Full averaging scheme for differential equation with maximum

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1 Contemporary Analysis and Applied M athematics Vol.3, No.1, , 215 Full averaging scheme for differential equation with maximum Olga D. Kichmarenko and Kateryna Yu. Sapozhnikova Department of Optimal Control and Economic Cybernetics, Odessa National I.I. Mechnikov University, Odessa, 6582, Ukraine olga.kichmarenko@gmail.com, kate.sapozhnikova@gmail.com Received: 14 January 215 Accepted: 1 March 215 Abstract. In this paper, existence and uniqueness of the solution of the Cauchy problem with maximum, justification of averaging method and an example are presented. Key words. Existence and uniqueness of the solution of the Cauchy problem, averaging method, differential equations with maximum, differential equations with delay. 1 Introduction Nowadays a big majority of different kinds of problems have been modeling as a systems of differential equations with maximum [5,14]. For example, this kind of equations is used in the theory of automatic control of various technical systems. It often happens that the law of regulation depends on the maximum values of some regulated state parameters over certain time intervals. In 1966, Popov [5] considered the system for regulating the voltage of a generator of constant current. The object of the experiment was a generator of constant current with parallel simulation and the regulated quantity was the voltage at the source electric current. The equation describing the work of the regulator involves the maximum of the unknown function and is as follows: T u t + ut + q max s [t h,t] us = ft, where T and q are constants characterizing the object, us the regulated voltage and fs the perturbed effect. 113

2 Full averaging scheme for differential equation with maximum Application of the averaging method for this kind of equations has been researched by Kichmarenko, Plotnikov [8], Bainov and the others [9 12]. In the present paper, theorem on the justification of the averaging method for differential equations with maximum and necessary condition for existence and uniqueness of the solution of the Cauchy problem are established. We consider the Cauchy problem with maximum on an interval with variable boundaries, thas a more general case comparing the problem in [1]. The Cauchy problem for the differential equation ẋt = f t, xt, max xτ, xt = x 1.1 τ [gt,γt] with maximum is considered. Here, xt R n is a phase vector, t [t, t 1 ] is time of the system existence, f : [t, t 1 ] R n R n R n is n-dimensional vector function, gt and γt are known functions with t gt γt t 1, and max xτ = max x 1τ,..., max x nτ. τ [gt,γt] τ [gt,γt] τ [gt,γt] Continuous function xt will be considered as a solution of problem 1.1. Note thaf gt = γt = t h, then 1.1 is a differential equation with constant delay. Moreover, if gt = γt, then 1.1 is a differential equation with variable delay. 2 Existence and uniqueness of the solution of the Cauchy problem with maximum We establish the following theorem on existence of system 1.1 thas a more general system comparing the one considered in [1]. The generality is contained in selecting of interval with variable boundaries. Theorem 2.1 Let the function ft, x, y be continuous in the neighborhood of the point t, x, x and ft, x, y ft, x, y L [ x x + y y ]. Then, the solution of Cauchy problem 1.1 exists and is unique in some neighborhood of the point t. 114

3 Olga D. Kichmarenko, Kateryna Yu. Sapozhnikova Proof. Let the function ft, x, y be defined for t t a. Problem 1.1 is associated with the integral equation It can be proved that xt = x + t F x = x + t f f τ, xτ, max xs. s [gτ,γτ] τ, xτ, max xs s [gτ,γτ] is a contraction operator in C [t,t +a] space [4]. We obtain that [ F u F v C = max [t,t +a] f τ, uτ, max us s [gτ,γτ] t ] f τ, vτ, max vs s [gτ,γτ] λ max [t,t +a] t uτ vτ + max us max vs s [gτ,γτ] s [gτ,γτ] t λa max ut vt + [t,t +a] max us max s [t,t] 2λa u v C. s [t,t] vs Here, λ R. Let λ = 1 2a, then on the interval [t, t + 1 2λ the solution of Cauchy problem 1.1 exists and unique. 3 Averaging method for differential equations with maximum Now, we consider the Cauchy problem for the differential equation. xt = εf t, xt, max xs, xt = x 3.1 s [gt,γt] with maximum and small parameter ε >. Let us consider the following averaged equation. yt = εf yt, max ys, yt = x 3.2 s [gt,γt] 115

4 Full averaging scheme for differential equation with maximum for equation 3.1. Here, f 1 T x, y = lim ft, x, ydt. 3.3 T T Note that system 3.2 is an autonomous system due to the right part of equation does not depend on t. Theorem 3.1 [2] Len Q = [, D D, D R n the following conditions hold: i ft, x, y is a continuous function on t and ft, x, y M, 3.4 ft, x, y ft, x, ȳ λ[ x x + y ȳ ], 3.5 ii gt and γt are evenly continuous functions with gt γt t, iii limit 3.3 exists evenly with respect to x, y, iv the solution of equation 3.2 for ε, ε 1 ], t, y D its ρ neighbourhood belongs to D. D together with Then, for any η >, L >, there exists ε η, L, ε 1 ] such that the following estimate holds: xt yt η, 3.6 s [gt,γt] ys where xt and yt are solutions of systems 3.1 and 3.2, accordingly x = y D. Proof. Note that f yt, max is a bounded function and satisfies the Lipschitz condition for both arguments. Using the integral equations for 3.1 and 3.2, we can write xt yt ε τ, f xτ, max f τ, yτ, max s [gτ,γτ] s [gτ,γτ] [ ] + ε f τ, yτ, max f yτ, max s [gτ,γτ] s [gτ,γτ] : = I + I. 3.7 For the uniform metric let us denote δt = max xs ys. 3.8 s [,t] 116

5 Olga D. Kichmarenko, Kateryna Yu. Sapozhnikova Then, using this denotation, Lipschitz properties of functions f f yt, max ys and 3.7, we get s [gt,γt] I ελ 2ελ [ xτ yτ + max xs s [gτ,γτ] t, xt, max xs, s [gt,γt] ] max ys s [gτ,γτ] ds δτ. 3.9 Now, we consider = i, i =, 1,..., m, m = Lε 1 and [, Lε 1 ] = m 1 [, +1 ]. Let t [t k, t k+1. Then, using the additive property of the integral, we obtain that k 1 +1 [ ] I ε f τ, yτ, max i= ys f yτ, max ys s [gτ,γτ] s [gτ,γτ] [ ] +ε f τ, yτ, max ys f yτ, max ys s [gτ,γτ] s [gτ,γτ] t k k 1 : = I i + I k. i= Let us estimate I k and I i for all i. Using the triangle inequality, we obtain that +1 [ ] I i ε f τ, y, max ys f y, max ys s [g,γ] s [g,γ] +ε +1 +ε +1 τ, f yτ, max ys f s [gτ,γτ] i= τ, y, max ys s [g,γ] yt f i, max ys f yτ, max ys s [g,γ] s [gτ,γτ] 3.1 : = J i + J i + J i Using 3.5, estimate 3.11, Theorem 3.1 i and ii, we get J i ελ +1 ελ +1 [ yτ y + max ys s [gτ,γτ] ε ε 2 λm τ 2 yx, f max s [gx,γx] ys ] max ys s [g,γ] dx + εm max{ωγ,, ωg, } + max{ωγ,, ωg, },

6 Full averaging scheme for differential equation with maximum where ωα, = sup t t αt αt is the continuity modulus of the function αt on the interval [, [6]. From the properties of the continuity modulus in paper [7], we reach k 1 L J i ελml i= λml m { ω γ, L, ω g, εm 2εm + max L + max {ω γ, L, ω g, L} 2 } L εm +ελml max {ω γ, L, ω g, L} In the similar way for derivation of estimate 3.13, one can prove that k 1 i= J i L 1 λml 2m + m + ε max {ω γ, L, ω g, L} From Theorem 3.1 i it follows that there exists a decreasing function θt such t that ε [ f τ, y, max ys f y, s [g,γ] ] max ys s [g,γ] εθ τ i θ τ i ε. Hence, for any η 1, there exists ε η 1 > such that for any ε ε η 1 the following inequality holds: t i+1 Ji ε f +ε f 2η 1. Using 3.4, is easy to see that τ, y, max ys f s [g,γ] τ, y, max ys f s [g,γ] y, max ys s [g,γ] y, max ys s [g,γ] 3.15 I k 2ML m From estimates , we conclude that I 2ML λ L2 m +2ελML max {ω γ, L, ω g, L} + 2mη 1 ε = νm, ε

7 Olga D. Kichmarenko, Kateryna Yu. Sapozhnikova For t [, τ] we can write Then, from 3.8 it follows that xt yt 2ελ δτ = max xs ys 2ελ sτ δsds + νm, ε τ 2ελ δsds + νm, ε τ δsds + νm, ε Applying the Gronwall-Bellman lemma, we have the following inequality: δt νm, εe 2ελt νm, εe 2ελL < η. Therefore, by appropriate choice of sufficiently large m and sufficiently small ε, the value νm, ε can be made as small as possible. Thus, Theorem 3.1 is proved. Note that, particularly, for 2π-periodic systems, estimation 3.6 could be xt yt Cε. 4 Example Let us consider the following illustrative example of system with small parameter and maximum: [ ] x 1 t = ε 2λx 1 sin t + max x 2s + µx 3 1 cos 3 t + x 2 sint + x 2, s [gt,γt] ] x 2 t = ε x 1 [ 2λx 1 sin t + max x 2s + µx 3 1 cos 3 t + x 2 cost + x 2 s [gt,γt] 4.1 with x 1 = 2, x 2 = π 2, gt = max{, t 1 2 }, γt = max{, t 1 4 }, λ =.7, µ =.2, ε > is the small parameter. The corresponding average system [ ] y 1 t = ε 2λy 1 cos t max y 2s, s [gt,γt] ] 4.2 y 2 t = ε y 1 [ 2λy 1 sin t max y 2s + 3 s [gt,γt] 8 µy3 1 also contains small parameter and maximum. However, is an autonomous differential equation. 119

8 Full averaging scheme for differential equation with maximum a ε =.1 b ε =.1 Figure 1: Trajectories of initial and average systems for some ε The figures of trajectories of initial and average systems for some ε are given above. In Table 1, the values max x 1 t y 1 t, max x 2 t y 2 t and max xt yt are presented. Table 1 ε max x 1 t y 1 t max x 2 t y 2 t max xt yt As we can see from Table 1 estimation 3.6 is executed. 5 Conclusion In this paper, differential equations with maximum are considered. Furthermore, a theorem on existence and uniqueness of the solution of the Cauchy problem is established. For this kind of equations with small parameter the theorem of justification of using average method is presented. In the last section, the full averaging theorem is used for specific differential equation with small parameter and maximum. 12

9 Olga D. Kichmarenko, Kateryna Yu. Sapozhnikova References [1] D. D. Bainov, S. G. Hristova, Differential Equations with Maxima, CRC Press Tylor and Francis Group, 211. [2] V. A. Plotnikov, O. D. Kichmarenko, A note on the averaging method for differential equations with maxima, Iranian Journal of Optimization [3] D. D. Bainov, H. D. Voulov, Differential Equations with Maxima, Sofia, [4] A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, [5] E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, 1966 in Russian. [6] B. Sendov, V. Popov, The Averaged Smoothness Moduli, Mir, Moscow, [7] V. A. Plotnikov, The Averaging Method in Control Oroblems, Lybid, Kiev-Odessa, 1992 in Russian. [8] V. A. Plotnikov, O. D. Kichmarenko, Averaging of differential equations with maxima, Vestnik Chernovitskogo Universiteta in Russian. [9] V. G. Angelov, D. D. Bainov, On the functional differential equations with maximums, Appl. Anal [1] D. D. Bainov, S. D. Milusheva, Justification of the averaging method for functional differential equations with maximums, Hardonic J [11] D. D. Bainov, A. I. Zahariev, Oscillating and asymptotic properties of a class of functional differential equations with maxima, Czechoslovak Math. J [12] S. Milusheva, D. D. Bainov, Justification of the averaging method for multipoint boundary value problems for a class of functional differential equations with maximums, Collect. Math [13] D. P. Mishev, Oscillatory properties of the solutions of hyperbolic differential equations with maximums, Hiroshima Math. J [14] A. R. Magomedov, Some question of differential equations with maximums, Izv. Akad. Nauk Azerb.SSR, Ser. Fiz.-Tek. i Mat. Nauk in Russian. 121

10 Full averaging scheme for differential equation with maximum [15] V. R. Petuhov, Questions of qualitative research solutions of differential equations with a maximum, Izv. Vuzov. Matem in Russian. [16] L. E. Elsgolts, S. B. Norkin, Introduction to the Theory of Differential Equations with Deviating Argument, Nauka, Moscow, 1971 in Russian. 122