On the global uniform asymptotic stability of time-varying dynamical systems

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1 Stud. Univ. Babeş-Bolyai Math ), No. 1, On the global unifom asymptotic stability of time-vaying dynamical systems Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Abstact. The objective of this wok is twofold. In the fist pat, we pesent sufficient conditions fo global unifom asymptotic stability and/o pactical stability in tems of Lyapunov-like functions fo nonlinea time vaying systems. Futhemoe, an illustative numeical example is pesented. Mathematics Subject Classification 010): 34D0, 37B5, 37B55. Keywods: Nonlinea time-vaying systems, Lyapunov function, asymptotic stability. 1. Intoduction The poblem of stability analysis of nonlinea time-vaying systems has attacted the attention of seveal eseaches and has poduced a vast body of impotant esults [1,, 13, 11] and the efeences theein. This fact motivated to study systems whose desied behavio is asymptotic stability about the oigin of the state space o a close appoximation to this, e.g., all state tajectoies ae bounded and appoach a sufficiently small neighbohood of the oigin [5] and efeences theein. Quite often, one also desies that the state appoaches the oigin o some sufficiently small neighbohood of it) in a sufficiently fast manne. To this end, the authos of [6] intoduce a concept of exponential ate of convegence and fo a specific class of uncetain systems they pesent contolles which guaantee this behavio. This popety is efeed to us as pactical stability see [4] fo the moe explanation and [3]). On the othe hand, the asymptotic stability is moe impotant than stability, also the desied system may be unstable and yet the system may oscillate sufficiently nea this state that its pefomance is acceptable, thus the notion of pactical stability is moe suitable in seveal situations than Lyapunov stability see [9], [10]). This pape aims, as fist objective, to povide sufficient conditions that ensue the global unifom pactical stability of system.1). A new quick poof fo the esults of [4] is also pesented. Next, sufficient conditions fo the GUPAS ae pesented.

2 58 Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Moeove, an example in dimensional two is given to illustate the applicability of the esult.. Definitions and tools Conside the time vaying system descibed by the following: ẋ = ft, x) + gt, x).1) whee f : R + R n R n and g : R + R n R n ae piecewise continuous in t and locally Lipschitz in x on R + R n. We conside also the associated nominal system ẋ = ft, x).) Fo all x 0 R n and t 0 R, we will denote by xt; t 0, x 0 ), o simply by xt), the unique solution of.1) at time t 0 stating fom the point x 0. Unless othewise stated, we assume thoughout the pape that the functions encounteed ae sufficiently smooth. We often omit aguments of functions to simplify notation,. stands fo the Euclidean nom vectos. We ecall now some standad concepts fom stability and pactical stability theoy; any book on Lyapunov stability can be consulted fo these; paticulaly good efeences ae [7, 8]: K is the class of functions R + R + which ae zeo at the oigin, stictly inceasing and continuous. K is the subset of K functions that ae unbounded. L is the set of functions R + R + which ae continuous, deceasing and conveging to zeo as thei agument tends to +. KL is the class of functions R + R + R + which ae class K on the fist agument and class L on the second agument. A positive definite function R + R + is one { that is zeo at the} oigin and positive othewise. We define the closed ball B := x R n : x. In ode to simplify the notation, we use the following notation. t V + x V ft, x) = V t + V x whee V : R R n R. Next, we need the definitions given below. Definition.1 unifom stability of B ). ft, x).3) 1. The solutions of system.1) ae said to be unifomly bounded if fo all α and any t 0 0 thee exists a βα) > 0 so that xt) < β fo all t t 0 wheneve x 0 < α.. The ball B is said to be unifomly stable if fo all ε >, thee exists δ := δε) such that fo all t 0 0 x 0 < δ = xt) < ε, t t 0..4) 3. B is globally unifomly stable if it is unifomly stable and the solutions of system.1) exist and ae globally unifomly bounded.

3 On the global unifom asymptotic stability 59 Definition. unifom attactivity of B ). B is globally unifomly attactive if fo all ε > and c > 0, thee exists T =: T ε, c) > 0 such that fo all t 0 0, xt) < ε t t 0 + T, x 0 < c. System.) is globally unifomly pactically asymptotically stable GUPAS) if thee exists 0 such that B is globally unifomly stable and globally unifomly attactive. Sufficient condition fo GUPAS is the existence of a class KL function β and a constant > 0 such that, given any initial state x 0, ensuing tajectoy xt) satisfies: xt) + β x 0, t), t t 0..5) If the class KL function β on the above elation.5) is of the fom β, t) = k λt, with λ, k > 0 we say that the system.) is globally unifomly pactically exponentially stable GUPES). It is also, woth to notice that if in the above definitions, we take = 0, then one deals with the standad concept of GUAS and GUES. Moeove, in the est of this pape, we study the asymptotic behavio of a small ball centeed at the oigin fo 0 xt), so that if = 0 in the above definitions we find the classical definition of the unifom asymptotic stability of the oigin viewed as an equilibium point see [8] fo moe details). In the sequel and in the ode to solve the poblem of GUAS and unifom exponential convegence to the ball B of the petubed system.1), we intoduce two technical lemmas, whee the poof of the second one is given in appendixes, that will be cucial in establishing the main esult of this wok. Lemma.3. [1] Let ϕ : [0, + ) [0, + ) be a continuous function, ε is a positive eal numbe and λ is a stictly positive eal numbe. Assume that fo all t [0, + ) and 0 u t, we have Then ϕt) ϕu) u λϕs) + ε) ds..6) ϕt) ε + ϕ0) exp λt)..7) λ Lemma.4. Let y : [0, + [ [0, + [ be a diffeentiable function, α be a class K function and c be a positive eal numbe. Assume that fo all t [0, + [ we have, ) ẏt) α yt) + c..8) Then, thee exists a class KL function β α such that ) yt) α 1 c) + β α y0), t..9) Poof. Let α of class K and c be such that.8) holds. Since α is a class K function, then thee exists some constant d such that αd) = c. Using this, one sees that ẏt) αyt)), wheneve yt) d. )

4 60 Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Let us intoduce the set S = {y : yt) d}. We claim that the set S is fowad invaiant. That is to say that if xt 0 ) S fo some t 0 0, then xt) S fo all t t 0. Indeed, suppose to the contay that thee exists a > t 0 such that ya) > d. Conside the set = {x < a R + such that yt) > d, t ]x, a[} Since t 1 and the continuity of y on R +, we have. Also, since is lowe bounded by t 0 then, m = inf is finite. We know that m, whee denotes the closue of. So that, we get by continuity ym) d, m t 0 moeove, we have yt) > d fo t ]m, a[. It follows that the inequality in the left hand side of ) holds fo each t ]m, a[, and theefoe that the absolutely continuous function yt) has a negative deivative almost eveywhee on the inteval ]m, a[. Thus ya) yt 0 ) d. This contadicts the fact that ya) > d. So S must indeed be fowad invaiant, as claimed. We continue now the poof of the Lemma. We distinguish the two possible cases: If y0) d, then yt) d fo all t 0 as claimed above. If y0) > d, then thee exists t 1 > 0 possibly t 1 = + ) such that yt) > d, fo all t [0, t 1 [ and yt) d, fo all t t 1, with the undestanding that the second case does not happen if t 1 = +. Now we apply Lemma. in [14], to obtain a class KL function β α such that yt) β α y0), t), t [0, t 1 [. So evey whee we have, ) ) yt) d + β α y0), t = α 1 c) + β α y0), t. which completes the poof. 3. Main esults In what follows, with the aid of the pevious Lemmas we give some new esults on GUAS and pactical stability Global unifom asymptotic stability In this section we suppose that the oigin x = 0 is equilibium point fo system.) and the petubation g vanishes, that is gt, 0) = 0, t 0. Conside the nonlinea system.) and intoduce a set of assumptions. Assume that: A1). Thee exists continuously diffeentiable V : [0, + [ R n [0, + [, such that c 1 x c V t, x) c x c 3.1a) t V + x V ft, x) c 3 x c x V gt, x) a x )bt) 3.1b) 3.1c)

5 On the global unifom asymptotic stability 61 whee c 1, c, c 3 and c ae stictly positive constants, a : [0, + [ R and b : [0, + [ R ae continuous functions satisfying: 0 bs) ds b lim t bt) = 0 3.) fo some constant b. A). Thee exist some constants k > 0 and 0, such that We state the following esult. a x ) k x c, fo all x. 3.3) Poposition 3.1. Unde assumptions A1) and A), the equilibium point x = 0 of.1) is globally unifomly asymptotically stable. Poof. We fist pove the fowad completeness of system.1) by poving that the finite escape time phenomenon does not occu, secondly we pove global unifom asymptotic stability in the sense of Lyapunov. No finite escape time: The time deivative of V t, x) along the tajectoies of system.1) is given by: V t, x) = t V + x V.ft, x) + x.gt, x) 3.4) Fom inequalities 3.1a), 3.1b) and 3.1c), we have xt) c 1 c 1 V 0, x0)) + 1 c 1 0 V s, xs)) ds Let and c c 1 x0) c + 1 c 1 0 α = sup a x ) x α b = sup bt). t 0 bs)a xs) ) ds. Fom 3.) and the continuity of a and b we have α and α b ae finite. It yields, by taking into account assumption A), xt) c c c 1 x0) c + α b c 1 0 α + k xs) c ) ds. An application of Gonwall s Lemma shows that the solutions exist fo all t 0. Global unifom asymptotic stability: Since lim bt) = 0, thee exists a time t 1 0, such that bt) c3 t k fo all t t 1. Hence, by 3.3), we obtain fo all x and fo all t t 1, V c 3 x c + c 3 k a x ) c 3 x c.

6 6 Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk So, V t, x) c 3 V t, x). This implies that the sphee S = {x : x } is globally attactive, that is, lim dxt), S) = 0 d is the t distance1 between x and S). Thus, boundedness of solutions follows. We invoke now Lemma.3 to show that the solution xt) goes to zeo. Put x 0 = xt 0 ). Since the solutions ae bounded, given any c > 0, thee exists β 1 > 0, such that fo all x 0 < c we have xt) < β 1 fo all t t 0. Fom now on, we fix an abitay constant c > 0 and x 0 such that x 0 < c. Define: M a = sup a x ). x <β 1 By continuity of a.) on R +, M a is finite. Obseve also that M a is independent of x 0 fo x 0 < c. Using 3.1) and 3.4), we obtain V t, x) c 3 c V t, x) + M a bt). 3.5) Let ɛ > 0 be abitay. We shall pove that thee exists a time T > 0 such that xt) < ɛ, fo all t T + t 0. Fist notice that, since lim t bt) = 0, thee exists a time T 1, such that M a bt) 1 c 3 c 1 c ɛ c, t T ) Integating inequality 3.5) fom u [T 1, t] to t T 1, on both sides of the inequality, we get ) c 3 V t, xt)) V u, xu)) V s, xs)) + ɛc c 1 ds 3.7) u c Using Lemma.3, the inequality above implies that, V t, xt)) ɛc c 1 + V T 1, xt 1 )) exp c ) 3 t T 1 ) 3.8) c ɛc c 1 + c xt 1 ) c exp c ) 3 t T 1 ) 3.9) c ɛc c 1 + c β1 c exp c ) 3 t T 1 ). 3.10) c with β 1 = xt 1 ). This is because xt) is bounded by β 1 fo all t t 0 0. On the othe hand, thee exists a time T which is independent of x 0 ), such that c β1 c exp c ) 3 t T 1 ) 1 c ɛc c 1, t T. 3.11) 1 dx, S) = inf x s s S

7 On the global unifom asymptotic stability 63 Using inequalities 3.8) and 3.11), we obtain Using the fact that, We have, V t, xt)) ɛ c c 1, t T = max T 1, T ). c 1 xt) c V t, xt)), t T + t 0. xt) ɛ, t T + t 0. Theefoe, the pesent poof is complete. Following the same analysis as above one can pove the following poposition in the case when we eplace c i x c, i = 1,, 3 by some K functions. Poposition 3.. Unde assumptions A 1) below and A), the oigin x = 0 of.1) is globally unifomly asymptotically stable equilibium point, whee A 1). Thee exists continuously diffeentiable V : R + R n R + such that α 1 x ) V t, x) α x ) t V + x V ft, x) α 3 x ) x V gt, x) a x )bt) fo some class K functions α i, i = 1,, 3 and a, b satisfying 3.). 3.. Global unifom pactical stability 3.1a) 3.1b) 3.1c) In this section, it is woth to notice that the oigin is not equied to be an equilibium point fo the system.). This may be in many situations meaningful fom a pactical point of view specially, when stability fo uncetain systems is investigated. As pointed out in [3], necessay conditions fo system.) to be globally unifomly pactically exponentially stable have been deived in [4] as follows. Theoem 3.3. [4] Conside system.). Let V : R + R n R be a continuously diffeentiable function, such that c 1 x V t, x) c x 3.13a) t V + x V ft, x) c 3 V t, x) b) fo all t 0, x R n, with c 1, c, c 3 ae positive constants and non negative constant. Then the ball B α is globally unifomly exponentially stable, with α =. The poof poposed in [4] is vey cleve but hee, using lemma.3, we can give a shote poof. Poof. Indeed, the time deivative of V along the tajectoies of system.1) is Using equation 3.13b) we get V t, x) = t V + x V ft, x). 3.14) V t, x) c 3 V t, x) )

8 64 Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Now, integating both sides the above inequality fom t 0 to u [0, t], we obtain V t, xt)) V u, xu)) By applying lemma.3 with λ = c 3 and ε =, it yields Which implies that, xt) u c 3 V s, xs)) + ) ds. 3.16) V t, xt)) c 3 + V 0, x0)) exp λt) 3.17) ) 1 V 0, x0)) exp λt) c 1 1 This completes the poof. The pevious theoem can be genealized as follows. V 0, x0)) exp λ c t), 1 1 c x0)) exp λ c t). 1 Theoem 3.4. Conside system.). Let V : R + R n R be a continuously diffeentiable function, such that c 1 x c V t, x) c x c 3.18a) t V + x V ft, x) c 3 V t, x) b) fo all t 0, x R n, with c, c 1, c, c 3 ae positive constants and a non negative constant. Then, ) 1 c 1. if c 1, the ball B α1, whee α 1 =, is globally unifomly exponentially stable. ) 1. if c 1, the ball B α, whee α = 1 c 1 c, is globally unifomly exponentially stable. Poof. Following a simila easoning as above, one can pove easily that ) 1 xt) + c c 1 1 x0) c c exp λt) c 1 c ) 1. If c 1, by using the fact that a + b) ε a ε + b ε, fo all a, b 0 and ε ]0, 1], one obtains xt) ) 1 c + c c 1 1 c 1 c c 3 x0) exp λ ) c t 3.0). If c 1. Since a + b) p p 1 a p + b p ), fo all a, b 0 and p 1, one can get the conclusion by using a simila easoning as above. This completes the poof. Anothe inteesting esult elying upon the notion of global unifom asymptotic pactical stability is the following theoem.

9 On the global unifom asymptotic stability 65 Theoem 3.5. Conside the nonlinea system.). Assume that thee exists a continuously diffeentiable function V : R + R n R satisfying α 1 x ) V t, x) α x ), 3.1a) t V + x V ft, x) α 3 V t, x)) + c 3.1b) fo some class K functions α 1, α, α 3, α 4 and c 0. Then the ball B is globally unifomly asymptotically pactically stable, whee = α1 1 α 1 3 c)). Poof. Let x 0 R n and conside the tajectoy xt) with the initial condition x0) = x 0 and define yt) = V t, xt)). Equation 3.1b) implies that ẏt) α 3 yt)) + c 3.) Hence, fom lemma.4 one one can deduce the existence of a class KL function β such that ) V t, xt)) α3 V 1 c) + β 0, x 0 ), t. 3.3) Put β d, s) = βα ), s) which is a class K function. Thus, using 3.1a), the inequality above implies that )) xt) α1 1 α3 1 c) + β d x 0, t 3.4) α1 1 )) α 1 3 c)) + α1 β 1 d x 0, t. 3.5) Hee, we use the following geneal fact, a weak fom of the tiangle inequality which holds fo any function γ of class K and any a, b 0 see Lemma 3): γa + b) γa) + γb). 3.6) It then holds that the GUPAS is fulfilled with = α1 1 α 1 3 c)) and ) )) β s, t = α1 β 1 d s, t. Lemma 3.6. let γ : [0, + [ [0, + [ a function of class K. Then, fo all x, y 0, we have: γx + y) γx) + γy). Poof. Clealy, if x = 0 o y = 0, then 3.6) holds. Let x > 0 and y > 0 and define the sets I x = [0, x], I y = [0, y] which ae compact fo evey fixed x and y since they ae closed and bounded subsets of [0, + [ R). If y x, the point y I x. As a consequence, we have γx + y) max s I x γx + s) := γ 1 x). γ 1 is well defined due to the continuity of γ and the compactness of I x. But, if s I x, we have s+x x. Since γ is a non deceasing function, then γ 1 x) = γx). If x y, the point x I y. By using the same agument as above, we conclude that γx + y) max t I y γy + t) := γ y) = γy).

10 66 Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk So, fo all x 0 and y 0, γx + y) γx) + γy). Example 3.7. We pesent now an example that implement the pevious theoem. Conside the following plana system: x1 + ɛ x1 exp x ) ) 1+x 1) ẋ = x 1 x + ɛ + at) x exp t ) x 1+x 1 whee x = x 1, x ) R, at) is a continuous bounded function and ɛ > 0. Choosing the quadatic Lyapunov function V t, x) = 1 x x. The time deivative of V along the tajectoies of the system is bounded by V t, x) V t, x) + ɛ. Let ɛ = 1, by application of theoem 3.3, this plana system is globally pactically exponentially stable. Moeove, the ball B 1 is globally unifomly pactically exponentially stable. Now, fo ɛ small enough, the state appoaches the oigin o some sufficiently small neighbohood of it) in a sufficiently fast manne. Conclusion. In this pape new sufficient conditions ae established to pove the global unifom pactical stability of a cetain class of time-vaying dynamical system. Moeove, a new poof fo the esult of [4] is pesented. The effectiveness of the conditions obtained in this pape is veified in a numeical example. Acknowledgement. The authos wish to thank the eviewe fo his valuable and caeful comments. Refeences [1] Ayels, O., Penteman, P., A new asymptotic stability citeion fo non linea time vaying diffeential equations, IEEE Tans. Aut. Cont, ), [] Bay, N.S., Phat, V.N., Stability of nonlinea diffeence time vaying systems with delays, Vietnam J. of Math, 41999, [3] BenAbdallah, A., Ellouze, I., Hammami, M.A., Pactical stability of nonlinea timevaying cascade systems, J. Dyn. Contol Sys., 15009), [4] Coless, M., Guaanteed Rates of Exponential Convegence fo Uncetain Systems, Jounal of Optimization Theoy and Applications, ), [5] Coless, M., Leitmann, G., Contolle Design fo Uncetain Systems via Lyapunov Functions, Poceedings of the 1988 Ameican Contol Confeence, Atlanta, Geogia, [6] Gaofalo, F., Leitmann, G., Guaanteeing Ultimate Boundedness and Exponential Rate of Convegence fo a Class of Nominally Linea Uncetain Systems, Jounal of Dynamic Systems, Measuement, and Contol, ), [7] Hahn, W., Stability of Motion, Spinge, New Yok, [8] Khalil, H., Nonlinea Systems, Pentice Hall, 00. [9] Lakshmikantham, V., Leela, S., Matynyuk, A.A., Pactical stability of nonlinea systems, Wold scientific Publishing Co. Pte. Ltd., 1990.

11 On the global unifom asymptotic stability 67 [10] Matynyuk, A.A., Stability in the models of eal wold phenomena, Nonlinea Dyn. Syst. Theoy, 11011), 7 5. [11] Pantely, E., Loia, A., On global unifom asymptotic stability of nonlinea time-vaying systems in cascade, Systems and Contol Lettes, ), [1] Pham, Q.C., Tabaeau, N., Slotine, J.E., A contaction theoy appoach to stochastic Incemental stability, IEEE Tansactions on Automatic Contol, 54009), [13] Phat, V.N., Global stabilization fo linea continuous time-vaying systems, Applied Mathematics and Computation, ), [14] Sontag, E.D., Smooth stabilization implies copime factoization, IEEE Tans. Autom. Contol, , Zaineb HajSalem Univesity of Sfax Faculty of Sciences of Sfax Depatment of Mathematics Stability and Contol Systems Laboatoy Mohamed Ali Hammami Univesity of Sfax Faculty of Sciences of Sfax Depatment of Mathematics Rte Souka BP1171, Sfax 3000, Tunisia MohamedAli.Hammami@fss.nu.tn Mohamed Mabouk Umm Alqua Univesity Faculty of Applied Sciences P. O. Box 14035, Makkah 1955, Saudi Aabia msmabouk@uqu.edu.sa

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

$ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},$ ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability