Impulsive Stabilization of certain Delay Differential Equations with Piecewise Constant Argument

Transcription

1 International Journal of Difference Equations ISSN , Volume3, Number 2, pp Impulsive Stabilization of certain Delay Differential Equations with Piecewise Constant Argument Xiaodi Li Xiamen University School of Mathematical Sciences Xia men, , P.R. China Abstract In this paper, we investigate the impulsive stabilization of certain delay differential equations with piecewise constant argument by using Lyapunov function and analysis methods. Some nonimpulsive systems can be stabilized by imposition of impulsive controls. We also give an example to demonstrate the effectiveness of the proposed control and stabilization methods. AMS Subject Classifications: 34K15. Keywords: Delay differential equation, piecewise constant argument, Lyapunov function, stabilization. 1 Introduction and Preliminaries In recent years, impulsive control and impulsive stabilization for delay differential equations have attracted a great deal of attention. It arises naturally from a wide variety of applications such as threshold theory in biology, ecosystems management, and orbital transfer of satellite. Recently, various results for the impulsive stabilization of delay differential equations are obtained via different approaches, for instance, see [2, 4, 8 11]. Impulses can make unstable systems stable and, stable systems can become unstable after impulses effects [1, 3 5, 7, 8]. However, to the best of author s knowledge, there are few works on impulsive stabilization of delay differential equations with piecewise constant argument. Received April 22, 2007; Accepted April 29, 2008 Communicated by Martin Bohner

2 268 X. Li In this paper, we consider the delay differential equation rtx t σ + tx δ t [t] τ i + ptx µ t = 0, t 0, xt = φt, τ t 0, x 0 = x 0 and the corresponding equation with impulses rtx t σ + tx δ t [t] τ i + ptx µ t = 0, t t k, t 0, xt = φt, τ t 0, x 0 = x 0, xt k = I k xt k, x t k = J k x t k, k Z +, where N is any given positive integer, τ = max 1iN τ i, [t] denotes the maximum of the set of integers that are smaller than or equal to t, and Z + is the set of all positive integers. The following assumptions will be needed throughout the paper: H 1 the sequence t k satisfies 0 t 0 < t 1 <... < t k <..., lim k t k = ; H 2 I k, J k : R R are continuous functions and I k 0 = J k 0 = 0, k Z + ; H 3 r,, p : [0, R are continuous functions, i = 1, 2,..., N; H 4 σ, µ, δ 1 are constants, τ 1; H 5 x denotes the right-hand derivative of x, i.e., x t k = x t + k = lim xt k + h xt + k. h 0 + h When = 0 and µ = 1, 1.2 reduces to the differential equation rtx t σ + ptxt = 0, t t k, t t 0, xt 0 = x 0, x t 0 = x 0, xt k = I k xt k, x t k = J k x t k, k Z Some properties of 1.3 have been extensively investigated in [6]. Furthermore, if σ = 1 in 1.3, then it reduces to the differential equation rtx t + ptxt = 0, t t k, t t 0, xt 0 = x 0, x t 0 = x 0, 1.4 xt k = I k xt k, x t k = J k x t k, k Z +. Oscillatory properties of 1.4 have been discussed in [10, 12]. In the present paper, we deal with the more general equation 1.1. We shall investigate the impulsive stabilization of 1.1 by using Lyapunov function and analysis methods. Some new results

3 Delay Differential Equations with Piecewise Constant Argument 269 are obtained here. It shows that the impulses do contribute to the equation s stability behavior. In the settings of this paper, according to [9], we can obtain global existence of the solution of 1.1. So, in this paper, we always assume the solutions of 1.1 and 1.2 exist globally. Definition 1.1. A function x : [ τ, a R, a > 0 is said to be a solution of system 1.2 through 0, φ if i x and x are continuous on [ τ, a\{t k ; k Z + } and are right continuous at t k ; ii x satisfies 1.1 almost everywhere in τ, a; iii x and x fulfill the third equality of 1.2 for each k Z +. Definition 1.2. The zero solution of 1.1 is said to be exponentially stabilized by impulses, if there exists α > 0, sequences {t k } k=1, I k, J k satisfying H 1 and H 2 such that for all ε > 0, there exists δ > 0 with the property that, when the solution x of 1.1 through 0, φ fulfills φ 2 τ +x 0 2 δ, 1.5 then x2 t + x t 2 ε exp αt, t 0, 1.6 where φ τ = sup φs. τs0 Definition 1.3. The zero solution of 1.1 is said to be exponentially stabilized by periodic impulses if there exists α > 0, a sequence {t k } k=1 satisfying H 1 and t k t k 1 = d, where d > 0 is a constant, I k, J k satisfying A 2 and I k x = Ix, J k x = Jx, x R, k Z + such that for all ε > 0, there exists δ > 0 such that the solution x of 1.1 through 0, φ with 1.5 fulfills 1.6. Remark 1.4. In the present paper, for convenience, we always suppose t 0 = 0. 2 Main Results In this section we shall establish theorems which provide sufficient conditions for exponential stabilization of 1.1 by impulses. Theorem 2.1. Assume that H 3 H 5 hold. Moreover, suppose H 6 there exist constants r,, p 0, i = 1, 2,..., N such that 1 rt r, pt p, t, t 0;

4 270 X. Li H 7 we have the inequalitiy 1 τ < r exp. Then the zero solution of 1.1 can be exponentially stabilized by impulses. Proof. Since condition H 7 holds, there exists α > 0 such that r τ i rτ exp 2α exp. 2.1 Considering τ 1, we may choose a sequence {t k } k=1 = {n} n=1. It is obvious that condition H 1 is satisfied and t k+1 t k = 1 t 0 = 0. On the other hand, we can choose a sequence {η k } k=1, η k 0, 1 that is such that the solution xt = xt, k, xk, φ beginning with t = k, and x2 k + x k 2 η k satisfies x2 t + x t 2 < 1, t [k, k Then let I k u = d k u, J k v = d k v, d k = min η 1 N k expα, T r τ i r, T = exp 2α exp. With what was mentioned above, in view of 2.1, it is obvious that d k 0. For any ε 0, 1, let δ = min η 0, ε, ε exp α 1 + r2 + r exp 1 2. Next we prove that for each solution xt = xt, 0, φ of 1.2 with φ 2 τ +x 0 2 δ,

5 Delay Differential Equations with Piecewise Constant Argument 271 we have x2 t + x t 2 ε exp αt, t 0. First, for t [0, 1, we choose the Lyapunov function V t = x 2 t + r 2 tx t 2σ + r Considering condition H 6 and 2.2, we have 0 t τ i x 2δ sds. V t x 2 t + x t 2 ; V t x 2 t + r 2 x t 2σ + r x 2 t + r 2 x t 2 + r 1 + r 2 + r τ i φ 2δ τ i τ i φ 2 τ x τ i t 2 + φ τ 2 ; and when t 0, 1, if we denote by V t the right upper derivative of V t along the solution of 1.2, then in view of H 4, 2.2 and the fact that a 2 + b 2 2 ab for any a, b R, we have V t = 2xtx t + 2rtx t σ rtx t σ r = 2xtx t + 2rtx t σ r x 2δ t τ i x 2 t + x t 2 + r r x 2δ t τ i = x 2 t + x t 2 + r x 2 t + x t 2 + r x 2δ t τ i tx δ t [t] τ i ptx µ t x t 2σ + x 2δ t τ i + rp x t 2σ + x 2µ t x t 2σ + rpx t 2σ + rpx 2µ t x t 2 + rpx t 2 + rpx 2 t

6 272 X. Li x a 2 i t + x t 2 V t, which implies that V t V 0 exp t, t [0, 1. So for t [0, 1, we get x 2 t + x t 2 V t V 0 exp t V 0 exp 1 + r 2 + r 1 + r 2 + r x τ i φ 2 τ exp ε 2 exp 2α ε 2 exp 2αt, τ i δ 2 exp i.e., x2 t + x t 2 ε exp αt, t [0, 1. In particular, we get x2 1 + x 1 2 ε exp α. Consequently, it follows that x2 1 + x 1 2 = d 1 x2 1 + x 1 2 d 1 ε exp α d 1 exp α = η 1, which implies that for t [1, 2, x2 t + x t 2 <

7 Delay Differential Equations with Piecewise Constant Argument 273 For t [1, 2, we choose the Lyapunov function V t = x 2 t + r 2 t x t 2σ + r 0 Then in view of H 4 and 2.3, we can obtain that for t [1, 2, x 2 t + x t 2 V t = V 1 exp V 1 exp t 1 t 1 τ i x 2δ sds. 0 x r 2 1x 1 2σ + r x 2δ sds exp τ i { } d 2 1 x r 2 1x 1 2σ + r τ i φ 2δ τ exp { { { d 2 1r 2 x x 1 2σ + r d 2 1r 2 ε 2 exp 2α + r d 2 1r 2 + r } τ i φ 2 τ exp } τ iˆδ2 exp } τ i ε 2 exp 2α exp T ε 2 exp 2α exp ε 2 exp 4α ε 2 exp 2αt. Hence, x2 t + x t 2 ε exp αt, t [1, 2. Arguing as before by induction hypothesis, we may prove, in general, that for k 1, x2 t + x t 2 ε exp αt, t [k, k + 1. Therefore, we finally obtain x2 t + x t 2 ε exp αt, t 0. The proof is complete.

8 274 X. Li Remark 2.2. In Theorem 2.1, we can choose linear functions I k u = d k u, J k v = d k v. In fact, from the procedure of the proof of Theorem 2.1, it is not difficult to find that we only need I k u, J k v satisfying I k u d k u, J k v d k v. Remark 2.3. In the case of δ = µ = σ = 1, assume that the conditions in Theorem 2.1 still hold. Then the zero solution of 1.1 can be exponentially stabilized by periodic impulses. Proof. Here we also choose the sequence {t k } k=1 = {n} n=1, t 0 = 0. Since δ = ρ = 1, we only need to let I k u = d u, J k v = d v, d = 1 N T r τ i, T = exp 2α exp, r ε δ = min ε, 1 + r2 + r exp α exp 1 2. Then the rest of the argument is the same as was employed in the proof of Theorem 2.1. Finally we can prove that each solution xt = xt, 0, φ of 1.2 with φ 2 τ +x 0 2 δ also satisfies x2 t + x t 2 ε exp αt, t 0, where φ τ = sup φs. τs0 3 Applications We shall give an example to illustrate that the unstable system can be exponentially stabilized by impulses. Example 3.1. Consider the equation 1 + e t x t x 3 t [t] x 3 t [t] ptx 4 t = 0, t 0, xt = φt = 3 δ, 0.02 t 0, x 0 = 1, 3.1

9 Delay Differential Equations with Piecewise Constant Argument 275 where δ is a positive constant, pt Γ, Γ = {st C[0, R : st 5 }. It is not 4 difficult to prove that when pt = 0 Γ, t 0, x t = δ t δ t, t e t 2 So once δ is given, it is obvious that xt, t. Hence the nonimpulsive equations 3.1 is unstable for pt = 0. However, considering the effect of impulses, we may choose α = 0.1, τ = 0.02, = 0.125, i = 1, 2, p = 5, r = 1. Then we have 4 1 rt 2. It is easy to check that τ = 0.02 < 2 exp 4 = r 1 2 exp 2. Therefore, the hypotheses in Theorem 2.1 are satisfied and hence the unstable equations 3.1 can be exponentially stabilized by impulses for all pt Γ. Remark 3.2. In Example 3.1, we can find that the solution of 3.1 without impulses effect is unstable. However, the stability is totally controlled by the function e 0.1t under proper impulses effect, which shows that the impulses do contribute to the equation s stability behavior. Acknowledgments The author thanks the referee for careful reading of the manuscript and useful suggestions that helped to improve the presentation. References [1] D. D. Bainov and P. S. Simeonov. Systems with impulse effect. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester, Stability, theory and applications. [2] Leonid Berezansky and Elena Braverman. Impulsive stabilization of linear delay differential equations. Dynam. Systems Appl., 52: , [3] Xilin Fu and Xiaodi Li. Oscillation of higher order impulsive differential equations of mixed type with constant argument at fixed time. Math. Comput. Modelling, 485-6: , [4] Xilin Fu and Xiaodi Li. W -stability theorems of nonlinear impulsive functional differential systems. J. Comput. Appl. Math., 2211:33 46, 2008.

10 276 X. Li [5] L. P. Gimenes and M. Federson. Existence and impulsive stability for second order retarded differential equations. Appl. Math. Comput., 1771:44 62, [6] Zhimin He and Weigao Ge. Oscillations of second-order nonlinear impulsive ordinary differential equations. J. Comput. Appl. Math., 1582: , [7] Xiaodi Li. Oscillation properties of higher order impulsive delay differential equations. Int. J. Difference Equ., 22: , [8] Xinzhi Liu. Impulsive stabilization and applications to population growth models. Rocky Mountain J. Math., 251: , Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology Edmonton, AB, [9] Xinzhi Liu and George Ballinger. Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal., 514, Ser. A: Theory Methods: , [10] Zhiguo Luo and Jianhua Shen. Impulsive stabilization of functional differential equations with infinite delays. Appl. Math. Lett., 165: , [11] Aizhi Weng and Jitao Sun. Impulsive stabilization of second-order delay differential equations. Nonlinear Anal. Real World Appl., 85: , [12] Chen Yong-shao and Feng Wei-zhen. Oscillations of second order nonlinear ODE with impulses. J. Math. Anal. Appl., 2101: , 1997.