchanged drug K e. It is satisfied: K = K m +K e ; 6) Denote by At) the drug quantity in the body at the moment t t 0. Generally, the substance quantity, located in the patient body cannot be determined experimentally at any time. Actually, it is possible to determine the concentration medication in some biological fluids, mostly blood. For mathematical modeling of the treatment process, it is convenient to introduce the volume, in which the drug is distributed. This value is called volume of distribution, denoted by V d and it is defined so that the following equality Ct) = At) V d, t t 0, is satisfied. Here Ct) is the drug concentration, measured in the blood or more generally in the plasma. We note that, the volume of distribution does not have a physiological sense. We could consider this as a fictitious volume and if quantity drug At) is distributed evenly, the concentration, measured in the plasma, will be Ct). In fact, a part of the drug is connected with the plasma and tissue proteins, so that its distribution is not uniformly. Therefore, the volume of distribution is possible to be different from the body s fluids volume;

43 3. APPLICATION: PHARMACOKINETIC MODEL 25 7) In the initial moment t 0, we assume that the drug concentration is C 0. In some cases it is considered C 0 = 0. Idealized mathematical model of the process above is described in Mihailova and Staneva-Stoytcheva [248] by means of the following initial value problem for impulsive differential equation: dc dt = KC, t t i, 1.33) Ct i +0) = Ct i )+ D i, i = 1,2,..., V d 1.34) Ct 0 ) = C ) The solution of the previous initial problem can be obtained easily: For t 0 t t 1, it is satisfied Ct) = C 0 exp K t t 0 )); For t 1 < t t 2, we obtain Ct) = C 0 exp Kt 1 t 0 ))+ D ) 1.exp Kt t 1 )) V d =C 0 exp Kt t 0 ))+ D 1 V d exp K t t 1 )); For t i < t t i+1, i = 2,3,..., the following equality Ct) =C 0 exp Kt t 0 ))+ D 1 V d exp Kt t 1 )) + D 2 V d exp Kt t 2 ))+ + D i V d exp Kt t i )), is valid, i.e. where Ct) = exp Kt) V d C i = 1 V d =C i exp Kt), j=0,1,...,i j=0,1,...,i D j expkt j ) D j expkt j ) and D 0 = C 0 V d. Various aspects of qualitative theory of impulsive models in pharmacokinetics are considered in Lin and Hui [221], Smith and Wahl [308].

44 26 1. EQUATIONS WITH FIXED MOMENTS OF IMPULSES We apply the results obtained in the previous sections to the model 1.33), 1.34), 1.35). Let us set ft,c) = fc) = KC and I i x) = D i V d. In addition, we assume that, the dose intervals T i are bounded from below, i.e. there exists a positive constant t, such that t i+1 t i = T i+1 t, i = 0,1,... This means that lim t i = andtherefore, theconditionh1.4holdstrue. i The conditions H1.1, H1.3, H1.5, H1.6 and H1.7 with constants L i = 0, i = 1,2,...) can be easily verified. It is natural to assume that, the drug concentration is bounded above, i.e., the unknown function C = Ct) is bounded. In fact, since the solution is a positive piecewise monotonous decreasing function, then the constant C max = C V d is one upper limit for C = Ct) if t 0 t T. In other words, we have 0 C C max for t 0 t T. Then we have t i T f t,c) = f C) = KC KC max = C f, 1.36) so condition H1.2 is also valid. This means that the conditions of Theorem 1.2 are fulfilled. Therefore, the solution of problem 1.33), 1.34), 1.35) depends continuously on the initial point and the impulsive moments. This fact has the following interpretation: The concentrations of pharmaceuticals in the patient body in two different treatment schemes are approximately equal, if the following conditions are fulfilled: The treatment schemes are applied to one and same patient; The manipulation period the period, during which the drugs are taken) is limited; The drugs concentrations in the initial moment of both treatment schemes differ insignificantly; The initial moments of both treatment schemes approximately coincide; The dosages for each discrete drug administration in both treatment schemes are equal; Dosing intervalsthe intervals between two consecutive moments of drug s intake) are bounded from below; D i

45 3. APPLICATION: PHARMACOKINETIC MODEL 27 The moments of the drug intakes approximately coincide in both treatment schemes. Using Remark 1.1, it is possible to determine that equation 1.33) is a gravitating with constant 1. In addition, if we find out that the solution C = Ct) is bounded above by constant 1, then from 1.36) it follows K that C f 1. Therefore, the solution of problem 1.33), 1.34), 1.35) is stable on the initial data and impulsive moments. The interpretation of this fact is similar to the described above, with the difference that the manipulation period may not be bounded. It is easy to check, that the estimate C = Ct) 1 K is valid if the following inequalities are fulfilled: C 0 1 K, exp KT i)+ KD i V d 1, i = 1,2,....

46

47 Chapter 2 Continuous Dependence and Diferentiability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects The main object in the second chapter is a class of nonautonomous nonlinear systems of ordinary differential equations with fixed moments of impulsive effect. The same class equations are investigated in the previous chapter. Numerous studies are devoted to the continuous dependence of the solutions of various classes impulsive differential equations on the initial condition. We note Akhmet [15], Bainov et al. [61], Bainov and Simeonov [71], Henderson and Thompson [170], Plotnikov and Kitanov [280], Samoilenko and Perestyuk [295]. Differentiability of the solutions of impulsive differential equation with respect of initial condition and parameters are considered in Akhmet [15], Akhmetov and Perestyuk [19], Dishliev and Bainov [113], Henderson and Thompson [170], Simeonov and Bainov [304]. Sufficient conditions under which the solutions of the system depend continuously on the impulsive perturbationsmore precisely, with respect to the size of impulses) have been found in the first section of the chapter. The question of differentiability of the solutions of described above impulsive systems has been studied in the second section in the chapter. We pay attention to the fact that we differentiate the solutions in respect to the impulsive effects. We note that continuous dependence and differentiability are specific concepts for the solutions of impulsive systems of differential equations. The results obtained have been applied to the logistic model of population dynamic. The impulsive logistic model from Section 2.3 has been examined in many research papers, list of which except mentioned results in the introduction) can be supplemented by the following: Berezansky and Braverman [83], Feng and Huang [143], Frigon 29

48 30 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY and O Regan [148], Gopalsamy [163], Liu and Chen [228], Sun and Chen [323], Zhang and Fan [365], Zhao and Tang [373]. The investigations in this chapter are based on the results obtained in Dishlieva [127].

49 1. CONTINUOUS DEPENDENCE Continuous Dependence of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects We consider the following initial problem for impulsive differential equations: where: dx dt = f t,x), t t i, 2.1) xt i +0) = xt i )+I i xt i ),µ i ), i = 1,2,..., 2.2) x0) = I 0 µ 0 ), 2.3) µ 0,µ 1,... are real parameters, µ 0,µ 1, m 1,m 2 ) = M R; f : R + D R n, n N, D is a domain in R n ; 0 = t 0 < t 1 < t 2 <...; I i : D M R n ; Id+I i ) : D M D and I 0 : M D. Id is the identity in R n M. The solution of problem 2.1), 2.2), 2.3) is a piecewise continuous function and we have: 1) For 0 = t 0 t t 1 the solution of problem 2.1), 2.2), 2.3) coincides with the solution of problem without impulses) 2.1), 2.3); 2) For t i < t t i+1, i = 1,2,..., the solution of problem 2.1), 2.2), 2.3) coincides with the solution of problem 2.1) with initial condition xt i +0) = Id+I i )xt i ),µ i ). Let xt;µ 0,µ 1,...) denote the solution of 2.1), 2.2), 2.3) for every choice of the parameters µ 0,µ 1, M. Then, we have: For 0 t t 1 the solution of this problem depends onthe initial condition parameter µ 0, only. Therefore, it does not depend on the impulsive functions I 1,I 2,... and parameters µ 1,µ 2,... We have xt;µ 0,µ 1,...) = xt;µ 0 ). For t i < t t i+1, i = 1,2,..., the solution depends onthe initial condition and impulsive functions I 1,I 2,...,I i, but it does not depends on the other impulsive functions I i+1,i i+2,..., i.e. xt;µ 0,µ 1,...) = xt;µ 0,µ 1...,µ i ).

50 32 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY In the following definitions we assume that the solution of the problem under consideration is defined and unique for t t 0 = 0. Definition 2.1. Let for t i < t t i+1, there exists the finite limit lim xt;µ 0,µ 1,...,µ i ) = xt;µ 0,µ 1,...,µ i ), i = 1,2,.... µ 0 µ 0 Then, the solution of problem 2.1), 2.2), 2.3) depends continuously on the initial point I 0 µ 0 ). Definition 2.2. Let for t i < t t i+1, there exists the finite limit lim x t;µ 0,µ 1,...,µ j 1,µ µ j µ j,µ ) j+1,...,µ i j = xt;µ 0,µ 1,...,µ j 1,µ j,µ j+1,...,µ i ), i = j, j +1,... Then, the solution of problem 2.1), 2.2), 2.3) depends continuously on the impulsive function I j. The main goal is to find the sufficient conditions for continuous dependence of the solution of initial problem 2.1), 2.2), 2.3) with respect to initial point I 0 µ 0 ) and impulsive functions I j, j = 1,2,... We use the following conditions: H2.1. f C[R + D,R n ]; H2.2. lim i t i = ; H2.3. For each point t 0,x 0 ) R + D, the equation 2.1) with initial condition xt 0 ) = x 0 has unique solution in the interval [t 0, ). We use without proof) the following statement. Theorem 2.1. Let the conditions H2.1, H2.2 and H2.3 hold. Then, the solution of problem with impulses 2.1), 2.2), 2.3) exists and it is unique for t 0. Let us add two conditions more: H2.4. There exists a constant L > 0, such that t,x ), t,x) R + D ) f t,x ) ft,x) L x x ; H2.5. I 0 C[M,R n ] and I i C[D M,R n ], i = 1,2,... Theorem 2.2. Let the conditions H2.1-H2.5 hold.

51 1. CONTINUOUS DEPENDENCE 33 Then, the solution of problem 2.1), 2.2), 2.3) depends continuously on the initial point I 0 µ 0 ) and impulsive functions I j, j = 1,2,... Proof. Let j {0,1,...}. First we verify that for t j < t t j+1, the solution depends continuously on the function I j. The interval t j,t j+1 ] is the interval of continuity of the solution, which is situated after the initial point for j = 0 or after the final impulsive effect for j = 1,2,... If j = 0, thesolutiondepends continuously ontheinitialpoint I 0 µ 0 )inthe interval t 0,t 1 ]. For j = 1,2,... it depends continuously onthe impulsive function I j in t j,t j+1 ]. Let µ j M. For t j < t t j+1 the solutions x t;µ 0,µ 1,...,µ j) and xt;µ0,µ 1,...,µ j ) coincide with the solutions of the initial problems respectively: and dx =f t,x), dt I 0 µ 0), if j = 0; xt j ) = xt j ;µ 0,...,µ j 1 ) +I j xtj ;µ 0,...,µ j 1 ),µ j), if j > 0 dx =f t,x), dt I 0 µ 0 ), if j = 0; xt j ) = xt j ;µ 0,...,µ j 1 ) +I j xt j ;µ 0,...,µ j 1 ),µ j ), if j > 0. Consequently, the following equalities are valid: x ) t;µ 0,...,µ j 1,µ j t I 0 µ 0 )+ f τ,xτ;µ 0 ))dτ, if j = 0; t 0 ) = xt j ;µ 0,...,µ j 1 )+I j xtj ;µ 0,...,µ j 1 ),µ j and xt;µ 0,...,µ j 1,µ j ) + t t j f )) τ,x τ;µ0,...,µ j 1,µ j dτ, if j > 0

52 34 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY t I 0 µ 0 )+ f τ,xτ;µ 0 ))dτ, if j = 0; t 0 = xt j ;µ 0,...,µ j 1 )+I j xt j ;µ 0,...,µ j 1 ),µ j ) t + f τ,xτ;µ 0,...,µ j 1,µ j ))dτ, if j > 0. t j Therefore, x t;µ 0,...,µ j 1,µ j) xt;µ0,...,µ j 1,µ j ) I 0 µ 0) I 0 µ 0 ) t + f τ,xτ;µ 0 )) f τ,xτ;µ 0)))dτ, if j = 0; t 0 ) I j xtj ;µ 0,...,µ j 1 ),µ j = I j xt j ;µ 0,...,µ j 1 ),µ j ) + t f τ,x τ;µ0,...,µ j 1,µ j)) t j f τ,xτ;µ 0,...,µ j 1,µ j )) ) dτ, if j > 0. Using condition H2.4 and the Gronwall s Inequality see Hartman [169], also Theorem 0.2 from Introduction), we get the following estimate xt;µ0,...,µ j 1,µ j ) xt;µ 0,...,µ j 1,µ j ) I 0 µ 0 ) I 0µ 0 ) + L xτ;µ 0) xτ;µ 0 ) dτ, if j = 0; t 0 ) Ij xti ;µ 0,...,µ j 1 ),µ j I j xt j ;µ 0,...,µ j 1 ),µ j ) + t t t j L ) x τ;µ0,...,µ j 1,µ j xτ;µ 0,...,µ j 1,µ j ) dτ, if j > 0

53 1. CONTINUOUS DEPENDENCE 35 I 0 µ 0) I 0 µ 0 ) e Lt t0), if j = 0; ) Ij xti ;µ 0,...,µ j 1 ),µ j I j xt j ;µ 0,...,µ j 1 ),µ j ) e Lt t j ), if j > 0. Therefore, the condition H2.5 yields lim x t;µ0,...,µ j 1,µ j) xt;µ0,...,µ j 1,µ j ) = 0, µ j µ j for t j < t t j+1. We assume that, there exists a number i, i j, such that lim x t;µ0,...,µ µ j µ j,...,µ i) xt;µ0,...,µ j,...,µ i ) = 0, j for t i < t t i+1, 2.4) i.e. solution of problem 2.1), 2.2), 2.3) depends continuously on the function I j for t i < t t i+1. Then for t i+1 < t t i+2, we have: x t;µ 0,...,µ j,...,µ ) i,µ i+1 =x t;µ0,...,µ j,...,µ ) i +I i+1 x t;µ0,...,µ j,...,µ ) ) i,µi+1 and + t t i+1 f x t;µ 0,...,µ j,...,µ i,µ i+1 ) =xt;µ0,...,µ j,...,µ i ) τ,x τ;µ0,...,µ j,...,µ i,µ i+1 )) dτ +I i+1 xt;µ 0,...,µ j,...,µ i ),µ i+1 ) + t t i+1 f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 ))dτ. Subtracting the last two equations, applying consecutively the condition H2.4 and the Gronwall s Inequality we achieve the following estimate xt;µ0,...,µ j,...,µ i,µ i+1 ) xt;µ 0,...,µ j,...,µ i,µ i+1 ) x t;µ0,...,µ j,...,µ i) xt;µ0,...,µ j,...,µ i ) + Ii+1 x t;µ0,...,µ j,...,µ ) ) i,µi+1 I i+1 xt;µ 0,...,µ j,...,µ i ),µ i+1 ) )e Lt t i+1).

54 36 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY Finally from the last inequality, the induction assumption 2.4), and continuity of the impulsive function I i+1 according to the condition H2.5, we derive the conclusion lim µ j µ j x t;µ0,...,µ j,...,µ i,µ i+1 ) The theorem is proved. xt;µ 0,...,µ j,...,µ i,µ i+1 ) = 0, for t i+1 < t t i+2.

55 2. DIFFERENTIABILITY Differentiability of the Solutions of Differential Equations with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects In this section we continue with investigation the problem 2.1), 2.2), 2.3). If for each t t 0, there exists the derivative µ 0 xt;µ 0,µ 1,...), then we say that the solution of problem is differentiable on the initial point. More precisely, we have the following definition. Definition 2.3. Let for t i < t t i+1, there exists the finite limit 1 lim µ 0 µ 0 µ 0 µ xt;µ 0,µ 1,...,µ i ) xt;µ 0,µ 1,...,µ i )), 0 where i = 1,2,... Then the solution of problem 2.1), 2.2), 2.3) is differentiable on the initial point I 0 µ 0 ). If for each t > t j, there exists the derivative µ j xt;µ 0,µ 1,...), then we say that the solution is differentiable on the impulsive function I j. We introduce the following definition. Definition 2.4. Let for t i < t t i+1, i = j,j +1,..., there exists the finite limit 1 lim x µ j µ j µ j µ t;µ0,...,µ j 1,µ j,µ ) j+1,...,µ i j xt;µ 0,...,µ j 1,µ j,µ j+1,...,µ i )). Then the solution of problem 2.1), 2.2), 2.3) is differentiable on the impulsive function I j. The main aim of this section is to find the sufficient conditions for differentiability of the solution of initial problem 2.1), 2.2), 2.3) with respect totheinitialpoint I 0 µ 0 )andimpulsive functions I j, j = 1,2,... Further, let R n R n be the set of square n n matrices. Let x R n be a vector and A R n R n be a matrix. Then x and A denote arbitrary coherent norms of the vector x and the matrix A, respectively.

56 38 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY We introduce the following conditions: H2.6. f x C[R+ D,R n R n )]. H2.7. There exist a constant F > 0, such that for each t,x) R + D the inequality x ft,x) F is fulfilled. H2.8. We have: H I 0 C 1 [M,R n ]; H I i C[D µ M,Rn ], i = 1,2,...; H I i C[D x M,Rn R n )], i = 1,2,... Theorem 2.3. Let the conditions H2.1-H2.8 hold true. Then: 1) The solution of problem 2.1), 2.2), 2.3) is continuously differentiable on the initial point I 0 µ 0 ) i.e. on the parameter µ 0 ) and on the impulsive functions I j i.e. on the parameters µ j ), j = 1,2,...; 2) For t 0 < t t 1 the derivative µ 0 xt;µ 0 ) satisfies the initial problem d dt µ 0 xt;µ 0 ) ) xt 0 ;µ 0 ) = µ 0 µ I 0µ 0 ) = ) x f t,xt;µ 0)) xt;µ 0 ), µ 0 3) For t j < t t j+1, j = 1,2,..., the derivative µ j xt;µ 0,µ 1,...,µ j ) satisfies the initial problem d dt µ j xt;µ 0,µ 1,...,µ j ) ) = x f t,xt;µ 0,µ 1,...,µ j )) ) xt;µ 0,µ 1,...,µ j ), µ j µ j xt j ;µ 0,µ 1,...,µ j ) = µ I jxt j ;µ 0,µ 1,...,µ j 1 ),µ j ). 4) For t j < t i < t t j+1, j = 0,1,..., the derivative µ j xt; µ 0, µ 1,...,µ j ) satisfies the initial problem ) d xt;µ 0,...,µ j,...,µ i ) = dt µ j x f t,xt;µ 0,...,µ j,...,µ i ))

57 2. DIFFERENTIABILITY 39 ) xt;µ 0,...,µ j,...,µ i ) µ j µ j xt i ;µ 0,...,µ j,...,µ i ) = µ j xt i+1 ;µ 0,...,µ j,...,µ i 1 ) + x I ixt i+1 ;µ 0,...,µ j,...,µ i 1 ),µ i ) µ j xt i ;µ 0,...,µ j,...,µ i 1 ). Proof. Let j {1,2,...} and µ j M. The case j = 0 is considered similarly. First we have to show that for t j < t t j+1 the derivative µ j xt;µ 0,...,µ j 1,µ j ) exists. For these values of argument t, both solutions xt;µ 0,...,µ j 1, µ j)andxt;µ 0,...,µ j 1,µ j )satisfythefollowingequations, respectively: x ) t;µ 0,...,µ j 1,µ j) =xtj ;µ 0,...,µ j 1 )+I j xtj ;µ 0,...,µ j 1 ),µ j and + t t j f )) τ,x τ;µ0,...,µ j 1,µ j dτ xt;µ 0,...,µ j 1,µ j ) =xt j ;µ 0,...,µ j 1 )+I j xt j ;µ 0,...,µ j 1 ),µ j ) + t t j f τ,xτ;µ 0,...,µ j 1,µ j ))dτ. Subtracting the last two equations, we obtain x ) ) t;µ 0,...,µ j 1,µ j x t;µ0,...,µ j 1,µ j =I j xtj ;µ 0,...,µ j 1 ),µ j) Ij xt j ;µ 0,...,µ j 1 ),µ j ) + t t j )) f τ,x τ;µ0,...,µ j 1,µ j f τ,xτ;µ0,...,µ j 1,µ j )) ) dτ = µ I ) j xtj ;µ 0,...,µ j 1 ),µ j) µ j µ j,

58 40 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY + Then, t t j ) x f τ,xτ;µ ) 0,...,µ j 1,µ j ))+φ j τ,µ j x τ;µ 0,...,µ j 1,µ j) xτ;µ0,...,µ j 1,µ j ) ) dτ. x t;µ 0,...,µ j 1,µ j) xt;µ0,...,µ j 1,µ j ) µ j µ j = µ I j xtj ;µ 0,...,µ j 1 ),µ j) t ) + x f τ,xτ;µ ) 0,...,µ j 1,µ j ))+φ j τ,µ j t j x ) τ;µ 0,...,µ j 1,µ j xτ;µ0,...,µ j 1,µ j ) µ j µ dτ, 2.5) j where the parameter µ j is between µ j and µ j. Therefore, it is fulfilled: ) f τ,x )) τ;µ φ j τ,µ 0,...,µ j 1,µ j f τ,xτ;µ0,...,µ j 1,µ j )) j = x τ;µ 0,...,µ j 1,µ j) xτ;µ0,...,µ j 1,µ j ) x f τ,xτ;µ 0,...,µ j 1,µ j )). For t j t t j+1, we consider the initial problem dy j t) dt = x f t,xt;µ 0,...,µ j 1,µ j )).y j t), y j t j ) = µ I jxt j ;µ 0,...,µ j 1 ),µ j ). The problem above can be rewritten in the following integral form y j t) = µ I jxt j ;µ 0,...,µ j 1 ),µ j ) + t t j x f τ,xτ;µ 0,...,µ j 1,µ j )).y j τ)dτ. 2.6) Subtracting the equations 2.5) and 2.6), we obtain x t;µ 0,...,µ j 1,µ j) xt;µ0,...,µ j 1,µ j ) µ j µ y j t) j

59 2. DIFFERENTIABILITY 41 = µ I j xtj ;µ 0,...,µ j 1 ),µ j) µ I jxt j ;µ 0,...,µ j 1 ),µ j ) t ) + x f τ,xτ;µ ) 0,...,µ j 1,µ j ))+φ j τ,µ j t j x ) τ;µ 0,...,µ j 1,µ j xτ;µ0,...,µ j 1,µ j ) µ j µ dτ j t t j = µ I j + + t φ j t j t t j x f τ,xτ;µ 0,...,µ j 1,µ j )).y j τ)dτ xtj ;µ 0,...,µ j 1 ),µ j) µ I jxt j ;µ 0,...,µ j 1 ),µ j ) τ,µ j ),yj τ)dτ ) x f τ,xτ;µ ) 0,...,µ j 1,µ j ))+φ j τ,µ j ) ) x τ;µ0,...,µ j 1,µ j xτ;µ0,...,µ j 1,µ j ) µ j µ y j τ) dτ. j Let us set z j t) = x ) t;µ 0,...,µ j 1,µ j xt;µ0,...,µ j 1,µ j ) µ j µ y j t), j Using the condition H2.7, we have z j t) µ I jxt j ;µ 0,...,µ j 1 ),µ i ) µ I jxt j ;µ 0,...,µ j 1 ),µ j ) t ) t ) ) + φj τ,µ yj j τ) dτ + F + φj τ,µ j zj τ) dτ. t j t j Let Y j be a positive constant such that y j t) Y j for t j t t j+1. Then, the last inequality and Gronwall s Inequality, yield: z j t) µ I jxt j ;µ 0,...,µ j 1 ),µ i )

60 42 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY t µ I jxt j ;µ 0,...,µ j 1 ),µ j ) + ) φ τ,µ.yj j dτ exp F + φj τ,µ j ) ) t tj ) ). 2.7) As µ j is between µ j and µ j, using condition H2.8 we get lim µ j µ j µ I jxt j ;µ 0,...,µ j 1 ),µ i ) µ I jxt j ;µ 0,...,µ j 1 ),µ j ) t j = ) From condition H2.6 and Theorem 2.1from which follows the continuous dependence of the solution xt;µ 0,...,µ j 1,µ j ) on the parameter µ j ), we obtain i.e. lim µ j µ j φ j t,µ j ) fτ,xt;µ 0,...,µ j 1,µ j )) fτ,xt;µ 0,...,µ j 1,µ j )) xt;µ 0,...,µ j 1,µ j ) xt;µ 0,...,µ j 1,µ j ) x f τ,xt;µ 0,...,µ j 1,µ j )) = 0, t j t t j ) = lim µ j µ j Using 2.7), 2.8) and 2.9) it follows that lim z j t) = 0, µ j µ j x t;µ 0,...,µ j 1,µ j) xt;µ0,...,µ j 1,µ j ) lim µ j µ j µ j µ j = y j t). It is shown that for t j < t t j+1, there exists a continuous derivative µ j xt;µ 0,...,µ j 1,µ j ) = y j t). This derivative satisfies the initial problem ) d xt;µ 0,...,µ j 1,µ j ) = dt µ j x f t,xt;µ 0,...,µ j 1,µ j )) ) xt;µ 0,...,µ j 1,µ j ), µ j µ j xt j ;µ 0,...,µ j 1,µ j ) = µ I jxt j ;µ 0,...,µ j 1 ),µ j ).

61 2. DIFFERENTIABILITY 43 Let for t j t i < t t i+1, the derivative µ j xt;µ 0,...,µ j,...,µ i ) satisfies the initial problem from the last statement of Theorem 2.3. If t i+1 < t t i+2, we get Hence, xt;µ 0,...,µ j,...,µ i,µ i+1 ) xt;µ 0,...,µ j,...,µ i,µ i+1 ) =x ) t i+1 ;µ 0,...,µ j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) +I i+1 x ti+1 ;µ 0,...,µ j,...,µ ) ) i,µi+1 I i+1 xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) + t t i+1 f τ,x τ;µ0,...,µ j,...,µ i,µ i+1 )) f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 )))dτ. 1 x µ j µ t;µ0,...,µ j,...,µ ) i,µ i+1 xt;µ0,...,µ j,...,µ i,µ i+1 ) ) j = 1 x µ j µ ti+1 ;µ 0,...,µ ) j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) ) j + I i+1 x ti+1 ;µ 0,...,µ j,...,µ ) ) i,µi+1 Ii+1 xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) x t i+1 ;µ 0,...,µ j,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) 1 x µ j µ ti+1 ;µ 0,...,µ ) j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) ) j + t t i+1 f τ,x τ;µ 0,...,µ j,...,µ i,µ i+1 )) f τ,xτ;µ0,...,µ j,...,µ i,µ i+1 )) x τ;µ 0,...,µ j,...,µ i,µ i+1 ) xτ;µ0,...,µ j,...,µ i,µ i+1 ) 1 µ j µ x τ;µ 0,...,µ ) j,...,µ i,µ i+1 j xτ;µ 0,...,µ j,...,µ i,µ i+1 ))dτ. 2.10) For t i+1 t t i+2 the initial problem is considered dy i+1 t) dt = x f t,xt;µ 0,...,µ j,...,µ i,µ i+1 )).y i+1 t), y i+1 t i+1 ) = µ j xt i+1 ;µ 0,...,µ j,...,µ i ) + x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) µ j xt i+1 ;µ 0,...,µ j,...,µ i ).

62 44 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY We rewrite the problem above in the integral form: y i+1 t) = µ j xt i+1 ;µ 0,...,µ j,...,µ i ) + x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) xt i+1 ;µ 0,...,µ j,...,µ i ) µ j + t t i+1 x f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 )).y i+1 τ)dτ. We introduce the notation z i+1 t) = 1 ) x µ j µ t;µ0,...,µ j,...,µ i,µ i+1 j xt;µ 0,...,µ j,...,µ i,µ i+1 )) y i+1 t). Subtracting the equations 2.10) and 2.11), we obtain 1 z i+1 t) = x µ j µ ti+1 ;µ 0,...,µ j,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) ) j 2.11) xt i+1 ;µ 0,...,µ j,...,µ i ) µ j Ii+1 x ti+1 ;µ 0,...,µ j +,...,µ ) ) i,µi+1 Ii+1 xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) x t i+1 ;µ 0,...,µ j,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) ) x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) 1 x µ j µ ti+1 ;µ 0,...,µ ) j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) ) j + x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) 1 x µ j µ ti+1 ;µ 0,...,µ ) j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) ) j ) xt i+1 ;µ 0,...,µ j,...,µ i ) µ j + + t t i+1 φ t t i+1 τ,µ j ) yi+1 τ)dτ x f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 ))+φ τ,µ ) ) j z i+1 τ)dτ, 2.12)

63 where φ t,µ j) = 2. DIFFERENTIABILITY 45 f τ,x τ;µ 0,...,µ j,...,µ i,µ i+1 )) f τ,xτ;µ0,...,µ j,...,µ i,µ i+1 )) x τ;µ 0,...,µ j,...,µ i,µ i+1 ) xτ;µ0,...,µ j,...,µ i,µ i+1 ) x f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 )). Using 2.12), the condition H2.7, and Gronwall s Inequality, we have { 1 z i+1 t) x µ j µ ti+1 ;µ 0,...,µ ) j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) ) j xt i+1 ;µ 0,...,µ j,...,µ i ) µ j I i+1 x ti+1 ;µ 0,...,µ j +,...,µ ) i),µi+1 Ii+1 xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) x t i+1 ;µ 0,...,µ j,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) 1 x µ j µ ti+1 ;µ 0,...,µ j,...,µ ) i xti+1 ;µ 0,...,µ j,...,µ i ) ) j + x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) 1 x µ j µ ti+1 ;µ 0,...,µ j,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) ) j xt i+1 ;µ 0,...,µ j,...,µ i ) µ j t ) + φ τ,µ Yi+1 j dτ exp F + ) ) φ τ,µ j t ti+1 ) ), 2.13) t i+1 where the constant Y i+1 is such that y i+1 t) Y i+1 for t i+1 t t i+2. Having in mind the condition H2.8 and Theorem 2.1 from which follows continuous dependence of the solution xt;µ 0,...,µ j,...,µ i ) on the parameter µ j ) for t = t i+1 we derive lim µ j µ j ) ) I i+1 x ti+1;µ 0,...,µ j,...,µ i,µi+1 Ii+1xt i+1;µ 0,...,µ j,...,µ i),µ i+1) x ) t i+1;µ 0,...,µ j,...,µi xti+1;µ 0,...,µ j,...,µ i) x Ii+1xti+1;µ0,...,µj,...,µi),µi+1) = 0.

64 46 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY The next limits exists by induction presumption: x t i+1 ;µ 0,...,µ j lim,...,µ i) xti+1 ;µ 0,...,µ j,...,µ i ) µ j µ j µ j µ j = xt i+1 ;µ 0,...,µ j,...,µ i ) µ j and x ) t i+1 ;µ 0,...,µ j,...,µ i xti+1 ;µ 0,...,µ j,...,µ i ) lim µ j µ j µ j µ j xt i+1 ;µ 0,...,µ j,...,µ i ) µ j = 0. As the function f is continuous according to Theorem 2.1, then for x t i+1 < t t i+2 it is fulfilled ) lim φ t,µ j = µ j µj f τ,x τ;µ 0,...,µ j,...,µ )) i,µ i+1 f τ,xτ;µ0,...,µ j,...,µ i,µ i+1 )) x τ;µ 0,...,µ j,...,µ ) i,µ i+1 xτ;µ0,...,µ j,...,µ i,µ i+1 ) x f τ,xτ;µ 0,...,µ j,...,µ i,µ i+1 )) = 0. lim µ j µj So, we find the four limits above. From the equation 2.13) we derive lim z i+1 t) = 0. µ j µ j In this way, we obtain the existence of a continuous derivative µ j xt;µ 0,...,µ j,...,µ i,µ i+1 ) = y i+1 t), t i+1 < t t i+2, which satisfies the problem ) d xt;µ 0,...,µ j,...,µ i,µ i+1 ) dt µ j = x f t,xt;µ 0,...,µ j,...,µ i,µ i+1 )) ) xt;µ 0,...,µ j,...,µ i,µ i+1 ), µ j µ j xt i+1 ;µ 0,...,µ j,...,µ i,µ i+1 ) = µ j xt i+1 ;µ 0,...,µ j,...,µ i )

65 2. DIFFERENTIABILITY 47 + x I i+1xt i+1 ;µ 0,...,µ j,...,µ i ),µ i+1 ) µ j xt i+1 ;µ 0,...,µ j,...,µ i ). The theorem is proved.

66 48 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY

67 3. APPLICATION Application: Continuous Dependence and Differentiability of the Solutions of Logistic Model with Fixed Moments of Impulses on the Initial Condition and Impulsive Effects As a model demonstrating the of results in this chapter, we examine a dynamic development of isolated populations, which is subjected to an external influence. Usually this effect is expressed in the removal or more rarely) in the addition of a certain quantities of biomass of the studied population. The duration of these external interventions are negligible, compared with the total period of the dynamic evolution of the population and therefore, it could be considered that, they are realized instantaneously in the form of impulses. It is natural to assume that, the amount of biomass which is removed or added at any single external influence is bounded below. Moreover, the range at which the population recovers, i.e. the continuation between two neighboring impulsive moments is bounded below. An adequate mathematical model of such processes is the impulsive logistic equation. The corresponding initial problem looks like this: where: dn dt = r K N K N), t t i, 2.14) Nt i +0) = Nt i ) p i Nt i )+µ i ), i = 1,2,..., 2.15) N0) = µ 0, 2.16) N = N t) > 0 is an amount of biomass at the moment t 0; t 1,t 2,... arethemoments, atwhichtheimpulsive perturbations, 0 < t 1 < t 2 <... take place; K =const> 0 is the capacity of environment saturation level); r =const> 0 is the reproductive potential of population; µ 0,µ 1, R,0 < µ 0 < K; p 1,p 2, R,0 < p i < 1,i = 1,2,...; N 0 is the amount of biomass at the initial moment t = 0, 0 < N 0 < K. It is known that, for any value of the initial point µ 0, such that 0 < µ 0 < K, i.e. µ 0 0,K) = D, the solution N t) of the equation without impulses 2.14) with initial condition N t 0 ) = µ 0 is defined for each t t 0 0 and it satisfies the relation: 0 < N t) < K N t) D. 2.17)

68 50 2. CONTINUOUS DEPENDENCE AND DIFERENTIABILITY In that manner, we conclude that the condition H2.3 is valid. We assume that lim t i =, i.e. the condition H2.2 is satisfied. The condition i H2.1 is obvious. If the parameters µ i are sufficiently large µ i > N t i ), i = 1,2,...), the impulsive interruptions are reduced to the removals, because in the model considered it is valid I i N t i ),µ i ) = p i N t i )+µ i ) < 0. We assume that µ i 0, i = 1,2,..., i.e. M = R +. Therefore then N t i +0) = N t i ) p i N t i )+µ i ) < N t i ) < K. 2.18) If the parameters p i are sufficiently small, for example, if p i µ i < N t i ) p i < N t i) 1 p i µ i +N t i ), i = 1,2,..., N t i +0) = N t i ) p i N t i )+µ i ) > ) In other words, from 2.17), 2.18) and 2.19) and some restrictions of the parameters µ i and p i, i = 1,2,..., it follows that the following inequalities 0 < N t i +0) < K, i = 1,2,..., are valid, i.e. Id+I i ) : D R + D, i = 1,2,... Here Id is the identity of D R + and Id+I i ) = N t i ) p i N t i )+µ i )). For N D = 0,K) the function f which coincides with the righthand side of equation 2.14)) is a Lipschitz function on the argument N with Lipschitz constant L = r, because N f t,n) = d dn f N) = r d K N K N) dn = r K 2N K r. Thus the condition H2.4 is valid. It is clear that, the conditions H2.5, H2.6 and H2.8 are satisfied. The condition H2.7 is fulfilled if constant F = r.

69 3. APPLICATION 51 Thus, we find that, the basic results obtained in the previous section are valid for the impulsive model of the logistic equation. The interpretation of the obtained results is: Let us consider two identical isolated populations species), which obey the one and the same logistic law of growth. Let the permissible difference in the development dynamics of these populations be in the primary amount of their biomasses, and or) in the size of the impulsive deprivations or additions to their biomasses. Recall that, these external impulsive interventions take place in the same moments. For the convenience, first population is called regular and other is perturbed. The following statements are valid. 1) The amount of biomasses in the regular and perturbed type are differ insignificantly during a limited period of time in the case of small differences in the starting quantities of biomasses in both experiments, which obey to the same mathematical model. Also, the differences in the biomasses of both populations are controllable small even assuming small differences in the size between the impulsive deprivation or addition) of biomasses. 2) The biomass quantity in the regular logistic impulsive species, which is described by the above mentioned model, depends smoothly the biomass amount is differentiable) with respect to the initial amount of biomass and the sizes of impulsive removal or addition).

70

71 Chapter 3 Continuous Dependence and Uniform Stability of the Solutions of the Differential Equations with Variable Moments of Impulses on the Impulsive Hypersurfaces and Impulsive Effects A relatively popular class of nonlinear nonautonomous systems of differential equations with impulses is studied in this chapter. The impulses take place at the moments, in which the integral curve of the corresponding initial problem meets some of the predefined hypersurfaces. These hypersurfaces named impulsive hypersurfaces) are situated in the extended phase space of the system and they do not intersept eachother. For this class of equations the impulse moments are not constant, they depend both on the initial point of the problem and the impulsive hypersurfaces, the impulsive effects, etc. The first results on the fundamental and qualitative theory of the equations, represented in this chapter, belong to A. Samoilenko and N. Perestyuk see monograph of Samoilenko and Perestyuk [295]). The integral curve of initial problem of impulsive systems to meets repeatedly including infinitely many times) one and the same impulsive hypersurface is possible. This phenomenon is called beating. If such phenomenon exists, it is possible that some integral curve does not overcome the given supersurface, i.e. for each argument of the solution domain the integral curve lyies in the same half space on the hypersurface. Moreover, if insuperable hypersurface is bounded in the phase space, we conclude that the impulsive moments have a point of condensation. In this case the condensation point is a limit point of the sequence of these moments. Therefore, the solution of the problem studied is not extended after the limit point considered. In this case, we have another phenomenon disappearance of the solution. In the 53

72 54 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES situation, described above the investigation of important questions of asymptotic theory of impulsive differential equations, such as: continuous dependence, stability, periodicity etc. is difficult and in most cases even impossible. We pay attention to the fact that first results on asymptotic properties of the solutions of impulsive differential equations with variable moments of impulses in the presence of the phenomenon beating are obtained in Dishliev and Bainov [115]. In Section 1, sufficient conditions for the absence of this phenomenon is considered. Numerous results are devoted on this problem, we will cite Akhmet [16], Bainov et al. [61], Bajo [73], Chen et al. [93], Dishliev and Bainov [112, 116], Dishliev and Dishlieva [121], Hu et al. [179], Karandjulov and Stoyanova [200], Lakshmikantham [210], Perestyuk and Chernikova[277], Rao and Tsakos[284], Rao et al.[286], Samoilenko et al. [296], Stoykov[322], Zhang and Jin[364]. First, who have found sufficient conditions for the absence of this phenomenon are A. Samoilenko and N. Perestyuk. Later we will formulate their results, published in Samoilenko and Perestyuk [294]. It is eassy to guess that a hypersurface, albeit limited, has overcome if the integral curve meets finite number of times this hypersurface. In Section 3.1 the sufficient conditions for overcoming the impulsive hypersurfaces are found. The results in this section are based on the papers Dishliev and Bainov [111, 116]. The sufficient condition for continuous dependence of the solutions of the considered systems with impulses on the impulsive hypersurfaces are found in Section 3.2. In other words, some restrictions are formulated for the systems under which relatively small changes in the impulsive sets reflect insignificantly on the the solutions values in limited in advance time periods. The results in this section are based on the article Dishliev and Bainov [114]. The same nonlinear nonautonomous systems of differential equations with variable moments of impulsive effects, as in the previous two sections of this chapter are studied in Section 3.3. Some new results, concerning the topological characteristics of the systems solutions and unlimited extension of their solutions are found. The main outcomes consist in finding the relationship between the asymptotic properties of the solutions of impulsive systems and their corresponding systems without impulses. More precisely, the sufficient conditions are found under which, if the zero solution of the system without impulses is strictly uniformly stable or Lipschitz stable, then zero solution of the impulsive system is

73 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES 55 uniformly stable on the initial condition and the impulsive effects. The results of this section are based on the article Bainov et al. [49].

74 56 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES

75 1. ABSENCE OF THE PHENOMENON BEATING Sufficient Conditions for the Absence of the Phenomenon Beating We investigate, in this section, system of differential equations with impulses, which we describe in detail as follows: 1) Let σ i : t = τ i x), i = 1,2, ) are hypersurfaces in R n+1, where t R + and x R n. For each x from the domain D R n we assume that the following inequalities are satisfied 0 < τ 1 x) < τ 2 x) < ) 2) Let P t be a mooving point with coordinates t,xt)) which does not leave the set S = {t,x);t 0,x D} = R + D R n+1. The motion law the function xt)) is determined by: 2.1) The set of hypersurfaces σ i, i = 1,2,...; 2.2) The set of functions I i : D R n, for which are fulfilled x+i i x) D, x D, i = 1,2,.... The functions I i, i = 1,2,..., arenamedimpulsive functions; 2.3) The system of ordinary differential equations dx dt = f t,x), t i 1 < t t i, i = 1,2,..., 3.3) where f : S R n ; 2.4) The points t i, i = 1,2,..., satisfying the inequalities 0 = t 0 < t 1 < t 2 <..., are moments at which, mapping point P t meets the hypersurfaces 3.1). These points are called impulsive moments. Note that it is not obligatory for point P t to meet hypersurface σ i at the moment t i ; The equations x t=ti = xt i +0) xt i ) = I ji xt i )), i = 1,2,..., 3.4) where j i is a number of hypersurface which the moving point meets at the moment t i ), describe the impulses. As we mention above the inequality i j i is possible. Moreover, in general the regulation of numbers i and j i has not been determined.

76 58 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES 3) The system of objects and relations between them: 2.1, 2.2, 2.3, 2.4 and 2.5, characterizing the motion of poin P t, is named impulsive system of ordinary differential equations with variable moments of impulses, coincide with the moments of meeting between the integral curve and impulsive hypersurfaces. In a shorter, in this chapter, this system is called the system with impulses or impulsive system. The motion law is called a solution of system with impulses. 4) The initial point depicting the position is 0,x 0 ), where x 0 D. 5) P t moves along the integral curve t,xt)) of the system 3.3) with the initial condition x0) = x 0 until the moment t 1, at which meets the hypersurface σ 1 in the point t 1,x 1 ) = t 1,xt 1 )) σ 1. Then mapping point instantaneously jumps from position t 1,x 1 ) into position ) t 1,x + 1 = t 1,x 1 +I 1 x 1 )). Then it goes on moving along the integral curve t,xt))ofthesystem 3.3)withtheinitial valuecondition xt 1 +0) = x + 1 until the moment t 2,t 2 > t 1, at which the mapping point integral curve) meets the hypersurface σ j2, where j 2 is some of the numbers 1,2,...After that, it jumps again and so on. 6) The following case is possible: At the moment t i, point P t meets the hypersurface σ ji at the point t i,x i ). After its jump it falls again on the hypersurface of 3.1), i.e. the point ) t i,x + i = t i,x i +I ji x i )) belong to the same or another hypersurface of 3.1), i.e. there exists number k such that t i = τ k xt i )+I ji xt i ))). Inthiscase, we consider thatpoint P t continues itsmoving along the integral curve of system 3.3) with the initial value condition xt i +0) = x + i, 3.5) i.e. the mapping point does not jump twice or more than once at the same moment. 7) The solution of system with impulses is a piecewice continuous function with first order discontinuities points, at which it is continuous to the left. In the interval t i,t i+1 ] the solution of the system with impulses coincides with the solution of system 3.3) with the initial value condition 3.5), where t i,x i ) is the point at which P t meets the hypersurface σ ji see Figure 3.1).

77 1. ABSENCE OF THE PHENOMENON BEATING 59 x 2 D σ 1 σ 2 σ 3 x + 1 x 2 x + 3 x 0 x 1 x 1 x + 2 x 3 t 1 t2 t 3 t Figure 3.1. Case j 1 = 1, j 2 = 3, j 3 = 3,... The initial problem for impulsive system of differential equations described above, may be written compactly: dx dt = f t,x), t i 1 < t t i, i = 1,2,..., x t=ti = xt i +0) xt i ) = I ji xt i )), i = 1,2,..., x0) = x 0. The following notations are used: xt) = xt;0,x 0 ) is the solution of initial problem above; xt;t,x ) is the solution of system with impulses 3.3), 3.4) with initial condition x i = xt i ); x + i = x i +I ji x i ). xt +0) = x ; Let aa 1,a 2,...,a n ),bb 1,b 2,...,b n ) R n and let A,B R n be two nonempty sets. Then, we will use the following notations: The scalar product: a,b = a 1 b 1 +a 2 b 2 + +a n b n ;

78 60 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES The Euclidean norm: a = a,a 1 2 = a 2 1 +a a 2 n; The Euclidean distance between the points a and b: ρ E a,b) = a b = a 1 b 1 ) 2 +a 2 b 2 ) 2 + +a n b n ) 2 ; The Euclidean distance between the sets A and B: ρ E A,B) = inf{inf{ρ E a,b),b B},a A}; B δ = {x R n, x < δ}, where δ is a positive constant; If A, then B δ A) = {x R n ; x a < δ,a A}, B δ ) = ; A is the closure of set A; A is the boundary of set A. Example 3.1. Let: 1) n = 1; D =, ) = R; 2) τ i x) = arctanx+iπ, x D, i = 1,2,...; 3) Function f satisfies the sufficient conditions for existence and uniqueness of solutions of the equation without impulses dx dt = f t,x) for t 0. 4) x D) I i x) > 0, i = 1,2,... Then the integral curve t,xt)) = t,xt;0,x 0 )) beats upon the first impulsive hypersurface in this case the first impulsive curve) σ 1 = {t,x) ; t = τ 1 x) = arctanc+π, x D}. Moreover, in this case, the solution of the equation with impulses is defined in [ 0, 3 2 π) and it is noncontinuable on the right. For f x) = 0, I 1 x) = 1 and x 0 = π a part of the graph of solution xt;0, π) is shown in Figure 3.2. The following lemma is a corollary of Bolzano Theorem, concerning the continuous function in a finite and closed interval. Lemma 3.1. Let the following conditions hold: 1) Let g 1,g 2 C[D,R], where D is a domain in R n. 2) The inequality g 1 x) < g 2 x) is valid for x D.

79 1. ABSENCE OF THE PHENOMENON BEATING 61 x x 3 x + 2 x 2 x + 1 π 2 3π 0 t 1 = π t 2 t 3 t 2 xt;0, π) x 0 = π σ 1 Figure 3.2 3) Let x C[[T 1,T 2 ],D], where T 1,T 2 are real constants and T 1 < T 2. 4) T 1 = g 1 xt 1 )) T 2 = g 1 xt 2 ))). 5) T 2 > g 2 xt 2 )) T 1 > g 2 xt 1 ))). Then, thereexistsapointt 0,T 1 < T 0 < T 2,suchthatT 0 = g 2 xt 0 )). The following conditions are used: H3.1. f C[S,R n ], where S = R + D. H3.2. There exists a constant C f > 0, such that t,x) S) f t,x) C f. H3.3 There exist positive constans L i,l i < 1 C f, i = 1,2,..., such that x,x D) τ i x ) τ i x ) < L i x x.

80 62 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES H3.4. There exists the limit lim τ ix) =, i uniformly with respect to x D. H3.5. t 0,x 0) R + D), the initial value problem without impulses dx dt = f t,x), xt 0) = x 0 possesses a unique solution, defined in the interval [t 0, ). H3.6. The following implications x D) τ i x+i i x)) τ i x), i = 1,2,... are valid. The following theorem gives the sufficient conditions, under which the integral curve t,xt)) = t,xt;0,x 0 )) meets each of the impulsive hypersurfaces not more than once. Theorem 3.1. Let the conditions H3.1-H3.6 be valid. Then the integral curve t,xt)) meets each one of the hypersurface 3.1) not more than once. Proof. Let the integral curve of the system with impulses meets the hypersurface σ jk = {t,x);t = τ jk x),x D} at the moment t k, i.e. t k = τ jk x k ) = τ jk xt k )) = τ jk xt k ;0,x 0 )). Let for 0 < t < t k the inequality t τ jk xt)) be valid. In other words t k is the moment of the first meeting between the integral curve t,xt)) and hypersurface σ jk. Further, we show that for t > t k the inequality t τ jk xt)) is satisfied. Thus Theorem 3.1 will be proved. Let us assume the contrary to what is claimed, i.e. there exists a point t, t > t k, such that t = τ jk x ), where x = xt ) = xt ;0,x 0 ). The following cases take place: Case 1. Let for t k < t < t the integral curve t,xt)) does not meet the hypersurfaces of 3.1), i.e. t τ i xt)) for t k < t < t and i = 1,2,... Therefore t = t k+1, x = x k+1 and j k = j k+1 see Figure 3.3).

81 1. ABSENCE OF THE PHENOMENON BEATING 63 x 2 D σ jk = σ jk+1 x + x k k+1 x k x 1 0 t k t = t k+1 t Figure 3.3 Since for t k < t < t = t k+1, the solution of the system with impulses coincides with the solution of the integral equation xt) = x k +I jk x k )+ t t k f θ,xθ))dθ = x + k + t t k f θ,xθ))dθ, 3.6) then we have xt) x + k Cf t t k ). For t = t k+1, the inequality above becomes xk+1 x + k Cf t k+1 t k ). 3.7) On the other hand, using H3.6 we obtain the inequality τ jk x + k) = τjk x k +I jk x k )) τ jk x k ) = t k, 3.8) where for the difference t k+1 t k, the following estimate holds true ) t k+1 t k τ jk x k+1 ) τ jk x + k Ljk xk+1 x k +. Using the inequality above and condition H3.3, we derive t k+1 t k < 1 xk+1 x + k. 3.9) C f Obviously, the inequality 3.7) conflicts with 3.9). Hence, our assumtion is false. Case 2. Let for t k < t < t the integral curve t,xt)) meets the hypersurfaces σ jk+1,σ jk+2,... at the moments t k+1,t k+2,..., t k < t k+1 <

82 64 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES t k+2 < < t, respectively. These meetings may be finite number or infinitely many times. The following equalities are valid t k+l = τ jk+l x k+l ), l = 1,2,... Assume that, σ jk+1 is the first impulsive hypersurface, which the integral curve t,xt)) meets for t > t k, hypersurface σ jk+2 is the second, etc. We show that j k+1 > j k. Let us suppose the contrary, i.e. j k+1 < j k. The equality j k+1 = j k was considered in the previous case. With the assumption above made we have τ jk+1 x) < τ jk x), x D. 3.10) Let t be an arbitrary point, such that t k < t < t k+1 see Figure 3.4). x 2 D x + k+1 σ jk+1 σ jk x 1 x k x + k 0 t k t t t k+1 t Figure 3.4 The function x = xt) is a solution of equation 3.6) in the closed interval [t,t k+1 ], then it is continuous in this interval. Therefore, t k+1 = τ jk+1 xt k+1 )). 3.11) Using 3.8) and condition H3.3, we have ) τ jk xt )) t k τ jk xt )) τ jk x + 1 k < xt ) x + k. 3.12) C f The equation 3.6) and condition H3.2 lead to the inequality xt ) x + k Cf t t k ).

83 1. ABSENCE OF THE PHENOMENON BEATING 65 By means of 3.12), we obtain: whence τ jk xt )) t k < t t k, τ jk xt )) < t. 3.13) By Lemma 3.1 and taking into acount 3.10), 3.11) and 3.13), we deduce that there exists a point t t,t k+1 ) t k,t k+1 ) such that, the following equality is valid t = τ jk+1 xt )), i.e. the integral curve t,xt)) meets the hypersurface σ jk at the moment t. The resulting conclusion conflicts to the assumption made before: σ jk+1 is the first hypersurface met by integral curve for t > t k. Therefore, the inequality j k+1 > j k is true. Similarly, we may show that j k+2 > j k+1 and so on, i.e. the inequalities: j k < j k+1 < j k+2 < ) are valid. Let for t k < t < t the integral curve t,xt)) meets finite number of hypersurfaces and let the last one be σ jk+l. Next impulsive moment after t k+l ) is t, at which t,xt)) meets the hypersurface σ jk. Therefore, it is satisfied j k > j k+1, which contradict 3.14). Let for t k < t < t the integral curve meets infinity many hypersurfaces. Then the following inequalities τ jk+l x k+l ) = t k+1 < t, l = 1,2, ) arevalid. UnderconditionH3.4thesequenceoffunctionsτ 1 x),τ 2 x),... increases unboundly uniformly with respect to x D). So, there exists a number k 0 such that τ i x) > t, x D 3.16) for each i > k 0. Let l be a sufficiently large integer, such that j k+l > k 0. By 3.5), we conclude that inequality τ jk+l x k+l ) < t is valid, which contradicts to 3.16). Therefore, the theorem is proved.

84 66 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES We will show that the result of A. Samoilenko and N. Perestyuk, published in Samoilenko and Perestyuk [294], is a consequence of the previous Theorem. Corollary 3.1 A. Samoilenko and N. Perestyuk). Let the following conditions hold: 1) The conditions H3.1, H3.2, H3.4 and H3.5 are satisfied; 2) Let I i C[D,R n ], i = 1,2,...; 3) Let τ i C 1 [D,R n ] and the following inequalities are valid τ i x) x 1, x D, i = 1,2,...; C f 4) The following inequalities are fulfilled τi x+θi i x)),i i x) 0, x D, i = 1,2,... x sup 0 θ 1 Then the integral curve t,xt)) meets each one of the hypersurfaces 3.1) no more than once. Proof. We will demonstrate that Theorem 3.1 yields Corollary 3.1. Indeed, condition H3.3 follows by condition 3 of the corollary. We may obtain condition H3.6 by condition 4 of the statements, as follows: Let the functions τ i, i = 1,2,..., be continuously differentiable in D. Assume that the next inequality is satisfied τ i x+i i x)) > τ i x). which is opposite of the inequality in condition H3.6). Let us introduce the function g i θ) = τ i x+θi i x)), defined for 0 θ 1. The function g i is continuously differentiable in its domain and the inequality g i 0) < g i 1) is hods true. Therefore, there exists a point θ, 0 θ 1, such that d dθ g iθ ) > 0, i.e. d dθ g τi x+θ I i x)) iθ ) =,I i x) > 0. x The last inequality contradicts condition 4 of Corollary 3.1. Therefore, the corollary is proved. Corollary 3.2. Let the following conditions be fulfilled: 1) The conditions H3.1, H3.2 and H3.5 are valid;

85 1. ABSENCE OF THE PHENOMENON BEATING 67 2) k N) : x,x D) τ k x ) τ k x ) < 1 C f x x ; 3) The point t 0,x 0 ) S satisfies the inequality τ k x 0 ) t 0. Then thefirsthypersurface of3.1), which ismet bytheintegralcurve t,xt;t 0,x 0 )) of the problem with impulses, has a number greater than k. The proof of Corollary 3.2 is a part of the proof of Theorem 3.1. Corollary 3.3. Let the following conditions be valid: 1) The conditions H3.1, H3.2 and H3.5 are valid; 2) j k N) : x,x D) τ jk x ) τ jk x ) < 1 C f x x. 3) The following inequalities are satisfied τ i+1 x) > τ i x+i i+1 x)), x D, i j k. 3.17) Then the integral curve t,xt)) of the system with impulses meets the hypersurface σ jk no more than once. Proof. Let t k be the first moment, at which integral curve t,xt)) meetsthehypersurfaceσ jk,i.e. for0 < t < t k theinequalityt τ jk xt)) is valid. Let σ jk+1 be the next hypersurface, met by the integral curve at the moment t k+1, t k+1 > t k. Using Corollary 3.2, we conclude that the inequality j k+1 > j k is fulfilled. From 3.17) we find τ jk x+ijk+1 x) ) < τ jk +1 x)) x+ijk+1 So, for x = x k+1, we have < < τ jk+1 1 x+ijk+1 x)) < τ jk+1 x). τ jk x + k+1) < tk ) The integral curve t,x t;t k+1,x + k+1)) of the system with impulses satisfies: it coincides with t,xt)) for t > t k+1 ; its initial point is t k+1,x + k+1) ; the inequality 3.18) is fulfilled for this initial point. Then, taking into account Corollary 3.2 again, we deduce that the next hypersurface, which the integral curve t,x t;t k+1,x + k+1)) meets, has a number greater than j k, i.e. j k+2 > j k. In fact, this is the hypersurface σ jk+2.

86 68 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES By induction we deduce: If the integral curve t,xt)) meets the hypersurface σ jk+1 at the moment t k+1, t k+1 > t k, and j k+1 > j k, then the next hypersurface, which the integral curve meets, has a number greater than j k. By the last implication, we reach the conclusion, that is the integral curve does not meet the hypersurface σ jk for t > t k. The corollary is proved. Corollary 3.4. Let the following conditions be fulfilled: 1) The conditions H3.1, H3.2 and H3.5 are valid. 2) j k N) : x,x D) τ jk x ) τ jk x ) < 1 C f x x. 3) The following inequality is satisfied sup{τ i x);x D} inf{τ i+1 x);x D} for i j k. Then the integral curve t,xt)) of the system with impulses meets the hypersurface σ jk no more than once. Corollary 3.5. Let the following conditions be fulfilled: 1) The conditions H3.1, H3.2 and H3.5 are valid. 2) The hypersurfaces 3.1) are hyperplanes and they have the form σ i = {t,x);t = τ i x) = a i,x +α i }, i = 1,2,..., where: 2.1) a i R n and a i < 1 C f ; 2.2) a i,x 0, x D; 2.3) 0 < α 1 < α 2,... and lim i α i = ; 2.4) 0 < τ 1 x) < τ 2 x) < 0 < a 1,x +α 1 < a 2,x + α 2 <..., x D. 3) The following inequalities are valid a i,i i x) 0,x D, i = 1,2,... Then the integral curve t,xt)) meets each one of the hypersurfaces 3.1) in the case - hyperplanes) no more than once. Proof. We will show the validity of conditions H3.3, H3.4 and H3.6. Therefore according to Theorem 3.1) the corollary will be proved. Let x and x be two points in D. Then

87 1. ABSENCE OF THE PHENOMENON BEATING 69 τ i x ) τ i x ) = a i,x x ) a i x x 1 C f x x, i.e., the condition H3.3 is fulfilled. It is obvious that lim τ ix) = lim a i,x +α i ) lim α i =, x D. i i i That is the condition H3.4 is valid. From condition 3 of the corollary, we derive i = 1,2,..., τ i x+i i x)) = a i,x+i i x) +α i = τ i x)+ a i,i i x) τ i x), So, the condition H3.6 is established. Thus the corollary is proved. i = 1,2,... Corollary 3.6. Let the following conditions be fulfilled: 1) The conditions 1, 2.1, 2.2, 2.3 and 3 of Corollary 3.5 are valid. 2) The implication x D) x < d holds true, where: d =, { if a 1 = a 2 =...; } αi+1 α i inf a i+1 a i ; a i+1 a i > 0, i = 1,2,..., if i N) : a i = a i+1. Then the integral curve t,xt)) meets each one of the hypersurfaces 3.1) no more than once. Proof. We show that the condition 2.4 of Corollary 3.5 is satisfied. Case 1. Let a 1 = a 2 =... Then τ i+1 x) τ i x) = a i+1,x +α i+1 a i,x α i =α i+1 α i > 0, i = 1,2,.... Case 2. Two subcases are possible: Case 2.1. The equality a i = a i+1 is valid. Then similarly to Case 1, we obtain that τ i x) < τ i+1 x). Case 2.2. Let a i a i+1. Then for each x D, i.e. x < d, it follows τ i+1 x) τ i x) = a i+1,x +α i+1 a i,x α i = a i+1 a i,x +α i+1 α i a i+1 a i x +α i+1 α i

88 70 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES ) αi+1 α i = a i+1 a i a i+1 a i x a i+1 a i d x ) > 0. According to Corollary 3.5, the integral curve of the considered intial problem of impulsive differential equations meets every one of the impulsive hypersurfaces 3.1) no more than once. The corollary is proved. Theorem 3.2. Let the following conditions be fulfilled: 1) The conditions H3.1-H3.6 are valid; 2) The following implications x D) τ i x) < τ i+1 x+i i x)), i = 1,2,... hold true. Then the integral curve t,xt)) meets each one of the hypersurfaces 3.1) just once. Proof. We show that t, xt)) meets the first impulsive hupersurface σ 1. Assume that the integral curve does not meet this hypersurface. Then the next inequality is valid t < τ 1 xt)), t 0. Indeed, assuming that the above inequality is not true, it follows that, there is a point t such that t τ 1 xt )). Let us set g 1 x) = 0, g 2 x) = τ 1 x) in the condition of Lemma 3.1. Then xt) coincides with the solution of the corresponding initial problem for the impulsive system; T 1 = 0 and T 2 = t. Therefore, the integral curve meets the hupersurface σ 1 at the some moment, between 0 and t. The conclusion obtained contradicts the assumption. We have t τ 1 x 0 ) < τ 1 xt)) τ 1 x 0 ) L 1 xt) x0) L 1 C f t. where t 0. Suppose, there exist t > 0 such that τ 1 xt)) τ 1 x 0 ). Then t τ 1x 0 ) 1 L 1 C f = const. The result contradicts the fact, that inequality t < τ 1 xt)) holds true for any t 0.

89 Hence, 1. ABSENCE OF THE PHENOMENON BEATING 71 τ 1 xt)) < τ 1 x 0 ), t > 0. So, if we choose t > τ 1 x 0 ), we get back to the conflict with the assumption made t < τ 1 xt)), t 0). Consequently, the integral curve of the impulsive system meets hypersurface σ 1. By analogy with the above-mentioned considerations, we conclude that the integral curve has infinitely many common points with the hupersurfaces of 3.1). Let t, xt)) meets consecutively the hupersurfaces σ j1,σ j2,... at the moments t 1,t 2,..., t 1 < t 2 <..., respectively. Since the condition of Theorem 3.1 are valid, then 1 < j 1 < j 2 <... Having in mind Theorem 3.1 again, it follows that the integral curve meets no more than once every hypersurface. Assume that, it does not meet the hypersurface σ j. Then there exists according to the inequality above) number k such that j k < j < j k+1. The integral curve t,x t;t k,x k)) +, coincides with t,xt)) for t > tk. For the initial point of the integral curve considered is satisfied ) ) τ j x + k > τj 1 x + k > > τ jk +1 x + k) = τjk +1x k +I jk x k )) > τ jk x k ) = t k. The following inequalities are valid, also t k+1 = τ jk+1 x k+1 ) > τ jk+1 1x k+1 ) > > τ j x k+1 ). Let: g 1 x) = τ j x) τ j x + k) +tk ; g 2 x) = τ j x); xt) = x t;t k,x k) + ; T 1 = t k and T 2 = t k+1. We will check all condition of Lemma 3.1. It is obvious that g 1,g 2 C[D,R] and x C[[T 1,T 2 ],D]. Moreover, we have ) g 1 x) = τ j x) τ j x + k +tk < τ j x) = g 2 x). Therefore, ) T 1 =t k = τ j x + k τj x + k) +tk )) =τ j x tk ;t k,x + k τj x + k) +tk ) =τ j xt k )) τ j x + k +tk

90 72 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES and =g 1 xt 1 )) T 2 =t k+1 > τ j x k+1 ) =τ j x tk+1 ;t k,x + k =g 2 xt k+1 )) =g 2 xt 2 )). According to Lemma 3.1 it follows that, there exists a point t, t k < t < t k+1, such that t = τ j xt )). The last equality contradicts the assumption made before. The theorem is proved. To continue our further investigations, we need some preliminary constructions. We introduce an algorithm, creating a sequence of points in R n. )) U y 1 = y y 2 = y 1 +Xy 1 ) y 3 = y 2 +Iy 2 ) y 4 = y 3 +Xy 3 ) y 5 = y 4 +Iy 4 ) Figure 3.5 Algorithm 3.1. Let y R n and U R n is a neighborhood of y y U). Let I,X : U R n. The sequence of points y 1,y 2,... is defined by the following rules:

91 1. ABSENCE OF THE PHENOMENON BEATING 73 1) y 1 = y ; 2) If y k U and k is odd, then y k+1 = y k +X y k ); 3) If y k U and k is even, then y k+1 = y k +Iy k ); 4) If y k / U, it does not generate more terms of the sequence and the algorithm ends see Figure 3.5). Lemma 3.2. Let the following conditions be satisfied: 1) y R n and U R n is a bounded neighborhood of y ; 2) There exist constants c and, c > 0, 0 < < 1, such that y U) c I y) c and c Xy) c. 3) It is fulfilled α = const,α > ) y 0 R n, y 0 = 1) : Then the sequence constructed by Algorithm 3.1, is finite. y U) α I y) y 0,Iy). y 1,y 2,..., 3.19) Proof. Assume the contrary, i.e. the sequence 3.19) is infinite. We will show that for every integer k 2 the following inequalities are valid: y 2k y ckα )+c and y 2k+1 y ckα ). We obtain the first inequality in 3.20) as follows: y 2k y y 2k y 2k 1 +y 2k 1 y 2k 2 + +y 2 y y 2k y 2k 1 +y 2k 2 y 2k 3 + +y 2 y y 2k 1 y 2k 2 +y 2k 3 y 2k 4 +y 3 y 2 = I y 2k 1 )+I y 2k 3 )+ +Iy 1 ) X y 2k 2 )+Xy 2k 4 )+ +Xy 2 ) y 0,Iy 2k 1 )+Iy 2k 3 )+ +Iy 1 )) X y 2k 2 ) Xy 2k 4 ) X y 2 ) = y 0,Iy 2k 1 ) + y 0,Iy 2k 3 ) + + y 0,Iy 1 ) X y 2k 2 ) Xy 2k 4 ) X y 2 ) α Iy 2k 1 ) + Iy 2k 3 ) + + Iy 1 ) ) X y 2k 2 ) + Xy 2k 4 ) + + X y 2 ) ) 3.20)

92 74 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES αkc k 1)c = ckα )+c. The second estimate in 3.20) holds true, because: y 2k+1 y = y 2k+1 y 2k +y 2k y 1 y 2k y 1 y 2k+1 y 2k ckα k 1) ) X y 2k ) ckα k 1) ) c ckα ). Since the neighborhood U is bounded and y U, it follows that C = const. > 0) : y R n, y y > C) y / U. If the integer k is sufficiently large, 3.20) yields: y 2k y > C and y 2k+1 y > C. It means that, the points y 2k and y 2k+1 does not lie in the neighborhood U. The last inequality is a contradiction to the assumption that the sequence 3.19) is infinite. The lemma is proved. The following theorem establishes the sufficient conditions, such that the integral curve t,xt)) meets finite number each one of the impulsive hypersurfaces 3.1). Theorem 3.3. Let the following conditions be satisfied: 1) The conditions H3.1, H3.2, H3.4 and H3.5 are valid. 2) There exist constants c i and C i, 0 < c i C i, i = 1,2,..., such that x D) c i I i x) C i. 3) There exist positive constants L i, c i 0 < L i < C f c i +C i ), i = 1,2,..., such that x,x D) τ i x ) τ i x ) < L i x x. 4) The inequalities τ i+1 x) > τ i x+i i+1 x)), x D, i = 1,2,... are valid. 5) For i = 1,2,..., and for each point x D, one of the both conditions is satisfied:

93 1. ABSENCE OF THE PHENOMENON BEATING ) τ i x+i i x)) τ i x); 5.2) There exist the neighborhoods U i x),v i x) and W i x) see Figure 3.6) of the point x such that: 5.2.1) x U i x) V i x) W i x); 5.2.2) V i x) is bounded; 5.2.3) ρr n \V i x),u i x)) c i +C i ; 5.2.4) ρr n \W i x),v i x)) c i +C i ; 5.2.5) x V i x)\u i x)) x W i x)\v i x)) τ i x ) τ i x ); 5.2.6) α i x),α i x) > i = C fl i C i c i 1 L i C i ) y i x) R n, y i x) = 1) : y V i x)) α i x) I i y) y i x),i i y). x x U i x) V i x)w i x) 0 σ i : t = τ i x) t Figure 3.6

94 76 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES Then the integral curve t,xt)) meets each one of the hypersurfaces 3.1) finite number times. Proof. Let the integral curve meets the hypersurface σ jk at the moment t k, i.e. the next equalities are satisfied t k = τ jk xt k )) = τ jk x k ). Let t k be the first moment, at which the integral curve meets the hypersurface σ jk. We show that t,xt)) meets σ jk a finite number times. Let σ jk+1 be the next hypersurface, which the integral curve meets at the moment t k+1, t k+1 > t k, i.e. we have t k+1 = τ jk+1 x k+1 ), t τ i xt)), t k < t < t k+1, i = 1,2,.... The following cases are possible: Case 1. Let j k+1 < j k. For the initial point of the integral curve t,xt;t k, x + k )) coinciding with a part of t,xt))) by condition 4 of the theorem, it follows that ) ) ) τ jk+1 x + k < τjk+1+1 x + k < < τjk 1 x + k = τ jk 1x k +I jk x k )) < τ jk x k ) = t k. 3.21) Using Corollary 3.2 and inequality 3.21), we derive that the hypersurface, met by the integral curve after σ jk, has a number, greater than j k. This conclusion contradicts the assumption made in this case. Case 2. Let j k+1 > j k. We consider the integral curve t,xt;t k+1, x + k+1 )), which coincides with t,xt)) for t > t k+1. According to condition 4 of Theorem 3.3, the initial point t k+1,x k+1) + satisfies the next inequalities τ jk x + k+1) < τjk +1 ) x + k+1) < < τjk+1 1 x + k+1 = τ jk+1 1 xk+1 +I x jk+1 k+1) ) < τ x jk+1 k+1) = t k ) By Corollary 3.2 and inequality 3.22) we obtain the conclusion that the next hypersurface of 3.1), which t,xt)) meets after σ jk+1, has a number j k+2, which is greater than j k. Let the next impulsive hypersurface of 3.1), which the integral curve meets after σ jk+2, has a number j k+3. Then, taking into account the case 1 the considerations are analogously, after the substitution j k and j k+1 by j k+2 and j k+3 respectively) we get that inequality j k+3 < j k+2 is impossible. Therefore, it is true the opposite inequality j k+3 j k+2, whence we find that j k+3 j k. By induction it can be shown, that the integral curve t,xt)) for t > t k

95 1. ABSENCE OF THE PHENOMENON BEATING 77 meets the impulsive hypersurfaces with numbers greater than j k. This means that t,xt)) meets σ jk only once at the moment t k ). Case 3. Let j k+1 = j k. It is clear that the variant 5.2 of condition 5 of Theorem is valid. If we assume that the condition 5.1 is satisfied, according to Theorem 3.1 the integral curve meets each hypersurface the most once, i.e. this option is excluded from the assumption made). In case 1 was shown that for t > t k the integral curve does not meet the impulsive hypersurface with number less than j k. In case 2 was found that, if for t > t k integral curvet,xt)) at a given moment meets a hypersurface with number greater than j k, then it does not meet the hypersurface σ jk for the second time. In additional we assume that the integral curve meets consistently infinitely many times hypersurface σ jk, i.e. the following equalities j k = j k+1 = j k+2 =... are valid. Let the moments of the meetings are t k,t k+1,t k+2,... consecutively and t k < t k+1 < t k+2 < ) Since for t k+l 1 < t t k+l, l = 1,2,..., the function xt) is a solution of the integral equation xt) = x k+l 1 +I jk x k+l 1 )+ then condition H3.2 gives us a reason to estimate t t k+l 1 f θ,xθ))dθ, xt) x k+l 1 I jk x k+l 1 ) C f t t k+l 1 ). From the previous result and using the fact that j k+l 1 = j k for t = t k+l, we obtain xk+l x + k+l 1 Cf t k+l t k+l 1 ). 3.24) Let t = τ jk x + k+l 1). Then t t k+l 1 = τ jk x k+l 1 +I jk x k+l 1 )) τ jk x k+l 1 ) L jk I jk x k+l 1 ) L jk C jk. 3.25) Using 3.24) and 3.25), we have t k+l t = ) τjk x k+l ) τ jk x + k+l 1 L xk+l jk x + k+l 1 Ljk C f t k+l t +t t k+l 1 ) L jk C f t k+l t +L jk C jk ).

96 78 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES Therefore, t k+l t C fl 2 j k C jk 1 C f L jk. 3.26) From 3.24), 3.25) and 3.26) we derive xk+l x + k+l 1 Cf t t k+l 1 + t k+l t ) C f L jk C jk + C fl 2 ) j k C jk 1 C f L jk i.e. = C fl jk C jk 1 C f L jk, xk+l x + k+l 1 jk c jk. 3.27) The inequality jk > 0 is valid, because L jk < c jk C f c jk +C jk ) < 1 C f 1 C f L jk > 0. It is easy to prove that jk < 1. Indeed, using the last inequality of condition 5.2 of the theorem, we have jk I jk y) <α jk x) I jk y) y jk x),i jk y) y jk x). I jk y) = I jk y). Now, weapplylemma3.2forthesequenceofpointsx k,x + k,x k+1,x + k+1,... For this purpose, we replace: y by x k ; the neighbourhood U by U jk x k ); the neighbourhood V by V jk x k ); the neighbourhood W by W jk x k ); constant by jk, constant c by c jk ; the function I by I jk ; X x k+l 1) + by xk+l x + k+l 1, l = 1,2,..., respectively. All conditions of Lemma 3.2 are satisfied according to: the condition 2 of the theorem; the last requirement of condition 5 of the theorem; inequality 3.27).

97 1. ABSENCE OF THE PHENOMENON BEATING 79 Therefore, there exists sufficiently large integer l such that x k+l / U jk x k ). Precisely, as x k U jk x k ), then there exists number l such that x k+l 1 U jk x k ). 3.28) As x k+l x k+l 1 xk+l x + k+l 1 + x + then from 3.27), and the inequality x + = Ijk x k+l 1) C jk, we get k+l 1 x k+l 1 k+l 1 x k+l 1, x k+l x k+l 1 C jk + jk c jk < c jk +C jk. 3.29) Using 3.28), 3.29) and condition 5 of the theorem, we obtain x k+l V jk x k )\U jk x k ). 3.30) We apply Lemma 3.2 again for the sequence x k+l,x + k+l,x k+l +1,x + k+l +1,... The following substitutions are made with this aim: y with x k+l ; the neighbourhood U with U jk x k ); the neighbourhood V with V jk x k ); the neighbourhood W with W jk x k ); the constant with jk, the constant c with c jk ; the function I with I jk ; X x + k+l +l 1) with xk+l +l x + k+l +l 1, l = 1,2,..., respectively. We find out, that there is a number l, such that the point x k+l +l / V jk x k ). Furthermore, it is possible to select the number l, so that x k+l +l W j k x k )\V jk x k ). 3.31) Taking into consideration 3.30), 3.31), and condition 5 of the theorem, we obtain the inequality τ jk x k+l +l ) τ j k x k+l ), or equivalently t k+l +l t k+l. The inequality above contradicts 3.23), i.e. the integral curve t, xt)) meets the impulsive hypersurface σ jk finite number of times. Theorem 3.3 is proved.

98 80 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES Corollary 3.7. Let the following conditions be satisfied: 1) The conditions 1, 2, 3 and 5 of Theorem 3.3 are valid. 2) The inequalities sup{τ i x);x D} < inf{τ i+1 x);x D}, i = 1,2, ) are fulfilled Then the integral curve t,xt)) meets each of the the hypersurfaces 3.1) finite number of times. We will discuss some examples, illustrating Theorem 3.3. Example 3.2. Let: 1) n = 1 and D =, ) = R; 2) The impulsive curves τ i are defined by τ i x) = τ x)+2i, where x D, i = 1,2,..., and τ x) = x 2l,x [2l 1,2l+1), l Z; 3) Let f t,x) = 0, t,x) S = R + R; 4) Let I i x) = I x) = 1, x D, i = 1,2,...; 4 5) The initial point is x 0 = 0. We will check that the initial problem in this example satisfies all conditions of Theorem 3.3. Indeed: Condition 1 of Theorem 3.3. The conditions H3.1, H3.4 and H3.5 are obvious. Condition H3.2 is fulfilled for constant C f = 1 4. Condition 2 of Theorem 3.3. The condition is fulfilled for c i = 1 4 and C i = 1 3, i = 1,2,...; Condition 3 of Theorem 3.3. The condition holds true for L i = 1 < 9 = c i, i = 1,2,...; 7 C f c i +C i ) Condition 4 of Theorem 3.3. For this condition is valid more restrictive inequality 3.32); Condition 5 of Theorem 3.3. Let k be an integer. We denote: If x 2k 1,2k ), then: U i x) = 2k 1 2,2k+ 1 ), 2 V i x) = 2k 3 2,2k+ 3 ), 2

99 1. ABSENCE OF THE PHENOMENON BEATING 81 W i x) = 2k 5 2,2k+ 5 ) ; 2 If x [ 2k + 1,2k ], then: U i x) = 2k 1 2,2k+ 5 ), 2 V i x) = 2k 3 2,2k+ 7 ), 2 W i x) = 2k 5 2,2k+ 9 ) ; 2 For every x D, y i x) is one-dimensional unit vector, i.e. y i x) = 1), x D, i = 1,2,...; α i x) = 1 > 2 3 = C fl i C i c i 1 L i C i ) = i. Using definition above, we check directly conditions 5.2. All conditions of Theorem 3.3 are valid and therefore the integral curve meets each one of impulsive curves σ i,i = 1,2,..., finite number of times. For example, the meetings points between the integral curve and first impulsive curve σ 1 are: T l = t l,x l ) = 2+ l 1 4, l 1 4 ), l = 1,2,...,5, consecutively. After that t,xt)) meets σ 2 in the point 19 T 6 = t 6,x 6 ) = 4, 5 ). 4 The inequality 5.1 of Theorem 3.3 is valid for T 6. The conditions of Theorem 3.1 are valid in this case too. This means that, the integral curve meets σ 2 only once at the point T 6 ) and so on see Figure 3.7). Example 3.3. Let us change the functions I i, i = 1,2,... in the conditions of Example 3.3. More precisely, let substitute function Ix) = from Example 3.2 with new one 1 4 Ix) = x) = I ix), i = 1,2,... Thenthecondition2ofTheorem3.3doesnotvalid. Itispossibletoverify directly that the integral curve meets infinity many times first impulsive hypersurface in the case impulsive curve). The direct calculations show

100 82 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES x T 6 T 3 T 4 T 5 T 2 0 = x 0 T 1 t σ 1 σ 2 Figure 3.7 that, the points of meetings between the integral curve t,xt)) and impulsive curve σ 1 are T i = t i,x i ) = i,1 2 1 i), i = 1,2,... It turns out that the first impulsive curve is insuperable and the solution is not continuable for t 3. Example 3.4. Ifwechangethefunctionτ intheimpulsivesystemin Example 3.2 with new one τ x) = x, then the conditions 5 of Theorem 3.3 could not be met. The direct calculation shows, that the integral curve meets infinity many times the first impulsive curve σ 1 = {t,x);t = x +2,x R}.

101 1. ABSENCE OF THE PHENOMENON BEATING 83 Example 3.5. If in the system with impulses from Example 3.2, we change only impulsive functions I i, i = 1,2,..., with 3x+1, if x > 0; I i x) = 2 3x 1, if x 0, 2 then the last implication of the conditions 5.2 of Theorem 3.3 will not be met. In this example the integral curve meets infinity many times the first impulsive curve σ 1 = {t,x);t = τ 1 x) = x 2l +2, x [2l 1,2l+1), l Z}.

102 84 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES

103 2. CONTINUOUS DEPENDENCE Continuous Dependence of the Solutions of the Differential Equations with Variable Impulsive Moments on the Impulsive Hypersurfaces We continue our investigation with the systems of impulsive differential equations with variable impulsive moments. Let us remind that, the impulses are realized in intersection points of integral curve corresponding to the initial problem of the system) with one of the given hypersurfaces σ i = { t,x);t = τ i x),t R +,x D }, i = 1,2, ) Here D is a domain of R n and τ i : D R +. We research again the same initial problem for the systems of nonlinear nonautonomous ordinary differential equations with impulses, which was considered in the previous section. dx dt = f t,x), t t i t τ i xt)), 3.34) xt) t=ti = xt i +0) xt i ) = I ji xt i )), i = 1,2,..., 3.35) x0) = x 0, 3.36) where: f : S R n,s = R + D; here and further on, t 1,t 2,..., 0 < t 1 < t 2 <..., 3.37) denote the moments, at which the integral curve t,xt)) of problem3.34),3.35),3.36) meets some hypersurfaces of3.33); j i is a number of hypersurface, which the integral curve of problem considered meets at the moment t i ; I i : D R n are so called impulsive functions; xt i ) = xt i 0),i = 1,2,...; x 0 D. The stated problem was described in detail in the Section 3.1. Here we remind only that in the general case i j i. We introduce some additional notations: τ 0 x) = 0 for x D and x + 0 = x 0; Ω i δ) = {t,x);τ i 1 x)+δ < t < τ i x) δ},i = 1,2,...,where δ is a positive constant. Theorem 3.4. Let the following conditions be satisfied: 1) The conditions H3.1 and H3.5 are valid.

104 86 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES 2) There exists a number i N, such that the points t and t satisfy the inequalities t i 1 < t t i and t i 1 < t t i. Then, the following inequality is valid for all i = 1,2,... τ i xt )) t )τ i xt )) t ) 0. Proof. Asdsume that the statement of the theorem is not true. Then, there exists a number k such that the inequality τ k xt )) t )τ k xt )) t ) < ) is valid. Assume that t < t the equality t = t contradicts 3.38)). We consider the function φt) = τ k xt)) t for t t t. The function φ is continuous in its domain according to condition 2 of Theorem 3.4. By 3.38)wegetφt )φt ) < 0. Therefore, thereexistsapointt,t < t < t, such that φt ) = 0, i.e. the integral curve t,xt)) ofproblem 3.34), 3.35), 3.36) meets the hypersurface σ k at the moment t. Therefore, there exists number i such that t i 1 < t < t i. This result contradicts 3.37) and the fact, that the integral curve meets the hypersurfaces of 3.33) only at the moments t 1,t 2,... The proof of theorem is completed. Theorem 3.5. Let the following conditions be satisfied: 1) The conditions H3.1-H3.5. are valid; 2) t,x ) Ω i 0); Thenthefirstmeetingofintegralcurvet,xt;t,x ))ofsystem3.34), 3.35) with initial condition xt ) = x for t > t is the hypersurface σ i. Proof. Assume that the first hypersurface, which the integral curve t,xt; t,x )) meets for t > t is σ k. Let the meeting be realized at the moment t, i.e. τ k xt ;t,x )) = t. 3.39) Assume also that k < i 1. As we have t,x ) = t,xt ;t,x )) Ω i 0), then τ i 1 xt ;t,x )) t < ) From 3.39) and the assumption made before, it follows that τ i 1 xt ;t,x )) t = τ i 1 xt ;t,x )) τ k xt ;t,x )) > 0.

105 2. CONTINUOUS DEPENDENCE 87 Taking into account 3.40), the inequality above, and the fact t k 1 < t < t = t k, we reach the contradiction with Theorem 3.4. Now, let us assume that k > i. Using the condition t,x ) Ω i 0) we obtain On the other hand, we have τ i xt ;t,x )) t > ) τ i xt ;t,x )) t = τ i xt ;t,x )) τ k xt ;t,x )) < 0. The inequality above and inequality 3.41) contradict Theorem 3.4. Let us suppose that k = i 1. Since for t t t, the solution of system 3.34), 3.35) with the initial condition xt ) = x coincides with the solution of the integral equation xt;t,x ) = xt ;t,x )+ then by condition H3.2 for t = t, we derive t t xθ;t,x )dθ, xt ;t,x ) xt ;t,x ) C f t t ). 3.42) On the other hand using condition H3.3 we find t t <τ i 1 xt ;t,x )) τ i 1 xt ;t,x )) L i 1 xt ;t,x ) xt ;t,x ) < 1 C f xt ;t,x ) xt ;t,x ), which contradicts inequality 3.42). Suppose thatfort > t, theintegralcurve t,xt;t,x ))doesnotmeet the hypersurface of 3.33). According to condition H3.5 we conclude that the solution is defined for any t t. We will show that under the assumption made for t > t, the following inequality is valid t < τ i xt;t,x )). 3.43) Indeed, if there exists point t,t > t, such that t = τ i xt ;t,x )), then it follows that the integral curve t,xt;t,x )) meets a hypersurface 3.33) for t > t, which contradicts the assumption made above. If there exists point t,t > t, such that t > τ i xt ;t,x )), then taking into consideration 3.41), we reach the contradiction with Theorem 3.4. By 3.43), using the conditions H3.2 and H3.3, we obtain t τ i xt ;t,x )) < τ i xt;t,x )) τ i xt ;t,x )) < L i C f t t ),

106 88 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES i.e. the following inequality holds t < τ ix ) t L i C f 1 L i C f = δ = const. The last inequality contradicts the assumption, according to which 3.43) is valid for every t > t including for t > δ). Thus the theorem is proved. Corollary 3.8. Let the conditions H3.1-H3.5 be satisfied. Then j 1 = 1, i.e. the integral curve t,xt)) = t,xt;0,x 0 )) of problem 3.34), 3.35), 3.36) meets first the hypersurface σ 1. Theorem 3.6. Let the conditions H3.1-H3.5 be satisfied. Then the integral curve t,xt)) of problem 3.34), 3.35), 3.36) meets infinity many hypersurfaces 3.33) for t R +. Proof. Assume that for t > 0, the integral curve t,xt)) meets consecutively the hypersurfaces σ 1,σ j2,...,σ jk at the moments t 1,t 2,...,t k, respectively. Moreover, let for t > t k the integral curve meets the hypersurfaces 3.33). If we assume that, there exists a point t, t > t k, such that, for number i the property t,xt )) Ω i 0) is true. According to Theorem 3.5, it will be follow that, the integral curve meets the hypersurface σ i for t > t > t k, which contradicts the assumption. Then, one of following two cases is fulfilled: Case 1. There exists number i such that for each t > t k, it is satisfied t,xt)) σ i, i.e. t = τ i xt)) for t > t k. Let t and t be two constants, such that t > t > t k. Then, by conditions H3.2 and H3.3, we get the following contadiction t t =τ i xt )) τ i xt )) L i xt ) xt ) < 1 C f xt ) xt ) t t. Case 2. There exists point t > t k, such that for i = 1,2,... the following inequalities are valid τ i xt )) < t = const. This result contradicts condition H3.4. The proof is completed.

107 2. CONTINUOUS DEPENDENCE 89 Theorem 3.7. Let the inequalities H3.1-H3.6 be satisfied. Then the integral curve t,xt)) of problem 3.34), 3.35), 3.36) is defined for each t R +. Proof. From Theorem 3.5 and Theorem 3.1 it follows that the sequence j 1,j 2,... of infinitely many numbers and moreover, the inequalities j 1 < j 2 <... are valid. Therefore, lim j i =. i From the last inequality and condition H3.4, we obtain lim t i = lim τ ji x i ) =. 3.44) i i From condition H3.5 it follows that the solution xt) is defined in intervals t i 1 < t t i, i = 1,2,... Therefore using the upper limited equality), we completed the proof of the theorem. Our goal below is to study the behavior of the solution under small perturbation of impulsive hypersurfaces. For this purpose let us introduce the following hypersurfaces σ i = { t,x);t = τ i x),t R +,x D }, i = 1,2, ) We called these hypersurfaces further perturbed impulsive hypersurfaces. Let us consider the initial problem dx dt = f t,x ), t t i t τ i x t)), 3.46) x t) t=t i = x t i +0) x t i) = I j i x t i)), i = 1,2,..., 3.47) x 0) = x ) Here: t 1,t 2,..., 0 < t 1 < t 2 <..., are moments, at which the integral curve t,x t)) of problem 3.46), 3.47), 3.48) meets some hypersurface of 3.45); ji is the number of the hypersurface, met by the integral curve at the moment t i, i = 1,2,...; We need the following notations: x t) = x t;0,x ) is the solution of problem 3.46), 3.47), 3.48); x i = x t i ); x + i = x i +I j i x i), i = 1,2,...

108 90 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES We consider the both problems 3.34), 3.35), 3.36) and 3.46), 3.47), 3.48). Let us note that the main difference in these two problems are the initial point and impulsive hypersurfaces see Figure 3.8). x 2 D x 0 x 0 x 1 t 1 t 1 t 2 t 2 t xt) x t) Figure 3.8 Definition 3.1. Wewillsaythatthesolutionxt)ofproblem3.34), 3.35), 3.36) depends continuously on the impulsive hypersurfaces3.33), if T > 0) ε > 0) η > 0) δ = δt,ε,η) > 0) : σ 1,σ 2, : τ i x) τ ix) < δ, x D, i = 1,2,...) x t) xt) = x t;0,x 0 ) xt;0,x 0 ) < ε for 0 t T, t t i > ηand x 0 = x 0. Further, we will use the following conditions: H3.7. The function f is uniformly Lipschitz continuous on t R + in the second argument in the domain D with corresponding Lipschitz constant L, i.e. t,x ),t,x ) S) f t,x ) f t,x ) L x x ; H3.8. Let I i C[D,R n ], i = 1,2,...; H3.9. The following inequalities are valid τ k x+i i x)) τ i x), x D, k = 1,2,..., i = 1,2,....

109 2. CONTINUOUS DEPENDENCE 91 Remark 3.1. Using the substitution x = xt) = ) xt;0,x 0 ) in the inequality of condition H3.9, we obtain t i τ k x + i, i = 1,2,..., k = 1,2,... Theorem 3.8. Let the following conditions be satisfied: 1) The conditions H3.1-H3.3, H3.5-H3.9 are valid; 2) t 1 < T < t 2. Then T, t 1 < T < t 2 ) ε > 0) δ = δt,ε) > 0, α = αt,ε) > 0, β = βt,ε) > 0) : x 0 D, x 0 x 0 < δ) 3.49) τ i C [ D,R +], τ i x) τ ix) < δ, x D, i = 1,2,... ) t 1 < T < t 2 ) and x t) xt) < αε for 0 t T and t t 1 > βε). Proof. First, we assume that t 1 < T < t 2 and j 1 = 1. Let moreover, the inequality t 1 t 1 be valid. The case t 1 > t 1 is considered similarly. The functions x t) = x t;0,x 0) and xt) = xt;0,x 0 ) satisfy the following two equations: x t) = x 0 + t f θ,x θ))dθ and 0 t xt) = x 0 + f θ,xθ))dθ, for 0 t t 1. Therefore t x t) xt) x 0 x 0 +L 0 0 x θ) xθ) dθ. From the inequality above and Gronwall s Inequality see also Theorem 0.2 from Introduction), we find the estimate x t) xt) x 0 x 0 explt) δexplt), 0 t t 1, 3.50)

110 92 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES where δ is an arbitrary positive constant we will define δ further). In addition we assume that x 0 x 0 < δ and τ i x) τ ix) < δ, x D, i = 1,2,.... Using conditions H3.2, H3.3, and Corollary 3.8 we obtain t 1 t 1 t 1 τ 1 x 1) + τ 1 x 1) t 1 = τ 1 x 1 ) τ 1x 1 ) + τ 1x 1 ) τ 1x 1 ) δ +L 1 x 1 x ) Inequalities 3.50) for t = t 1 ) and 3.51) yield x 1 x 1 x 1 x t 1 ) + x t 1 ) x 1 From the last inequality we deduce = x t 1 ) x t 1 ) + x t 1 ) xt 1 ) t 1 f θ,x θ))dθ +δexplt) t 1 C f t 1 t 1 +δexplt) δexplt)+c f δ +L 1 x 1 x 1 ). x 1 x 1 explt)+c f 1 C f L 1 δ. 3.52) Let ε be an arbitrary positive constant. From condition H3.8 and inequality 3.52), it follows that there exists a constant δ > 0 such that I 1 x 1) I 1 x 1 ) < ε. 3.53) We fix δ < ε. Then from 3.52) and 3.53) we receive x + 1 x + 1 = x 1 +I 1 x 1 ) x 1 I 1 x 1 ) explt)+c f 1 C f L 1 δ +ε <α 1 ε, 3.54) where is a constant. α 1 = explt)+c f C f L C f L 1

111 2. CONTINUOUS DEPENDENCE 93 By the inequalities 3.52) and 3.54) we have 1 x + 1 xt 1 ) t = x + 1 x + 1 f θ,xθ))dθ t 1 x + 1 x + 1 +Cf t 1 t 1) C f δ +L 1 x 1 x 1 )+α 1 ε ) explt)+c f C f 1+L 1 δ +α 1 ε < α 2 ε, 3.55) 1 C f L 1 where L 1 explt)+1 α 2 = C f +α 1 1 C f L 1 is a constant. The functions x t) and xt) in the interval t 1 t T satisfy the following two integral equations: t x t) = x 1 + f θ,x θ))dθ and t 1 t xt) = xt 1 )+ f θ,xθ))dθ, t 1 respectively. Using the Gronwall s Inequality and inequality 3.55) we reach the estimate where x t) xt) α 2 explt) =αε, t 1 t T. 3.56) On the other hand, inequalities 3.51) and 3.52) imply t 1 t 1 L 1expLT)+1 δ 1 C f L 1 =βε, 3.57) β = L 1expLT)+1 1 C f L 1 is a constant. From inequality α > explt) see 3.50) and 3.56)), we obtain x t) xt) αε for 0 t t 1 and t 1 t T. 3.58)

112 94 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES The implication 3.49) follows from inequalities 3.58) and 3.57). To complete the proof show that if δ is sufficiently small, then the assumption made in the begining is fulfilled, i.e. the inequalities t 1 < T < t 2 and j 1 = 1 are valid. We show that the integral curve t,x t)) of problem 3.46), 3.47), 3.48) for t > 0 meets firstly hypersurface σ 1. Case 1. Assume that for t > 0, independently of δ value, the integral curve t,x t)) does not meet hypersurface of 3.45). Let us look at the hypersurface σ = {t,x);t = τ 1 x)+δ,x D}, where δ is an arbitrary positive constant. From Corollary 3.8 it follows that the integral curve t,x t)) meets the hypersurface σ. Let this meeting is realisez at the moment t. The function x t) for 0 t t is bounded. Therefore, there exists a bounded domain D 1,D 1 D, such that x t) D 1 for 0 t t. The sets: σ 0 D1 ) = { t,x);t = 0,x D1 } and σ 1 D1 ) = { t,x);t = τ1 x),x D 1 } are bounded and closed, whence we conclude that ρ E σ0 D1 ),σ1 D1 )) = δ > 0. Let 0 < δ < min { δ, δ 2 }. Then we find 0 < τ 1 x) δ < τ 1 x) < τ 1x)+δ < τ 1 x)+δ,x D ) Therefore, it follows that Additionally 0 < τ 1 x 0 ) = τ 1 x 0)). 3.60) 0 = τ 1 x t ))+δ t > τ 1 x t )) t, 3.61) because 3.59) and t = τ 1 x t ))+δ. The function φ 1 t) = τ 1 x t)) t is continuous in the interval 0 t t. By means of the inequalities 3.60) and 3.61), we receive that φ 1 0) > 0 and φ 1 t ) < 0. Therefore, there exists a point t, 0 < t < t, such that φ 1 t ) = 0, i.e. τ 1 x t )) = t. The last equality shows that the integral curve t,x t)) meets the hypersurface σ 1 at the moment t, which contradicts the assumption. Case 2. Assume that, there exists a number sequence δ 1,δ 2,... such that:

113 2. CONTINUOUS DEPENDENCE 95 δ k > 0,k = 1,2,...; δ 1 δ 2...; lim k δ k = 0; j 1 = j 1 δ k) > 1, k = 1,2,..., i.e. if the inequalities τ i x) τ ix) < δ k, x D, i = 1,2,..., are valid, then the integral curve t,x t)) of problem 3.46), 3.47), 3.48) for t > 0 meets firstly the hypersurface σ j 1 of 3.45). The meeting is realized at the moment t 1 = t 1 δ k). The number of the hypersurface, which is met first is j 1 > 1. Let us consider the following initial problem without impulses) dy dt = f t,y), y0) = x ) Let t be the moment, at which the integral curve t,yt)) of problem 3.62)meetsthehypersurface σ accordingtocorollary3.8suchmoment exists and it does not depend on constant δ). We will note that the integral curves t,yt)) and t,x t)) coincide for 0 t t 1. If the inequality t t 1 is valid, then similarly to Case 1, we obtain a contradiction. Indeed, there exists positive constant δ k, δ k < min { } δ, δ 2, where δ and δ are defined in Case 1 of the proof. We achieve again the inequalities 3.60) and 3.61). Using the both inequalities for the function φ 1 t) = τ1 x t)) t, which is continuous for 0 t t, we find that φ 1 0) > 0 and φ 1 t ) < 0, leading to contradiction. Let the inequality t > t 1 be satisfied. Then the function x t) D 1 for 0 t t 1. Note δ = min { ρ E σ0 D1 ),σ1 D1 )),ρe σ1 D1 ),σ2 D1 ))}. We will point, that domain D 1 does not depend on constant δ. Let δ k < min { } δ, δ 2. Then the following inequalities are fulfilled 0 <τ 1 x) δ k < τ 1 x) < τ 1x)+δ k < τ 2 x) δ k τ j 1 x) δ k τ j 1 x), x D 1, j 1 = j 1 δ k ). 3.63) Let us consider the function φ 2 t) = τ 1 x t)) t, defined in the interval 0 t t 1. The functions φ 1 and φ 2 coincide in the common part of its domains. By inequality 3.63) we find 0 < τ 1 x 0))

114 96 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES and 0 = τ j 1 x t 1)) t 1 > τ 1 x t 1)) t 1, i.e. φ 2 0) > 0 and φ 2 t 1 ) < 0. Using last two inequalities, and the continuity of function φ 2 for 0 t t 1, it follows that there exists a point t, 0 < t < t 1, such that φ 2 t ) = 0, i.e. the integral curve t,x t)) meets the hypersurface σ1 at the moment t,t < t 1, which contradicts the assumption made. Thus we show that for sufficiently small constant δ the integral curvet,x t)) of problem 3.46), 3.47), 3.48) for t > 0 meets firstly the hypersurface σ1 of 3.45). Since t 1 < T, taking into consideration inequality 3.57), we find that for sufficiently small constant δ, the inequality t 1 < T is valid. We will point that in the proof of inequality 3.57) of all the assumptions made at the beginning of the proof, we used only the fact j 1 = 1, which was established above). From the inequalities 3.54) and 3.57), we get ) )) ρ E t1,x + 1, t 1,x + 1 < α1 +β)ε. 3.64) According Theorem 3.5 and condition H3.9 more precisely, see Remark 3.1) it is satisfied ) t 1,x + 1 Ωj2 0). Then, we have ) ρ E t1,x + 1,R n \Ω j2 0) ) = δ iv > ) Let 0 < δ < δ iv. Then from 3.65) we find ) ρ E t1,x + 1,R n \Ω j2 δ) ) = δ v > ) Using 3.64) and 3.66) it follows that for sufficiently small value of ε, i.e. for sufficiently small value of δ, the relation ) t 1,x + 1 Ωj2 δ) is fulfilled. The meaning of the last relation is τ j 2 1 x + ) +δ < t 1 < τ j 2 1 x + ) δ. 3.67) If the integral curve of problem 3.46), 3.47), 3.48) for t > t 1 does not meet the hypersurface of 3.45), then the theorem is proved. Assume that there exists the moment t 2, i.e. the integral curve t,x t)) meets the hypersurface of 3.45) for t > t 1. We assume also that j 2 < j 2. Then 0 =τ j 2 xt 2 )) t 2 <τ j 2 xt 2))+δ t 2 τ j2 1xt 2 ))+δ t ) Consider the function { ) τ j2 1 x + φ 3 t) = 1 +δ t 1, if t = t 1, τ j2 1x t))+δ t, if t 1 < t t 2..

115 2. CONTINUOUS DEPENDENCE 97 Obviously φ 3 t) is continuous in its domain. From the first inequality of 3.67) it follows that φ 3 t 1 ) < 0. By 3.68) we derive φ 3t 2 ) > 0. Therefore, there exists a point t iv,t 1 < t iv < t 2, such that φ 3 t iv ) = 0 τ j2 1 x t iv)) +δ t iv. 3.69) We apply concequtively condition H3.2, equality 3.69), first of the inequalities 3.67) and condition H3.3. So we reach the contradiction x t iv) ) x + 1 Cf t iv t 1 <C f τj2 1 x t iv)) ) ) +δ τ j2 1 x + 1 δ C f L j2 1 x t iv) x + 1. Therefore, j 2 j 2. Then we have The function ϕ 4 t) = 0 =τ j 2 xt 2)) t 2 >τ j 2 xt 2)) δ t 2 τ j2 xt 2)) δ t ) { ) τ j2 x + 1 δ t 1, if t = t 1, τ j2 x t)) δ t, if t 1 < t t 2 is continuous for t 1 t t 2. The second inequality of 3.67) and 3.70) imply the inequalities φ 4 t 1) > 0 and φ 4 t 2) < 0, respectively. From the last two inequalities it follows that the integral curve t,x t)) meets the hypersurface σ = {t,x);t = τ j2 x) δ,x D} at the moment t v, which satisfies the inequality Similarly to 3.57) we obtain the inequality t v < t ) t 2 t v < β 1 δ. 3.72) The idea of the proof of 3.72) coincides with the proof of 3.57). The difference is that the hypersurfaces σ 1 and σ1, and initial moment 0 are replaced by the hypersurfaces σ j2 and σ and the moment t 1 ). Since T < t 2, then from 3.72) it follows that for sufficiently small δ the inequality T < t v is fulfilled, whence using 3.71) we deduce that T < t 2. Thus the theorem is proved.

116 98 3. EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES Theorem 3.9. Let the conditions H3.1-H3.9 be satisfied. Then the solution of problem 3.34), 3.35), 3.36) depends continuously on the impulsive hypersurfaces of 3.33). Proof. Let T be an arbitrary positive constant. From the equality 3.44), it follows that the integral curvet, xt)) of problem3.34),3.35), 3.36) for 0 t T meets finite number of times the hypersurfaces of 3.33). Assume that the following inequalities are valid 0 < t 1 < t 1 < < t p T < t p+1. We introduce the notations: θ 1 = 0,θ k = 1 t 2 k 1 +t k ), k = 2,3,...,p and θ p+1 = T. Let x 0 = x 0 and ε and η be arbitrary positive constants. Let δ p+1 = ε. From the previous theorem for each one of the intervals [θ k,θ k+1 ], k = 1,2,...,p, it follows that there exists constant δ k > 0 such that if the inequalities x θ k ) xθ k ) < δ k and τ i x) τ i x) δ k for x D, i = 1,2,..., are valid, then we have: x t) xt) < ε,θ k t θ k+1, t t k > η; θ k 1 < t k < θ k < t k+1 ; x θ k+1 ) xθ k+1 ) < δ k+1. We define the constants δ k, k = 1,2,...,p+1, concequtively in the order: δ p+1,δ p,...,δ 1. Let δ = min{δ 1,δ 2,...,δ p+1 }. Therefore, if for x D and i = 1,2,..., then τ i x) τ ix) < δ x t) xt) < ε for 0 t T and t t i > η, i = 1,2,...,p. Thus the proof is complete.

117 3. UNIFORM STABILITY OF THE SOLUTIONS Uniform Stability of the Solutions of the Differential Equations with Variable Impulsive Moments on the Initial Condition and Impulsive Perturbations We will discuss some properties, related to uniform stability of the solutions of differential equations with variable impulsive moments on the initial condition and impulsive perturbations. Again, we consider the following inital problem Here: dx dt = f t,x), t t i t τ i xt)), 3.73) xt) t=ti = xt i +0) xt i ) = I ji xt i )), i = 1,2,..., 3.74) x0) = x ) f : S R n,s = R + D and D is a domain of R n ; τ i : D R + ; I i : D R n ; x 0 D; t 1,t 2,..., 0 < t 1 < t 2 <..., are the momets, at which the integral curve t, xt)) of problem 3.73), 3.74), 3.75) meets some of the hypersurfaces σ i = { t,x);t = τ i x),t R +,x D }, i = 1,2,...; 3.76) j i is a number of hypersurface, which the integral curve meets at the moment t i in the general case i j i ) and xt i 0) = xt i ). We investigate two systems: basic one 3.73), 3.74) and perturbed impulsive system dx dt = f t,x ), t t i t τ ix t)), 3.77) x t) t=t i = x t i +0) x t i ) = I ji x t i )), i = 1,2,..., 3.78) where: Ii : D R n ; t 1,t 2,..., 0 < t 1 < t 2 <..., are the moments, at which the integral curve t,x t)) of problem 3.77), 3.78), 3.75) meets some of the hypersurfaces 3.76) and ji is a number of hypersurface of 3.76), which the integral curve of the perturbed problem meets at the moment t i. The difference between the basic and perturbed problem is in the impulsive perturbations effects) see Figure 3.9). We introduce the following conditions: H3.10. There exists a constant, > 0, such that Id+I i ) : D D\B D), i = 1,2,..., where Id is the identity in R n ;

118 EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES x 2 x 0 D xt) x t) x 1 σ 1 σ j2 0 t 1 = t 1 t t 2 2 t Figure 3.9 H3.11. There exists a constant d, d > 0, such that x D) τ i x+i i x))+d < τ i x), i = 1,2,... Theorem Let the following conditions be satisfied: 1) The conditions H3.1-H3.5, H3.10 and H3.11 are valid. 2) The following inequalities are valid x D) I i x) I i x) < min{,c f d}, i = 1,2,... Then for any point t 0,x 0 ) R + D, the solution x t;t 0,x 0 ) of system 3.77), 3.78) with initial condition is continuable for each t t 0. x t 0 ) = x ) Proof. Condition H3.10 and condition 2 of the theorem imply Id+ Ii ) : D D. The last relation and condition H3.5 lead to the conclusion that the solution x t;t 0,x 0 ) exists and does not leave the domain D in its definition set. We show that the studied solution is continuable for each t t 0. The following two cases are possible: Case 1. The integral curve t,x t;t 0,x 0 )) meets finite number impulsive hypersurfaces of 3.76) for t > t 0. Let the meetings be realized at the moments t 1,t 2,...,t k. Obviously, we have

119 it is satisfied 3. UNIFORM STABILITY OF THE SOLUTIONS 101 x + k = x k +I j k x k ) = x t k ;t 0,x 0 )+I j k x t k ;t 0,x 0 )) D; condition H3.5 is valid; we have x t;t 0,x 0 ) = x ) t;t k,x + k, for t t k. Therefore, x t;t 0,x 0 ) D and the solution of perturbed problem is continuable for each t t k. That is why, the solution of problem 3.77), 3.78), 3.79) is continuable for each t t 0. Case 2. The integral curve t,x t;t 0,x 0 )) meets infinity many hypersurfaces of 3.76), for t > t 0. First, we show that j 1 < j 2 <..., 3.80) i.e., that the sequence of numbers of hypersurfaces, which the integral curve t,x t;t 0, x 0 )) meets concequtively is strictly increasing. From conditions H3.3, H3.11, and condition 2 of Theorem 3.10 we obtain t i τ j i x + i ) =τj i x i ) τ j i x i +I j i x i ) ) x i +I j i x i )) +τ j i x i +I j i x i )) =τ j i x i ) τ ji ) τ j i x i +I ji x i ) d τ j i x i +I j i x i) ) τ j i x i +Ij i i)) x Ij d L j i i x i) Ij i x i) d L j i C f d > 0, for each i = 1,2,... From the last inequality and 3.2) we find that ) t i,x + i Ωk 0), where k > ji. From Theorem 3.5 it follows that, the first hypersurface, which the integral curve t,x )) t;t i,x + i meets for t t i is σ k. Since )) t,x t;t i,x + i = t,x t;t 0,x 0 )) for t t i, then it follows that, the hypersurface σ k = σ j i+1, i.e. j i+1 = k > j i, by which the validity of inequalities 3.80) is proved. Then, having in mind

120 EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES that the constants j i are natural numbers, we find that lim i j i =. Through condition H3.4 we have lim i t i = lim τ j i x i i ) =. Using the limit above and the fact, that the solution of problem 3.77), 3.78), 3.79) is defined in each of the intervals t i 1 i],t,i = 1,2,..., we conclude that assertion of the theorem is also true in this case. Thus the theorem is proved. We introduce the conditions: H3.12. The equality f t,0) = 0,t R + is satisfied; H3.13. The equalities I i 0) = 0, i = 1,2,... are fulfilled. From conditions above, we obtain t0 R +) xt;t 0,0) = 0,t t 0. Definition 3.2. We will say that zero solution trivial solution) of system 3.73), 3.74) is uniformly stable on the initial condition, if ε > 0) δ = δε) > 0) : t0,x 0 ) R + B δ D) ) xt;t 0,x 0 ) < ε, t t 0. Definition 3.3. We will say that zero solution trivial solution) of system 3.73), 3.74) is uniformly stable on the impulsive perturbations effects), if ε > 0) δ = δε) > 0) : t0 R +) I i : D R n, I i x) I i x) < δ, x D, i = 1,2,...) x t;t 0,0) < ε, t t 0. Definition 3.4. We will say that zero solution trivial solution) of system 3.73), 3.74) is uniformly stable on the initial condition and the impulsive perturbations effects), if ε > 0) δ = δε) > 0) : t0,x 0 ) R + B δ D) ) I i : D R n, I i x) I ix) < δ, x D, i = 1,2,...) x t;t 0,x 0 ) < ε, t t 0.

121 3. UNIFORM STABILITY OF THE SOLUTIONS 103 From Definition 3.2 it follows that, for any ε > 0 infinity many numbers of constants correspond for which the definition is fulfilled. Furthermore, each constant δ, corresponding to ε satisfies the inequality δ ε. Further by δε) we denote the upper limit of all δ, satisfying Definition 3.2. For each ε > 0, we can form an infinite sequence of constants: δ 1 = δε),δ 2 = δδ 1 ),...,δ i = δδ i 1 ),... The following inequalities are valid δ 1 δ 2 > 0. Therefore, the sequence δ 1,δ 2,... converges. Let δ = δ ε) 0 be its limit. Definition 3.5. We will say that zero solution trivial solution) of system 3.73) without impulses) is strictly uniformly stable on the initial condition, if ε > 0) δ ε) > 0. Lemma 3.3. Let the following conditions be satisfied: 1) The zero solution of system 3.73) is uniformly stable; 2) δ > 0) : t 0,x 0 ) R + B δ D)) xt;t 0,x 0 ), i.e. the norm of solution is monotonically decreasing function. Then the zero solution of system without impulses 3.73) is strictly uniformly stable on the initial condition. The proof of Lemma 3.3 is trivial. Indeed, condition 2 of the lemma implies that for each ε > 0, it is satisfied δ ε) = ε > 0. Lemma 3.4. Let the following conditions be satisfied: 1) The conditions H3.1, H3.5, H3.12 are valid; 2) The zero solution of system without impulses 3.73) is strictly uniformly stable on the initial condition. Then t0 R +) ε > 0) x 0 B δ D,δ = δ ε)) xt;t 0,x 0 ) < δ,t t 0. Proof. Let ε > 0,δ 1 = δε),δ 2 = δδ 1 ),..., lim i δ i = δ and the initial point t 0,x 0 ) R + B δ D). Then xt;t 0,x 0 ) < δ i 1 for t t 0, i = 1,2,..., δ 0 = ε,

122 EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES because x 0 < δ i. Therefore for t t 0. Thus Lemma is proved. xt;t 0,x 0 ) < δ Further we use the following definitionsee Dannan and Elaydi [102]). Definition 3.6. We will say that the zero solution of system without impulses 3.73) is uniformly Lipschitz stable on the initial condition, if G = const > 0) g = const > 0) : t0,x 0 ) R + B g D) ) xt;t 0,x 0 ) < G x 0,t t 0. Theorem Let the following conditions be satisfied: 1) The conditions H3.1-H3.5, H3.10-H3.13 are valid; 2) The zero solution of system without impulses 3.73) is strictly uniformly stable on the initial condition; 3) There exists positive constant C such that x D) x+i i x) 1 x, i = 1,2, C Then the zero solution of system 3.73), 3.74) is uniformly stable on the initial condition and impulsive perturbations. Proof. Let t 0 0,ε > 0 and δ = δε) ε. Recall that δ is the corresponding constant of the definition for strict uniform stability of the zero solution of system 3.73). Let δ = min {,C f d, } C 1+C δ. Let the functions Ii, i = 1,2,..., satify the inequalities I i x) I i x) < δ, x D. Then see Theorem 3.10) for each point x 0 D solution x t;t 0,x 0 ) is continuable for any t > t 0. Let x 0 < δ δ. Therefore see Lemma 3.4) we obtain the estimate x t;t 0,x 0 ) = xt;t 0,x 0 ) < δ ε, t 0 t t 1.

123 3. UNIFORM STABILITY OF THE SOLUTIONS 105 From condition 3 of the theorem and the estimate above for t = t 1 we find = x 1 +I ji x 1 ) x + 1 x 1 +I j i x 1 ) + I j i x 1 ) I j i x 1 ) < 1 1+C x 1 +δ 1 1+C δ + C 1+C δ =δ. Again, using Lemma 3.4 we derive x t;t 0,x 0 ) = ) x t;t 1,x + 1 = ) x t;t 1,x + 1 < δ ε, t 1 < t t 2. In general, we obtain the following inequality by induction: x t;t 0,x 0 ) < δ ε, t i 1 < t t i, i = 1,2,... Thus the theorem is proved. Theorem Let the following conditions be satisfied: 1) The conditions H3.1-H3.5, H3.10-H3.13 are valid. 2) The zero solution of system without impulses 3.73) is uniformly Lipschitz stable on the initial condition with Lipschitz constants G and g see Definition 3.5). 3) There exists constant C, C > 0, such that x D) x+i i x) 1 x, i = 1,2,... G+C Then the zero solution of system 3.73), 3.74) is uniformly Lipschitz stable on the initial condition and impulsive perturbations. Proof. Let t 0 R +, and let ε be an arbitrary positive constant, { δ = min,c f d,g, ε }. G Let the functions Ii, i = 1,2,..., satisfy the inequalities Ii x) I i x) < C δ, x D. G+C

124 EQUATIONS WITH VARIABLE MOMENTS OF IMPULSES ThenfromTheorem3.10,itfollowsthatforanypointx 0 D thesolution x t;t 0,x 0 ) is continuable for any t t 0. Let x 0 < δ g. Moreover, and Therefore x t;t 0,x 0 ) = xt;t 0,x 0 ) <Gδ ε, t 0 t t 1. x + x 1 = 1 +Ij i x 1 ) x 1 +I j i x 1) I + j i x 1) I j i x 1) < 1 G+C x 1 + C G+C δ δ g. x t;t 0,x 0 ) = ) x t;t 1,x + 1 = ) x t;t 1,x + 1 <Gδ ε, t 1 < t t 2. Finally, we obtain the following estimates by induction x t;t 0,x 0 ) < ε, t i 1 < t t i, i = 1,2,... Thus the theorem is proved.

125 Chapter 4 Continuous Dependence of the Solutions of Differential Equations with Variable Moments of Impulses on the Initial Condition and Barrier Curves The basic object of investigation in the present chapter is one specific class of nonlinear nonautonomous differential equations with variable moments of impulsive effects. The impulsive moments coincide with the moments ofmeetingsbetween theintegralcurve andsomeoftheso called barrier curves. The barrier curves are situated in the extended phase space of the considered equations and coincide with the contour of the set, in which lie the integral curves of the studied initial problems. The impulsive moments for this class of equations are not fixed. Directly we can see that they depend on the initial point and any other parameter of the equation e.g. the right hand side of the equation, the barrier curves, the size of the impulsive effects, etc.). In other words, if one of the parameters of the equation is changed, then the moments at which the impulses take place also change in the general case. In the first section in this chapter we find the sufficient conditions, under which the solution of such type of equations depends continuously on the perturbations with respect to the initial condition and barrier curves. The mentioned concepts of barrier curves and continuous dependence of the solutions are introduced and investigated for the first time in Dishlieva [128]. Inthesecondsection ofthechapter 4theobtainedresults areapplied on a mathematical model of population dynamics, more precisely to the impulsive model of Gompertz. During the last years, the impulsive differential equations with variable moments of impulsive effects provoke a serious research interest 107

126 BARRIER CURVES with respect to their various applications. We cite the following results: Akhmet [16], Bainov et al. [49], Benchohra et al. [81], Chellaboina et al. [90], Chen et al. [93], Chukleva [97], Chukleva et al. [98], Dishliev and Bainov [114, 115, 116], Dishliev and Stoykov [125, 126], Frigon and O Regan [148], Lu and Wang [234], Lu et al. [235], Shuai et al. [302], Stoykov [322], Tang et al. [324], Wu and Meng [339], Xiao et al. [342], Zhang et al. [363], Zhang and Jin [364]. Numerous studies are devoted to Gompertz Impulsive Model of population dynamics, the latest of which are: Braverman and Mamdani [89], Cordova-Lepe [100], Jia and Li [186], Wang et al. [334], Zeng [359], Zhang and Xiu [368].

127 1. NOT FIXED MOMENTS Continuous Dependence of the Solutions of Differential Equations with Not Fixed Moments of Impulses on the Initial Condition and Barrier Curves The basic object in the present chapter is the following initial problem for nonlinear impulsive ordinary differential equations: dx dt = f t,x), χ 1t) < xt) < χ 2 t), 4.1) xt+0) = xt)+i 1 t,xt)), xt) = χ 1 t), 4.2) xt+0) = xt)+i 2 t,xt)), xt) = χ 2 t), 4.3) xt 0 ) = x 0, 4.4) where: f : S R; S = {t,x);t 0,X 1 t) < x < X 2 t)}; χ 1,χ 2,X 1,X 2 : [0, ) R; X 1 t) < χ 1 t) < χ 2 t) < X 2 t), t 0; I 1 : S R +, I 2 : S R ; χ 1 t)+i 1 t,χ 1 t)) < χ 2 t) and χ 2 t)+i 2 t,χ 2 t)) > χ 1 t) for t 0; t 0 0 and χ 1 t 0 ) < x 0 < χ 2 t 0 ). We introduce the curves: γ 1 = {t,x);x = χ 1 t),t 0} and γ 2 = {t,x);x = χ 2 t),t 0}, which are called barrier curves, the corresponding functions χ 1 and χ 2 are named barrier functions. It is natural to assume that the barrier functions are continuous in its domain. It is possible, for t 0, the integral curve of problem 4.1), 4.2), 4.3), 4.4) to meet one of the curves γ 1 or γ 2. Let the meetings be realized at the moments t 1,t 2..., for which the following inequalities 0 t 0 < t 1 < t 2... are valid. From nowonwewillcallthemimpulsive moments. Thesolutionoftheproblem under consideration is a piecewise continuous function. It holds: 1) For t 0 t t 1 the solution of problem 4.1), 4.2), 4.3), 4.4) is identical with the solution of problem without impulses) 4.1), 4.4); 2) If xt i ) = χ 1 t i ), then for t i < t t i+1, i = 1,2,..., the solution of problem 4.1), 4.2), 4.3), 4.4) coincides with the solution of equation 4.1) with the initial condition xt i +0) = Id+I 1 )t i,xt i )), 4.5) where Id is the identity in S;

128 BARRIER CURVES 3) If xt i ) = χ 2 t i ), then for t i < t t i+1, i = 1,2,..., the solution of the problem is identical with the solution of equation 4.1) with the initial condition xt i +0) = Id+I 2 )t i,xt i )). 4.6) The equalities 4.5) and 4.6) will be called theoutside) impulsive effects. Thesolution oftheproblem will bedenoted by xt;t 0,x 0 ). Assume that the following inequalities are valid: where: χ 1 t) xt;t 0,x 0 ) χ 2 t), t t 0. Let us consider the corresponding perturbed problem: dx dt = f t,x ), χ 1 t) < x t) < χ 2 t), 4.7) x t+0) = x t)+i 1 t,x t)), x t) = χ 1t), 4.8) x t+0) = x t)+i 2 t,x t)), x t) = χ 2 t), 4.9) x t 0 ) = x 0, 4.10) χ 1,χ 2 : [0, ) R; X 1 t) < χ 1t) < χ 2t) < X 2 t),t 0; χ 1 t)+i 1t,χ 1 t)) < χ 2 t) and χ 2 t)+i 2t,χ 2 t)) > χ 1 t 0; t 0 0 and χ 1t 0) < x 0 < χ 2t 0). t) for The moments, at which the integral curve of the perturbed problem 4.7), 4.8), 4.9), 4.10) meets consecutively some of the barrier curves: γ1 = {t,x);x = χ 1 t),t 0} or γ 2 = {t,x);x = χ 2 t),t 0}, are t 1, t 2,... The inequalities 0 t 0 < t 1 < t 2 <... are valid. The differences between the basic problem and its corresponding perturbed problem are in the initial point and in the barrier functions. Let x t;t 0,x 0) denote the solution of problem 4.7), 4.8), 4.9), 4.10) see Figure 4.1). Let B η t ) = t η,t +η). Definition 4.1. We say that the solution of problem 4.1), 4.2), 4.3), 4.4) depends continuously on the initial point t 0,x 0 ) and the barrier functions χ 1 and χ 2, if: ε > 0) η > 0) T > 0) δ = δε,η,t) > 0) : t 0 R +, t 0 t 0 < δ ) x 0 R, x 0 x 0 < δ)

129 x x 0 x 0 1. NOT FIXED MOMENTS 111 X 2 t) χ 2 t) χ 2 t) x t;t 0,x 0 ) xt;t 0,x 0 ) 0 χ 1 t) χ 1 t) X 1 t) t 0 t 0 t 1 t 1 t 2 t 2 t Figure 4.1 χ 1,χ 2 C 1 [[0,T],R], χ 1t) χ 1 t) < δ, χ 2t) χ 2 t) < δ x t;t 0,x 0) xt;t 0,x 0 ) < ε, t [t 0,T]\ i=0,1,... for t [0,T]) B η t i ). The requirements for the continuous differentiation of the barrier functions in their domain for the perturbed problem see the definition above) is desirable, although not compulsory. Actually, in the following considerations, the barrier functions of basic problem 4.1), 4.2), 4.3), 4.4) have to be C 1 functions. We will note also, that it is natural for the barrier functions and their corresponding perturbed barrier functions to be of one and the same class, i.e. they must have same degree of smoothness. This is the reason to assume that χ 1,χ 2 C 1 [[0,T],R] in the definition above. Let us introduce the following conditions: H4.1. Let f C[S,R]; H4.2. There exists a constant C f > 0 such that t,x) S) f t,x) C f ;

130 BARRIER CURVES H4.3. Let χ 1,χ 2,X 1,X 2 C[R +,R]andlet foranyt 0 thefollowing inequalities X 1 t) < χ 1 t) < χ 2 t) < X 2 t) hold true; H4.4. There exist the positive constants χ, L χ1 and L χ2 such that: H For each t 0, we have χ 2 t) χ 1 t) χ ; H For any two points t,t R + the following inequality is valid χ 1 t ) χ 1 t ) L χ1 t t ; H For any two points t,t R + the following inequality is satisfied χ 2 t ) χ 2 t ) L χ2 t t ; H4.5. Let I 1 : γ 1 R +, I 2 : γ 2 R ; H4.6. There exist a constants I and I, 0 < I < I < 1, such that for any t R + the following inequalities are valid I I 1t,χ 1 t)) χ 2 t) χ 1 t) I, I I 2t,χ 2 t)) χ 1 t) χ 2 t) I ; H4.7. For each point t,x ) S the equation 4.1) with initial condition xt ) = x has a unique solution, defined for t 0. Theorem 4.1. Assume that: 1) The conditions H4.1-H4.7 hold true; 2) The inequalities χ 1 t 0 ) < x 0 < χ 2 t 0 ) are satisfied. Then: 1) The solution xt;t 0,x 0 ) of the problem with impulses 4.1), 4.2), 4.3), 4.4) exists and it is unique for t t 0 ; 2) The following inequalities are valid χ 1 t) xt;t 0,x 0 ) χ 2 t), t t 0 ; 3) If the impulsive moments t 1,t 2,... are infinity many in number, then lim i t i =. Proof. We will consider four different cases, regarding the number of common point of the integral curve and the barier curves.

131 1. NOT FIXED MOMENTS 113 Case 1. For t t 0, the integral curve t,xt;t 0,x 0 )) does not meet the barrier curves γ 1 and γ 2. Therefore, the solution of the problem with impulses coincides with the solution of problem without impulses 4.1), 4.4) and in accordance with condition H4.7 the solution exists and it is unique for t t 0. Using the inequalities: χ 1 t 0 ) < x 0 < χ 2 t 0 ), xt;t 0,x 0 ) χ 1 t), xt;t 0,x 0 ) χ 2 t), and the continuity of the functions χ 1 and χ 2, we deduce that χ 1 t) < xt;t 0,x 0 ) < χ 2 t), t t 0. Case 2. For t t 0 the integral curve meets a finite number of times the barrier curves. Assume that the moments of these meetings are t 1,t 2,...,t k. It is clear that for 0 t 0 t t 1 the unique solution of the problem satisfies the inequalities of the second statement of theorem. For t i < t t i+1, i = 1,2,...,k 1, the solution of problem 4.1), 4.2), 4.3), 4.4) coincides with the solution of the initial problem dx =f t,x), 4.11) dt χ 1 t i )+I 1 t i,χ 1 t i )), xt i +0) =x + if xt i = i ;t 0,x 0 ) = χ 1 t i ); 4.12) χ 2 t i )+I 2 t i,χ 2 t i )), if xt i ;t 0,x 0 ) = χ 2 t i ). If xt i ;t 0,x 0 ) = χ 1 t i ), i.e., if the integral curve meets the barrier curve γ 1 at the moment t i, taking into account condition H4.6, we obtain I χ 2 t i ) χ 1 t i )) I 1 t i,χ 1 t i )) I χ 2 t i ) χ 1 t i )). Therefore, it is satisfied X 1 t i ) <χ 1 t i ) < 1 I )χ 1 t i )+I χ 2 t i ) χ 1 t i )+I 1 t i,χ 1 t i )) = x + i 1 I )χ 1 t i )+I χ 2 t i ) <χ 2 t i ) <X 2 t i ). That is why ) t i,x + i S. Similarly, it is easy to see that, if xt i ;t 0,x 0 ) = χ 2 t i ) i.e. if the integral curve meets the barrier curve γ 2 at the moment t i ), then again by condition H4.6) I χ 1 t i ) χ 2 t i )) I 2 t i,χ 2 t i )) I χ 1 t i ) χ 2 t i )).

132 BARRIER CURVES Therefore, X 1 t i ) <χ 1 t i ) < I χ 1 t i )+1 I )χ 2 t i ) χ 2 t i )+I 2 t i,χ 2 t i )) = x + i I χ 1 t i )+1 I )χ 2 t i ) <χ 2 t i ) <X 2 t i ). That is why, the inclusion ) t i,x + i S remains true. From condition H4.7 it is clear that, the solution of problem without impulses 4.11), 4.12) is defined and unique for t i < t t i+1. So, the solution xt;t 0,x 0 ) of the studied initial problem exists and it is unique in this interval. Since χ 1 t i ) < x + i < χ 2 t i ) and the functions χ 1 and χ 2 are continuous, then the following inequalities are satisfied: χ 1 t) < xt;t 0,x 0 ) < χ 2 t), t i < t t i+1, i = 1,2,...,k 1. As before, it is easy to see the inclusion t k,x + k) S. Therefore, the solution of problem 4.1), 4.2), 4.3), 4.4) exists and it is unique for t > t k. It satisfies the inequality of the second statement of the theorem. Case 3. For t t 0 the integral curve of problem 4.1), 4.2), 4.3), 4.4) meets the barrier curves γ 1 and γ 2 infinity many times at the moments t 1,t 2,... and furthermore, lim t i =. Then, as in the previous case, the solution xt;t 0,x 0 ) is unique in each of the intervals i t i 1 < t t i, i = 1,2,..., and it satisfies the inequalities of statement 2 of the theorem. Taking into account, that lim t i =, we conclude i that the solution of the impulsive problem is unique and continuable for t 0. It satisfies the second statement of the theorem in this interval. Case 4. For t t 0 the integral curve t,xt;t 0,x 0 )) meets infinity many times the barrier curves and lim t i = T <, i.e. the solution is i not continuable for t T. For every two consecutive impulsive moments t i and t i+1 one of the following four possibilities: Subcase 4.1. xt i ;t 0,x 0 ) = χ 1 t i ), xt i+1 ;t 0,x 0 ) = χ 1 t i+1 ); Subcase 4.2. xt i ;t 0,x 0 ) = χ 2 t i ), xt i+1 ;t 0,x 0 ) = χ 2 t i+1 ); Subcase 4.3. xt i ;t 0,x 0 ) = χ 1 t i ), xt i+1 ;t 0,x 0 ) = χ 2 t i+1 ); Subcase 4.4. xt i ;t 0,x 0 ) = χ 2 t i ), xt i+1 ;t 0,x 0 ) = χ 1 t i+1 ).

133 1. NOT FIXED MOMENTS 115 We study the behaviour of the solution for any of the subcases above. Subcase 4.1. For t i < t t i+1 the solution of the basic initial problem satisfies the integral equation xt;t 0,x 0 ) = xt i ;t 0,x 0 )+I 1 t i,xt i ;t 0,x 0 )) Therefore, from equation 4.13) at t = t i+1, we get + t χ 1 t i+1 ) = χ 1 t i )+I 1 t i,χ 1 t i ))+ t i f τ,xτ;t 0,x 0 ))dτ. 4.13) t i+1 t i f τ,xτ;t 0,x 0 ))dτ. Using the conditions H4.2, H4.6, and the second inequality of condition H4.4, we obtain the inequality L χ1 t i+1 t i ) I χ 2 t i ) χ 1 t i )) C f t i+1 t i ), whence with the first inequality of condition H4.4 we find the estimate t i+1 t i I. χ L χ1 +C f. Subcase 4.2. The details in this subcase are similar to those in the previous case. We have t i+1 t i I χ L χ2 +C f. Subcase 4.3. In this subcase, for t i < t t i+1 the solution of problem 4.1), 4.2), 4.3), 4.4) satisfies the integral equation 4.13). We have at t = t i+1 : That is why, χ 2 t i+1 ) = χ 1 t i )+I 1 t i,χ 1 t i ))+ t i+1 t i+1 t i f τ,xτ;t 0,x 0 ))dτ χ 2 t i+1 ) χ 2 t i )) t i f τ,xτ;t 0,x 0 ))dτ. =χ 2 t i ) χ 1 t i )) I 1 t i,χ 1 t i )) 1 I )χ 2 t i ) χ 1 t i )).

134 BARRIER CURVES Using the last inequality we get i.e. C f +L χ2 )t i+1 t i ) 1 I ) χ, t i+1 t i 1 I ) χ. L χ2 +C f Subcase 4.4. Similarly, we consider the variant 4.4. The next estimate holds true t i+1 t i 1 I ) χ. L χ1 +C f Finally, let us set t = min { I. χ L χ1 +C f, } I. χ, 1 I ) χ, 1 I ) χ. L χ2 +C f L χ2 +C f L χ1 +C f In such a way we estimate the distance between two adjacent impulsive moments, i.e. the inequalities t i+1 t i t > 0, i = 1,2,... Therefore, lim i t i =, i.e. the subcase 4.4 is impossible. Thus, the theorem is proved. We have to continue, adding three conditions more: H4.8. Let χ 1,χ 2 C 1 [R +,R]; H4.9. The inequalities f t,χ 1 t)) < d dt χ 1t) and f t,χ 2 t)) > d dt χ 2t) for t 0 are fulfilled; H4.10. Let I 1 C[S,R + ] and I 2 C[S,R ]. Theorem 4.2. Assume that: 1) The conditions H4.8 and H4.9 are fulfilled. 2) The inequalities χ 1 t 0 ) < x 0 < χ 2 t 0 ) are valid. 3) The integral curve t,xt)) of the problem without impulses 4.1), 4.4) meets the barrier curve γ 1 the barrier curve γ 2 ) at the moment t 1, i.e. Xt 1 ) = χ 1 t 1 ) Xt 1 ) = χ 2 t 1 )). Then, for every t > t 1 the inequality is valid. X t) < χ 1 t) Xt) > χ 2 t))

135 1. NOT FIXED MOMENTS 117 Proof. Let the integral curve of problem4.1),4.4) meets the barrier curve γ 1 at the moment t 1. The considerations of the meeting with the barrier curve γ 2 are similarly. Let t 1 be thefirst moment of their meeting, i.e. X t) > χ 1 t) for t 0 t < t 1 and X t 1 ) = χ 1 t 1 ). We consider three cases. Case 1. Let for every t > t 1 the inequality Xt) χ 1 t) be satisfied. Then the differentiable function F 1 t) = X t) χ 1 t), t 0 t <, has a minimum at the point t 1. Therefore, the equality d dt F 1t 1 ) = 0 is valid. On the other hand, it holds i.e. d dt F 1t 1 ) = d dt Xt 1) d dt χ 1t 1 ) =f t 1,Xt 1 )) d dt χ 1t 1 ) =f t 1,χ 1 t 1 )) d dt χ 1t 1 ), 4.14) f t 1,χ 1 t 1 )) d dt χ 1t 1 ) = 0. The last equality contradicts the first inequality of the condition H4.9. Case 2. Assume that, there exists a sequence of real numbers {τ i } such that τ i > t 1, i = 1,2,...; lim i τ i = t 1 and X τ i ) χ 1 τ i ). We derive consecutively Xτ i ) X t 1 ) =Xτ i ) χ 1 t 1 ) χ 1 τ i ) χ 1 t 1 ) d dt Xτ i)τ i t 1 ) d dt χ 1τ i )τ i t 1 ) f τ i,xτ i)) d dt χ 1τ i ). Here the points τ i and τ i are between t 1 and τ i. It is fulfilled lim f d i τ i,xτ i )) lim i dt χ 1τ i ) f t 1,Xt 1 )) d dt χ 1t 1 )

136 BARRIER CURVES f t 1,χ 1 t 1 )) d dt χ 1t 1 ). Again, the last inequality contradicts condition H4.9. Case 3. There exists a point t + 1,t+ 1 > t 1, such that X t) < χ 1 t),t 1 < t < t + 1 and X ) ) t + 1 = χ1 t + 1. Then, d dt F ) 1 t + d 1 = dt F 1 t ) F 1 t) F 1 = lim t t t t + 1 = lim t t F 1 t) t t + 1 t + 1 ) = lim t t X t) χ 1 t) t t + 1 On the other hand, similarly to 4.14) we obtain d dt F ) )) 1 t + 1 = f t + 1,χ 1 t + d 1 dt χ ) 1 t + 1. Therefore, f )) t + 1,χ 1 t + d 1 dt χ ) 1 t + 1 0, i.e. the result contradicts again condition H4.9. The theorem is proved. 0. Theorem 4.3. Assume that: 1) The conditions H4.1-H4.10 hold. 2) The inequalities χ 1 t 0 ) < x 0 < χ 2 t 0 ) are valid. Then, the solution of problem 4.1), 4.2), 4.3), 4.4) depends continuously on the initial point t 0,x 0 ) and the barrier functions χ 1 and χ 2. Proof. Let T, ε and η be arbitrary positive constants. For the sake of convenience, we split the proof into several parts: Part 1. In accordance with condition H4.2, since χ 1 t 0 ) < x 0 < χ 2 t 0 ), we have δ 1 > 0) : t 0 R+, t 0 t 0 < δ 1 ) x 0 R, x 0 x 0 < δ 1 ) χ 1,χ 2 C[[0,T],R], χ 1 t) χ 1t) < δ 1, χ 2 t) χ 2t) < δ 1, for t [0,T]) χ 1 t 0 ) < x 0 < χ 2 t 0 ) and X 1t) < χ 1 t) < χ 2 t) < X 1t), 0 t T.

137 1. NOT FIXED MOMENTS 119 Part 2. From condition H4.6 for 0 < t T it follows that: χ 1 t) < χ 1 t)+i 1 t,χ 1 t)) < χ 2 t) and χ 1 t) < χ 2 t)+i 2 t,χ 2 t)) < χ 2 t). Using the inequalities above, we reach the conclusion δ 2,0 < δ 2 δ 1 ) : χ 1,χ 2 C[[0,T],R], χ 1t) χ 1 t) < δ 2, χ 2 t) χ 2t) < δ 2, 0 t T) χ 1 t) < χ 1 t)+i 1t,χ 1 t)) < χ 2 t), χ 1 t) < χ 2 t)+i 2t,χ 2 t)) < χ 2 t),0 t T. In other words, after an impulsive perturbation, the integral curves of the investigated problem and the corresponding perturbed problem start of points, which are located between their corresponding barriers curves. Part 3. Let for t 0 < t T, the integral curve of problem 4.1), 4.2), 4.3), 4.4) does not meet the barrier curves γ 1 and γ 2. We introduce the notation λ = min{ xt;t 0,x 0 ) χ 1 t), xt;t 0,x 0 ) χ 2 t), t 0 t T} > 0. Let δ 3 = min { δ 2, λ 2}. Assume that χ 1t) χ 1 t) < δ 3, χ 2t) χ 2 t) < δ 3, 0 t T. 4.15) Further, we use the theorem of continuous dependence on the initial condition of the solution of the initial problem without impulses, for brevity we call Theorem of Continuous Dependence see Theorem 7.1, Section 7, I, Coddington and Levinson [99], see also Theorem 0.1 in the introduction). We have δ 4,0 < δ 4 δ 3 ) : t 0 R+, t 0 t 0 < δ 4 ) x 0 R, x 0 x 0 < δ 4 ) x t;t 0,x 0 ) xt;t 0,x 0 ) < λ 2, max{t 0,t 0} = t max 0 t T. From 4.15) and 4.16) it is clear that λ xt;t 0,x 0 ) χ 1 t) = xt;t 0,x 0 ) x t;t 0,x 0)+x t;t 0,x 0) χ 1t)+χ 1t) χ 1 t) 4.16) xt;t 0,x 0 ) x t;t 0,x 0) + x t;t 0,x 0) χ 1t) + χ 1t) χ 1 t)

138 BARRIER CURVES < λ 2 + x t;t 0,x 0) χ 1t) + λ 2. Therefore, x t;t 0,x 0 ) χ 1 t) > 0, tmax 0 t T. 4.17) If t 0 t 0 is satisfied, then 4.17) is equivalent to the inequality x t;t 0,x 0 ) χ 1 t) > 0, t 0 t T. 4.18) Let t 0 < t 0. Then for every t,t 0 t t 0 = t max 0, we obtain x t;t 0,x 0 ) χ 1 t) x t;t 0,x 0 ) χ 1t) χ 1 t) χ 1 t) t x 0 + f τ,x τ;t 0,x 0 ))dτ t 0 χ 1 t 0 )+χ 1t 0 ) χ 1t) δ 3 t x 0 χ 1t 0 ) f τ,x τ;t 0,x 0 ))dτ t 0 χ 1 t 0) χ 1 t) δ 3 x 0 χ 1 t 0) C f t 0 t 0 L χ1 t 0 t 0 δ 3 x 0 χ 1t 0 ) C f +L χ1 )δ 4 δ 3. Since according to Part 1) the inequality x 0 χ 1t 0 ) > 0 is valid, then it is possible to determine the constants δ 3 andδ 4 such that, the following estimate is valid This means that x 0 χ 1t 0 ) C f +L χ1 )δ 4 δ 3 > 0. x t;t 0,x 0) χ 1t) > 0,t 0 t t 0 = t max 0, whence, considering 4.17) we reach again the inequality 4.18). Similarly it can be shown, that the next inequality is satisfied x t;t 0,x 0 ) χ 2 t) > 0, t 0 t T. 4.19) From the inequalities 4.18) and 4.19) it follows the conclusion that for t 0 < t T, the integral curve of the perturbed initial problem does not meet the barrier curves, i.e. the solution of problem4.7),4.8), 4.9), 4.10) is not subject to the impulsive effects and therefore, it coincides with the solution of problem without impulses 4.7), 4.10). In this case

139 1. NOT FIXED MOMENTS 121 we note that the assertion of Theorem 4.3 follows from the theorem of continuous dependence Theorem 0.1 in Introduction). We assume further, that the integral curve of the studied initial problem meets at least one of the barrier curves for t 0 < t T. Part 4. We will show that a finite number impulsive moments are contained in the interval t 0 < t T. Indeed, if the meetings between the integral curve of the initial problem and barrier curves are finite number for t 0 < t <, then the assertion of this part is trivial. If the curves γ 1 and γ 2 are met by the integral curves infinitely many times, then according to Theorem 4.1 we have that lim t i = and therefore the i meetings are a finite number for t 0 < t T. Let be true: t 0 < t 1 <... < t k T < t k+1. Part 5. Let the first meeting of the integral curve of the regular problem with the barrier curves be at the moment t 1,t 0 < t 1 < T. Let γ 1 be the first met barrier curve, i.e. χ 1 t) < xt;t 0,x 0 ) < χ 2 t) for t 0 t < t 1 and xt 1 ;t 0,x 0 ) = χ 1 t 1 ). Similarly we consider the case xt 1 ;t 0,x 0 ) = χ 2 t 1 ). We will show that under the assumption above and for sufficiently small perturbations of the initial condition and barrier curves then the integral curve t,x t;t 0,x 0 )) meets the perturbed barrier curve γ 1 for t 0 < t < T. Two cases are possible: Case 1. χ 1 t 1 ) < χ 1t 1 ). Assume that for every initial point x 0, χ 1t 0) < x 0 < χ 2t 0), theintegral curve oftheperturbedproblemdoes notmeet theperturbedbarriercurveγ1 fort 0 t t 1, i.e. x t;t 0,x 0 ) > χ 1 t). Then from the theorem of continuous dependence it follows that δ 5,0 < δ 5 δ 4 ) : t 0 R+, t 0 t 0 < δ 5 ) x 0 R, x 0 x 0 < δ 5 ) x t;t 0,x 0 ) xt;t 0,x 0 ) χ 1 t 1) χ 1 t 1 ), max{t 0,t 0} = t max 0 t t 1, whence for t = t 1 we find x t 1 ;t 0,x 0 ) xt 1;t 0,x 0 ) χ 1 t 1) χ 1 t 1 ). 4.20) We consider the function F1 t) = x t;t 0,x 0 ) χ 1 t), which is continuous for t 0 t t 1. The following inequality is true F 1 t 0 ) = x t 0 ;t 0,x 0 ) χ 1 t 0 ) = x 0 χ 1 t 0 ) > 0.

140 BARRIER CURVES Using 4.20) we derive the estimate F 1 t 1) = x t 1 ;t 0,x 0 ) χ 1 t 1) xt 1 ;t 0,x 0 ) χ 1 t 1 ) = 0, whence, it follows that there exists a point t 1,0 < t 1 t 1, such that F1 t 1 ) = 0, i.e. x t 1 ;t 0,x 0 ) = χ 1 t 1 ). The last equality contradicts the assumption we made. Therefore, if the initial points of the basic and perturbed problems are sufficiently close, then the integral curve of problem 4.7), 4.8), 4.9), 4.10) meets the curve γ1 at the moment t 1,t 0 < t 1 t 1 < T. Case 2. χ 1 t 1 ) χ 1t 1 ). Denote by Xt;t 0,x 0 ) the solution of problem without impulses) 4.1), 4.4). It is clear that for t 0 t t 1, the equality X t;t 0,x 0 ) = xt;t 0,x 0 ) is valid. According to Theorem 4.2 the inequality X t;t 0,x 0 ) < χ 1 t) is satisfied for every t 1 < t T. Let point t be an arbitrary in the interval t 1,T]. Then Xt ;t 0,x 0 ) < χ 1 t ). Assume that for t 0 t t and for every initial point x 0,χ 1t 0) < x 0 < χ 2 t 0 ) the integral curve of the perturbed problem does not meet the perturbed barrier curve γ1, i.e. x t;t 0,x 0 ) > χ 1 t). Note It is obvious that λ = χ 1 t ) X t ;t 0,x 0 ) > 0. δ 6,0 < δ 6 δ 5 ) : t 0 R+, t 0 t 0 < δ 6 ) x 0 R, x 0 x 0 < δ 6 ) x t;t 0,x 0 ) Xt;t 0,x 0 ) λ 2, tmax 0 t t, From implication above at t = t, we receive Let δ 7 = min { δ 6, λ 2}. Assume that: x t ;t 0,x 0) Xt ;t 0,x 0 ) λ 2. t 0 R+, t 0 t 0 < δ 7, x 0 x 0 < δ 7, χ 1 t) χ 1t) < δ 7, χ 2 t) χ 2t) < δ 7 for 0 t T. Again, considering the continuous function F 1 t) = x t;t 0,x 0 ) χ 1 t) for t 0 t t, we obtain successively F 1 t ) =x t ;t 0,x 0 ) χ 1 t ) =x t ;t 0,x 0 ) Xt ;t 0,x 0 )+Xt ;t 0,x 0 ) χ 1 t )+χ 1 t ) χ 1t ) x t ;t 0,x 0) Xt ;t 0,x 0 ) χ 1 t ) Xt ;t 0,x 0 ))

141 + χ 1 t ) χ 1 t ) 1. NOT FIXED MOMENTS 123 λ 2 λ+ λ 2 =0. Since F1 t 0 ) > 0, then there exists a point t 1, t 0 < t 1 t T, such that F1 t 1 ) = 0, i.e. x t 1 ;t 0,x 0 ) = χ 1 t 1 ). In other words, we reach the conclusion, that the integral curve t,x t;t 0,x 0 )) of perturbed problem 4.7), 4.8), 4.9), 4.10) meets the curve γ1 at the point t 1,t 0 < t 1 < T. Part 6. We will estimate the difference between the impulsive moments t 1 and t 1. Let t 1 t 1. The case t 1 > t 1 is considered analogously. Let ε 1 be an arbitrary positive number. From Theorem 0.1 in the Introduction it follows δ 8,0 < δ 8 δ 7 ) : t 0 R+, t 0 t 0 < δ 8 ) x 0 R, x 0 x 0 < δ 8 ) x t;t 0,x 0) xt;t 0,x 0 ) ε 1, t max 0 t t ) Let δ 9 = min{δ 8,ε 1 } and let us assume χ 1 t) χ 1t) < δ 9, χ 2 t) χ 2t) < δ 9, 0 t T. 4.22) We denote F t) = χ 1 t) t f τ,xτ;t 0,x 0 ))dτ, t 1 t t 1. Therefore F t 1 ) F t 1 ) = χ 1 t 1 ) t 1 χ 1 t 1 t 1 t 1 t 1 ) =xt 1 ;t 0,x 0 ) f τ,xτ;t 0,x 0 ))dτ t 1 t 1 f τ,xτ;t 0,x 0 ))dτ f τ,xτ;t 0,x 0 ))dτ χ 1 t 1 ) t 1 =xt 1 ;t 0,x 0 ) χ 1 t 1 ) x t 1 ;t 0,x 0 )+x t 1 ;t 0,x 0 ) =xt 1 ;t 0,x 0 ) x t 1 ;t 0,x 0 )+χ 1 t 1 ) χ 1t 1 ) 2ε 1.

142 BARRIER CURVES There exists a point t,t 1 < t < t 1, such that F t 1 ) F t 1) = d dt F t )t 1 t 1) 2ε 1 ) d dt χ 1t ) f t,xt ;t 0,x 0 )) t 1 t 1) 2ε 1 ) d dt χ 1t ) f t,χ 1 t ))+f t,χ 1 t )) f t,xt ;t 0,x 0 )) t 1 t 1 ) 2ε 1. From the last inequality we obtain where m+f t,χ 1 t )) f t,xt ;t 0,x 0 )))t 1 t 1 ) 2ε 1, 4.23) { } d m = min dt χ 1t) f t,χ 1 t)), 0 t T. By the implications above and the first inequality in condition H4.9 we obtain that m > 0. Since the function f is continuous it follows that there exists constant such that if χ 1 t ) xt ;t 0,x 0 ) <, then From the continuity of the functions f t,χ 1 t )) f t,xt ;t 0,x 0 )) < m ) φ 1 t) = f t,χ 1 t)) d dt χ 1t) < 0and φ 2 t) = f t,χ 2 t)) d dt χ 2t) > 0, for 0 t T. Therefore, we reach the conclusion, that δ 10,0 < δ 10 δ 9 ) : χ 1,χ 2 C 1 [[0,T],R], χ 1t) χ 1 t) < δ 10, χ 2t) χ 2 t) < δ 10, 0 t T) φ 1t) = f t,χ 1t)) d dt χ 1t) < 0, φ 2 t) = f t,χ 2 t)) d dt χ 2 t) > 0, 0 t T. In other words, if the curves γ 1 and γ1 on one hand and γ 2 and γ2 on the other hand are close enough to each other, the condition H4.9 is valid not only on the curves γ 1 and γ 2, but also one the curves γ1 and γ 2. This allows us to apply Theorem 4.2 to the solution X t;t 0,x 0 ) of problem

143 1. NOT FIXED MOMENTS 125 without impulses 4.7), 4.10) and the perturbed barrier curve γ1. We conclude that for every t > t 1, the following inequality is satisfied In particular we have X t;t 0,x 0 ) < χ 1 t). X t ;t 0,x 0 ) < χ 1 t ). 4.25) We set δ 11 = min{δ 10, } and assume that χ 1 t) χ 1t) < δ 11, χ 2 t) χ 2t) < δ 11 for 0 t T. From 4.25) and Theorem 0.1 in Introduction it follows that δ 12,0 < δ 12 δ 11 ) : t 0 R +, t 0 t 0 < δ 12 ) x 0 R, x 0 x 0 < δ 12 ) X t ;t 0,x 0) xt ;t 0,x 0 ) < χ 1t ) X t ;t 0,x 0). If the inequality X t ;t 0,x 0 ) xt ;t 0,x 0 ) is valid, then from the estimate above we derive xt ;t 0,x 0 ) X t ;t 0,x 0 ) < χ 1 t ) X t ;t 0,x 0 ) xt ;t 0,x 0 ) < χ 1t ). If the inequality X t ;t 0,x 0 ) > xt ;t 0,x 0 ) is fulfilled, then by 4.25) once again we receive the validity of the next inequality xt ;t 0,x 0 ) < χ 1t ). As t 0 < t < t 1, it is satisfied χ 1 t ) < xt ;t 0,x 0 ). Therefore we find χ 1 t ) < xt ;t 0,x 0 ) < χ 1t ) 0 < xt ;t 0,x 0 ) χ 1 t ) < χ 1 t ) χ 1 t ) δ 11, whence, taking into consideration 4.24), it follows that f t,χ 1 t )) f t,xt ;t 0,x 0 )) < m 2. According to 4.23) and the last inequality we derive that m m ) t 1 t 1 2 ) 2ε 1 t 1 t 1 4ε 1 m. 4.26) Remind that in the estimate above ε 1 is an arbitrary positive constant. Consequently, without loss of generality, we may assume that if

144 BARRIER CURVES the initial points and barrier curves of the basic and perturbed problem are sufficiently close, then the estimate t 1 t 1 < η is valid. Part 7. Again, without loss of generality, we may assume that t 1 t 1. We will estimate the difference between the solutions of the original and perturbed initial problems with impulses at the moment t 1 +0 = t max It holds x t 1 +0;t 0,x 0 ) xt 1 +0;t 0,x 0 ) t1 = x t 1 ;t 0,x 0 )+I 1x t 1 ;t 0,x 0 ))+ f τ,x τ;t 0,x 0 ))dτ t 1 t 1 xt 1 ;t 0,x 0 ) f τ,xτ;t 0,x 0 ))dτ I 1 xt 1 ;t 0,x 0 )) t 1 x t 1;t 0,x 0) xt 1;t 0,x 0 ) + I 1 x t 1;t 0,x 0)) I 1 xt 1 ;t 0,x 0 )) t 1 t 1 + f τ,x τ;t 0,x 0 ))dτ + f τ,xτ;t 0,x 0 ))dτ. 4.27) t 1 t 1 The first addend in the right hand side of 4.27) we estimate by 4.21). More precisely, it is valid x t 1 ;t 0,x 0 ) xt 1 ;t 0,x 0 ) < ε ) To estimate the second addend in the right hand side of 4.27), first we will estimate the difference x t 1 ;t 0,x 0 ) xt 1;t 0,x 0 ) = χ 1 t 1 ) χ 1t 1 ) χ 1 t 1 ) χ 1t 1 ) + χ 1t 1 ) χ 1t 1 ), from where with 4.22), condition H4.4 and estimate 4.26) we find x t 1;t 0,x 0) xt 1 ;t 0,x 0 ) ε 1 +L χ1 t 1 t 1 ε ). 4.29) m Then, using the continuity of the function I 1 and estimate 4.29), we obtain that ε 2 > 0) ε 1 ε 2 ) > 0) : x t 1 ;t 0,x 0 ) xt 1;t 0,x 0 ) ε m )) I 1 x t 1;t 0,x 0)) I 1 xt 1 ;t 0,x 0 )) < ε )

145 1. NOT FIXED MOMENTS 127 By means of condition H4.1 and 4.26) we estimate the third addend in the right hand side of 4.27) as follows t 1 f τ,x τ;t 0,x 0 ))dτ C f t 1 t 1 4C f m ε ) t 1 Analogously, we get t 1 f τ,xτ;t 0,x 0 ))dτ t 1 4C f m ε ) Finally, by the inequalities 4.27), 4.28), 4.30), 4.31) and 4.32) we receive x t 1 +0;t 0,x 0 ) xt 1 +0;t 0,x 0 ) 2 m 2+m+4C f)ε 1 +ε ) Part 8. The results of the previous parts of the proof of the theorem can be summarized as follows: T > 0) ε > 0) η > 0) δ1 > 0) δ0 = δ0t,ε,η,δ 1) > 0) : t 0 R +, t 0 t ) 0 < δ0 x 0 R, x 0 x 0 < δ0 ) χ 1,χ 2 C1 [[0,T],R], χ 1 t) χ 1t) < δ 0, χ 2 t) χ 2t) < δ 0 for 0 < t < T) we have: χ 1 t 0 ) < x 0 < χ 2 t 0 ),X 1t) < χ 1 t) < χ 2 t) < X 1t),0 t T Part 1); χ 1 t 1 ) < xt 1 +0;t 0,x 0 ) < χ 1 t 1 ),χ 1t 1) < x t 1 +0;t 0,x 0) < χ 1t 1) Part 2); if χ 1 t) < xt;t 0,x 0 ) < χ 1 t) for t 0 t T, then χ 1 t) < x t;t 0,x 0 ) < χ 1 t) for t 0 t T see Part 3); if t,xt;t 0,x 0 )) meets the barrier curve γ 1 or γ 2 ) for t 0 < t < T, then the integral curve t,x t;t 0,x 0 )) meets the perturbed barrier curve γ1 or γ2) respectively for t 0 < t < T Part 5); t 1 t 1 < η Part 6); x t 1 +0;t 0,x 0 ) xt 1 +0;t 0,x 0 ) < δ1 Part 7); x t;t 0,x 0 ) xt;t 0,x 0 ) < ε,max{t 0,t 0} = t max 0 t t min 1 = min{t 1,t 1} inequality 4.21)).

146 BARRIER CURVES t min i We set = min{t i,t i}, t max i By induction for i = 1,2,...,k consecutively we receive: T > 0) ε > 0) η > 0) δi+1 > 0 ) ) ) δ i = δi T,ε,η,δ i+1,0 < δ i δi 1 : x i = x t max i = max{t i,t i}, i = 1,2,...,k and t min k+1 = T. +0;t 0,x 0 ), x i xtmax i +0;t 0,x 0 ) < δ i ) χ 1,χ 2 C1 [[0,T],R], χ 1 t) χ 1t) < δ i, χ 2 t) χ 2t) < δ i, for t [0,T]) we have if the integral curve t,xt;t 0,x 0 )) meets the barrier curve γ 1 or γ 2 ) for t max i < t < T, then the integral curve t,x t;t 0,x 0 )) of the perturbed problem meets the perturbed barrier curve γ1 or γ2 ) respectively for tmax i < t < T; t i+1 t i+1 < η; ) ) x t max i+1 +0;t 0,x 0 x t max i+1 +0;t 0,x 0 < δ i+1 ; x t;t 0,x 0 ) xt;t 0,x 0 ) < ε,t max i < t t min i+1. Thus, we obtain formally the sequence of constants: δ0, δ 1,...,δ k ; δ 0 < δ 1 < < δ k ; δ0 = δ 0 δ 1 ), δ 1 = δ 1 δ 2 ),...,δ k = ) δ k δ k+1. First we fix the constant δk+1 = ε, then we define the constants δ k,δ k 1,...,δ 0, consecutively. From the induction made above, for the obtained in this way constant δ0, it follows T > 0) ε > 0) η > 0) δ0 = δ0t,ε,η) > 0) : ) t 0 R +, t 0 t 0 < δ0 x 0 R, x 0 x 0 < δ0) χ 1,χ 2 C1 [[0,T],R], χ 1 t) χ 1t) < δ0, χ 2 t) χ 2t) < δ0, x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t max i < t t min i+1 ; t max i t min i for t [0,T]) < η, i = 0,1,...,k x t;t 0,x 0 ) xt;t 0,x 0 ) < ε,t [t 0,T]\ B η t i ). The theorem is proved. i=0,1,...

147 2. APPLICATION Application: Continuous Dependence of the Solutions of the Gompertz Model with Non Fixed Moments of Impulses on the Initial Condition and Barrier Curves Gompertz Model is a mathematical model for a time series, where growth is slowest at the start and end of a time period. This model is an adequate mathematical model for studying numerous real processes of evolution of an isolated population. In sixties A.K. Laird see Laird [209]) successfully used the Gompertz model to fit data of growth of tumors. In fact, tumors are cellular populations growing in a confined space where the availability of nutrients is limited. Let us note the future works of Steel, Wheldon et al see Steel [320], Wheldon [336]), where one may find many applications and discussions of Gompertz model and its applicability in medicine. On the other hand, some isolated populations develop optimally if the amount of their biomasses is maintained within certain limits. The optimality of their development is in a certain sense, such that: growth is relatively more intensive or a population growth is highest for fixed period as in Laird model), etc. As a rule these quantitative restrictions are highly dependent upon food supplies, environment, competitions, etc. To support the population biomass in these optimal limits it is possible to carry out external discrete effects perturbations), which consist of the removal or addition of certain quantities of biomass. One of the possibilities is the duration of these external effects to be negligibly small compared with the total duration of the process development of the isolated species. In this variant of maintaining the optimal levels of the population biomass, the effects are made instantaneously, in the form of impulses. An appropriate mathematical apparatus for studying the mentioned above processes are impulsive differential equations. It is natural to assume that the impulsive effects consist in adding or removal of the certain biomass amount. They are realized when the biomass quantity reaches the predefined boundaries, which define the optimal amount of biomass. An adequate mathematical model of such processes is the Impulsive Equation of Gompertz. The corresponding initial problem has the type: dn dt = N r γlnn), χ 1 < N t) < χ 2, 4.34) N t+0) = N t)+i 1 t,n t)), N t) = χ 1, 4.35) N t+0) = N t)+i 2 t,n t)), N t) = χ 2, 4.36)

148 BARRIER CURVES N 0) = N 0, 4.37) where: N = N t) is the amount of biomass at the moment t 0; r = const > 0 is the reproductive potential of population; γ = const > 0 is a coefficient of intra specific competition coefficient of self-poisoning), specific to the concrete type); χ 1 t) = χ 1 = const > 0 and χ 2 t) = χ 2 = const > 0 are respectively lower and upper barrier functionsconstants), defining the optimal limit values of the amount of biomass. In other words, thebiomassn = N t)oftheisolatedspecieswhichdevelopment dynamics is described by the impulsive problem 4.34), 4.35), 4.36), 4.37), is optimal if the following restrictions are valid χ 1 N t) χ 2, t 0; Assume that the next inequalities are satisfied ) r 0 < χ 1 < χ 2 < exp. γ We will pay ) attention to the fact, that the constants X 1 = 0 are X 2 = exp are zeros of the right sides of the equation 4.34) and r γ therefore they are singular solutions. It is clear that these solutions are unattainable if the initial point is between them; I 1 = I 1 t,n) = const > 0 is the size of impulsive addition of biomass if it reaches the lower barrier value of biomass. We assume that 0 < I 1 < χ 2 χ 1 is satisfied; I 2 = I 2 t,n) = const < 0 is the size of impulsive removal of biomass if it reaches the upper barrier value of biomass. We have χ 1 χ 2 < I 2 < 0; N 0 is an amount of biomass at the initial moment t = 0. The inequalities χ 1 < N 0 < χ 2 are satisfied. Discrete effects are realized in the above model at the moments at which the biomass of the isolate population issue is equalized with some of the barrier quantities. The moments of impulsive influences effects) are denoted by t 1,t 2,... and the following inequalities 0 = t 0 < t 1 < t 2 <... arevalid. Ifatthemoment t i theequality N t i ) = χ 1 isfulfilled, thenthe magnitude of the impulsive effect coincides with I 1 t i,n t i )) = I 1 > 0 and we have a discrete addition of the biomass at this moment. If at the moment t i the equality N t i ) = χ 2 is fulfilled, then the magnitude of the impulsive effect coincides with I 2 t i,n t i )) = I 2 < 0 and we have a discreteremovalofthebiomass, i = 1,2,... Thepurposeofthesediscrete

149 2. APPLICATION 131 interventions is to maintain the biomass within the optimal range, i.e. for each t 0 the inequalities χ 1 N t) χ 2 to be fulfilled. The right hand side of equation 4.34) is positive, because the values of the phase variable N are between the barrier functions in this case between the barrier constants), then. Therefore, in the model of Gompertz, the biomass amount increases between two neighboring impulsive moments. Consequently only upper barrier constant can be reach, i.e. in the considered model only discrete removals are made at the moments at which the biomass quantity of the studied isolate population equalizes with the upper barrier quantity χ 2. It is known that for each value of the initial ) point N, satisfying the r inequalities 0 = X 1 t) < N < X 2 t) = exp, the solution N t) of γ the equation without impulses 4.14) with initial condition N t ) = N is defined for any t t 0 and it is fulfilled 0 = X 1 < N t) < X 2 = exp ) r. γ That means the condition H4.5 is true. We have that the function Moreover and f t,n) = f N) = N r γlnn) C [ X 1,X 2 ),R +]. lim f t,n) = lim N r γlnn) = 0 N X 1 =0 N 0 lim f t,n) = lim N r γlnn) = 0, N X 2 =exp γ) r N exp γ) r whence it follows that f is bounded in the open interval X 1,X 2 ). Consequently, condition H4.2 is satisfied. Since the inequalities ) r 0 = X 1 < χ 1 < χ 2 < X 2 = exp γ are valid, then condition H4.3 is obvious. We introduce the symbols χ = χ 2 χ 1 and L χ1 = L χ2 = 0. Then condition H4.4 is valid too. We denote I = min { I1 χ 2 χ 1, I 2 χ 1 χ 2 Then the following inequalities are fulfilled: I I 1 χ 2 χ 1 I, I } {, I I1 = max, χ 2 χ 1 I 2 χ 1 χ 2 I. I 2 χ 1 χ 2 }.

150 BARRIER CURVES So, condition H4.6 is fulfilled. As χ 1,χ 2,I 1,I 2 are constants, then they are continuous functions and therefore the conditions H4.8 and H4.10 are satisfied too. It holds f t,χ 2 t)) =f χ 2 ) = χ 2 r γlnχ 2 ) ) r rγ =γχ 2 γ lnχ 2 > γχ 2 ln exp =0 = d dt χ 2 = d dt χ 2t), ))) r γ i.e. the second inequality in condition H4.9 is valid. Since the barrier constant χ 1 is out of reach from the solution of the model impulsive problem, then the first inequality in condition H4.7 does not need to be satisfied. As a result of made above considerations in this section we find that the conditions H4.1-H4.10 are satisfied from the impulsive model of Gompertz 4.34), 4.35), 4.36), 4.37). Therefore, the results, which are obtained in the previous section in the present chapter are valid for the studied model. This means: 1) The isolated biological species whose evolution is subordinated to the impulsive model of Gompertz, according to Theorem 4.1 continues its existence indefinitely long time; 2) Easytoseethatthesolutionoftheimpulsive modelofgompertz reaches the upper barrier constant infinitely many times. If the moments of these meetings are t 1,t 2,..., then lim i t i = is fulfilled according to Theorem ) If we change slightly the biomass amount N 0 at the initial moment t 0 = 0 and the values of upper and lower barrier constants, then the new impulsive model will be called perturbed. Then, according to Theorem 4.3, the amounts of biomasses in the regular and perturbed model are approximately equal for a limited period of time except relatively small intervals. These small periods of time are located between the corresponding moments of impulsive perturbations for both models.

151 Chapter 5 Orbital Hausdorff Continuous Dependence of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects The basic object of investigation in this chapter is impulsive nonlinear autonomous systems differential equations. The impulsive moments are non fixed. More precisely, the impulses are realized when the trajectory of the corresponding initial problem intersects the so called impulsive set, situated in the phase space of the system. Assume that the impulsive set is a smooth surface. In the first section in the chapter, the concept orbital Hausdorff continuous dependence on the initial point and impulsive effects is introduced. Sufficient conditions are found, under which the solutions possess this property. A more detailed, the Hausdorff continuous dependence on the initial point and impulsive effects means that, relatively small perturbations of the initial point and impulses lead to small differences between the trajectories of the basic initial problem and the corresponding perturbation problem. The trajectories are defined in the limited time interval fixed in advance. The distance between the trajectories is in the terms of Hausdorff metric. The selected metric is very convenient in determining the distances between discontinuous functions, especially when the points of discontinuity of both functions are different in time as the solutions of impulsive differential equations). For example, using the uniform metric in the previous two chapters, closeness between the solutions of both problems the regular and its perturbed) is not required in the neighborhood of impulsive moments, although the sum of the lengths of these surroundings could be an arbitrary small. Consequently, using 133

152 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE pedantically the uniform metric for impulsive systems with variable impulsive moments, we could not argue that the small changes in the system parameters lead to the small changes in its solution in terms of uniform metric. This inconvenience in the Hausdorff metric falls. In the second section of the chapter, the theoretical results are applied to the impulsive mathematical model of Lotka-Volterra, describing the evolutionary dynamics of the predator-prey type, which is subjected to the short term external effects. The research in this chapter is based on the paper of Dishliev and Dishlieva [123]. The first studies, devoted to the impulsive equations using Hausdorff metric, belong to B. Ahmad and S. Sivasundaram: Ahmad and Sivasundaram [8, 9]. Numerous studies are dedicated on the above-mentioned impulse model of Lotka-Volterra, we will cite the following: Akhmet et al. [18], Chen et al. [92], Dai et al. [101], Dong et al. [130], Elaydi and Yakubu [135], Gnana and Lakshmikantham [161], Guo and Chen [166], Huo [181], Jin et al. [191], Liu et al. [222, 223, 225], Liu and Chen [227], Liu et al. [230], Meng and Chen [242], Meng et al. [243, 245], Saito [289], Shen and Li [298], Xia [340], Zeeman and Zeeman [356], Zhao et al. [374].

153 where: 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE Orbital Hausdorff Continuous Dependence of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects We consider the following initial problem: dx = f x), φxt)) 0, 5.1) dt xt+0) = xt)+ixt)), φxt)) = 0, 5.2) x0) = x 0, 5.3) f : D R n ; φ : D R; n N, n 2; D is a domain in R n, D R n ; x 0 D. The set of points x D, satisfying the equality φx) = 0, is named impulsive set in the case, this is impulsive surface in D). This set is denoted by Φ, i.e. Φ = {x D;φx) = 0}. The function I : Φ R n is called impulsive function. Assume that Id+I) : Φ D, where Id is the identity in R n. The moments, at which the trajectory of the problem above meets consecutively the impulsive surface, are denoted by t 1,t 2,...,0 < t 1 < t 2 <... The solution xt;x 0 ) of problem 5.1), 5.2), 5.3) is a piecewise continuous function. It is fulfilled: 1.1. For 0 t < t 1, the solution of problem 5.1), 5.2), 5.3) coincides with the solution of problem without impulses) 5.1), 5.3) and the inequality φxt;x 0 )) 0 is valid; 1.2. At the moment t 1, the equalities xt 1 ;x 0 ) = xt 1 0;x 0 ) = x 1 and φxt 1 ;x 0 )) = φx 1 ) = 0 are satisfied; 1.3. We have xt 1 +0;x 0 ) =xt 1 ;x 0 )+Ixt 1 ;x 0 )) =Id+I)xt 1 ;x 0 )) =Id+I)x 1 ) =x + 1 ;

154 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 2.1. For t 1 < t < t 2, the solution of problem 5.1), 5.2), 5.3) coincides with the solution of system 5.1) with initial condition xt 1 +0) = x + 1 and the inequality φxt;x 0 )) 0 is valid; 2.2. At the moment t 2, the equalities xt 2 ;x 0 ) = xt 2 0;x 0 ) = x 2 and φxt 2 ;x 0 )) = φx 2 ) = 0 are fulfilled; 2.3. We have xt 2 +0;x 0 ) =xt 2 ;x 0 )+Ixt 2 ;x 0 )) =Id+I)x 2 ) =x + 2 ; and so on... We will consider the following perturbed form of the problem 5.1), 5.2), 5.3): dx dt = f x ), φx t)) 0, 5.4) x t+0) = x t)+i x t)), φx t)) = 0, 5.5) x 0) = x 0, 5.6) where I : Φ R n and Id+I ) : Φ D, x 0 D. It is obvious that the differences between problem 5.1), 5.2), 5.3) and perturbed problem 5.4), 5.5), 5.6) are in the initial condition and impulsive function. Let x t;x 0 ) denotes the solution of perturbed problem 5.4), 5.5), 5.6). Let t 1, t 2,..., 0 < t 1 < t 2 <... be the moments at which the perturbed trajectory meets the impulsive surface Φ. We use the following notations: x i =xt i ;x 0 ) = xt i 0;x 0 ), x i =x t i;x 0) = x t i 0;x 0), x + i =xt i +0;x 0 ) = xt i ;x 0 )+Ixt i ;x 0 )) x + =Id+I)x i ), i =x t i +0;x 0 ) = x t i ;x 0 )+I x t i ;x 0 )) =Id+I )x i ), i = 1,2,... WedenotebyXt;x 0 )andx t;x 0 )thesolutionofproblemswithout impulses 5.1), 5.3) and 5.4), 5.6), respectively. For the trajectories of problems 5.1), 5.2), 5.3) and 5.4), 5.5), 5.6), we introduce the notations: γx 0 ;[0,T]) = {xt;x 0 );0 t T}

155 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 137 x x 0 0 x + 2 x + 2 x + 1 x + 1 γx 0 ;[0,T]) x 2 x 1 x 1 Φ γ x 0 ;[0,T]) D x 2 Figure 5.1 and γ x 0;[0,T]) = {x t;x 0);0 t T}, respectively, where T is a positive constant see Figure 5.1). Let aa 1,a 2,...,a n ), bb 1,b 2,...,b n ) R n be two points. Then their dot product, Euclidean norm and Euclidean distance are: a,b =a 1 b 1 +a 2 b 2 + +a n b n, a = a,a 1 2 = a 2 1 +a a2 n, ρ E a,b) = a 1 b 1 ) 2 +a 2 b 2 ) 2 + +a n b n ) 2. It is clear that a b = ρ E a,b). If A and B are two nonempty subsets in R n, then Euclidean and Hausdorff distances between them are: and ρ E A,B) =inf{inf{ρ E a,b),b B},a A} ρ H A,B) =max{sup{inf{ρ E a,b),b B},a A},

156 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE respectively. The inequality sup{inf{ρ E a,b),a A},b B}}, ρ E A,B) ρ H A,B) is obvious. The closure and the contour of set A are denoted by Ā and A. Remark 5.1. We have: ρ E γx 0 ;[0,T]),γ x 0;[0,T])) = inf{inf{ρ E x t ;x 0),xt;x 0 )),0 t T},0 t T}; ρ H γx 0 ;[0,T]),γ x 0 ;[0,T])) = max{sup{inf{ρ E x t ;x 0 ),xt;x 0)),0 t T},0 t T}, sup{inf{ρ E x t ;x 0 ),xt;x 0)),0 t T},0 t T}}; ρ E γx 0 ;[0,T]),γ x 0 ;[0,T])) ρ H γx 0 ;[0,T]),γ x 0 ;[0,T])) sup{ρ E x t;x 0 ),xt;x 0)),0 t T} =sup{ x t;x 0 ) xt;x 0),0 t T}. Definition 5.1. We will say that the solution of problem 5.1), 5.2), 5.3) depends orbital Hausdorff continuously on the initial point x 0 and impulsive function I, if: ε > 0) T > 0) x 0 D) I : Φ R n ) δ = δε,t,x 0,I) > 0) : x 0 D,ρ Ex 0,x 0) < δ) I : Φ R n,ρ E I x),ix)) < δ for x Φ) ρ H γ x 0;[0,T]),γx 0 ;[0,T])) < ε. Our goal in the present section is to find the sufficient conditions for orbital Hausdorff continuous dependence of the solution of problem 5.1), 5.2), 5.3) on the initial point and impulsive function. Our basic conditions are: H5.1. f C[D,R n ]; H5.2. If the impulsive moments are infinity many, then lim i t i = ; H5.3. For every point x 0 D, the problem without impulses 5.1), 5.3) has a unique solution, and it is defined for t 0; H5.4. The impulsive set Φ = {x D;φx) = 0} satisfies Φ\Φ D\D;

157 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 139 H5.5. φ C 1 [D,R] and for every point x Φ gradφx),f x) > 0; H5.6. There exists a positive constant C f such that, for every point x D the inequality is valid; H5.7. I C[Φ,R n ]; H5.8. The inequality is valid. f x) C f ρ E Φ,Id+I)Φ)) > 0. Theorem 5.1. Let the conditions H5.1, H5.2 and H5.3 be satisfied. Then for every point x 0 D the solution of problem with impulses 5.1), 5.2), 5.3) exists and it is unique for t 0. The proof of Theorem 5.1 is obvious and we omit it. Theorem 5.2. Let the conditions H5.1-H5.8 be satisfied. Then the solution of problem 5.1), 5.2), 5.3) depends orbital Hausdorff continuously on the initial point x 0 and impulsive function I. Proof. Let ε and T be arbitrary positive constants. There are two cases: the trajectory γ does not intersect the impulsive set Φ, or trajectory intersects the impulsive set. We consider these two cases separately. Case 1. Let the trajectory γx 0 ;[0,T]) = {xt;x 0 );0 t T} does not intersect the impulsive set Φ, i.e. Assume that γx 0 ;[0,T]) Φ =. ρ E γx0 ;[0,T]), Φ ) = 0. Then, since γx 0 ;[0,T]) is a compact set and Φ is a close set, then we find out that γx 0 ;[0,T]) Φ. Whence, we deduce that γx 0 ;[0,T]) Φ\Φ.

158 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE By condition H5.4 we reach the conclusion γx 0 ;[0,T]) D\D. In other words, the trajectory of problem 5.1), 5.3) meets the contour of the domain D. This implies that the solution of the problem without impulses is not continuable from some placethe moment of meeting with D) further. The conclusion above contradicts condition H5.3. In this way we find out that ρ E γx0 ;[0,T]), Φ ) > 0. Therefore, there exists a constant d > 0 such that ρ E γx 0 ;[0,T]),Φ) = d > ) By Theorem 0.1 from Introduction, we obtain δ 1 = δ 1 x 0,T) > 0) : x 0 D, x 0 x 0 < δ 1 ) x t;x 0 ) xt;x 0) < min{ε,d}, 0 t T sup{ x t;x 0 ) xt;x 0),0 t T} < min{ε,d}. 5.8) From the estimate above, and Remark 5.1, it follows that ρ E γ x 0;[0,T]),γx 0 ;[0,T])) < d. The last inequality and 5.7) yield ρ E γ x 0;[0,T]),Φ) > 0. Therefore, the trajectory γ x 0;[0,T]) as well as the trajectory γx 0 ;[0, T])) does not intersect the impulsive set Φ. Now 5.8) yields ρ H γ x 0 ;[0,T]),γx 0;[0,T])) < ε. The last inequality proves the theorem in Case 1. Case 2. The trajectory γx 0 ;[0,T]) of the impulsive problem intersects it is possible many times) the impulsive set Φ. We will split the proof of the theorem in Case 2 into several parts: Part 2.1. We show that, the trajectory γ x 0;[0,T]) of perturbed problem 5.4), 5.5), 5.6) intersects also the impulsive set Φ for 0 < t < T. Let the moment of the first meeting of the trajectory γx 0 ;[0,T]) with the impulsive set Φ be t 1, 0 < t 1 < T. We substitute: F t) = φxt;x 0 )), F t) = φx t;x 0)), 0 < t < T. Here the functions Xt,x 0 ) and X t;x 0) are the solutions of the problems without impulses 5.1), 5.3) and 5.4), 5.6), respectively. Then it is fulfilled F t 1 ) = φxt 1 ;x 0 )) = φxt 1 ;x 0 )) = 0.

159 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 141 By condition H5.5 we obtain gradφx),f x) > 0, x Φ. We consider the case gradφx),f x) < 0, for x Φ similarly. The following is true d dt F t 1) = d dt φxt 1;x 0 )) = gradφx t 1 ;x 0 )),f X t 1 ;x 0 )) >0, whence, we deduce that there exists a constant t > 0 such that: and F t 1 τ) = φxt 1 τ;x 0 )) < 0 F t 1 +τ) = φxt 1 +τ;x 0 )) > 0 for 0 < τ < t. We determine the magnitude of t, further. From the continuity of the function φ, it follows that, there exists a constant d > 0, such that: and x D,ρ E x,x t 1 t;x 0 )) < d) φx) < 0 x D,ρ E x,xt 1 + t;x 0 )) < d) φx) > 0. Let ε 1 be an arbitrary positive number, which we will specify further. The theorem of continuous dependence Theorem 0.1) yields δ 2 = δ 2 x 0,d,ε 1, t),0 < δ 2 < δ 1 ) : x 0 D, x 0 x 0 < δ 2 ) X t 1 t;x 0 ) Xt 1 t;x 0 ) < d, X t 1 + t;x 0) Xt 1 + t;x 0 ) < d, X t;x 0 ) Xt;x 0) < ε 1 2, 0 t t 1 + t. 5.9) From the first inequality in 5.9), it follows that whence, we find Similarly, we find ρ E X t 1 t;x 0 ),Xt 1 t;x 0 )) < d, F t 1 t) = φx t 1 t;x 0 )) < 0. ρ E X t 1 + t;x 0 ),Xt 1 + t;x 0 )) < d see Figure 5.2), where from we derive the inequality F t 1 + t) = φx t 1 + t;x 0)) > 0.

160 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE ϕx) < 0 Φ d Xt 1,x 0 ) X t 1,x 0 ) ϕx) > 0 d Xt 1 +,x 0 ) X t 1 +,x 0 ) Figure 5.2 Then from the continuity of the function F it follows that, there exists a point t 1, t 1 t < t 1 < t 1 + t, such that F t 1 ) = 0 φx t 1 ;x 0 )) = 0, i.e. the trajectory γ x 0 ;[0, )) intersects the impulsive surface Φ in the point t 1, where t 1 t 1 < t. Part 2.2. We find the estimate for the distance Let us set t min i ρ H γ x 0 ;[0,t 1 ]),γx 0;[0,t 1 ])). = min{t i,t i From the last estimate in 5.9) it follows that }, tmax i = max{t i,t i }, i = 1,2,... x t;x 0) xt;x 0 ) = X t;x 0) X t;x 0 ) < ε 1 2, 0 t t min ) Without loss of generality we may assume that the inequality t 1 t 1 is satisfied. Then using condition H5.6, the following estimate is valid x t ;x 0) xt 1 ;x 0 ) x t ;x 0) x t 1 ;x 0) + x t 1 ;x 0) xt 1 ;x 0 )

161 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 143 t < t 1 f x τ;x 0 )) dτ + t t 1 )C f + ε 1 2 tc f + ε 1 2 for every t,t 1 t t 1. We specify the magnitude of the constant t. Suppose that tc f < ε 1 2. Using the last inequality we get x t ;x 0) xt 1 ;x 0 ) < ε 1, t min 1 = t 1 t t 1 = t max ) In particular, setting t = t max 1 = t 1 From 5.10), we receive, that in 5.11), we have x t 1;x 0) xt 1 ;x 0 ) < ε ) sup{inf{ρ E x t ;x 0 ),xt;x 0)),0 t t 1 },0 t t 1} sup{inf{ρ E x t ;x 0 ),xt;x 0)),0 t t 1 },0 t t 1 } sup{ρ E x t;x 0 ),xt;x 0)),0 t t 1 } =max{ x t;x 0 ) xt;x 0),0 t t 1 } ε ) Again, by inequalities 5.10) and 5.11) we reach sup{inf{ρ E x t ;x 0 ),xt;x 0)),0 t t 1 },0 t t 1 } =max { sup { } inf{ρ E x t ;x 0),xt;x 0 )),0 t t 1 },0 t t 1 = t min 1, sup { }} inf{ρ E x t ;x 0),xt;x 0 )),0 t t 1 },t min 1 = t 1 t t 1 max{sup{ρ E x t ;x 0),xt ;x 0 )),0 t t 1 }, sup{ρ E x t ;x 0),xt 1 ;x 0 )),t 1 t t 1}} max{max{ x t ;x 0 ) xt ;x 0 ),0 t t 1 }, max{ x t ;x 0 ) xt 1;x 0 ) < ε 1,t 1 t t 1 { } ε1 } max 2,ε 1 = ε ) Then, from 5.13) and 5.14) we conclude that ρ H γ x 0 ;[0,t 1 ]),γx 0;[0,t 1 ])) < ε 1. ε 1 2

162 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE ) Part 2.3. We will estimate the distance ρ E x + 1,x + 1. Let ε2 be an arbitrary positive number. In accordance with condition H5.7 and inequality 5.12), it is fulfilled ε 1 = ε 1 ε 2,I) > 0) : x t 1 ;x 0 ) xt 1;x 0 ) < ε 1 ) Ix t 1 ;x 0 )) I xt 1;x 0 )) < ε ) Again, using 5.12), 5.15), and the additional assumption, that for any x Φ the following estimate is valid we get ρ E x + 1,x + 1 I x) I x) = ρ E I x),ix)) < δ 2, ) = x + 1 x + 1 = x t 1 +0;x 0 ) xt 1 +0;x 0 ) = x t 1 ;x 0 )+I x t 1 ;x 0 )) xt 1;x 0 ) Ixt 1 ;x 0 )) x t 1 ;x 0 ) xt 1;x 0 ) + I x t 1 ;x 0 )) I xt 1;x 0 )) <ε 1 + I x t 1 ;x 0 )) Ix t 1 ;x 0 )) + Ix t 1 ;x 0 )) I xt 1;x 0 )) <ε 1 +δ 2 +ε 2. Part 2.4. We will estimate the difference t 2 t 1 from below, i.e. the difference between the second and the first impulsive moment of the basic problem 5.1), 5.2), 5.3). According to condition H5.8 there exists a positive constant d, such that Since x 2 = xt 2 ;x 0 ) Φ and x + 1 d ρ E x2,x + 1 ρ E Φ,Id+I)Φ)) = d > 0. Id+I)Φ), then t 2 ) x2 x 1 + = t 2 Therefore, we obtain the estimate t 1 t 1 f x τ;x + 1 )) dτ )) f x τ;x + dτ 1 t2 t 1 )C f. t 2 t 1 d C f. Part 2.5. We will show that the trajectory γx 0 ;[0,T]) meets finite number of times times the impulsive set Φ. Assume that the meetings

163 1. ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE 145 are infinity many. By analogy with part 2.4 we find the estimates t i+1 t i d C f, i = 1,2,... The last inequalities show that lim i t i =, hence finite numbers of these impulsive moments belong to the finite interval [0,T], which proves the statement in this part. Let 0 < t 1 < t 2 < < t k < T t k+1. Part 2.6. Using the previous parts of the current proof, we come to the following conclusions: δ 1 > 0) δ 0,0 < δ 0 < δ 1) : x 0 D,ρ E x 0,x 0 ) < δ 0) I : D R n,ρ E I x),ix)) < δ0, for x D). We have: 1.1. The trajectory γ x 0;[0,T]) meets the impulsive set Φ at the moment t 1 and the inequality t 1 t 1 < δ1 is valid; 1.2. ρ H γ x 0 ;[0,t ) 1 ]),γx 0;[0,t 1 ])) < δ1 ; 1.3. ρ E x + 1,x + 1 < δ 1. Similarly: δ2 > 0) δ 1,0 < δ 1 < δ 2 ) : ) ) x + 1 D,ρ E x + 1,x + 1 < δ 1 I : D R n,ρ E I x),ix)) < δ1, for x D) 2.1. The trajectory γ x 0 ;[0,T]) meets the impulsive set Φ at the moment t 2 and t 2 t 2 < δ2 ; 2.2. ρ H γ x 0 ;[0,t ) 2 ]),γx 0;[0,t 2 ])) < δ2 ; 2.3. ρ E x + 2,x + 2 < δ 2 etc. Finally, we obtain the conclusion δ k+1 > 0 ) ) ) ) δk,0 < δk < δk+1 : x + k D,ρ E x + k,x+ k < δ k I : D R n,ρ E I x),ix)) < δk, for x D) k.1. The trajectory γ x 0 ;[0,T]) does not meet the impulsive set Φ for t k < t < T; k.2. ρ H γ x 0;[0,T]),γx 0 ;[0,T])) < δk+1. After all, if we substitute ε = δk+1 and define the monotonously decreasing sequence of constants δk,δ k 1,...,δ 0, we find ε > 0) δ 0 = δ 0 ε,t,x 0,I) > 0) : x 0 D,ρ Ex 0,x 0) < δ 0 ) I : D R n,ρ E I x),ix)) < δ 0, for x D) ρ H γ x 0;[0,T]),γx 0 ;[0,T])) < ε. The theorem is proved.

164 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE

165 2. APPLICATION Application: Orbital Hausdorff Continuous Dependence of the Solutions of Lotka-Volterra Model with Non Fixed Moments of Impulses on the Initial Condition and Impulsive Effects The Lotka-Volterra mathematical model also known as the predatorprey model, initially proposed in 1910), is a pair of nonlinear, differential equations used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. Let us underline that in the classical Lotka-Volterra model it is supposed that there are not any external influences. Historically, there are many works devoted to this model. Let us mention the works of Kolmogorov and so-called Kolmogorov predator-prey model see F. Hoppensteadt, Predator-prey model, Scholarpedia, 1, No )), which is a more general framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease, and mutualism. The Lotka-Volterra mathematical model is also a practical model describing many economical processes see M. Desai, P. Ormerod, Richard Goodwin: A short appreciation, The Economic Journal, 108, No ), ). In this section a community of the type predator-prey is studied with additional assumption: it is subjected to the external influences, usually due to the human intervention. These effects are expressed in the removal or addition of certain quantities of biomass both from the prey and predator. It is natural to require the following restrictions of the external influences: The duration of each of these perturbations is negligible compared with the total duration of the process so that it is possible to assume that, these effects are instantaneous, in the form of impulses; The influences take placeinthemoments atwhich thebiomasses of the prey and the predators reach the certain quantitative characteristics. From mathematical point of view the impulsive removals or additions of biomass take place at the meeting between the system trajectory and a pre-fixed set, called impulsive set, which is located in the phase space of the system. Usually, the impulsive set is a smooth curve of the space of permitted states of the system; It is practically impossible to add or to remove biomass under a certain minimum. Therefore, we conclude that the sizes of the impulsive perturbations are limited from below;

166 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE The sizes of the impulsive perturbations are limited from above because: If the influence is a biomass removal the most frequent case which is investigated), it is impossible for the amount taken away to be more than the amount available in the moment of the impulsive effect; If theinfluence isabiomass additionthisis ararecase), then from an financial point of view it is not appropriate for the supplement amount of biomass to be above an economically justified maximum; Very often, due to the objective reasons, it is impossible to separate the prey and predator. In these cases the removed taken) biomass is a mix of biomasses from both of the prey and predator. Moreover, the removed volumes from both species are proportional to the amounts of their biomasses at the moment of the removal. More precisely, if the biomass quantities of the prey and predator at the moment of impulsive effects are m and M, respectively, then the removed quantities of biomass are.m and.m, respectively. The function = m,m) is defined for every point of the impulsive set and the inequalities 0 m,m) 1 take place. Frequent case is = const; It turns out that the exploitation of the considered above co - associations is facilitated if they possess a periodic law of evolution. Therefore, after a removal type impulsive perturbation is desirable for the amounts of both species biomasses to be in such volumes that, they lie on the same trajectory, which is described before the impulsive moment. The generalized Lotka-Volterra impulsive model with an initial condition, satisfying the requirements above, has the form: dm dt = ṁ = F mm,m) = mr 1 q 1 M), M t) kmt); 5.16) dm = dt Ṁ = F M m,m) = M r 2 q 2 m), M t) kmt); 5.17) mt+0) = 1 )mt), M t) = kmt); 5.18) M t+0) = 1 )M t), M t) = kmt); 5.19) m0) = m 0, M 0) = M 0, 5.20) where: m = mt) > 0isthebiomassquantityofthepreyatthemoment t 0;

167 2. APPLICATION 149 M = M t) > 0 is the biomass quantity of the predator at the moment t 0; The constants r 1 > 0 and r 2 > 0 are specific coefficients of the relative growth of the first species prey) and of second species predator), respectively; The constants q 1 > 0 and q 2 > 0 are the coefficients, reflecting interspecies competition for the prey and the predator, respectively; The set of points m,m) which belong to the acceptable states of the co-association R + R + and satisfy the equality M = km, is called an impulsive set. In this case the set is a ray with origin coinciding with the Cartesian system origin and a slope k > 0, which will be specified further); mt) and M t) are the quantities of biomass of the prey and the predator respectively, which are removed in the form of impulses. The function = m,m) will be specified further. The impulsive moments coincide with the moments at which the ratio of the quantities of biomasses of the predator and prey reaches the value k. The constants m 0 > 0 and M 0 > 0 are the biomasses quantities of both species at the initial moment t = 0. It is known that system 5.16), 5.17) possesses: Unstable stationary point 0,0), the origin is a saddle point); r2 Stable stationary point m 00,M 00 ) =, r ) 1 ; q 2 q 1 First integral of the form where U m,m) =q 1 M +q 2 m r 1 lnm r 2 lnm +r 1 ln r 1 1 )+r 2 ln r ) 2 1 q 1 q 2 =W m,m) W m 00,M 00 ), W m,m) = q 1 M +q 2 m r 1 lnm r 2 lnm; For every point m,m) R + R +,m,m) m 00,M 00 ), the inequality U m,m) > 0 is valid. It is fulfilled U m 00,M 00 ) = 0; For any constant c 0, the implicitly given curve γ c = {m,m) : U m,m) = c}

168 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE is a trajectory of the system 5.16), 5.17) with a properly chosen initial condition it is sufficient to assume that U m 0,M 0 ) = c); For any constant c > 0, the set D c = {m,m) : U m,m) < c} is a connected domain, located in R + R +, with a contour D c = γ c ; For any constant c > 0, it is satisfied m 00,M 00 ) D c ; If 0 < c 1 < c 2, then γ c1 D c2. The obtained basic results in this chapter are applicable to the model 5.16)-5.20). For more clarity, we separate the corresponding considerations in this section into several parts: Part 1. We will assume that the domain D, where is located the trajectory of the problem with impulses of the population dynamics, is situated between two border trajectories γ c1 and γ c2, where the constants c 1 and c 2 satisfy the inequalities 0 < c 1 < c 2, i.e. D = D c2 \D c1 see Figure 5.3). Part2. Wewillassumethatthetrajectoryoftheimpulsive problemis apartofthetrajectoryγ c0, where theconstant c 0 satisfies theinequalities c 1 < c 0 < c 2. Part 3. The impulsive surface in this case the impulsive segment) is a part of the straight line l : M = km. Hence, it is fulfilled φm,m) = M km. The slope k of the line l is specified as follows. Let the points m 00,M00), 1 m 1 00,M 00 ) γ c1, as constants m 1 00 and M00 1 are the larger solutions of the following equations, respectively: U m,m 00 ) = c 1 q 2 m r 2 lnm = c 1 r 2 ln r ) 2 1, q 2 U m 00,M) = c 1 q 1 M r 1 lnm = c 1 r 1 ln r 1 1 q 1 We find approximately the solutions of the equations above. Suppose that M 00 k M1 00. m 1 00 m 00 We define the impulsive set impulsive segment) in this way: ). Φ = {m,m);m km = 0,m > m 00,M > M 00 } D c2 \D c1. Having in mind the choice of the slope k we conclude that the endpoints of Φ i.e. the contour Φ of Φ) belong to the trajectories γ c1 and γ c2

169 2. APPLICATION 151 U c 2 c 1 m m 00 D M 00 γ c1 γ c2 M Figure 5.3 respectively, i.e. on the contour D of the domain D. This means that Φ\Φ = Φ D = D\D, i.e. the condition H5.4 is valid. Part 4. We assume that the trajectory of impulsive problem 5.16)- 5.20) of the population dynamics is periodical. This assumption is imperative because the existence of cycles in the development and a periodicity in the biomass removal of the investigated community in terms of exploitation is preferred. This assumption will be satisfied if a mapping point of theprocess m,m), which ismoving onthetrajectory γ c0 ofthe considered problem after an impulsive effect falls again on the same trajectory. Thus, we attain the conclusion that, the impulsive vector has a form I m,m) = Im,km) = m,k m ), where m and m are solution of the following system: U m,km) = c 0, U m m,km m )) = c 0, m > 0.

170 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE The above system consists of one and the same equation but with different variables). This equation has a more detailed type q 1 km+q 2 m r 1 lnkm) r 2 lnm = c 0 r 1 ln r 1 q 1 1 ) r 2 ln r ) 2 1 q 2 q 1 k +q 2 )m r 1 +r 2 )lnm = c 00, 5.21) where c 00 = c 0 r 1 ln r 1 1 ) r 2 ln r ) 2 1. kq 1 q 2 Let the approximate solutions it is impossible to find the exact solutions ingeneral case) oftheequation 5.21)bem 0 = m 0c 0 )andm 0 = m 0c 0 ), as m 0 < m 0. Then m = m 0 and M = km 0 are the approximate quantities of the prey and predator biomasses, respectively in the moment of impulsive perturbation. The sizes of impulsive effects removals) according to the prey and predator are: m = m c 0 ) = m 0 c 0) m 0 c 0) and k m = k m c 0 ) = km 0 c 0) m 0 c 0)). Clearly, the impulsive function I = I 1,I 2 ) = m c 0 ),k m c 0 )) is continuous on c 0, where c 1 < c 0 < c 2, i.e. on the impulsive set Φ. Consequently the solution H5.7 is valid see Figure 5.4). Part 5. According to the choice of the impulsive function in the previous section it follows that Id+I)Φ) = {m,m);m km = 0,m < m 00,M < M 00 } D c2 \D c1. Then, ρ E Φ,Id+I)Φ)) = m 1 m 1 1+k 2 > 0, where m 1 and m 1 are two different solutions of the equation q 1 km+q 2 m r 1 lnkm) r 2 lnm = c 1 r 1 ln r 1 1 ) r 2 q 1 q 1 k +q 2 )m r 1 +r 2 )lnm = c 11, where c 11 = c 1 r 1 ln r 1 1 ) r 2 ln r ) 2 1. kq 1 q 2 ln r 2 q 2 1 )

171 M 1 00 M 2. APPLICATION 153 Φ = {m,m); M km = 0} km 0 k m M 00 γ γ c1 c0 γ c2 km 0 0 m 0 m m 00 m Figure 5.4 Thus, we show that condition H5.8 is fulfilled. Part 6. Suppose that for t = 0, it is satisfied: m0) = m 0 = m 0, M 0) = M 0 = km 0. 0 m 1 00 The given assumption is negligible, because of requirement that the initial point m 0,M 0 ) belongs to set Id+I)Φ). Part 7. Since the solution is a periodic, then there exists a positive constant t which is equal to the period of the solution of problem with impulses) such that: t i = i t, i = 1,2,... are true. Therefore, lim t i =, i i.e. condition H5.2 is fulfilled. m

172 ORBITAL HAUSDORFF CONTINUOUS DEPENDENCE Part 8. The condition H5.1 follows by the form of the right hand side of the equations 5.16) and 5.17). It is known that the problem without impulses 5.16), 5.17), 5.20) possesses a unique solution which trajectory coincides with γ c0, where c 0 = U m 0,M 0 ). Consequently Theorem 5.1 is valid, i.e. the solution of problem 5.16)-5.20) exists and it is unique for every t 0. Part 9. Since then we deduce Hence φm,m) = M km and M km C 1 D c2 \D c1,r), φ C 1 D,R). On the other hand, we have gradφm,m) = m M km), M gradφm,m),f m,km) = k,1), ) M km) = k,1). mr 1 q 1 km), kmr 2 q 2 m)) =km q 2 m r 2 )+q 1 km r )) 1 q 2 =M q 2 m m 00 )+q 1 M M 00 )), 5.22) Then from Part 3 it follows that for every point m,m) Φ the inequalities m > m 00 and M > M 00 are fulfilled. Therefore, using 5.22), it follows that gradφm,m),f m,km) > 0, i.e. condition H5.5 is valid. Part 10. It is trivial to verify the condition H5.6. Part 11. The considerations made above assure that the Lotka- Volterra generalized impulsive model satisfies the conditions of Theorem 5.2. Therefore, the periodical solution of problem 5.16)-5.20) depends orbital Hausdorff continuously on the initial point m 0,M 0 ) and the impulsive function I. The following interpretation of the result obtained is possible. Let: The initial quantities of the biomasses of the prey and predator in the perturbed co-association are not significantly different from the initial quantities of the biomasses of the prey and the predator in the original co-association; q 1

173 2. APPLICATION 155 The impulsive removals in the perturbed and the original coassociation are approximately equal. Then for a finite time interval independently how large it is) the changes of the biomass in the both biosystems communities develop approximately under one and same law, i.e. the regular and the perturbed problem possess neighboring trajectories for a finite time interval.

174

175 Chapter 6 Orbital Hausdorff Stability of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition The basic object of investigation in the previous and present chapters is a class of autonomous nonlinear differential equations with variable impulsive effects. We remind that the impulses are realized at the moments, in which the trajectory of the corresponding initial problem meets the so called impulsive set, situated in the phase space of the system. In the studies in this chapter we assume unlike to Chapter 5), that the impulsive set coincides with a part of hyperplane. For the problem, described above, for the first time the concepts orbital gravitation and orbital Hausdorff stability on the initial condition are introduced. The term orbital gravitation refers to the systems without impulses. We will say that one system of differential equations possesses this property with a constant κ, if the Hausdorff distance between its two arbitrary trajectories is κ times less than the Euclidean distance between them. In the first section of Chapter 6, sufficient conditions are found, under which if the corresponding system without impulses is orbital gravitating then the solution of the basic problem with impulses is orbital Hausdorff stable on the initial condition. In the second section of the chapter, the classical without impulses) model of Lotka-Volterra of the population dynamics is considered. The sufficient conditions are found for the orbital gravitation and Hausdorff stability on the initial point. The results, obtained in this chapter are new. During the last years many studies have been devoted to the qualitative theory of the differential equations without impulses using Hausdorff metric: Georgiou et al. [159], Gnana and Lakshmikantham [161], Mönch 157

176 ORBITAL HAUSDORFF STABILITY and von Harten [256], Nieto [266]. Here this metric is used to research impulsive differential equations.

177 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION Orbital Hausdorff Stability of the Solutions of Autonomous Differential Equations with Non Fixed Moments of Impulses on the Initial Condition Throughout the present secton, we consider the following initial problem dx dt = f x), a,xt) a 0, 6.1) xt+0) = xt)+i xt)), a,xt) = a 0, 6.2) x0) = x 0, 6.3) where: f : D R n ; a = a 1,a 2,...a n ) R n, a = 1; a 0 R; n N; D is a domain in R n ; x 0 D. The set of points x D, satisfying the equality a,x = a 0, is named impulsive set in the case this is a part of hyperplane in D). We denote this set by α, i.e. α = {x D; a,x = a 0 }. The function I : α R n is called impulsive function. Assume that Id+I) : α D, where Id is identity in R n. The moments, at which the trajectory of the problem abovemeetsconsecutively theimpulsiveset, aredenotedbyt 1,t 2,...,0 < t 1 < t 2 <... The solution xt;x 0 ) of this problem is a piecewise continuous function. Moreover: 1.1. For 0 t < t 1, the solution of problem 6.1), 6.2), 6.3) coincides with the solution of problem without impulses) 6.1), 6.3) and the inequality a,xt;x 0 ) a 0 is valid; 1.2. At the moment t 1, the equalities: xt 1 ;x 0 ) = xt 1 0;x 0 ) = x 1 and a,xt 1 ;x 0 ) = a,x 1 = a 0 are satisfied; 1.3. xt 1 +0;x 0 ) = xt 1 ;x 0 )+Ixt 1 ;x 0 )) = Id+I)xt 1 ;x 0 )) = Id+I)x 1 ) = x + 1 ; 2.1. For t 1 < t < t 2, the solution of problem 6.1), 6.2), 6.3) coincides with the solution of the system 6.1) with the initial condition xt 1 +0) = x + 1 and the inequality a,xt;x 0) a 0 is valid; 2.2. At the moment t 2 the equalities xt 2 ;x 0 ) = xt 2 0;x 0 ) = x 2 and a,xt 2 ;x 0 ) = a,x 2 = a 0

178 ORBITAL HAUSDORFF STABILITY are satisfied; 2.3. xt 2 +0;x 0 ) = xt 2 ;x 0 )+Ixt 2 ;x 0 )) = Id+I)xt 2 ;x 0 )) = Id+I)x 2 ) = x + 2; and so on. Along with the problem 6.1), 6.2), 6.3), we consider also the corresponding perturbed initial problem 6.1), 6.2) with the initial condition x0) = x 0, 6.4) where x 0 D. Let xt;x 0 ) denote the solution of the perturbed problem 6.1), 6.2), 6.4) and t 1,t 2,..., 0 < t 1 < t 2 <... are the moments at which the trajectory of this problem meets the impulsive hyperplane α. We use the notations: x i =xt i ;x 0 ) = xt i 0;x 0 ), x i =xt i ;x 0 ) = xt i 0;x 0 ), x + i =xt i ;x 0 )+I xt i ;x 0 )) = Id+I)x i ) = xt i +0;x 0 ), x + i =xt i;x 0)+Ixt i;x 0)) = Id+I)x i) = xt i +0;x 0), where i = 1,2,... The following additional notations are also used: We denote by and γx 0 ;[0, )) = {xt;x 0 ),0 t < }, γx 0 ;[0, )) = {xt;x 0 ),0 t < }. the trajectories of problems 6.1), 6.2), 6.3) and 6.1), 6.2), 6.4), respectively; The solutions and trajectories of the problems without impulses 6.1), 6.3) and 6.1), 6.4) we denote by: X t;x 0 ), Γx 0 ;[0, )), X t;x 0), and Γx 0;[0, )), respectively; The parts of the trajectories: γx 0 ;[0, )), γx 0 ;[0, )), Γx 0;[0, )), Γx 0 ;[0, )), defined for 0 a t b, are denoted by: γx 0 ;[a,b]), γx 0;[a,b]), Γx 0 ;[a,b]), Γx 0;[a,b]), respectively;

179 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 161 Definition 6.1. We say that the system 6.1) is orbital gravitating in the domain D with a constant κ 1, if: x 0,x 0 D) ρ H Γx 0;[0, )),Γx 0 ;[0, ))) κρ E Γx 0;[0, )),Γx 0 ;[0, ))) max{sup{inf{ρ E X t ;x 0 ),Xt;x 0)),t 0},t 0}, sup{inf{ρ E Xt ;x 0 ),Xt;x 0)),t 0},t 0}} κinf{inf{ρ E Xt ;x 0 ),Xt;x 0)),t 0},t 0}. Definition 6.2. We will say that the solution of problem 6.1), 6.2), 6.3) is orbital Hausdorff stable if: ε > 0) x 0 D) δ = δε,x 0 ) > 0) : x 0 D,ρ Ex 0,x 0) < δ) ρ H γx 0 ;[0, )),γx 0;[0, ))) < ε. The main purpose of the present section is to find the sufficient conditions for orbital Hausdorff stability of the solution of problem 6.1), 6.2), 6.3). Further we will use the following conditions: H6.1. f C[D,R n ]; H6.2. There exist two positive constants C f and C f, such that for every point x D the following inequalities are fulfilled: C f f x) C f ; H6.3. For every point x 0 D, the problem without impulses 6.1), 6.3) possesses a unique solution defined for t 0; H6.4. The set α = {x D; a,x = a 0 }, where a = a 1,a 2,...a n ) R n, a = 1 and a 0 R, is bounded; H6.5. There exists a constant C a, 0 < C a 1, such that for every point x α the next inequality is fulfilled a,f x) C a f x) ; H6.6. There exists a constant C Id+I, such that the next inequality is valid ρ E α,id+i)α)) C Id+I > 0. H6.7. There exists a positive constant C I, such that for every two points x,x α the next inequality is true ρ E x +I x ),x +Ix )) C I.ρ E x,x ). Theorem 6.1. Assume that:

180 ORBITAL HAUSDORFF STABILITY 1) The conditions H6.1 and H6.3 are valid; 2) If the impulsive moments are infinitely many, then lim i t i =. Then for every point x 0 D there exists a unique solution of the initial problem with impulses 6.1), 6.2), 6.3) for t 0. We omit the proof of Theorem 6.1. From the theorem above it follows that Theorem 6.2. Assume that: x 0 D) γx 0 ;[0, )) D). 1) The conditions H6.1 H6.7 are valid; 2) The system 6.1) is orbital gravitating in the domain D with constant κ 1; 3) The inequality is valid. C I < C a κ1+c a ). Then for every initial point x 0 / α, the solution of problem 6.1), 6.2), 6.3) is orbital Hausdorff stable on the initial condition. Proof. Let ε be an arbitrary positive constant. There are two cases: the trajectory does not intersect the impulsive set, or trajectory intersects the impulsive set. We will consider these two cases separately. Case 1. Let the trajectory γx 0 ;[0, )) does not intersect the impulsive set α, i.e. γx 0 ;[0, )) = Γx 0 ;[0, )). Assume that ρ E γx 0 ;[0, )),ᾱ) = 0. According tothe conditionh6.4 theset ᾱ iscompact. Since γx 0 ;[0, )) is a closed set, then we deduce that whence we find out that γx 0 ;[0, )) ᾱ, γx 0 ;[0, )) ᾱ\α. Taking into account that ᾱ\α D\D,

181 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 163 then we deduce that γx 0 ;[0, )) D\D. In other words, the trajectory of problem considered6.1),6.3) meets the contour of the domain D. This means that the solution of the problem without impulses is not continuable from some place the moment of meeting) further. This result contradicts condition H6.3. Therefore, the inequality ρ E γx 0 ;[0, )),ᾱ) > 0. is valid. Let ρ E γx 0 ;[0, )),α) = r, 6.5) where r is a positive constant. Let δ = 1 minε,r). Then having in mind κ that system 6.1) is orbital gravitating in the domain D with coefficient κ 1, for every x 0 D and ρ E x 0,x 0 ) < δ we have ρ H Γx 0;[0, )),Γx 0 ;[0, ))) κρ E Γx 0;[0, )),Γx 0 ;[0, ))) κρ E x 0,x 0) < κδ =minε,r). 6.6) Taking into account equality 6.5) and estimate 6.6), we find ρ H Γx 0 ;[0, )),α) ρ H Γx 0 ;[0, )),Γx 0;[0, ))) +ρ H Γx 0 ;[0, )),α) > r+r =0. Therefore, the trajectory Γx 0 ;[0, )) of the perturbed problem without impulses 6.1), 6.4) does not meet the impulsive set α, which means that γx 0;[0, )) = Γx 0;[0, )) and the perturbed solution is not subjected to the impulsive perturbations as well as the solution of the basic problem). Using again the estimate 6.6) it follows that x 0 D,ρ E x 0,x 0 ) < δ) ρ H γx 0;[0, )),γx 0 ;[0, ))) =ρ H Γx 0;[0, )),Γx 0 ;[0, ))) <ε, i.e. the solution of problem 6.1), 6.2), 6.3) is orbital Hausdorff stable with respect to the initial point x 0.

182 ORBITAL HAUSDORFF STABILITY Case 2. The trajectory γx 0 ;[0, )) intersects it is possible many times) the impulsive set α. We will split the proof of the theorem in this case into several parts: Part 2.1. We will show that, if the distance between the points x 0 and x 0 is sufficiently small, then the trajectory γx 0 ;[0, )) of the perturbed problem 6.1), 6.2), 6.4) intersects also the impulsive set α. In accordance with condition H6.3 the solutions X t;x 0 ) and Xt;x 0 ) of the problems without impulses 6.1), 6.3) and 6.1), 6.4) are defined for every t 0. They are not subjected to the impulsive perturbations whether or not meet the impulsive set). In view of the condition H6.5 we assume that a,f x) C a f x), x α. We consider the case a,f x) C a f x), x α, similarly. Using conditions H6.2 we obtain that f x),a C a f x) C a C f > 0, x α. Let µ, 0 < µ < 1, be an arbitrary constant we will define it later). From the inequality above we obtain that f x),a µc a f x) 1 µ)c a f x) 1 µ)c a C f >0, x α. The left hand side of this inequality is a continuous function, which is greater than a positive constant and according to condition H6.4 the impulsive set α is bounded. Then it follows that, there exists a positive constant = µ) such that for every constant d, 0 d, it is fulfilled: The initial point x 0 / Bα,d) = {x D,a 0 d a,x a 0 +d}; The inequality is valid. f x),a µc a f x), x Bα,d). 6.7) We will note that the set Bα,d) consists of all the points of D, situated between the hyperplanes: α d = {x R n ; a,x = a 0 d} and α +d = {x R n ; a,x = a 0 +d}.

183 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 165 Assumethatthemomentoffirstmeetingofthetrajectoryγx 0 ;[0, )) with the impulsive set α is t 1 > 0. We consider the function At) = a,xt;x 0 ) a 0, t 0, for which is fulfilled At 1 ) = a,xt 1 ;x 0 ) a 0 = a,xt 1 ;x 0 ) a 0 = a,x 1 a 0 =0. 6.8) It means that the point Xt 1 ;x 0 ) α Bα,d). On the other hand, since the point x 0 = X0;x 0 ) / Bα,d), then it follows that there exists a point t d,0 < t d < t 1, such that X t d ;x 0 ) α d D. By estimate 6.7) for every point t t 1, for which d At) d, i.e.x t;x 0 ) Bα,d), the next inequality holds d dt At) = d dt a,x t;x 0) a 0 ) = a,f Xt;x 0 )) µc a f Xt;x 0 )) µc a C f = const > ) From 6.8) and 6.9) it is clear that there exists a point t +d > t 1, such that At +d ) = d, i.e. Xt +d ;x 0 ) α +d D see Figure 6.1). x 2 D α d α α +d X +d Xt +d ;x 0) Xt d ;x 0) Xt d ;x 0 ) Xt 1 ;x 0 ) 0 x 0 x0 x 1 X d Xt +d ;x 0 ) t Figure 6.1

184 ORBITAL HAUSDORFF STABILITY We deduce that: µ,0 < µ < 1) = µ) > 0) : d,0 d ) x 0 D\Bα,d)) t d,t +d R,0 < t d < t 1 < t +d ) : Xt d ;x 0 ) α d and Xt +d ;x 0 ) α +d. Since a is a unit vector, i.e. a = 1, it is satisfied d = ρ E α,α d ) = ρ E α,α +d ) > 0. In accordance with condition 2 of the theorem, the system 6.1) is orbital gravitating in the domain D with coefficient κ 1. Let d < minε,r) and δ = d. Then from the next inequality κ x 0 x 0 = ρx 0,x 0) < d κ and the fact Xt d ;x 0 ) α d, it follows that ρ E Γx 0;[0, )),α d ) ρ E Γx 0;[0, )),Xt d ;x 0 )) sup{ρ E Γx 0;[0, )),Xt;x 0 )),t 0} =sup{inf{ρ E Xt ;x 0),Xt;x 0 )),t 0},t 0} ρ H Γx 0 ;[0, )),Γx 0;[0, ))) κρ E Γx 0 ;[0, )),Γx 0;[0, ))) κρ E x 0,x 0) κ d κ = d. 6.10) In similar way we get ρ E Γx 0 ;[0, )),α +d) d. 6.11) From the inequalities 6.10) and 6.11) it follows that there exist points t d and t +d, 0 < t d < t +d, such that the following estimates are valid: ρ E X t ;x 0),α d ) < d, ρe X t + ;x 0),α+d ) < d. 6.12) We consider the function A t) = a,x t;x 0 ) a 0, t 0. Let X d and X +d be the orthogonal projections of the points X ) t d ;x 0 and X t +d 0) ;x respectively on the hyperplane α. Then from 6.12) it follows that, both vectors X d X ) ) t d 0)) ;x and X t +d ;x 0 X+d are unidirectional. The case when the vector a is unidirectional with these vectors will be considered. The other case is similar. Something

185 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 167 more, upon the admission to the beginning of Part 2.1 of the proof, i.e. in case of the inequality a,f x) C a f x), x α it can be shown that the vectors: X d X )) ) t d ;x 0 and X t +d ;x0) X+d are unidirectional. The following inequalities are valid under the given assumption: A t d) = a,x t d ;x 0) a0 = a,x d + a,x t d;x0) X d a0 = a,x d X ) t d;x 0 = X d X t d ;x 0) < 0; A t +d) = a,x t +d ;x 0) a0 = a,x +d + a,x t +d 0) ;x X+d a0 = a,x ) t +d ;x 0 X+d = X t +d ;x 0) X+d > 0. From the continuity of function A and the previous two inequalities, it follows that there exists a point t 1, 0 < t d < t 1 < t +d, such that A t 1) = 0 a,x t 1;x 0) a 0 = 0 Xt 1;x 0) α, i.e. the trajectory Γx 0;[0, )) trajectory γx 0;[0, )) of perturbed problem 6.1), 6.2), 6.4), respectively) intersects the impulsive set α. Part 2.2. We introduce a function T : R + R +, defined in the following way: t 0) T = T t) 0) : ρ E XT ;x 0 ),X t;x 0)) = ρ E Γx 0 ;[0, )),Xt;x 0)); ρ E Γx 0 ;[0,T )),Xt;x 0 )) < ρ E Γx 0 ;[0, )),Xt;x 0)), or which is the same: ρ E X T ;x 0 ),Xt;x 0)) < ρ E X t ;x 0 ),Xt;x 0)), 0 t < T = T t). In other words, for every moment t 0 and its corresponding point Xt;x 0 ), the moment T 0 is the first moment in which the distance between the points X T ;x 0 ) and Xt;x 0) is equal to the Euclidian distance between the trajectory Γx 0 ;[0, )) and point Xt;x 0) see Figure 6.2).

186 ORBITAL HAUSDORFF STABILITY x 0 x 0 XT,x 0 ) Figure 6.2 Part 2.3. The system 6.1) is gravitating for every t 0. Therefore: ρ E XT t);x 0 ),Xt;x 0)) = ρ E Γx 0 ;[0, )),Xt;x 0)) =inf{ρ E X t ;x 0 ),Xt;x 0)),0 t < } sup{inf{ρ E Xt ;x 0 ),Xt;x 0)),0 t < }, 0 t < } ρ H Γx 0 ;[0, )),Γx 0;[0, ))) κρ E Γx 0 ;[0, )),Γx 0;[0, ))) κρ E x 0,x 0) = k x 0 x 0 < κδ = d <ε. The last inequality yields ρ E Γx 0 ;[0, )),x 1) =ρ E X T t 1 );x 0 ),x 1) = X T t 1 );x 0 ) x 1 < d. Part 2.4. We find the following estimate see Part 2.3) ρ E x 1,x 1) = ρ E xt 1 ;x 0 ),xt 1;x 0 )). Let the point Xp be the orthogonal projection of point X T t 1 );x 0 ) onto the hyperplane α see Figure 6.3). Then X p X T t 1 );x 0) x1 XT t 1 );x 0)

187 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 169 x 0 x 0 XT t 1 );x 0 ) α x 1 x p x 1 Figure 6.3 On the other hand x 1 X T t 1 );x 0),a x 1 XT t 1 );x 0) =ρ E XT t 1 );x 0 ),x 1) d. t 1 f X τ;x 0))dτ,a T = t 1 ) t 1 f Xτ;x 0))dτ T t 1 ) t 1 f Xτ;x 0 )),a dτ T = t 1 ) t 1 f Xτ;x 0))dτ T t 1 ) t )) dτ 1 µc a f Xτ;x 0 T t 1 ) t 1 f X τ;x 0))dτ =µc a. T t 1 )

188 ORBITAL HAUSDORFF STABILITY Therefore, x 1 XT t 1 );x 0) 1 µc a x 1 X p +X p XT t 1 );x 0),a moreover, or = 1 µc a x 1 X p,a + X p XT t 1 );x 0 ),a ) = 1 µc a X p X T t 1 );x 0) d µc a, x 1 Xp x 1 XT t 1 );x 0 ) d, µc a ρ E x 1,x 1 ) = x 1 x 1 x 1 Xp + X p x 1 x 1 Xp + X T t 1 );x 0) x 1 d +d = d 1+ 1 ). 6.13) µ.c a µ.c a Part 2.5. We find an estimate using the previous results for the distance between the points x + 1 and x + 1. For this purpose, we use the fact that the operatorid + I) is compressing see condition H6.7). Using the same condition H6.7 and inequality 6.13) we obtain the following estimate ) ρ E x + 1,x + 1 =ρe xt 1 ;x 0 )+Ixt 1 ;x 0 )),xt 1;x 0 )+Ixt 1 ;x 0 ))) C I ρ E xt 1 ;x 0 ),xt 1;x 0 )) =C I ρx 1,x 1) dc I 1+ 1 ). 6.14) µc a From condition 3 of Theorem 6.2, it is obvious that the next inequalities are valid κc I 0 < C a 1 κc I ) < 1. We assume, without loss of generality, that κc I C a 1 κc I ) µ < 1, because µ is an arbitrary constant, satisfying the inequality 0 < µ < 1. Therefore, µc a C I κ1+µc a ).

189 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 171 Finally, from 6.14) and the last inequality, we get the estimate ) ρ E x + 1,x + d 1 κ. Part 2.6. We can assume that d C I+Id, because d is an arbitrary constant, satisfying the inequality 0 d min{ε,r, }. Then, using condition H6.6, we have ρ E α,x + 1 ) =ρe α,x 1 +I x 1 )) ρ E α,id+i)α)) C Id+I d, where x + 1 is a given point. Therefore x + 1 / Bα,d) = {x D,a 0 d a,x a 0 +d}. Part 2.7. Let the trajectory of the initial problem of impulsive differential equations meets repeatedly the impulsive set α. We will estimate the difference t 2 t 1 of two consecutive impulsive moments. It is valid: ) ) d ρ E α,x + 1 ρe x2,x + 1 = x2 x 1 + = xt2 ;x 0 ) xt 1 +0;x 0 ) t 2 )) = x τ;x + 1 dτ t 2 )) f x τ;x + dτ 1 C f t 2 t 1 ). Hence t 1 f t 1 t 2 t 1 d C f. If the impulsive moments are infinitely many, similarly we get the estimates t i+1 t i d Cf, i = 1,2,.... Therefore, lim i t i =. That is why, according to Theorem 6.1, we deduce that the solution of the initial problem 6.1), 6.2), 6.3) is continuable for every t > 0 independently of the choice of the initial point x 0 of the domain D.

190 ORBITAL HAUSDORFF STABILITY Part 2.8. Let t d be the moment for which Γx 0 ;[0, )) α d = X t d ;x 0 ) and Γx 0 ;[0,t d )) α d = see Case 1). Obviously 0 < t d < t 1. From Part 2.3 it follows that Therefore, ρ E X T t);x 0),Xt;x 0 )) < d for t 0. ρ E XT t);x 0),α) ρ E X T t);x 0),Xt;x 0 )) because, the inequality holds for 0 t t d. +ρ E Xt;x 0 ),α) > d+d = 0 Γx 0 ;[0,T t))) α = T t) < t 1, 0 t t d, ρ E Xt;x 0 ),α) d Part 2.9. We have see Part 2.8) sup{inf{ρ E xt ;x 0 ),xt;x 0)),0 t T t d )},0 t t d } =sup{inf{ρ E X t ;x 0 ),Xt;x 0)),0 t T t d )},0 t t d } =sup{ρ E Γx 0 ;[0,T t d ))),Xt;x 0 )),0 t t d } =sup{ρ E Γx 0;[0, )),Xt;x 0 )),0 t t d } sup{ρ E Γx 0;[0, )),Xt;x 0 )),0 t < } ρ H Γx 0;[0, )),Γx 0 ;[0, ))) κρ E Γx 0;[0, )),Γx 0 ;[0, ))) κ x 0 x 0 κδ = d < ε. Part In this part we consider the difference t 1 t d. Let point X p be the orthogonal projection of point X t d ;x 0 ) onto the hyperplane α. It is satisfied X p Xt d ;x 0 ) = d.

191 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 173 Moreover, we have x 1 Xt d ;x 0 ),a x 1 Xt d ;x 0 ) t 1 f Xτ;x 0 ))dτ,a t d = t 1 f X τ;x 0 ))dτ t d t 1 f X τ;x 0 )),a dτ t d = t 1 f Xτ;x 0 ))dτ t d t 1 µc a f Xτ;x 0 )) dτ t d t 1 f Xτ;x 0 ))dτ t d =µc a, or x 1 Xt d ;x 0 ) 1 µc a x 1 X p +X p X t d ;x 0 ),a = 1 µc a x 1 X p,a + X p Xt d ;x 0 ),a ) = 1 µc a X p X t d ;x 0 ) d µc a. From condition H6.2 it follows that the function f does not change its sign in the domain. Then t 1 t 1 1 t 1 t d = dτ f X τ;x 0 )) dτ C f t d t d = 1 t 1 C f f X τ;x 0 ))dτ t d

192 ORBITAL HAUSDORFF STABILITY = 1 C f Xt 1 ;x 0 ) X t d ;x 0 ) = 1 C f x 1 Xt d ;x 0 ) d µc a C f. Part Using the estimate from Part 2.10, we obtain ρ E Γx 0 ;[t d,t 1 ]),X T t d );x 0 )) =inf{ρ E X t;x 0 ),XT t d );x 0 )),t d t t 1 } { =inf ρ E t Xt d ;x 0 )+ f Xτ;x 0 ))dτ,xt t d );x 0), t d t d t t 1 } =inf{ t X t d;x 0 )+ f X τ;x 0 ))dτ X T t d );x 0 ), t d Xt d ;x 0 ) X T t d );x 0 ) t +inf f Xτ;x 0 ))dτ,t d t t 1 t d ρ E Xt d ;x 0 ),XT t d );x 0 )) t +inf f X τ;x 0 )) dτ,t d t t 1 t d d+ Cf d µc a C f = µc ac f +C f µc a C f d. t d t t 1 }

193 1. ORBITAL HAUSDORFF STABILITY ON THE INITIAL CONDITION 175 Part First, we get that sup{inf{ρ E X t ;x 0 ),Xt;x 0)),0 t t 1 },0 t t 1} sup{inf{ρ E X t ;x 0 ),Xt;x 0)),0 t T t d ) t 1 },0 t t 1} max{sup{inf{ρ E X t ;x 0),Xt;x 0 )),0 t T t d )}, 0 t t d }, sup{inf{ρ E X t ;x 0),Xt;x 0 )),t = T t d )},t d < t t 1 }} max{d,sup{ρ E X T t d );x 0),X t;x 0 )),t d < t t 1 }} = µc ac f +C f µc a C f d. 6.15) Similarly: sup{inf{ρ E Xt ;x 0 ),Xt;x 0)),0 t t 1 },0 t t 1 } Finally from 6.15) and 6.16) it follows that µc ac f +C f µc a C f d. 6.16) max{sup{inf{ρ E Xt ;x 0),Xt;x 0 )),0 t t 1},0 t t 1 }, sup{inf{ρ E Xt ;x 0),Xt;x 0 )),0 t t 1 },0 t t 1}} =ρ H {Γx 0;[0,t 1]),Γx 0 ;[0,t 1 ])} =ρ H {γx 0;[0,t 1]),γx 0 ;[0,t 1 ])} µc ac f +C f µc a C f d. Part From the previous parts of the proof we obtain ) κc I ε > 0) µ = const; C a 1 κc I ) µ < 1 = µ) > 0) { }) µca C f ε d;0 < d < min µc a C f +C f, µ),c I+Id δ;0 < δ < d ) κ x 0 D,ρ Ex 0,x 0) < δ) ) ρ E x + 1,x + d 1 < see Part 2.5); κ x + 1 / Bα,d) see Part 2.6); lim i t i = see Part 2.7); ρ H γx 0 ;[0,t 1 ]), γx 0;[0,t 1 ])) < ε see Part 2.12).

194 ORBITAL HAUSDORFF STABILITY Part The following cases are possible: The trajectory γx 0 ;[0, )) meets only once the impulsive set α. Then from , , by analogy with Case 1 it follows that: ρ H γx 0;[t 1, )),γx 0 ;[t 1, ))) =ρ H Γx 0;[t 1, )),Γx 0 ;[t 1, ))) <ε. In this case the theorem follows from the last inequalities and The trajectory γx 0 ;[0, )) meets two times the impulsive set α. Then similarly to , and we get: ρ E x + 2,x + 2 x + 2 / Bα,d), ) < d κ, ρ H γx 0;[t 1,t 2]),γx 0 ;[t 1,t 2 ])) < ε, 6.17) from where analogously to Case 1 we find the following inequality ρ H γx 0 ;[t 2, )),γx 0;[t 2, ))) < ε. In this case the theorem follows from last inequalities, and 6.17) The trajectory γx 0 ;[0, )) meets finite numbers times the impulsive set α. We prove the theorem in this case using an induction The trajectory γx 0 ;[0, )) meets infinitely many times the impulsive set α. Then using an induction we get the following estimates: ρ H γ x 0 ; [ t i 1,t i]),γx0 ;[t i 1,t i ]) ) < ε, i = 1,2,..., where t 0 = t 0 = 0. Having in mind that lim t i = part 2.7), i we deduce that Thus the theorem is proved. ρ H γx 0;[0, )),γx 0 ;[0, ))) < ε.

195 2. LOTKA-VOLTERRA MODEL Orbital Hausdorff Stability of the Solutions of Lotka-Volterra Model without Impulses on the Initial Condition We consider the following initial problem: dm dt =ṁ = F mm,m) = mr 1 q 1 M), 6.18) dm =Ṁ dt M m,m) = M r 2 q 2 m), 6.19) m0) =m 0, M 0) = M 0, 6.20) where as Section 5.2): m = mt) > 0 and M = M t) > 0 are the biomasses quantities of the prey and the predator respectively at the moment t 0; r 1 = const. > 0 and r 2 = const. > 0 are specific coefficients of the relative growth of the first species prey) and the second species predator), respectively; q 1 = const. > 0 and q 2 = const. > 0 are the coefficients reflecting interspecies competition for the prey and the predator, respectively; m 0 > 0 and M 0 > 0 are the quantities of biomasses of both species at the initial moment t = 0. We recall some basic facts from Section 5.2, related to the Lotka- Volterra model without impulses. It is known that system 618), 6.19) possesses: r2 Stable stationary point m 00,M 00 ) =, r ) 1 ; q 2 q 1 A first integral of the form where U m,m) =q 1 M +q 2 m r 1 lnm r 2 lnm +r 1 ln r 1 )+r 2 ln r ) 2 1 q 1 q 1 =W m,m) W m 00,M 00 ), W m,m) = q 1 M +q 2 m r 1 lnm r 2 lnm; For any constant c 0, the implicitly curve γ c = {m,m) : U m,m) = c} is a trajectory of the system 6.18), 6.19) with a properly chosen initial condition it is sufficient to assume that U m 0,M 0 ) = c);

196 ORBITAL HAUSDORFF STABILITY For any constant c > 0 the set D c = {m,m) : U m,m) < c} is a connected domain, located in R + R +, with a contour D c = γ c ; For any constant c > 0 it is satisfied m 00,M 00 ) D c ; If 0 < c 1 < c 2, then γ c1 D c2 ; The Euclidean and the Hausdorff distance between the trajectories γ c0 and γ c 0 satisfy the following equalities ρ E γc 0,γ c0 ) = inf { inf { ρe m,m ),m,m)),m,m ) γ c 0 }, m,m) γ c0 }, ρ H γc 0,γ c0 ) = max { sup { inf { ρ E m,m ),m,m)),m,m ) γ c 0 },m,m) γc0 }, sup { inf{ρ E m,m ),m,m)),m,m) γ c0 },m,m ) γ c 0 }}, respectively. We will paraphrase the definitions for orbital gravitation and orbital Hausdorff stability of the solutions of initial problem for the Lotka- Volterra system in the next two definitions. Definition 6.3. We say that system 6.18), 6.19) is orbital gravitating in the domain D with a constant κ 1, if: c 0,c 0 R +) : γ c 0,γ c 0 D ) ρ H γc 0,γ c0 ) κ.ρe γc 0,γ c0 ) max { sup { inf { ρ E m,m ),m,m)),m,m ) γ c 0 },m,m) γc0 }, sup { inf{ρ E m,m ),m,m)),m,m) γ c0 },m,m ) γ c 0 }} κ.inf { inf { ρ E m,m ),m,m)),m,m ) γ c 0 },m,m) γc0 }. Definition 6.4. We say that the solution of problem 6.18), 6.19), 6.20) is orbital Hausdorff stable on the initial point if: ε > 0) m 0,M 0 ) R + R +) δ = δε,m 0,M 0 ) > 0) : m 0,M 0) R + R +,ρ E m 0,M 0),m 0,M 0 )) < δ ) ρ H γc 0,γ c0 ) < ε, where c 0 = U m 0,M 0 ) and c 0 = U m 0,M 0 ).

197 2. LOTKA-VOLTERRA MODEL 179 In the sequel, we will use the following two theorems although they clearly have an independent interest). Theorem 6.3. Assume that: 1) The constants c 0 and c 0 satisfy the inequality 0 < c 0 < c 0 ; 2) The domain D = D c 0 \D c0. Then for every point m,m) γ c0 there exists a point m,m ) γ c 0 such that µ = {m µ,m µ );m µ =1 λ)m+λm, M µ =1 λ)m +λm,0 λ 1} D. Proof. Let m,m) γ c0. We consider the half-line sl = {m+λ m m 00 ),M +λ M M 00 )),λ 0}. Obviously, there exists a constant λ > 0 such that m+λ m m 00 ),M +λ M M 00 )) = m,m ) γ c 0. For the segment µ with endpoints m,m) γ c0 and m,m ) γ c 0 the equalities µ ={1 λ)m+λm,1 λ)m +λm ),0 λ 1} ={m+λ m m 00 ),M +λ M M 00 )),0 λ λ } are valid, where λ = λ λ. Our goal is to show that µ D see Figure 6.4). One of the following cases is valid: Case 1. m m 00, M M 00 ; Case 2. m < m 00, M > M 00 ; Case 3. m > m 00, M < M 00 ; Case 4. m m 00, M M 00, where m 00 = r 2 and M 00 = r 1. q 2 q 1 We will consider Case 1, only. We have: We set q 2 r 2 m q 2 q 2 = q 2 r 2 r m 2 = 0, 00 q 2 q 1 r 1 M q 1 q 1 = q 1 r 1 r M 1 = q 1 F λ ) = U m+λ m m 00 ),M +λ M M 00 ))), 0 λ λ.

198 ORBITAL HAUSDORFF STABILITY M M 00 γ c 0 γ c0 M M Sl 0 mm m 00 m Figure 6.4 Then because and d dλ F λ ) = d dλ U m+λ m m 00 ),M +λ M M 00 ))) = d dλ W m+λ m m 00 ),M +λ M M 00 ))) = q 2 + 0, q 1 m m 00 0, q 2 M M 00 0, q 1 r 2 m+λ m m 00 ) r 1 M +λ M M 00 ) ) m m 00 ) ) M M 00 ) r 2 m+λ m m 00 ) q 2 r 2 m 0, r 1 M +λ M M 00 ) q 1 r 1 M 0.

199 2. LOTKA-VOLTERRA MODEL 181 i.e. Therefore F 0) F λ ) F λ ), c 0 =U m,m) = F 0) F λ ) =U m+λ m m 00 ),M +λ M M 00 ))) F λ ) = U m,m ) =c 0, 0 λ λ. From inequalities above, we conclude that m+λ m m 00 ),M +λ M M 00 ))) D, 0 < λ < λ µ D. Thus the theorem is proved. Theorem 6.4. Assume that: 1) The constants c 0 and c 0 satisfy the inequalities 0 < c 0 < c 0 ; 2) γ D = D c 0 \D c0 ; 3) For every points m,m) γ c0 and m,m ) γ c 0 the segment µ = {m µ,m µ );m µ =1 λ)m+λm, intersects the set γ, i.e. γ µ. M µ =1 λ)m +λm, 0 λ 1} Then for every point m,m) γ c0 there exists two points m,m ) γ c 0 and m,m ) γ µ such that, the vectors m m,m M) and gradu m,m ) are collinear see Figure 6.5). Proof. Let m,m) γ c0. Similarly to the proof of the previous theorem, there are four typical cases: Case 1. m m 00, M M 00 ; Case 2. m < m 00, M > M 00 ; Case 3. m > m 00, M < M 00 ; Case 4. m m 00, M M 00. Let us consider Case 1. We denote: γ c 0 =γ c 0 {m,m ),m m,m M}, γ =γ {m,m ),m m,m M}.

200 ORBITAL HAUSDORFF STABILITY M M M M gradum,m ) γ c 0 γ γ c0 0 m m m m Figure 6.5 For each point m,m ) γ it is fulfilled: U m gradu m,m,m ) ) = m = q 2 r 2 m,q 1 r 1 M ),, U ) m,m ) M where for the first coordinate of the vector above we have: q 2 r 2 m q 2 r 2 m q 2 r 2 m 00 =q 2 r 2 r 2 q 1 = 0. Similarly, for the second coordinate we have q 1 r 1 M 0.

201 2. LOTKA-VOLTERRA MODEL 183 Case 1.1. Let the inequalities m < m 00 and M < M 00 be valid. Then the following strict inequalities are satisfied: q 2 r 2 < 0, q m 1 r 1 < 0. M Let µ be the segment with endpoints m,m) γ c0 and m,m ) γ c 0. For each point m,m,m,m ) γ c γ µ) we consider the 0 function F m,m,m,m ) = m m q 2 r M M 2 q m 1 r, 1 M Firs, let the point m 1,M 1 ) = m,m 1 ) γ c. Then it is clear that 0 M1 < M and therefore F m 1,M 1,m,M ) =F m,m 1,m,M ) = m m q 2 r M 1 M 2 q m 1 r < 0. 1 M Further, if the point m 2,M 2 ) = m 2,M) γ c, then m 0 2 < m. We receive F m 2,M 2,m,M ) =F m 2,M,m,M ) = m 2 m q 2 r M M 2 q m 1 r > 0. 1 M Since the function Fis continuous on the connected set γ c γ µ), 0 intersection and Cartesian product of connected sets), then there exists a point m,m,m,m )from this set such that F m,m,m,m ) = 0 m m q 2 r 2/ m The case 1.1 is completed. = M M q 1 r 1/ M. Case 1.2. Let m = m 00 and M M 00. We suppose that m = m. Then, the following equalities are valid: Hence m = m = m 00 = m, q 2 r 2 m = 0. m m,m M) = 0,M M) and gradu m,m ) = 0,q 1 r ) 1. M From the last equalities it follows that the vectors m m,m M) and gradum, M ) are collinear.

202 ORBITAL HAUSDORFF STABILITY Case1.3. Letm m 00 andm = M 00. Then, similarlytotheprevious case we get the equalities: m m,m M) = m m,0) and gradu m,m ) = q 2 r ) 2 m,0. Therefore, these vectors are collinear. Thus, the statement of the theorem is proved in this case. The remaining three cases: m < m 00, M > M 00 ; m > m 00, M < M 00, and m m 00, M M 00 are considered similarly. We omit them. The theorem is proved. The main results in the present section are the next two theorems. Theorem 6.5. Assume that: 1) The constants c 1 and c 2 satisfy the inequality 0 < c 1 < c 2 ; 2) D = D c2 \D c1. Then the system 6.18), 6.19) is orbital Hausdorff gravitating in the domain D with a constant κ = sup{ gradu m,m ),m,m ) D} inf{ gradu m,m ),m,m ) D}. Proof. Let γ c0,γ c 0 D. Then the following inequalities are valid: c 1 < c 0 < c 2 and c 1 < c 0 < c 2. Further we consider the trajectories: γ c0 ={m,m) : U m,m) = c 0 } ={m,m) : q 1 M +q 2 m r 1 lnm r 2 lnm + r 1 ln r 1 1 )+r 2 ln r ) } 2 1 = c 0, q 1 q 2 γ c 0 ={m,m) : U m,m) = c 0 } ={m,m) : q 1 M +q 2 m r 1 lnm r 2 lnm +r 1 ln r 1 1 )+r 2 ln r ) } 2 1 = c 0. q 1 q 2 Assume that c 0 < c 0. Let us note that the proof of the case c 0 > c 0 is similar and the case c 0 = c 0 is trivial.

203 2. LOTKA-VOLTERRA MODEL 185 Let m,m) γ c0, m,m ) be a point such that m,m ) γ c 0 and the segment µ with the endpoints m,m) and m,m ) belongs to D c 0 \D c0 see Theorem 6.3). We consider a function F λ) = U m+λm m),m +λm M)), 0 λ 1. Obviously: F 0) = U m,m) = c 0, F 1) = U m,m ) = c 0 and F is continuously differentiable function on the interval [0, 1]. Then there exists at least one constant λ 0 = λ 0 m,m,m,m ), 0 < λ 0 < 1, such that c 0 c 0 = F 1) F 0) = F λ 0 ) = d dλ U m+λ 0m m),m +λ 0 M M)) = m U m+λ 0m m),m +λ 0 M M))m m) + M U m+λ 0m m),m +λ 0 M M))M M) = gradu m+λ 0 m m),m +λ 0 M M)), We set: m m,m M) = gradu m+λ 0 m m),m +λ 0 M M)), m m M M, m m) 2 +M M) 2 m m) 2 +M M) 2 m m) 2 +M M) ) γ = {m,m );m,m ) = m+λ 0 m m),m +λ 0 M M)), where: m,m) γ c0, m,m ) γ c 0 ; 1 λ)m+λm,1 λ)m +λm ) D c 0 \D c0 for 0 < λ < 1, c 0 c 0 = F λ 0 ). In other words, we have: m,m ) µ;

204 ORBITAL HAUSDORFF STABILITY µ = {m µ,m µ );m µ = 1 λ)m+λm,m µ = 1 λ)m+λm, 0 λ 1}; m,m) γ c0,m,m ) γ c 0 ; µ D c 0 \D c0 ; c 0 c 0 = gradu m,m ),m m,m M). It is clear that γ is a connected set and for every two points m,m) γ c0 and m,m ) γ c 0 the segment µ intersects the set γ. According to Theorem 6.4, it is possible to choose two points: m,m ) γ c 0, m,m ) = m+λ 0 m m),m +λ 0 M M)) γ µ such that the vectors: and m m,m M), gradu m,m ) = gradu m+λ 0 m m),m +λ 0 M M)) are collinear. Then from 6.21) it follows that Therefore c 0 c 0 = gradu m+λ 0 m m),m +λ 0 M M)), gradu m+λ 0 m m),m +λ 0 M M)) gradu m+λ 0 m m),m +λ 0 M M)) m m) 2 +M M) 2 ρ E m,m),m,m )) = gradu m+λ 0 m m),m +λ 0 M M)) ρ E m,m),m,m )). = c 0 c 0 gradu m+λ 0 m m),m +λ 0 M M)), whence we obtain c 0 c 0 sup{ gradu m,m ),m,m ) D} = const 1 ρ E m,m),m,m )) const 2 = c 0 c 0 inf{ gradu m,m ),m,m ) D}.

205 2. LOTKA-VOLTERRA MODEL 187 From the last inequalities we find: const 1 inf { inf { ρ E m,m ),m,m)),m,m ) γ c 0 },m,m) γc0 } =ρ E γc 0,γ c0 ) 6.22) and sup { inf { ρ E m,m ),m,m)),m,m ) γ c 0 },m,m) γc0 } Similarly to 6.23), we deduce that const ) sup { inf{ρ E m,m ),m,m)),m,m) γ c0 },m,m ) γ c 0 } Using 6.23) and 6.24) we get const ) ρ H γ c 0,γ c0 ) =max{sup{inf{ρ E m,m ),m,m)),m,m ) γ c 0 }, m,m) γ c0 }, sup{inf{ρ E m,m ),m,m)),m,m) γ c0 }, m,m ) γ c 0 }} const ) From 6.22) and 6.25) it shows that const 1 ρ E γc 0,γ c0 ) ρh γc 0,γ c0 ) const2, From where, finally, we derive the estimate ) ρ H γc 0,γ c0 ) const 2 = sup{ gradu m,m ),m,m ) D} ρ E γc 0,γ c0 const 1 inf{ gradu m,m ),m,m ) D} = κ. The proof is complete. As corollary of the last theorem we obtain the following result. Theorem 6.6. The solution of problem 6.18), 6.19), 6.20) is orbital Hausdorff stable on the initial point. Proof. Let: ε be an arbitrary positive number; The constant c 0 = U m 0,M 0 ); The constants c 1 and c 2 satisfy the inequalities 0 < c 1 < c 0 < c 2. For example let c 1 = 1 2 c 0 and c 2 = 3 2 c 0.

206 ORBITAL HAUSDORFF STABILITY Then, according to Theorem 6.5, it follows that the system 6.18), 6.19) is orbital gravitating in the domain D = D c2 \D c1 with a constant κ = κd) = κc 1,c 2 ) = κc 0 ) = κm 0,M 0 ). Then for every point m 0,M 0 ) D, such that it is valid ρ E m 0,M 0 ),m 0,M 0 )) < ε 2, ρ H γc 0,γ c0 ) κρe γc 0,γ c0 ) κρe m 0,M 0),m 0,M 0 )) < ε, The theorem is proved. c 0 = U m 0,M 0 ).

207 3. HARMONIC OSCILLATOR Orbital Hausdorff Stability of the Solutions of Model of Harmonic Oscillator Consider the equation, describing the vertical vibrations of a particle material point) with nonzero mass, hanged on a perfect spring agile in an isolated environment, i.e. without any friction and absence of permanently active environmental forces. Let us denote by y = yt) the point deviation from its equilibrium position y = 0. Then dy 2 dt 2 +µ2 y = 0, where the parameter µ depends on material point mass and spring qualities. The general real solution of the equation above has the form yt) = C 1 cosµt+c 2 sinµt yt) = Ccosµt+ν), where C = C1 2 +C2 2 > 0, ν = arctan C 2 C 1, C 1 and C 2 are arbitrary real constants. We will remind that C, µ and ν are called amplitude magnitude), frequency and an initial phase, respectively. We introduce the functions: x 1 t) = yt) = Ccosµt+ν) and x 2 t) = dyt) = Cµsinµt+ν), dt which expressed the moving of the material point and its speed, respectively. These functions are solution of the linear system dx 1 dt = x 2, with the initial condition dx 2 dt = µ2 x 1 x 1 0) = Ccosν, x 2 0) = Cµsinν That is why, the problem above can be rewritten as: dx dt = Ax, x0) = x 0, where: A = 01 µ 2 0, x = x 1, and x 0 = Ccosν Cµsinν. x 2 The system trajectory is the ellipse: µ 2 x x2 2 = µ2 C 2, i.e. it is satisfied Γx 0 ;[0, )) ={xt;x 0 ),0 t < } = { x = x 1,x 2 ) : µ 2 x 2 1 +x2 2 = µ2 C 2}.

208 ORBITAL HAUSDORFF STABILITY Similarly, if where C,ν R, then x 0 = C cosν C µsinν Γx 0 ;[0, )) ={x t;x 0 ),0 t < }, = { x = x 1,x 2) : µ 2 x 1) 2 +x 2) 2 = µ 2 C ) 2}. Furthermore, we will assume that the constant µ satisfies the inequality 0 < µ < 1. Then, we directly verify that and Hence ρ H Γx 0 ;[0, )),Γx 0;[0, ))) = C C ρ E Γx 0;[0, )),Γx 0 ;[0, ))) = µ C C. ρ H Γx 0 ;[0, )),Γx 0;[0, ))) 1 µ ρ E Γx 0 ;[0, )),Γx 0;[0, ))), i.e. the system dx dt = Ax is orbital gravitating with a constant κ = 1 µ. Suppose that the material point is subjected to the impulsive effect in the moment at which it passes through its equilibrium position x 1 = 0, moving bottom up. The effect consists of an instantly moving of the material point and the cancelation of its speed. The movement and speed of the material point are described by the initial problem of the impulsive systems dx dt =Ax, a,xt) 0, x 2t) > 0, xt+0) =xt)+ixt)), a,xt) = 0, x 2 t) > 0, x0) =x 0, where we suppose that: At the moments of the impulsive effects it is fulfilled x 2 t) > 0, i.e. the material point is moving bottom - up ; The domain D is bounded by two ellipses: D = { x 1,x 2 ) : λ 2 µ 2 C 2 µ 2 x 2 1 +x 2 2 µ 2 C 2}, λ 0,1); The vector a = 1,0) and the impulsive set α coincides with a part of x 2 axis: α = Ox + 2 D = {x 1,x 2 ) : x 1 = 0,λµC x 2 µc};

209 3. HARMONIC OSCILLATOR 191 The impulsive function I : α R 2 is described by the equality p x 2 ) Ix) = Ix 1,x 2 ) = µ +p 1)λC, x = x x 1,x 2 ) α, 2 where p is a constant, satisfying the inequality 0 < p < 1 2 µ2 see Figure 6.6). x 2 µc α λµc C λc 0 ν λc C x 1 λµc µc Figure 6.6 Under the assumptions above, the considered impulsive system takes the form: dx 1 dt = x 2, dx 2 dt = µ2 x 1, for x 1 t),x 2 t)) / α, x 1 t+0) = p x 2 µ +p 1)λC, x 2 t+0) = 0, for x 1 t),x 2 t)) α. We will show that this system satisfies the conditions of Theorem 6.2. The conditions H6.1, H6.3 and H6.4 are obvious. The condition

210 ORBITAL HAUSDORFF STABILITY H6.2 follows by the inequalities C f =µ 2 C = µ 2 µ 2 x 2 1 +x 2 2) µ 4 x 2 1 +x 2 2 = f x) µ 2 x 2 1 +x 2 2 = µc Let x α, then =C f. a,f x) = 1,0), x 2, µ 2.0 ) = x 2 = x µ2.0) 2 = f x) = C α f x), i.e. condition H6.5 is valid with a constant C α = 1. Ley Therefore Id+I)α) {x 1,x 2 ) : C x 1 λc,x 2 = 0} ρ E α,id+i)α)) <ρ E {x 1,x 2 ) : x 1 = 0, µc x 2 λµc}, {x 1,x 2 ) : C x 1 λc,x 2 = 0}) =λc 1+µ 2 = C I+Id >0, i.e. condition H6.6 is valid. Let x = 0,x 2 ), x = 0,x 2 ) and x,x α. Then ρ E x +I x ),x +Ix )) ) ) ) )) =ρ E p x 2 µ +1 p)λc,0, p x 2 µ +1 p)λc,0 = p µ x 2 x 2 = p µ.ρ Ex,x ) = C I.ρ E x,x ). So, the correctness of the condition H6.7 with a constant C I = p ) is µ established. Finally, the inequality in the third condition of Theorem 6.2 has the form C a C I < p κ1+c a ) µ < µ p < µ2. Thereforeifp 0, 1 2 µ2), thenassertionoftheorem6.2holdstrue. Hence the solution of the initial problem, describing the vertical vibrations of

211 3. HARMONIC OSCILLATOR 193 a material point with a nonzero mass, hanged of the perfect spring agile in an isolated environment is orbital Hausdorff stable.

212

213 Chapter 7 Optimization Problems in Population Dynamics In the first section of this chapter we consider Verhulst logistic equation. A typical application of the logistic equation is a common model of population growth, where the rate of reproduction is proportional to both the existing population and the amount of available resources. Let us note, it is supposed that there are not any external influences in the classical Verhulst model. In the chapter, some Verhulst type models are studied with additional assumption: the models are subjected to the external influences, usually due to the human intervention. A frequently occurring situation is that, when the population is subjected to the outer effects, anthropogenic impact included. We consider the case when the outer effects are discrete in time and are expressed in instantaneous taking away or supplement of certain quantities of biomass. Such a process is described fairly adequately by means of differential equations with impulses at fixed moments. The moments of impulsive effects and their quantities are determined optimally, so that the time for the reproduction of taken away biomass is minimal. An optimal regime of reproduction of the taken away amount of biomass is described in the second section, using the results obtained in Section 7.1. The optimal levels of taking away and the moments at which they take place are obtained. We use the word optimum here with respect to the minimum time of reproduction. Such studies can be found in: Akhmet et al. [17], Alzabut [21], Bai et al. [31], Braverman and Braverman [88], Dong et al. [131], Eloe et al. [139], Gao and Hung [156], Guo et al. [167], Hung et al. [180], Ignatiev [182], Ignatyev and Ignatyev [184, 185], Jian [188], Lu and Wang [233, 234], Lu et al. [235], Nie et al. [264], Özbekler and Zafer [271], Shuai et al. [302], Tang et al. [324], Tang and Chen [325], Wang et al. [328, 329, 330, 331], Zeng et al. [357], Zeng [359], Zhang et al. [367], Zhang and Xiu [368], Zhao and Zhang [370], Zhao et al. [371, 372]. 195

214 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS A class of optimization problems with respect to the impulsive control, is studied in the third section. The necessary and sufficient conditions for existence of optimal impulsive control for the initial value problem of dynamic systems are found. In the last section of the chapter, the results for the impulsive control are applied to some impulsive analogues of the classical problems of population dynamics the impulsive Verhulst and Gompertz models for the isolated populations). Similar results were obtained in Liu and Chen [229]. The main ideas in the first two sections are published in Bainov and Dishliev [42] and the results in the last two sections of the chapter are obtained in Nenov [261].

215 1. TIME REQUIRED FOR REPRODUCTION Minimization of the Time Required for Reproduction of an Isolated Population Let us consider the following Verhulst equation: dn dt = µ N K N), 7.1) K which describes the development dynamics of many isolated populations. We use the following notations: N = N t) > 0 is the biomass of the population at the moment t 0; K > 0 is the capacity of the environment; µ > 0 is the difference between the birth-rate and death-rate in the considered population. We assume that at the initial moment t = 0, we have N 0) = N 0, 7.2) where 0 < N 0 < K 2. We note that, in many cases, the specific types of populations are subjected to the outer short-term interventions during their development. Often, they are caused by the human intervention. Here, we consider the case, when the outer interventions are discrete in time and they are expressed in instantaneous removal or rarely) adding a certain amount of biomass to the population. Consider the impulsive analogue of the Verhulst equation 7.1): dn dt = µ K N K N), t τ i, 7.3) N t) t=τi = N τ i +0) N τ i ) = I i, i = 1,2,..., 7.4) where: τ 1,τ 2,...0 < τ 1 < τ 2 <...) are the moments, at which the outer discrete interventions are realized. They are called the moments of impulsive effect; I 1,I 2,... are the quantities of biomass, which are taken away if I 1 > 0,I 2 > 0,...) or added if I 1 < 0,I 2 < 0,...) in the moments τ 1,τ 2,..., respectively. We consider the important applications in terms of the biotechnologies in the case of taking away of biomass, i.e. we assume that I 1 0,I 2 0,... We note that the solution of problem 7.3), 7.4), 7.2) is a piecewise continuous function with points of discontinuity τ 1,τ 2,..., at which

216 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS the solution is continuous on the left. We note again that for τ i < t τ i+1,i = 1,2,..., the solution of the problem with impulses 7.3), 7.4), 7.2) coincides with the solution of equation 7.1) with an initial condition. N τ i +0) = N τ i ) I i. 7.5) Since for t > τ i, i = 1,2,..., the solutions of problems 7.1), 7.5) are strictly monotonically increasing functions, then for strictly defined time intervals after the moments of impulsive effects τ i, the population will recover its biomass. The lengths of these time intervals are called times of recovering regeneration) of biomass taking away at the moments τ i, i = 1,2,... Let τ i denote these intervals. It is obvious that, if τ i+1 τ i = τ i, then the next equality is valid N τ i + τ i ) = N τ i ). Let a certain quantity of biomass be planned for taking away from the population by several consecutive discrete outer influences. This amount is denoted by I. It is natural to suppose that the quantities of biomass, taken away at the impulsive moments are bounded below. Assume that, there exists a constant I 0,0 < I 0 < K, such that I 0 I i, i = 1,2,.... The purpose of this section is to define the moments τ 1,τ 2,... of impulsive effects and the biomass quantities I 1,I 2,..., which are taken away at these moments so that the sum of the times of regeneration of the discrete amounts taken away to be minimal. In the following lemmas we formulate some necessary inequalities. Lemma 7.1. Let 0 < I 1 < K and 0 < I 2 < K. Then K + I 1 +I 2 2 K +I 1 )K +I 2 ) K I 1 )K I 2 ) K I 1 +I 2 2 The proof of the lemma is trivial and we omit it. Lemma 7.2. Let I i > 0, i = 1,2,..., and I 1 +I 2 + +I n < K. Then K +I 1 +I 2 + +I n K +I 1)K +I 2 )...K +I n ) K I 1 I 2 I n K I 1 )K I 2 )...K I n ). 2.

217 1. TIME REQUIRED FOR REPRODUCTION 199 Lemma 7.2 is proved by induction. Note that the equality holds for n = 1, too. Lemma 7.3. Let 0 < I < K and 1 m < n. Then K + I m K + I n m K I > n K I. 7.6) m n Proof. Let a = I. Then 0 < a < 1 and 7.6) is reduced to the K inequality ) m ) n m+a n+a >. m a n a Obviously, if we show that function φx) = X ln X +a X a, X > 1, is strictly monotonically decreasing, then the validness of inequality above is trivial. For this aim, we calculate: dφ dx X +a = ln X a 2aX < 0, X 1. X 2 a2 and d 2 φ dx = 4a 3 2 X 2 a 2 2 > 0, X 1. ) Then, the first derivative is strictly monotonically increasing for X 1. Moreover, dφx) lim X dx = lim ln X +a X X a 2aX ) = 0. X 2 a 2 Therefore, dφx) < 0, X 1, dx That is why, the function φ is strictly monotonically decreasing. The lemma is proved. Lemma 7.4 see Natanson [259]). Let f : R R and for every two points I 1,I 2 R it is valid ) I1 +I 2 f I 1 )+f I 2 ) 2f. 2

218 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS Then, we have ) I1 +I 2 + +I n f I 1 )+f I 2 )+ +f I n ) nf, n for any n N and any I 1,I 2,...,I n R. The proof of this result can be found in Natanson [259]. Remark 7.1. Equation 7.1) is Bernoulli differential equation. Then it is solvable by quadratures: Nt) = K ) K 1 N 0, t 0 7.7) e µt +1 Remark 7.2. The inflection point of the function N = Nt) has coordinates ) 1 K t,n) = µ ln 1, K ). N 0 2 Remark 7.3. Since function 7.7) is strictly monotonically increasing for t 0, then the inverse function exists: t = tn) = 1 µ ln K N 0 1 K N 1, 0 < N < K. 7.8) The equality ψn) = tn +I) tn) = 1 K µ ln 1 N 7.9) 1. K N+I follows from 7.8), where 0 < I < K and 0 < N < K I. In particular, for N = K I, we have 2 ) K I ψ = 2 K +I ln 2 µ K I. 7.10) Theorem 7.1. Let 0 I < K and 0 < N < K I. Then ) K I ψ ψn) 2

219 1. TIME REQUIRED FOR REPRODUCTION 201 ) ) K +I K I t t tn +I) tn). 7.11) 2 2 Proof. From 7.9) it follows that the function ψ = ψn) achieves its minimum at the point N = K I, which immediately implies 7.11). 2 Remark 7.4. Theorem 7.1 allows us to assert that the time of regeneration of the biomass quantity I i, taken away at the moment τ i, is minimal if the moment of impulsive effect τ i satisfies the equality K 2 =1 2 K +Ii 2 + K I i 2 = 1 2 N τ i) I i +N τ i )) ) = 1 2 N τ i +0)+N τ i )) =N τ i ) I i 2, i.e., if the moment of impulsive effect τ i satisfies the equality N τ i ) = K +I i, i = 1,2,... 2 The impulsive effects I i,i = 1,2,..., satisfying the last equalities is named centered. Theorem 7.2. Let I i > 0, i = 1,2,...,n and I 1 +I 2 + +I n = I < K. Then for each N, 0 < N < K I, we have [ ) )] K +I1 K I1 tn +I) tn) t t 2 2 [ ) )] K +I2 K I2 + t t + + [ t 2 2 ) K +I1 K I1 t 2 2 )]. 7.12) Proof. Using Theorem 7.1 for the left-hand side of 7.12) we obtain ) ) K +I K I tn +I) tn) t t. 2 2 From the last inequality and 7.10), we have tn +I) tn) 2 µ K +I ln, 0 < N < K I. 7.13) K I

220 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS or For the right-hand side of 7.12), we receive [ ) )] [ ) )] K +I1 K I1 K +I2 K I2 t t + t t [ ) )] K +In K In + + t t 2 2 = 2 µ ln K +I 1 K I µ ln K +I 2 K I µ ln K +I n K I n = 2 µ ln K +I 1)K +I 2 )...K +I n ) K I 1 )K I 2 )...K I n ). 7.14) Therefore, using Lemma 7.2, we obtain K +I K I K +I 1)K +I 2 )...K +I n ) K I 1 )K I 2 )...K I n ), 2 K +I ln µ K I 2 µ ln K +I 1)K +I 2 )...K +I n ) K I 1 )K I 2 )...K I n ). From the inequality above, 7.13), and 7.14), we receive 7.12). Thus the theorem is proved. Remark 7.5. Let: 1) An isolated population develops according to the Verhulst equation; 2) I is an arbitrary fixed biomass volume of this population, taken away by the discrete outer influence; 3) is the time for regeneration of the biomass, taken away by an arbitrary in time) single outer influence; 4) is the time for recovery of biomass by several centered outer influences. Then Theorem 7.2 yields the inequality see Figure 7.1 and Figure 7.2). Theorem 7.3. Let 0 < I 1 < K, 0 < I 2 < K. Then [ ) )] [ ) )] K +I1 K I1 K +I2 K I2 t t + t t [ ) )] K +I1 +I 1 )/2 K I1 +I 2 )/2 2 t t. 7.15) 2 2

221 1. TIME REQUIRED FOR REPRODUCTION 203 K I K 2 N 0 0 t Figure 7.1 K I 2 I 1 K 2 N 0 0 t Figure 7.2. I = I 1 +I 2, Proof. By Lemma 7.1 we have K +I 1 )K +I 2 ) K + I 1 +I 2 ) K I 1 )K I 2 ) 2 K I 1 +I 2 )2 2.

222 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS K I 1 K 2 N 0 0 t Figure 7.3. I 1 = I 2 = I 2, After taking the logarithm of both sides of the last inequality we get I 1 +I 2 ) 2 µ ln K +I 1) K I 1 ) + 2 µ ln K +I 2) K I 2 ) 4 K + µ ln 2 K I 1 +I 2 ), from where by 7.9) and 7.10), we obtain 7.15). Remark 7.6. Let: 1) An isolated population develop according to the Verhulst equation; 2) I be an arbitrary fixed biomass volume of the population, taken away from it, between two discrete outer effects; 3) be the time for regeneration the biomass, taken away by two centered outer influences with volumes I 1 and I 2, where I = I 1 +I 2 in the common case I 1 I 2 ); 4) be the time for recovery the biomass by two centered outer effects with equal volumes, i.e. I 1 = I 2 = I 2. Then Theorem 7.3 yields see Figure 7.2 and Figure 7.3). Theorem 7.4. Let: 1) 0 < I 0 I i < K and 0 < N i < K I i for i = 1,2,...,m; 2) n N) : ni 0 I < n+1)i 0, where I = I 1 +I 2 + +I m.

223 1. TIME REQUIRED FOR REPRODUCTION 205 Then [tn 1 +I 1 ) tn 1 )]+[tn 2 +I 2 ) tn 2 )]+ +[tn m +I m ) tn m )] [ ) )] K +I/n K I/n n t t. 7.16) 2 2 Proof. From Theorem 7.1 it follows that [tn 1 +I 1 ) tn 1 )]+[tn 2 +I 2 ) tn 2 )]+ +[tn m +I m ) tn m )] [ ) )] [ ) )] K +I1 K I1 K +I2 K I2 t t + t t [ ) )] K +Im K Im + + t t. 7.17) 2 2 We consider the function f : 0,K) R +, defined by the equality ) ) ) K X K +X K X f X) = ψ = t t, 0 < X < K By Theorem 7.3 we have ) I1 +I 2 f I 1 )+f I 2 ) 2f, 2 whence, in view of Lemma 7.4, we deduce that ) I1 +I 2 + +I m f I 1 )+f I 2 )+ +f I m ) mf = mf 2 By 7.17) and the definition of f, we obtain the inequality ) I. m [tn 1 +I 1 ) tn 1 )]+[tn 2 +I 2 ) tn 2 )]+ +[tn m +I m ) tn m )] [ ) )] K +I/m K I/m m t t. 7.18) 2 2 Using 7.10) we derive [ ) )] K +I/m K I/m m t t 2 2 = 2 ) m K +I/m µ ln. 7.19) K I/m It is easy to guess that m n. In view of Lemma 7.3 we conclude that ) m ) n K +I/m K +I/n. K I/m K I/n Therefore, 2 µ ln ) m K +I/m 2 K I/m µ ln ) n K +I/n. K I/n

224 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS By the last inequality, 7.18) and 7.19) we obtain 7.16). Remark 7.7. Let: 1) An isolated population develop according to the Verhulst equation; 2) I be arbitrary fixed biomass volume of the population, taken away from it by the discrete outer effects; 3) I 0 betheminimum quantity canbetakenawayfromthebiomass at a single outer influence; 4) I 0 I; 5) Let n be the greatest integer, which is less than or equal to the constant I I 0, i.e. n N and n I I 0 < n+1; 6) is the time for biomass recovering, taken away in several e.g. the number of k) outer influences with volumes I 1,I 2,...,I k, where I 1 I 0,I 2 I 0,...,I k I 0 and I = I 1 +I 2 + +I k ; 7) n is the time for recovering of the biomass, taken away with volume I inthenumber ofncentered outerinfluences withequal volumes, i.e. I 1 = I 2 = = I n = I n I 0. Then, using Theorem 7.4, we derive n. In other words, under the conditions 1-4 of Remark 7.7 the minimal time for regeneration the quantity I, taken away from the biomass of the population, is realized by n centered discrete outer effects with one and the same volume I n.

225 2. APPLICATION Application: A Model of Optimal Regime of Outer Effects We describe the optimal regime of outer effects. The optimality here and throughout the preceding section) is understood in the sense of the most quickest regeneration of the biomass amount, taken away from the population. The assumptions here coincide with the conditions of Theorem 7.4. Let us formulate them: 1) 0 < I 0 I i < K,i = 1,2,... each one impulsive removal of the biomass is bounded below by the positive constant I 0 and above by the capacity of the environment - K); 2) I 1 + I 2 + = I,I 0 < I the sum of the biomass quantities from all impulsive removals is equal to the pre-fixed constant I,I > I 0, which is called a planned production); 3) n N) : ni 0 I < n+1)i 0 the natural n is called optimal number of impulsive removals); 4) 0 < N 0 < K the initial quantity of biomass is less than half of 2 the capacity of environment). Then using Theorem 7.4, it is clear that the optimal regime of the impulsive effects on the dynamics of population development in order production a certain biomass amount is described by the following initial problem of impulsive equations dn dt = µ K N K N), t τ i, 7.20) N t) t=τi = N τ i +0) N τ i ) = I, i = 1,2,...,n, n 7.21) N 0) = N 0, 7.22) where the moments of impulsive effects τ 1,τ 2,...,τ n are defined in the following way: First, we find τ 1 using the formula ) K +I/n τ 1 = t = 1 2 µ ln K N 0)nK +I). N 0 nk I) Furthermore, we have τ i+1 = τ i + τi = τ i + τ = τ 1 +i. τ, i = 1,2,...,n, where ) ) K +I/n K I/n τ = t t 2 2 = 1 ) 2 nk +I µ ln. nk I

226 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS Consequently τ i = 1 µ ln [ K N 0 N 0 ) ] 2i+1 nk +I, i = 1,2,...,n. nk I Finally, we note that under restrictions 1-4, imposed above, the minimal time for regeneration of biomass I, taken away from a population which dynamics is determined by the initial problem of Verhulst impulsive equation 7.20), 7.21), 7.22) is n τ = 2n nk +I ln µ nk I, where n is the greatest integer, less than the quotient I I 0 and I 0 is the minimal biomass quantity, which can be taken away at a single moment. Consider the following numerical example. Example 7.1. Let: µ = 0.03, K = 100, N 0 = 15, I 0 = 15 and I = 50. Then for these initial data, it is obtained: n = 3, τ 1 69,034, τ 2 91,466, τ 3 113,897, I 1 = I 2 = I 3 = I 3 16,67. The minimal time for regeneration of the biomass, taken away is 3 τ 67,294. The graph of the optimal solution of initial problem 7.20), 7.21), 7.22) is given on Figure 7.4. K K 2 + I 6 K 2 K I 2 6 N 0 0 τ τ τ τ 1 τ 2 τ 3 t Figure 7.4

227 3. LAGRANGE S METHOD Impulsive Controllability and Optimization Problems. Lagrange s Method In this section we will discuss some problems to the impulsive discrete) control on the solutions of sufficiently smooth dynamical systems. We can consider that the instant outer influences on the evolutionary systems are impulsive control. Let: m,n N, m n, x m) R m, x n) R n, x m) = x 1,x 2,...,x m), x n) = x 1,x 2,...,x n), and h C 1 [R n,r]. The following notation are introduces x m)h ) x n)) h x n) =, h ) x n),..., h ) ) x n). x 1 x 2 x m In particular for m = n it is valid gradh x n)) = h x n)) = x n)h x n)) ) h x n) =, h ) x n),..., h ) ) x n). x 1 x 2 x n The main object of investigation is the following initial problem of impulsive differential equations with fixed moments of impulsive effects dx dt = f t,x), t R+ \T, 7.23) xt+0) = It,x), t T, 7.24) xt 0 ) = x 0, 7.25) where: f R + R n R n ; I : T R n R n ; t 0,x 0 ) R + R n. Assume that the number of the impulsive effects is p. The moments, at which the influences are realized we denote by t 1,t 2,...,t p, where t 0 < t 1 < t 2 < < t p. The set of impulsive moments are denoted by T, i.e. T = {t 1,t 2,...,t p }. We define the concept set of permissible controls: Definition 7.1. Let:

228 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS 1) h 1,h 2 C 1 [R n,r]; 2) x R n,h l x) = 0) h l x) 0), l = 1,2; 3) The sets D l = {x R n ;h l x) < 0},l = 1,2, satisfy the inclusions D 2 D 2 D 1 {x R n ;h 2 x) 0} {x R n ;h 1 x) < 0} x R n,h 2 x) 0) h 1 x) < 0); 4) x 0 D 1 and 0 t 0 < T; 5) The set U consists of all ordered pairs T,I), for which the set of points T = {t 1,t 2,...,t p } and the function I : T R n R n satisfy the following restrictions: 5.1. t 0 < t 1 < t 2 < < t p T; 5.2. I C 1 [T R n,r n ]; 5.3. xt;t 0,x 0 ) D 1 for t 0 t T; 5.4. xt;t 0,x 0 ) D 2. Then U is called a set of permissible impulsive controls. Let T,I) U. Denote: a i = xt i ;t 0,x 0 ), b i = I t i,a i ), i = 1,2,...,p, a 0 = b 0 = x 0, a p+1 = xt;t 0,x 0 ), A = {a 1,a 2,...,a p }, B = {b 1,b 2,...,b p }. 7.26) Definition 7.2. Let: 1) π : U R 2p+1)n and πt,i) = A,B,a p+1 ); 2) F C 1[ R 2p+1)n,[0,1] ] ; 3) The function Φ : U [0,1] and ΦT,I) = F πt,i)). Then Φ is called target impulsive function. We formulate the problem for optimal impulsive control: Problem 7.1. Find a ordered pair T 0,I 0 ) of permissible impulsive controls U, such that Φ T 0,I 0) = min{φt,i);t,i) U}. 7.27) The following conditions are introduced: H7.1. f C[R + R n,r n ];

229 3. LAGRANGE S METHOD 211 H7.2. There exists a constant C f > 0, such that t,x) R + R n) f t,x) C f. H7.3. For any point t 0,x 0 ) R + R n, problem without impulses 7.23), 7.25) has a unique solution, defined for t t 0 ; H7.4. For any point t,x), for which t 0 < t < T and x D 1, it is valid only one of the two inequalities: or f t,x), h 1 x) = f t,x),gradh 1 x) ) f t,x), h 1 x) = f t,x),gradh 1 x) ) Let d l C 1 [R n,r] be characteristic functions corresponding to the domains D l, l = 1,2. More precisely, let the next equalities be valid { 0, if x D l, d l x) = exp h 1 l x) ), if x R n \D l, l = 1,2. We denote: X = {x 1,x 2,...,x p }, Υ = {y 1,y 2,...,y p }, 7.30) where: x 1,...,x p,y 1,...,y p R n. We introduce the functions p d : R p+1)n R +, dx,x p+1 ) = d 2 x p+1 )+ d 1 x i ). It is clear that, function d is continuously differentiable in R p+1)n and it cancels if and only if x 1,x 2...,x p D 1, and x p+1 D 2. Let {ψ 1 t,x),ψ 2 t,x),...,ψ n t,x)} be a set of n independent first integrals of system 7.23) in a domain, containing the closed set [t 0,T] D 1. Let us consider the following problem: where Problem 7.2. Find i=1 minf X,Υ,x p+1 ) 7.31) dx,x p+1 ) = 0, 7.32) ψ j θ i,x i ) = ψ j θ i 1,y i 1 ), 7.33) i = 1,2,...,p + 1; j = 1,2,...,n; θ 0 = t 0 ; θ 1,θ 2,...θ p R; θ p+1 = T; Θ = {θ 1,θ 2,...θ p } and x 0 = y 0 D 1.

230 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS Weprovetheequivalence ofproblems7.1and7.2inthenexttheorem. Theorem 7.5. Let the conditions H7.1-H7.4 be valid. Then: 1) The following two statements are equivalent: 1.1) There exists a permissible impulsive control T 0,I 0 ) U, for which equality 7.27) is valid; 1.2) Problem 7.31), 7.32), 7.33) has a solution Θ 0,X 0,Υ 0, x 0 p+1 ); 2) If θ 0 i < θ 0 i+1, i = 1,2,...,p 1, Θ0 = T 0, X 0 = A, Υ 0 = B and x 0 p+1 = a p+1. Proof. Assume that, inequality7.28) is satisfiedif7.29) holds true, then theproofis similar). Let Θ 0,X 0,Υ 0,x 0 p+1) beasolutionof Problem 7.2 i.e. problem 7.31), 7.32), 7.33)). Let the inequalities θ 0 i < θ 0 i+1, i = 1,2,...,p 1 be fulfilled. Denote T 0 = Θ 0, i.e. t 0 1 = θ0 1, t0 2 = θ0 2,...,t0 p = θ0 p, I0 C [ T 0 R n,r n], I 0 t 0 i,x0 i) = y 0 i, i = 1,2,...,p, where X 0 = { x 0 1,x0 2,...,x0 p} and Υ 0 = { y 0 1,y0 2,...,y0 p}. We prove that the solution x = xt;t 0,x 0 ) of the initial problem with impulses dx dt = f t,x), t [t 0,T]\T 0, xt+0) = I 0 t,x), t T 0, xt 0 ) = x 0 satisfies the inclusions xt;t 0,x 0 ) D 1 for t 0 t T and xt;t 0,x 0 ) D 2. Assume that, there exists a point t,t 0 < t < T, such that xt ;t 0,x 0 ) R n \D ) It is clear that, there exists a number i {0,1,...,p} such that t 0 i t < t 0 i+1. Denote t = inf { t;t 0 i t < t0 i+1,xt;t 0,x 0 ) R n \D 1 }. 7.35) We will consider two cases: Case 1. Let t / Θ 0 = T 0, i.e. t 0 i < t < t 0 i+1. From condition 7.32) it follows that x 0 i = x t 0 i;t 0,x 0 ) D1.

231 3. LAGRANGE S METHOD 213 Then, taking into consideration 7.34) and 7.35), we deduce that xt ;t 0,x 0 ) = x t ;t 0 i,x 0 i) D1. By condition H7.4 we get the conclusion, that: { xt;t0,x 0 ),t 0 i < t < t } D1 and { } xt;t0,x 0 ),t < t < t 0 i+1 R n \D 1. From the last conclusion, it follows that x 0 i+1 = x t 0 i+1 ;t ) 0,x 0 = x t 0 i+1 ;t 0 i i),x0 = lim x ) t;t 0 t t 0 i+1 0 i,x0 i R n \D 1, which contradicts 7.32). Case 2. Let t = θi 0 = t 0 i, where i {1,2,...,p}. The following variants are possible: Case 2.1. y 0 i D 1. As y 0 i = lim t t 0 i +0 xt;t 0,x 0 ) = lim t t +0 xt;t 0,x 0 ), is fulfilled, then there exists = const > 0 such that, for any t,t t t +, it is satisfied xt;t 0,x 0 ) D 1. This inclusion contradicts the choice of t see 7.35)). Case 2.2. y 0 i R n \D 1. From condition H7.4 and in view of the fact that xt ;t 0,x 0 ) R n \D 1, we find that x 0 i+1 R n \D 1. Thus, we reach again the contradiction with 7.32). We have {xt;t 0,x 0 ),t 0 < t < T} D 1. From the definition of functions d 2 = d 2 x p+1 ) and d = dx,x p+1 ) and equality 7.32), it follows that x 0 p+1 = xt;t 0,x 0 ) D 2. Then, accordingtodefinition7.1wehaveθ 0,I 0 ) U. SinceΘ 0,X 0,Υ 0, x 0 p+1) is a solution of extreme problem 7.31), then equality 7.27) is satisfied. Let the ordered pair T 0,I 0 ) U satisfies equality 7.27), i.e. this is a point of minimum for the target impulsive function Φ or in other words, it is an optimal impulsive control. Then from the definition of the set U and the function d = dx,x p+1 ), it follows that Θ 0,X 0,Υ 0,xp+1) 0 is a solutionof problem 7.2, where Θ 0 = T 0,X 0 = A, Υ 0 = Bandx 0 p+1 = a p+1 see 7.26)). Thus the theorem is proved.

232 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS Using Theorem 7.5 and the closeness of set Θ,X,Υ,x p+1 ), for which the equalities 7.32) and 7.33) are valid, it follows solvability of the impulsive control problem. More precisely the next Theorem is correct. Theorem 7.6. Let the following conditions be valid: 1) The conditions H7.1-H7.4 hold; 2) The domain D 1 is bounded. Then, there exists a permissible impulsive control T 0,I 0 ) U, satisfying the equality 7.27), i.e. Problem 7.1 has a solution. We continue with the next theorem. Theorem 7.7. Let the following conditions be valid: 1) The conditions H7.1-H7.4 hold; 2) The ordered pair T 0,I 0 ) U satisfies equality 7.27), i.e. this is a solution of Problem 7.1. Then, there exist the numbers ξ and η j i, i = 1,2,...,p + 1,j = 1,2,...,n, such that: ξ + p+1 i=1 n η j 0, 7.36) j=1 xi F A 0,B 0,ap+1) 0 = ξ xi d ) n A 0,a 0 p+1 η j i xψ j t 0 i,ai) 0, 7.37) yi F ) n A 0,B 0,a 0 p+1 = η j i+1 xψ j t 0 i,ai) 0, 7.38) n j=1 Here: η j i d dt ψ j j=1 t 0 i,a 0 i) = n j=1 η j i+1 i j=1 d dt ψ ) j t 0 i,b 0 i. 7.39) a 0 i = x t 0 i ;t 0,x 0 ) ; b 0 i = I 0 t 0 i,a0 i), i = 1,2,...,p; A 0 = { a 0 1,a0 2,...,a0 p} ; B 0 = { b 0 1,b0 2,...,b0 p} ; a 0 p+1 = xt;t 0,x 0 ). Proof. From Theorem 7.5, it follows that under certain conditions Problems 7.1 and 7.2 are equivalent. Then, we can reformulate the theorem in this way:

233 3. LAGRANGE S METHOD 215 The equalities 7.36), 7.37), 7.38), 7.39) are necessary conditions for existence the solution of Problem 7.2. We consider the Langrange function for problem7.31), 7.32),7.33): L Θ,X,Υ,x p+1,ξ,η1,...,ηp) 1 n =F X,Υ,xp+1 )+ξdx,x p+1 ) p n + η j i ψ j θ i,x i ) ψ j θ i 1,y i 1 )). i=1 The derivatives of the Langrange function are: d n ) L = η j d i dθ i dt ψ jθ i,x i ) η j d i+1 dt ψ j θ i,y i ), 7.40) d dx k i d dy k i j=1 j=1 L = d F X,Υ,x dx k p+1 )+ξ d dx,x i dx k p+1 )+ i L = d F X,Υ,x dyi k p+1 ) n d η j i+1 dy k j=1 i n d η j i dx k j=1 i ψ j θ i,x i ), 7.41) ψ j θ i,y i ), 7.42) where k = 1,2,...,n, x i = x 1 i,x 2 i,...,x n i) and y i = yi,y 1 i,...,y 2 i n ). The equalities 7.37), 7.38) and 7.39) follow from equalities 7.41), 7.42) and 7.40), respectively, and the Lagrange Theorem for conditional extreme). Thus the theorem is proved. Theorem 7.8. Let the following conditions be valid: 1) The conditions H7.1-H7.3 hold; 2) It is fulfilled D 1 = D 2 = R n ; 3) The ordered pair T 0,I 0 ) U satisfies the equality 7.27). Then xi F ) ) A 0,B 0,a 0 p+1,f t 0 i,a 0 i = yi F ) ) A 0,B 0,a 0 p+1,f t 0 i,b 0 i, i = 1,2,...,p ) Proof. Since ψ j,j = 1,2,...,n, are first integrals of the system considered 7.23), then the following equalities are true d n dt ψ d j t,x)+ f k t,x) dx kψ jt,x) = 0, k=1

234 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS where f t,x) = f 1 t,x),f 2 t,x),...,f n t,x)) and x = x 1,x 2,...,x n). Consecutively, we find η j i d dt ψ j n t 0 i,ai) 0 = η j i f ) k t 0 i,a 0 d i k=1 dx k i ) ψ j t 0 i,a 0 i = η j i xψ j t 0 i,a 0 i),f t 0 i,a 0 i). After adding the above equalities over j = 1,2,...,n the result is n η j d i dt ψ n j t 0 i,ai) 0 = η j i ) xψ j t 0 i,ai) 0,f t 0 i,a 0 i j=1 j=1 n = η j i xψ j t 0 i,ai) 0,f t 0 i,ai) 0. j=1 By equality 7.37) and the fact that, the derivatives of function d in domain D 1 are equals 0, we find consecutively n η j d i dt ψ ) j t 0 i,a 0 i = xi F ) A 0,B 0,a 0 p+1 +ξ xi d ) A 0,a 0 p+1,f t 0 i,ai) 0 j=1 = xi F A 0,B 0,a 0 p+1),f t 0 i,a 0 i). 7.44) Similarly, we find the next formula n η j d i+1 dt ψ ) j t 0 i,b 0 i = yi F ) ) A 0,B 0,a 0 p+1,f t 0 i,b 0 i. 7.45) j=1 In view of equalities 7.44) and 7.45) and equality 7.39) from the previous theorem, we derive 7.43). Thus the theorem is proved. The next result improves the preceding results in the terms of necessary and sufficient conditions for optimal impulsive control of the studied problem. In order to simplify the research, we assume that the domain of the problem coincides with R n. Theorem 7.9. Let the following conditions be valid: 1) The conditions H7.1-H7.4 hold; 2) It is satisfied D 1 = D 2 = R n ; 3) All first integrals ψ j t,x), j = 1,2,...,n, are concave down.

235 3. LAGRANGE S METHOD 217 Then theordered pair T 0,I 0 ) U satisfies equality 7.27)if andonly if there exist nonnegative numbers η j i, i = 1,2,...,p+1, j = 1,2,...,n, such that: 1) xi F ) n A 0,B 0,a 0 p+1 = η j i xψ j t 0 i,a 0 i); Here: j=1 2) yi F ) n A 0,B 0,a 0 p+1 = η j i+1 xψ j t 0 i,a 0 i); 3) j=1 n η j i dψ dt j t 0 i,a0 i ) = n η j i+1 dψ dt j t 0 i,b0 i ); j=1 j=1 4) ψ j θ i,x i ) ψ j θ i 1,y i 1 ); 5) η j i ψ j θ i,x i ) ψ j θ i 1,y i 1 )) = 0. a 0 i = x t 0 i ;t 0,x 0 ) ; b 0 i = I t 0 i,a0 i), i = 1,2,...,p; A 0 = { a 0 1,a0 2,...,a0 p} ; B 0 = { b 0 1,b0 2,...,b0 p} ; a 0 p+1 = xt;t 0,x 0 ). The proof is similarly to the proof of Theorem 7.7 using the Kuhn- Tucker Theorem, so the details are omitted. In the last theorem of the section we assume that the dimension of the phase space is n = 1. We consider the initial problem where: f R + R R; I : T R R; T = {t 1,t 2,...,t p }; t 0,x 0 ) R + R. dx dt = f t,x), t R+ \T, 7.46) xt+0) = It,x), t T, 7.47) Let the following conditions hold: xt 0 ) = x 0, 7.48) H7.5. f C 1 [R + R,R]; H7.6. There exists a constant C f > 0 such that t,x) R + R ) f t,x) C f ; H7.7. For any point t 0,x 0 ) R + R, the problem without impulses 7.46), 7.48) has a unique solution, defined for t t 0 ; H7.8. D 1 = D 2 = R is satisfied; H7.9. The function ψ = ψt,x) is the first integral of equation 7.46).

236 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS We consider the optimal impulsive problem on the solution x = xt;t 0,x 0 ) of Problem 7.46), 7.47), 7.48) with the optimization function p F A,B,a p+1 ) = a p+1 + a i b i ). 7.49) Theorem Let the following conditions be valid: The conditions H7.5-H7.9 hold. The ordered pair T 0,I 0 ) U satisfies equality 7.27). The optimization function F is defined by the equality 7.49). Then: ψ t 0 i,bi) 0 = ψ t 0 i 1,ai 1) 0, i = 1,2,...,p+1, 7.50) d dt ψ ) t 0 d i,b0 i = dt ψ ) t 0 i,a0 i 1, i = 1,2,...,p, 7.51) f t 0 i,ai) 0 = f t 0 i,bi) 0, i = 1,2,...,p. 7.52) Proof. From formulas 7.37) and 7.38) of Theorem 7.7, we get d 1 = η i dx ψ ) t 0 d i,a0 i, 1 = ηi+1 dx ψ ) t 0 i,a0 i, i = 1,2,...,p, where η 1,η 2,...,η p+1 R. Therefore, η i = η i+1. Then, in view of 7.39), we obtain the equality 7.51). We note that from condition 3 of the theorem considered and the equalities 7.37) and 7.38), it is clear that η i 0,i = 1,2,...,p + 1. The equality 7.52) follows from Theorem 7.8. Thus the theorem is proved. i=1

237 4. APPLICATION Application: Impulsive Controllability and Optimization Problems in Population Dynamics Many mathematical models are devoted to the dynamics of isolated species and they are based on the differential equations of the type dn dt = Nf t,n)+gt,n), 7.53) where: The solution N = N t) refers to biomass amount of population at the moment t > 0; The function f = f t,n) characterizes the rate of change of quantity of population at the moment t; The function g = gt,n) describes effects of external factors, which are continuous in time. Different variants of functions f and g produce various models of isolated populations. Let us point out following examples: If f t,n) = µ K N), we obtain the Verhulst equation K dn dt = µ N K N)+gt,N), 7.54) K if there is a continuous external intervention. The meaning of the positive coefficients µ and K is described in Section 7.1; If f t,n) = r γlnn, we get the Gompertz Model dn = N r γlnn)+gt,n) 7.55) dt with continuous external effects. The sence of the positive coefficients µ and K is described in section 3.2; If f t,n) = f N), then 7.53) is an evolutionary model of the stationary population; If gt,n) = 0, then 7.53) is an evolutionary model of the isolated population. Let N = N t;t 0,N 0 ) be a solution of differential equation 7.53) with initial condition N t 0 ;t 0,N 0 ) = N ) Let t 1,t 2,...,t p,t 0 < t 1 < t 2 <,...,< t p, be the moments of external impulsive effects on the solution of studied model. These outer effects, which occur instantaneously, consist in adding or removal a certain biomass amount of the population. Then N t i +0;t 0,N 0 ) = It i,n t i ;t 0,N 0 )), i = 1,2,...,p, 7.57)

238 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS where: N t i +0;t 0,N 0 ) = lim t t i +0 N t;t 0,N 0 ), i = 1,2,...,p; The function I = It, N) characterizes the external effects, which realize at the moments t 1,t 2,...,t p. For instance, if and It i,n) = N I i I i > 0, then at the moment t i, the biomass is removed and its quantity is I i. We note that, using the impulsive systems as an apparatus for managing the population development, is achieved by identifying both the optimal impulsive parameters and the moments of impulsive effects. Let us investigate the Verhulst impulsive model of 7.54), 7.57), 7.56), where It i,n) = N I i, i = 1,2,...,p. As an example of the optimization problem, we consider: Problem 7.3. Let T T > t 0 ), be the end of time interval. Find: 1) Optimal number p of impulsive effects; 2) Optimal impulsive moments t 1,t 2,...,t p ; 3) Optimal volumes of removals I 1,I 2,...,I p. so that the quantity of biomass I max = I 1 + I 2 + +I p the so called yield production)) to be maximal for t [t 0,T]. Remark 7.8. We interpret equation 7.52) of Theorem 7.10 in the terms of mathematical model of an independent population. Additionally, we assume that the target function has the form 7.49). Then from 7.52) it follows that, if impulsive control is optimal, then for each one of the impulsive jumps the value on the right side of the equation at the moment of jump is equal to the value of the same right hand side immediately after it. We give some geometrical interpretations of Theorem 7.10 under the following additional conditions: H7.10. The constants χ 1 and χ 2,χ 1 < χ 2 exist, such that f t,χ 1 ) = f t,χ 2 ) = 0;

239 4. APPLICATION 221 H7.11. For any t, t 0 < t < T, the function φx) = f t,x) has a unique maximum for χ 1 < x < χ 2 ; H7.12. The number of impulsive effects p = 1 and t 0 < t 1 < T. Remark 7.9. We note that the right hand side of the Verhulst equation 7.54) for isolated population, i.e. the function f t,n) = f N) = µ N K N) K satisfies the conditions H7.10 and H7.11. It is fulfilled χ 1 = 0, χ 2 = K and only one a maximum of the right side is achieved at the point K 2. Remark Let χ 1 < x 0 < χ 2, a 1 = xt 1 ;t 0,x 0 ) and c = f t 1,a 1 ). From condition H7.10, it follows that the set [t 0,T] [χ 1,χ 2 ] is invariant with respect to equation 7.46). Therefore, χ 1 < a 1 < χ 2. By condition H7.11, it is seen that f t,x) > 0 for t 0 t T and χ 1 < x < χ 2. Again, from the same condition it follows that, there exist the functions γ 1,γ 2 C 1 [[t 0,T],[χ 1,χ 2 ]], such that: γ 1 t) < γ 2 t) and f t,γ 1 t)) = f t,γ 2 t)) = c see Figure 7.5). It is valid a 1 = γ 2 t 1 ). We define b 1 = γ 1 t 1 ) and thus, we determine theimpulsive function I C 1 [γ 2 t),γ 1 t)]andiγ 2 t)) = γ 1 t),t 0 t T. In particular it is satisfied I a 1 ) = b 1. In view of formula 7.52), we find out that under the restrictions H7.10-H7.12, the ordered pair t 1,I) = t 0 1,I0 ) is an optimal impulsive control for initial problem 7.46), 7.47), 7.48). Example 7.2. Let the evolution of an isolated population be described by the Verhulst equation dx dt = µ xk x), 7.58) K where µ = 0,03 is the reproductive potential of the population and K = 100 is the capacity of environment. Let an impulsive removal of the population biomass take place three times p = 3) in the time interval 0 = t 0 t T 100. Let the quantity of the isolated population be 15 at the initial moment t 0 = 0). We determine the moments of removal t 1,t 2 and t 3, so that the quantity of produced biomass be maximum. The impulsive analogue of the initial problem of the Verhulst equation, describing the dynamics of an

240 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS f x c 0 x 0 t 1 T χ1 γ 1 t xt;t 0,x 0 ) χ 2 γ 2 Figure 7.5 isolated population has the type dx dt = µ K xk x), t t 1,t 2,t 3, xt i +0) = I t i,xt i )), i = 1,2,3, x0) = 15. The optimization function has the form F x 1,x 2,x 3,y 1,y 2,y 3 ) = 3 x i y i ). i=1 From the previous theorem, it follows that, if t 0 1,t 0 2,t0 3,a0 1,a0 2,a0 3,b0 1,b0 2,b0 3,a0 4) is a solution of the studied optimization problem, then where ψ t 0 i,ai) 0 = ψ t 0 i 1,bi 1) 0, i = 1,2,3,4, d dt ψ ) t 0 i,a 0 d i = dt ψ ) t 0 i 1,b 0 i 1, i = 1,2,3, f t 0 i i),a0 = f t 0 i,bi) 0, i = 1,2,3, ψt,x) = x exp 0,03t)

241 4. APPLICATION 223 is the first integral of differential equation 7.58). We solve the system above using numerical method for example Newton s method). The initial value conditions are: t 0 1 = 55, t0 2 = 85, t0 3 = 100, a 0 1 = 60, a 0 2 = 60, a 0 3 = 60, b 0 1 = 40, b0 2 = 40, b0 3 = 0. The solutions with an accuracy of fourth decimal place are: t 0 1 = 66,2851, a 0 1 = 56,3004, b 0 1 = 43,6996, t 0 2 = 83,1760, a0 2 = 56,3004, b0 2 = 43,6996, t 0 3 = 100, a0 3 = 56,2436, b0 3 = 0. The total biomass amount, taken away, i.e., maximum yield is F A 0,B 0) = F a 0 1,a0 2,a0 3,b0 1,b0 2,b0 3) = 81,4455. Example 7.3. We examine the Gompertz equation, describing the dynamics of an isolated population dx = xr γlnx), dt where: r = 0,03; x 0 = 0,2; t 0 = 0 and T = 5. Deprivations of the population biomass are made twice p = 2) in the form of impulses. The purpose is to determine the moments of removal t 1 and t 2 so that the total amount of the biomass, taken away, to be maximum. The optimization function has the type F x 1,x 2,y 1,y 2 ) = x 1 y 1 )+x 2 y 2 ). The first integral of the Gompertz equation has the form From the equality ψt,x) = t+ln0,03 lnx). d 2 dtdt ψt,x) d 2 dtdx ψt,x) d 2 dxdt ψt,x) d 2 dxdx ψt,x) = 0, it follows that the first integral ψ is a convex function. Therefore, we may apply Theorem 7.9. The solution of the corresponding system with

242 OPTIMIZATION PROBLEMS IN POPULATION DYNAMICS 1 x x 0 = t 1 x a 0 1 = a x 0 = 0.2 b t = t 0 2 t Figure 7.6 an accuracy of fourth decimal place is t 0 1 = 2,2012, a 0 1 = 0,8595, b 0 1 = 0,0523,

243 4. APPLICATION 225 t 0 2 = 5, a0 2 = 0,8595, b0 2 = 0. The total amount of biomass yield with the assumptions, made in the considered Example 7.3) is F A 0,B 0) = F a 0 1,a 0 2,b 0 1,b 0 2) = 1,6666. From Theorem 7.9, it follows that there exists only one global solution of the studied problem. The graphs of integral curves of the Gompertz equation and its impulsive analogue are shown on Figure 7.6.

244

245 Chapter 8 Continuous Dependence of the Solutions of Differential Equations with Variable Structure and Non Fixed Moments of Impulses with Respect to the Switching Functions One specific class of nonlinear nonautonomous systems of ordinary differential equations with variable structure and impulses is studied in this chapter. The changing of the right hand side of the system and impulsive effects of the solution are performed at the moments, at which the so-called switching functions become zero. Frequently, after these impulsive effects the process continues its development obeyed to the new rules and laws, different from the previous ones. Applications of the differential equations with variable structure are mainly in the control theory: Filippov [146], Gao and Hung [156], Hung et al. [180], Mu and Tang [258], Paden and Sastry [272], Shevitz and Paden [300], Utkin [327]. The impulsive equations are used mostly for describing and investigating the species development, which are subjected to the discrete effects: Ahmad and Stamov [12], Bainov and Dishliev [42], Benchohra et al. [80], Jiang and Lu [189], Lakshmikantham et al. [211], Li and Xing [220],LiuandYe[224],Meng etal.[246,247],padenandsastry[272],shi et al. [301], Shuai et al. [302]. The equations with variable structure and impulses are used for investigation the dynamics of the hydraulic valve stopper in article Dishliev and Bainov [118]. The moments when, the impulsive effects take place and the structure changes can be determined in different ways, which define different classes of the systems considered. We quote the following: The switching moments are fixed in advance; The switching moments coincide with the moments at which the integral curve trajectory) cancels the predefined functions, determinedintheextended phasespaceorphasespace)ofthesys- 227

246 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE tem of differential equations. These functions are called switching; The switching moments coincide with the moments, at which the trajectory of the system considered meets the predefined sets, situated in the extended phase space in general, these sets are hypersurfaces); The switching moments coincide with the moments, at which the solution minimizes a functional; The switching moments are random in their nature, etc. In this chapter, the switching moments are of second type. The sufficient conditions for continuously dependence of the solution of the corresponding initial problem for described above systems differential equations on the initial condition and switching functions are obtained in the first section of this chapter. The results are published in Chukleva et al. [98]. The dynamics of the shutter back blow-down valve is described in the second section of chapter 8 by the systems differential equations with variable structure and impulses. The structure of the modeling system varies in the relation to the states of the shutter valve: closed and open. The impulses reflect to: The instantaneous change in the speed of the shutter valve in transition from open to closed position; The jump-like movement of the shutter valve in transition from closed to open state. The main result in Section 2 is to find the conditions, similar to those in the preceding section, which ensure the continuous dependence of system solutions in terms of changes in the initial condition and switching functions. The results obtained in this section are published in Chukleva [97].

247 1. CONTINUOUS DEPENDENCE Continuous Dependence of the Solutions of the Differential Equations with Variable Structure and Non Fixed Moments of Impulses with Respect to the Switching Functions The main object of investigation in this section is the following initial problem of the system nonlinear nonautonomous ordinary differential equations with variable structure and impulses at non fixed moments: where: dx dt = f it,x), φ i xt)) 0, t i 1 < t < t i, 8.1) φ i xt i )) = 0, i = 1,2,..., 8.2) xt i +0) = xt i )+I i xt i )), 8.3) xt 0 ) = x 0, 8.4) f i : R + D R n, f i = f 1 i,f 2 i,...f n i ), phase space D is nonempty domain of R n ; φ i : D R; I i : D R n and Id+I i ) : D D, where Id is an identity in R n ; The initial point t 0,x 0 ) R + D. The solution of the initial problem is left continuous function at the moments t 1,t 2,... Moreover, this solution is a differentiable function in each open intervals t i 1,t i ), i = 1,2,... It is satisfied: 1.1. For t 0 t t 1, the solution of problem 8.1), 8.2), 8.3), 8.4) coincides with the solution of initial problem 8.1), 8.4) with invariable structure and without impulses), i.e. it coincides with the solution of the problem dx dt = f 1t,x),xt 0 ) = x 0 ; 8.5) 1.2. For any t, t 0 < t < t 1, it is satisfied φ 1 x 1 t)) 0, where x 1 t) is the solution of initial problem 8.5); 1.3. Let t 1 be the first moment after t 0, for which the next equation is satisfied φ 1 x 1 t 1 )) = 0; 1.4. At the moment t 1, the right hand side of the problem changes and impulsive effect on the solution takes place, i.e. it is satisfied

248 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE the equality 8.3) is satisfied for i = 1. We have xt 1 +0) = x 1 t 1 )+I 1 x 1 t 1 )) = Id+I 1 )x 1 t 1 )); 2.1. For t 1 < t t 2, the solution of problem 8.1), 8.2), 8.3), 8.4) coincides with the solution of initial problem dx dt = f 2t,x),xt 1 +0) = Id+I 1 )x 1 t 1 )); 8.6) 2.2. For any t,t 1 t t 2, the inequality φ 2 x 2 t)) 0 is valid, where x 2 t) is the solution of the initial problem 8.6); 2.3. Let t 2 be the first moment after t 1, for which it is fulfilled the equality φ 2 x 2 t 2 )) = 0; 2.4. At themoment t 2, animpulsive effect takes place, besides changing the system structure, i.e. equality 8.3) is satisfied for i = 2. We have xt 2 +0) = x 2 t 2 )+I 2 x 2 t 2 )) = Id+I 2 )x 2 t 2 )) etc. The solution of the research problem is continuous to the left at the moments t 1,t 2,... In general for example, when the functions I i x) 0 for x D, i = 1,2,...) this solution has right discontinuity at the points, indicated above, i.e. there is finite jump discontinuity. Note that, the points t 1,t 2,... are called switching moments, the functions I i,i = 1,2,..., are impulsive functions and φ i,i = 1,2,..., are switching functions. Further, the solution of problem 8.1), 8.2), 8.3), 8.4) we denote by xt;t 0,x 0, φ 1,φ 2,...). More precisely, we have xt;t 0,x 0,φ 1,φ 2,...) xt;t 0,x 0 ), t 0 < t < t 1 ; xt;t 0,x 0,φ 1 ), t 1 < t < t 2 ; =. xt;t 0,x 0,φ 1,φ 2,...φ i ), t i < t < t i+1 ;. 8.7)

249 1. CONTINUOUS DEPENDENCE 231 Together with the basic problem, we study the so called perturbed initial problem: dx dt = f it,x ), φ i x t)) 0, t i 1 < t < t i, 8.8) φ i x t i )) = 0, i = 1,2,..., 8.9) x t i +0) = x t i )+I ix t i )), 8.10) x t 0 ) = x 0, 8.11) where: The switching functions φ i : D R; The initial point t 0,x 0 ) R+ D. Thesolutionoftheproblemaboveisdenotedbyx t;t 0,x 0,φ 1,φ 2,...). Like 8.7) there is x t;t 0,x 0 ), t 0 < t < t 1 ; x t;t 0,x 0,φ 1), t 1 < t < t 2; x t;t 0,x 0,φ 1,φ 2,...) =. x t;t 0,x 0,φ 1,φ 2,...φ i), t i < t < t i+1;. Definition 8.1. We say that the solution of problem 8.1), 8.2), 8.3), 8.4) depends continuously on the initial condition and the switching functions, if: ε = const > 0) η = const > 0) T = const > t 0 ) δ = δε,η,t) > 0) : t 0 R +, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ) φ i C[D,R], φ i x) φ i x) < δ for x D,i = 1,2,...) x t;t 0,x 0,φ 1,φ 2,...) xt;t 0,x 0,φ 1,φ 2,...) < ε for t max 0 t T and t t i > η, i = 1,2,...), where t max 0 = max{t 0,t 0}. Note that the existence of proximity between the both solutions of the problem considered and the corresponding perturbed problem) it not required in the pre-fixed neighborhoods t i η,t i +η),i = 1,2,..., at the switching moments of the basic problem For convenience, we introduce the symbols: Φ i = {x D;φ i x) = 0},i = 1,2,..., are switching hypersurfaces of the basic problem;

250 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE Φ i = {x D;φ i x) = 0},i = 1,2,..., are switching hypersurfaces of the perturbed problem; I 0 x) = 0 for x D. The equality Id+I 0 )x) = x, x D, is valid; γt 0,x 0 ) = {t,xt;t 0,x 0,φ 1,φ 2,...)),t 0 t T} is a trajectory of the problem considered for t 0 t T; γ t 0,x 0 ) = {t,x t;t 0,x 0,φ 1,φ 2,...)),t 0 t T} is a trajectory of the perturbed problem for t 0 t T;. and.,. are the Euclidean norm and the dot product in R n ; B δ x 0 ) = {x R n ; x x 0 < δ}isδ-neighborhoodofthepoint x 0. The following conditions are introduced: H8.1. f i C[R + D,R n ], i = 1,2,...; H8.2. There exists a constant C f > 0, such that t,x) R + D ) f i t,x) C f, i = 1,2,...; H8.3. φ i C 1 [D,R]; H8.4. There exists a constant C gradφ > 0, such that x D) gradφ i x) C gradφ, i = 1,2,...; H8.5. I i C[D,R n ] and Id+I i ) : Φ i D; H8.6. There exists a constant C φid+i) > 0, such that x Φ i ) φ i+1 Id+I i )x)) = φ i+1 x+i i x)) C φid+i), H8.7. The next inequalities are satisfied: i = 1,2,...; φ i+1 Id+I i )x)) gradφ i+1 x),f i+1 t,x) < 0,t,x) R + D, i = 1,2,...; H8.8. There exists a constant C gradφ,f > 0, such that t,x) R + D ) gradφ i x),f i t,x) C gradφ,f, i = 1,2,...; H8.9. For any point t 0,x 0 ) R + D and for each i = 1,2,..., there exists a unique solution of the initial problem for t t 0 dx dt = f it,x), xt 0 ) = x ) Theorem 8.1. Let the conditions H8.1-H8.7 hold. Then:

251 1. CONTINUOUS DEPENDENCE 233 1) If the trajectory γt 0,x 0 ) of problem 8.1), 8.2), 8.3), 8.4) meets consecutively the switching hypersurfaces Φ i and Φ i+1, then for the corresponding switching moments t i and t i+1, the following estimate is valid t i+1 t i C φid+i) C gradφ C f. 2) If the trajectory γt 0,x 0 ) meets all the switching hypersurfaces Φ i, i = 1,2,..., then the switching moments increase infinitely, i.e. lim i t i =. Proof. Under conditions H8.7, the functions φ i+1 Id+I i )x)) and gradφ i+1 x),f i+1 t,x) do not cancel in the domain and they have opposite signs for any point t,x) R + D. Without loss of generality, we assume that the following inequalities are valid: and φ i+1 Id+I i )x)) < 0, x D 8.13) gradφ i+1 x),f i+1 t,x) > 0, t,x) R + D. 8.14) We consider the function ϕ : [t 1,t i+1 ] R, defined by the equality φ i+1 xt i +0;t 0,x 0,φ 1,...φ i )) = φ ϕt) = i+1 xt i ;t 0,x 0,φ 1,...φ i 1 ) 8.15) +I i xt i ;t 0,x 0,φ 1,...φ i 1 ))), t = t i ; φ i+1 xt;t 0,x 0,φ 1,...φ i )), t i < t t i+1. According to condition H8.6 and inequality 8.13), we have ϕt i+1 ) ϕt i ) =φ i+1 xt i+1 ;t 0,x 0,φ 1,...φ i )) φ i+1 xt i +0;t 0,x 0,φ 1,...φ i )) =0 φ i+1 xt i ;t 0,x 0,φ 1,...φ i 1 ) +I i xt i ;t 0,x 0,φ 1,...φ i 1 ))) = φ i+1 Id+I i )xt i ;t 0,x 0,φ 1,...φ i 1 ))) = φ i+1 Id+I i )xt i ;t 0,x 0,φ 1,...φ i 1 ))) C φid+i). 8.16)

252 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE On the other hand, using 8.14) and the conditions H8.1, H8.2, H8.3 and H8.4, we obtain consecutively: ϕt i+1 ) ϕt i ) = d dt ϕθ)t i+1 t i ) = d dt φ i+1xθ;t 0,x 0,φ 1,...φ i ))t i+1 t i ) = φ i+1 xθ;t 0,x 0,φ 1,...φ i ))fi+1 1 x θ,xθ;t 0,x 0,φ 1,...φ i )) 1 + φ i+1 xθ;t 0,x 0,φ 1,...φ i ))f x i+1θ,xθ;t 2 0,x 0,φ 1,...φ i )) ) φ i+1 xθ;t 0,x 0,φ 1,...φ i ))fi+1 n x θ,xθ;t 0,x 0,φ 1,...φ i )) n t i+1 t i ) = gradφ i+1 xθ;t 0,x 0,φ 1,...φ i )),f i+1 θ,xθ;t 0,x 0,φ 1,...φ i )) t i+1 t i ) gradφ i+1 xθ;t 0,x 0,φ 1,...φ i )) f i+1 θ,xθ;t 0,x 0,φ 1,...φ i )) t i+1 t i ) C gradφ C f t i+1 t i ). The following estimate is achieved from the inequality above t i+1 t i 1 C gradφ C f ϕt i+1 ) ϕt i )), whence, by means of the inequality 8.16), it follows that t i+1 t i C φid+i) C gradφ C f. If the trajectory of the basic problem meets infinitely many switching hypersurfaces, then using the previous estimate, we achieve the conclusion lim t i = lim t i t i 1 )+t i 1 t i 2 )+ +t 1 t 0 )+t 0 ) i i lim i i C φid+i) C gradφ C f +t 0 =. Thus the theorem is proved. Theorem 8.2. Let the following conditions hold:

253 1. CONTINUOUS DEPENDENCE 235 1) The conditions H8.1-H8.8 are valid; 2) For any point t,x) R + D, the next inequality is satisfied φ 1 x 0 ) gradφ 1 x),f 1 t,x) < 0. Then the trajectory of problem 8.1), 8.2), 8.3), 8.4) meets every one of the hypersurfaces Φ i, i = 1,2,... Proof. First of all, we show that, the trajectory of the basic problem meets the hypersurface Φ 1. One of the following two cases is valid from the condition 2: Case 1. φ 1 x 0 ) < 0, gradφ 1 x),f 1 t,x) > 0 for t,x) R + D; Case 2. φ 1 x 0 ) > 0, gradφ 1 x),f 1 t,x) < 0 for t,x) R + D. Here we discuss the first case. Another case can be considered in a similar way. We introduce the function ϕt) = φ 1 xt;t 0,x 0 )) for t t 0. We have ϕt 0 ) = φ 1 xt 0 ;t 0,x 0 )) = φ 1 x 0 ) < 0. Under the condition H8.5, it is satisfied Using the facts d dt ϕt) = gradφ 1xt;t 0,x 0 )),f 1 t,xt;t 0,x 0 )) = gradφ 1 xt;t 0,x 0 )),f 1 t,xt;t 0,x 0 )) C gradφ,f = const > 0. ϕt 0 ) < 0 and d dt ϕt) = const > 0, t > t 0, it follows that, there exists a point t 1 > t 0, such that φ 1 xt 1 ;t 0,x 0 )) = ϕt 1 ) = 0. This means that at the moment t 1, the trajectory γt 0,x 0 ) meets the hypersurface Φ 1. Assume that the trajectory of the problem considered meets consecutively the hypersurfaces Φ 1,Φ 2,...,Φ i at the moments t 1,t 2,...,t i, respectively. Then we prove that γt 0,x 0 ) meets the hypersurface Φ i+1. Once again, without loss of generality, we assume that the inequalities 8.13) and 8.14) are valid. As in the previous Theorem, we consider the function ϕ, defined by 8.15). We have ϕt i +0) =φ i+1 xt i ;t 0,x 0,φ 1,...φ i 1 )+I i xt i ;t 0,x 0,φ 1,...φ i 1 ))) =φ i+1 Id+I i )xt i ;t 0,x 0,φ 1,...φ i 1 ))) < )

254 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE For t > t i, it is satisfied d dt ϕt) = d dt φ i+1xt;t 0,x 0,φ 1,...φ i )) = gradφ i+1 xt;t 0,x 0,φ 1,...φ i )),f i+1 t,xt;t 0,x 0,φ 1,...φ i )) = gradφ i+1 xt;t 0,x 0,φ 1,...φ i )),f i+1 t,xt;t 0,x 0,φ 1,...φ i )) C gradφ,f = const > ) From 8.17) and 8.18), it follows that, there exists a point t i+1 > t i such that ϕt i+1 ) = 0 φ i+1 xt i+1 ;t 0,x 0,φ 1,...φ i )) = 0. The interpretation of this equality is that the trajectory of problem 8.1), 8.2), 8.3), 8.4) meets the hypersurface Φ i+1. The proof of this theorem follows by induction. Thus the theorem is proved. Using Theorem 8.1 and condition H8.9, we deduce the validity of the next theorems: Theorem 8.3. Let the following conditions H8.1-H8.7 and H8.9 hold. Then the solution of problem 8.1), 8.2), 8.3), 8.4) exists and it is unique for t 0 t <. Theorem 8.4. Let the following conditions hold: 1) The conditions H8.1-H8.7 and H8.9 are valid; 2) For any point t,x) R + D, the next inequality is satisfied φ 1 x 0 ) gradφ 1 x),f 1 t,x) < 0. 3) Thetrajectoryγt 0,x 0 )ofproblem8.1),8.2),8.3),8.4)meets the hypersurface Φ 1 at the moment t 1. Then δ = const > 0) : t 0 R+, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ) φ 1 C[D,R], φ 1 x) φ 1x) < δ for x D) γ t 0,x 0 ) Φ 1.

255 1. CONTINUOUS DEPENDENCE 237 Proof. Under condition 2, the following inequalities take place: φ 1 x 0 ) =φ 1 x 0 +I 0 x 0 )) =φ 1 Id+I 0 )x 0 )) 8.19) 0, gradφ 1 x),f 1 t,x) 0, t,x) R + D. Assume that: φ 1 x 0 ) = φ 1 xt 0 ;t 0,x 0 )) < 0, φ 1 xt 1 ;t 0,x 0 )) = ) and φ 1 xt;t 0,x 0 )) < 0, t 0 < t < t 1. The case φ 1 x 0 ) > 0 is considered in the same way. Assume that for any t t 0, it is satisfied φ 1 xt;t 0,x 0 )) 0. Then t 1 is a point of maximum of the function ϕt) = φ 1 xt;t 0,x 0 )). Therefore, it is fulfilled 0 = d dt ϕt 1) = d dt φ 1xt 1 ;t 0,x 0 )) = gradφ 1 xt 1 ;t 0,x 0 )),f 1 t 1,xt 1 ;t 0,x 0 )). This equality contradicts to the second of the inequalities 8.19). Therefore, there exists a point θ > t 1, such that ϕθ) = φ 1 xθ;t 0,x 0 )) > 0. From the inequality φ 1 x 0 ) < 0 and the continuity of the function φ 1, it follows that there exists a positive constant δ, such that for any point x B δ x 0 ), the inequality φ 1 x) < 0 is satisfied. By analogy, using the inequality φ 1 xθ;t 0,x 0 )) > 0 and the continuity of the function φ 1, we obtain the existence of the positive constant δ, such that for each point x B δ xθ;t 0,x 0 )), we have φ 1 x) > 0. We denote = min{ φ 1 x), x B δ x 0 )}, = min{ φ 1 x), x B δ xθ;t 0,x 0 ))}, In view of Theorem 0.1 from Introduction, it follows that: δ = const, 0 < δ < δ ) : t 0 R+, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ) x t;t 0,x 0 ) xt;t 0,x 0 ) < δ for t max 1 t θ).

256 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE We assume also, that for an arbitrary chosen continuous function φ 1 : D R the next extra inequality is valid φ 1 x) φ 1x) < min{, } for x D. For the part of the trajectory γ t 0,x 0), locked between the points x 0 and x θ;t 0,x 0), we obtain the following restriction: 1) For the initial point x 0, it is satisfied x 0 x 0 < δ x 0 x 0 < δ x 0 B δ x 0). Then we have φ 1 x 0 ) =φ 1 x 0 ) φ 1x 0 )+φ 1x 0 ) φ 1 x 0 ) φ 1x 0 ) +φ 1x 0 ) = φ 1x 0) φ 1 x 0) φ 1 x 0) < = ) 2) For the final point x θ;t 0,x 0), it is valid x t;t 0,x 0) xt;t 0,x 0 ) < δ for t max 1 t θ x θ;t 0,x 0 ) xθ;t 0,x 0 ) < δ x θ;t 0,x 0 ) B δ xθ;t 0,x 0 )). As a result, we ascertain that φ 1x θ;t 0,x 0)) =φ 1x θ;t 0,x 0)) φ 1 x θ;t 0,x 0))+φ 1 x θ;t 0,x 0)) φ 1 x θ;t 0,x 0)) φ 1 x θ;t 0,x 0 )) φ 1x θ;t 0,x 0 )) > = ) From 8.21) and 8.22) for the functions ϕ t) = φ 1 x t;t 0,x 0 )), we derive: ϕ t 0 ) = φ 1 x t 0 ;t 0,x 0 )) = φ 1 x 0 ) < 0 and ϕ θ) = φ 1 x θ;t 0,x 0 )) > 0. Using the continuity of the considered function ϕ, we deduce that there exists a point t 1,t 0 < t 1 < θ, such that ϕ t 1 ) = 0 φ 1 x t 1 ;t 0,x 0 )) = 0. The meaning of the last equality is that the trajectory γ t 0,x 0 ) of perturbed problem 8.8), 8.9), 8.10), 8.11) meets the hypersurface Φ 1 at the moment t 1. Thus the theorem is proved.

257 1. CONTINUOUS DEPENDENCE 239 Theorem 8.5. Let the following conditions hold: 1) The conditions H8.1-H8.7 and H8.9 are valid; 2) For any point t,x) R + D, the next inequality is satisfied ϕ 1 x 0 ) gradϕ 1 x),f 1 t,x) < 0; 3) Thetrajectoryγt 0,x 0 )ofproblem8.1), 8.2), 8.3), 8.4)meets the hypersurface Φ 1 at the moment t 1. Then η = const > 0) δ = δη) > 0) : t 0 R +, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ) φ 1 C[D,R], φ 1x) φ 1 x) < δ for x D) t 1 t 1 < η. Proof. Using previous theorem, it is easy to see that there exists a positive constant δ > 0, such that t 0 R +, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ ) φ 1 C[D,R], φ 1 x) φ 1x) < δ for x D) it follows that the trajectory γ t 0,x 0 ) of problem 8.8), 8.9), 8.10), 8.11) intersects the perturbed hypersurface Φ 1 at the moment t 1. We assume that, the inequalities 8.20) are valid see condition 2 of Theorem 8.5). Let η be an arbitrary constant and 0 < η < min{t 1 t 0,t 2 t 1 }. Then the following inequalities are valid: ϕ 1 xt;t 0,x 0 )) < 0 for t 0 t t 1 η and ϕ 1 xt+η;t 0,x 0 )) > 0. We set = min{ ϕ 1 xt;t 0,x 0 )), t 0 t t 1 η}, = ϕ 1 xt 1 +η;t 0,x 0 )). From the first of the both inequalities above for t = t 1 η, it follows that ϕ 1 xt 1 η;t 0,x 0 )) > 8.23) Using the theorem of continuous dependence Theorem 0.1 from Introduction), it follows that δ = const, 0 < δ < δ ) : t 0 R +, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ ) x t;t 0,x 0) xt;t 0,x 0 ) < 1 2C gradφ min{, } for

258 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE Let for the function ϕ 1 C[D,R], we have ) t max 1 t t 1 +η. 8.24) φ 1 x) φ 1x) < 1 2 min{, } for x D. 8.25) Under condition H8.2, the gradients of the functions φ i, i = 1,2,..., are bounded above. Hence, we deal with Lipschitz continuous functions, i.e., the following inequalities are valid: ϕ i x ) ϕ i x ) C gradϕ x x, x,x D, i = 1,2. Using the estimates 8.23), 8.24) and 8.25), we obtain: ϕ 1x t 1 η;t 0,x 0)) = ϕ 1x t 1 η;t 0,x 0)) ϕ 1 x t 1 η;t 0,x 0)) +ϕ 1 x t 1 η;t 0,x 0)) ϕ 1 xt 1 η;t 0,x 0 ))+ϕ 1 xt 1 η;t 0,x 0 )) ϕ 1 x t 1 η;t 0,x 0 )) ϕ 1x t 1 η;t 0,x 0 )) +C gradϕ x t 1 η;t 0,x 0 ) xt 1 η;t 0,x 0 ) +ϕ 1 xt 1 η;t 0,x 0 )) < C gradϕ 2C gradϕ = ) On the other hand, we have φ 1 x t 1 +η;t 0,x 0 )) =φ 1 x t 1 +η;t 0,x 0 )) φ 1x t 1 +η;t 0,x 0 )) +φ 1 x t 1 +η;t 0,x 0 )) φ 1xt 1 +η;t 0,x 0 )) +φ 1 xt 1 +η;t 0,x 0 )) > φ 1 x t 1 η;t 0,x 0 )) φ 1x t 1 η;t 0,x 0 )) C gradφ x t 1 η;t 0,x 0 ) xt 1 η;t 0,x 0 ) +φ 1 xt 1 η;t 0,x 0 )) > 1 2 C gradφ 1 2C gradφ = ) We rewrite the inequalities 8.26) and 8.27) in more compact form: φt 1 η) < 0 and φt 1 +η) > 0, respectively. From the continuity of the function φ t) = φ 1 x t;t 0,x 0 )) for t 1 η t t 1 + η it follows that there exists a point t 1, t 1 η < t 1 < t 1 +η η < t 1 t 1 < η t 1 t 1 < η,

259 1. CONTINUOUS DEPENDENCE 241 such that ϕ t 1) = φ 1x t 1;t 0,x 0)) = 0. Thelastequalitymeansthatthetrajectoryγ t 0,x 0 )oftheperturbed problem intersects the hypersurface Φ 1 at the moment t 1, for which the inequality t 1 t 1 < η is satisfied. Thus the theorem is proved. The main result in this paragraph is contained in the following theorem. Theorem 8.6. Let the conditions H8.1-H8.7 and H8.9 hold true. Then the solution of problem 8.1), 8.2), 8.3), 8.4) depends continuously on the initial condition and the switching functions. Proof. Let ε and η be arbitrary positive constants and T > t 0. The following cases are possible: Case 1. The trajectory γt 0,x 0 ) of the problem meets no one of the switching hypersurfaces for t 0 t T. In this case the assertion of the theorem follows from Theorem 0.1 for the continuous dependence. Case 2. The trajectory γt 0,x 0 ) meets only one hypersurface Φ 1 for t 0 t < T. Then, it is fulfilled: 2.1. According to Theorem 8.4, we have δ i = const > 0 ) : t 0 R+, t 0 t 0 < δ i) x 0 D, x 0 x 0 < δ i) φ 1 C[D,R], φ 1x) φ 1 x) < δ i for x D ) γ t 0,x 0) Φ 1, i.e. the trajectory γ t 0,x 0 ) meets the switching hypersurface Φ 1 at the moment t 1 ; 2.2. In accordance with Theorem 8.5, it is satisfied δ iii, 0 < δ iii < η ) δ ii = const > 0 ) : t 0 R +, t 0 t 0 < δ ii) x 0 D, x 0 x 0 < δ ii) φ 1 C[D,R], φ 1x) φ 1 x) < δ ii for x D ) t 1 t 1 < δ iii, where constant δ iii we be specified later. From the last inequality, it follows that t 1 t 1 < η; 2.3. From the Theorem 0.1 for continuous dependence, we have δ v, 0 < δ v < ε) δ iv = const > 0 ) :

260 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE t 0 R +, t 0 t 0 < δ iv) x 0 D, x 0 x 0 < δ iv) x t;t 0,x 0 ) xt;t 0,x 0 ) < δ v < ε, t max 0 t t min 1, where t min 1 = min{t 1,t 1}. We will determine the constant δ v further; 2.4. We assume that t min 1 = t 1 and t max 1 = t 1. The considerations in the second case are similar. Using the condition H8.2, we deduce that t 1 x t 1;t 0,x 0) xt 1 ;t 0,x 0 ) = x t 1 ;t 0,x 0)+ f 1 τ,x τ ;t 0,x 0))dτ t 1 xt 1 ;t 0,x 0 ) δ v + t 1 t 1 f 1 τ,x τ ;t 0,x 0 )) dτ δ v +C f t 1 t 1 =δ v +C f δ iii From the continuity of the function I 1 and taking into account the previous point, it follows that δ vi = const > 0 ) δ iii > 0, δ v > 0 ) : x t 1 +0;t 0,x 0 ) xt 1 +0;t 0,x 0 ) x t 1;t 0,x 0) xt 1 ;t 0,x 0 ) + I 1 x t 1+0;t 0,x 0)) I 1 xt 1 ;t 0,x 0 )) < δ vi, where the constant δ vi we will determine later Again with the theorem for continuous dependence Theorem 0.1), it follows that δ iii > 0 ) δ vi > 0 ) : t 1, t 1 t 1 < δ iii) x t 1 +0;t 0,x 0 ), x t 1 +0;t 0,x 0 ) xt 1 +0;t 0,x 0 ) < δ vi) x t;t 0,x 0,φ 1 ) xt;t 0,x 0,φ 1 ) < ε, t max 1 t T We specify the constants in the following sequence: From 2.1, we specify δ i ; From 2.6, we specify δ iii and δ vi ; From 2.5, we determine δ v and further refine δ iii ;

261 1. CONTINUOUS DEPENDENCE From 2.3, we find δ iv ; From 2.2, refine δ iii δ iii < η) and determine δ ii Let δ = min{δ i,...,δ vi }. The obtained result can be summarized as: t 0 R +, t 0 t 0 < δ ) x 0 D, x 0 x 0 < δ) φ 1 C[D,R], φ 1x) φ 1 x) < δ for x D) we have: x t;t 0,x 0 ) xt;t 0,x 0 ) < ε, t max 0 t t min 1 see 2.3); x t;t 0,x 0,φ 1) xt;t 0,x 0,φ 1 ) < ε, t max 1 t T see 2.6); t 1 t 1 = t max 1 t min 1 < η see 2.2). We rewrite the inequalities 2.8.1, and more compact in the form: x t;t 0,x 0,φ 1,φ 2,...) xt;t 0,x 0,φ 1,φ 2,...) < ε Thus the theorem is proved in this case. for t max 0 t T and t t 1 > η. Case 3. The trajectory of the problem meets a finite number hypersurfaces for t 0 t < T. Let the following inequalities be fulfilled for the moments of these meetings: We use the notations: t 0 < t 1 < t 2 < < t k < T < t k+1 <. T 0 = t max 0, T 1 = 1 2 t 1+t 2 ), T 2 = 1 2 t 2+t 3 ),...,T k 1 = 1 2 t k 1+t k ), T k = T. Similarly to the previous case, we have: 3.1. δ 2, 0 < δ 2 < ε) δ 1, 0 < δ 1 < ε): t 0 R +, t 0 t 0 < δ 1 ) x 0 D, x 0 x 0 < δ 1 ) φ 1 C[D,R], φ 1x) φ 1 x) < δ 1 for x D) x t;t 0,x 0 ) xt;t 0,x 0 ) < δ 2, T 0 t t min 1 ; x t;t 0,x 0,φ 1 ) xt;t 0,x 0,φ 1 ) < δ 2 for t min 1 t T and t max 1 t min 1 < η;

262 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE 3.2. δ 3, 0 < δ 3 < ε) δ 2, 0 < δ 2 < ε): x t;t 0,x 0,φ 1), x T 1 ;t 0,x 0,φ 1) xt 1 ;t 0,x 0,φ 1 ) < δ 2 ) φ 2 C[D,R], φ 2x) φ 2 x) < δ 2 for x D) x t;t 0,x 0,φ 1 ) xt;t 0,x 0,φ 1 ) < δ 3, T 1 t t min 2 ; x t;t 0,x 0,φ 1,φ 2 ) xt;t 0,x 0,φ 1,φ 2 ) < δ 3 for t min 2 t T 2 and t max 2 t min 2 < η;. 3.k-1. δ k, 0 < δ k < ε) δ k 1, 0 < δ k 1 < ε): x t;t 0,x 0,φ 1,...,φ k 2 ), x T 1 ;t 0,x 0,φ 1,...,φ k 2 ) xt 1;t 0,x 0,φ 1,...,φ k 2 ) < δ k 1 ) φ k 1 C[D,R], φ k 1 x) φ k 1x) < δ k 1 for x D ) x t;t 0,x 0,φ 1,...,φ k 2 ) xt;t 0,x 0,φ 1,...,φ k 2 ) < δ k, T k 2 t t min k 1 ; x t;t 0,x 0,φ 1,...,φ k 1 ) xt;t 0,x 0,φ 1,...,φ k 1 ) < δ k for t min k 1 t T k 1 and t max k 1 tmin k 1 < η; 3.k. δ k, 0 < δ k < ε): x t;t 0,x 0,φ 1,...,φ k 1 ), x T 1 ;t 0,x 0,φ 1,...,φ k 1 ) xt ) 1;t 0,x 0,φ 1,...,φ k 1 ) < δ k φ k C[D,R], φ k x) φ kx) < δ k for x D) x t;t 0,x 0,φ 1,...,φ k 1) xt;t 0,x 0,φ 1,...,φ k 1 ) < ε, x t;t 0,x 0,φ 1,...,φ k ) xt;t 0,x 0,φ 1,...,φ k ) < ε for t min k t T k and t max k T k 1 t t min k ; t min k < η;

263 1. CONTINUOUS DEPENDENCE 245 The constants δ i, i = 1,2,...,k, we define in reverse order: first, we identify δ k, after that we determine δ k 1, etc. Finally, we find δ 1. Let δ = δ 1. The result of the items k can be summarized as: t 0 R+, t 0 t 0 < δ) x 0 D, x 0 x 0 < δ) φ i C[D,R], φ i x) φ ix) < δ for x D and i = 1,2,...) x t;t 0,x 0,φ 1,φ 2,...) xt;t 0,x 0,φ 1,φ 2,...) < ε for t max 0 t T and t t i > η, i = 1,2,... ). The theorem is proved in this case. Case 4. The trajectory of problem 8.1), 8.2), 8.3), 8.4) meets infinity many switching hypersurfaces for t 0 t T. The inequalities t 0 < t 1 < t 2 <... < T are valid in this case. The last inequalities contradict to the second statement of Theorem 8.1. Therefore, this case is impossible. Thus the proof is complete.

264 DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE

265 2. MODELLING Modelling by the Differential Equations with Variable Structure and Non Fixed Moments of Impulses The dynamic mathematical model of hydraulic blow down back pressure valve is investigated. The model is borrowed from Dishliev and Bainov [118]. Some optimization properties of its parameters and the continuous dependence of the solution of the corresponding modeling system with respect to these parameters are studied in this section. The main elements of the valve see Figure 8.1) are: 1) Intake manifold with volume V; 2) Mitreshutter valvewithananglemeasure 2αoftheshutter crest and mass M c ; 3) Screw-shaped spiral) spring with a mass M s and specific constant of elasticity C s ; 4) Exhaust collector. Figure 8.1 The following notations are introduced: 1) = t) 0 is the shifting distance) from a shutter valve in vertical direction) to its bed; 2) P 1 = P 1 t) 0 and P 2 = P 2 t) 0 are the pressure of the intake and exhaust collector, respectively; 3) Q 1 = Q 1 t) 0 and Q 2 = Q 2 t) 0 are the incoming and outgoing flow of the fluid, respectively;