Razumikhin-Type Stability Criteria for Differential Equations with Delayed Impulses

This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.


Introduction
During the last decades, the stability theory of impulsive delay differential systems has been undergoing fast development due to its important applications in various areas such as population management, disease control, image processing, and secure communication ( [5], [9], [13], [14], [19], [20]).For general impulsive delay differential equations, existence and uniqueness results of solutions were obtained in [2] and [8]; uniform stability and uniform asymptotic stability criteria were established in [4] and [12]; sufficient conditions on exponential stability were discussed in [1], [3], [10], [11], [15]- [18], and [20].However most of the current research on stability analysis has been focused on the impulsive delay differential equations with time delay occurred only in the differential equations.Recently, an impulsive delay differential model with delayed impulses has been investigated in impulsive synchronization of chaotic systems in secure communication where time delays appeared in both differential and difference equations of the error dynamics due to the presence of transmission delays in the process [6,7].This type of equations also have potential applications in other fields.For instance, time delays may be considered in the difference equation of the population growth model since the amount of harvesting or stocking may depend on the past and current population when the time needed for reproduction is not negligible; while in disease control models, time delays maybe be introduced into the impulses if the time needed for drugs to take effect is taken into consideration.
In this paper, we establish some global exponential stability criteria for impulsive delay differential equations with delayed impulses based on the the results and methods developed in [15]- [18].The two impulsive stabilization results released the lower bounds for the length of impulsive intervals and the result dedicated to keeping the good stability property of systems subject to certain type of impulsive perturbations is less restrictive compared to some known results in that the magnitude of impulsive disturbances may be larger than the magnitude of the states before perturbation [12].Generally speaking, the stability analysis of impulsive delay differential systems with delay in both differential and difference equations is more challenging than that of impulsive delay differential systems whose time delays only appear in the differential equations.
The rest of this paper is organized as follows.In Section 2, we introduce some notation and definitions, and then present several global exponential stability criteria for the general differential systems with delayed impulses in Section 3. Finally, some examples with numerical simulations are given to illustrate the effectiveness of our results in Section 4.
Consider the following impulsive system where F, I k : R In this paper, we assume that functions F, I k , k ∈ N satisfy all necessary conditions for the global existence and uniqueness of solutions for all t ≥ t 0 ( [2]).Denote by x(t) = x(t, t 0 , φ) the solution of (2.1) such that x t 0 = φ.We further assume without loss of generality that all the solutions x(t) of (2.1) are continuous except at t k , k ∈ N, at which x(t) is right continuous (i.e., x(t + k ) = x(t k ), k ∈ N) and the left limit x(t − k ) exists.

Definition 2.1 Function
) exists; and ii) V (t, x) is locally Lipschitzian in all x ∈ R n , and for all t ≥ t 0 , V (t, 0) ≡ 0. Definition 2.2 Given a function V : R + ×R n → R + , the upper right-hand derivative of V with respect to system (2.1) is defined by EJQTDE, 2013 No. 14, p. 3

Definition 2.3
The trivial solution of system (2.1) is said to be globally exponentially stable, if there exist some constants α > 0 and M ≥ 1 such that for any initial data 3 Lyapunov-Razumkhin method In this section, we shall present some Razumikhin-type theorems on global exponential stability for system (2.1) based on the Lyapunov-Razumikhin method and mathematical induction.Our results show that impulses play an important role in stabilizing some differential systems with delayed impulses.
Proof.Let x(t) = x(t, t 0 , φ) be a solution of system (2.1) and v(t) = V (t, x(t)).We shall show where d 0 = e 0 = 0. Let Next we show that Q(t) ≤ 0 for t ∈ [t 0 , t 1 ).To this end, we let α > 0 be any arbitrary constant and prove that Q(t) ≤ α for t ∈ [t 0 , t 1 ).Suppose not, then there exists some t ∈ [t 0 , t 1 ) so that Thus we have D + v(t * ) ≤ −cv(t * ) by condition (ii).And then we obtain which contradicts the definition of t * , and hence we have By condition (iii) with ϕ(s) = x(t − m + s) and s ∈ [−τ, 0], we have, where m ∈ N and m ≤ m.
EJQTDE, 2013 No. 14, p. 5 For any given α > 0, we show that Q(t) ≤ α for t ∈ (t m , t m+1 ).Suppose not.Let Hence for s ∈ [−τ, 0], we have Therefore, by condition (ii), we have D + v(t * ) ≤ −cv(t * ) and Again this contradicts the definition of t * , which implies Thus by the method of mathematical induction, we have By conditions (i) and (iii), we have Thus the proof is complete.
Remark 3.1.Condition (iii) of Theorem 3.1 is less restrictive compared to some known results (see [12] for example) in that it allows the solution to jump up at the impulsive moments since 1 + d k > 1 and e k > 0. This obviously cannot guarantee the stability of a delay differential system.Our result gives sufficient conditions on keeping the good stability property of the system under impulsive perturbations.EJQTDE, 2013 No. 14, p. 6 Theorem 3.2 Assume that there exist a function V ∈ ν 0 , constants p, c 1 , c 2 , λ > 0 and α > 1 such that (i) c 1 x p ≤ V (t, x) ≤ c 2 x p , for any t ∈ R + and x ∈ R n ; (ii) the upper right-hand derivative of V with respect to system (2.1) satisfies whenever qV (t, ϕ(0)) ≥ V (t + s, ϕ(s)) for all s ∈ [−τ, 0], where q ≥ αe λτ is a constant; where d k , e k (∀k ∈ N) are positive constants; (iv) Then the trivial solution of the impulsive system (2.1) is globally exponentially stable with convergence rate λ p .
By the continuity of v(t) in the interval [t m , t m+1 ), we have ; if t+s ∈ [t m , t), we again obtain that v(t + s) ≤ M φ p τ e −λ(t+s−t 0 ) from the definition of t.Therefore we have in both cases, by conditions (ii) and (iv), that Thus we have v(t + s) ≤ qv(t) for all s ∈ [−τ, 0] and t ∈ [t * , t].It follows from condition (ii) that D + v(t) ≤ 0, for t ∈ [t * , t], which implies that v(t * ) ≥ v( t), i.e., M φ p τ e −λ(t m+1 −t 0 ) ≥ M φ p τ e −λ( t−t 0 ) , which contradicts the fact that t < t m+1 .This implies that the assumption is not true, and hence (3.2) holds for k = m + 1.Thus by mathematical induction, we obtain that Hence by condition (i), we have where This implies that the trivial solution of system (2.1) is globally exponentially stable with convergence rate λ p .Remark 3.2.It is well-known that, in the stability theory of functional differential equations, the condition D + V (t, x) ≤ 0 can not even guarantee the asymptotic stability of a functional differential system (see [9,11]).However, as we can see from Theorem 3.2, impulses can contribute to the exponential stabilization a functional differential system.Theorem 3.3 Assume that there exist a function V ∈ ν 0 and constants α > τ , p, c 1 , c 2 > 0, and λ ≥ c > 0 such that , where q ≥ e λ(2α+τ ) is a constant; EJQTDE, 2013 No. 14, p. 9 where d k , e k (∀k ∈ N) are positive constants; (iv) Then the trivial solution of the impulsive system (2.1) is globally exponentially stable and the convergence rate is λ p .
For the sake of contradiction, suppose (3.16) is not true.Then we define From (3.15) and (3.17), we have v(t m ) < e −λα M φ p τ e −λ(t m+1 −t 0 ) < v( t), which implies that there exists some t * ∈ (t m , t) such that ) by the definition of t.By conditions (ii) and (iv), we have 14, p. 11 Therefore in both cases we have v(t + s) ≤ qv(t).From condition (ii), we have that D + v(t) ≤ cv(t).Since λ ≥ c, we have v( t) ≤ v(t * )e αc = e −λα M φ p τ e −λ(t m+1 −t 0 ) e αc ≤ M φ p τ e −λ(t m+1 −t 0 ) < v( t).This contradiction implies the assumption is not true.Thus (3.9) holds for k = m + 1 and by mathematical induction, we have Then by condition (i), we get where 1 p }, this implies that the trivial solution of system (2.1) is globally exponentially stable with convergence rate λ p .Remark 3.3.It is well-known that, in the stability theory of delay differential equations, the condition D + V (t, x) ≤ cV (t, x) allows the derivative of Lyapunov function to be positive which may not even guarantee the stability of a delay differential system (see [9,10,11] and Example 4.2).However, as we can see from Theorem 3.3, impulses have played an important role in exponentially stabilizing a delay differential system.
Remark 3.4.The above stabilization theorems released the lower bounds for the length of impulsive intervals as required in the stability theorems in [10], [11], [15]- [17] and therefore the conditions are less restrictive.Our results are more applicable in that they deal with systems with time delays in both states and impulses.

Examples and Simulations
In this section, we give two examples and their numerical simulations to illustrate our results.
Example 4.1 Consider the impulsive nonlinear delay differential equation with time delays in both differential and difference equations This shows condition (ii) of Theorem 3.1 holds. Moreover, We see condition (iii) of Theorem 3.1 holds with d k = 1 2 k and e k = ( 23 ) k .Thus by Theorem 3.1, the trivial solution of system (4.1) is globally exponentially stable with convergence rate 2. The numerical simulations of this example are given in Figure 4 (impulse-perturbed system) and Figure 2 (nonimpulsive system).As we can see from the simulation, the system keeps the global exponential stability property under relatively small impulsive perturbations.
Furthermore, we obtain that Thus by Theorem 3.3, we obtain that the trivial solution of (4.3) is globally exponentially stable with convergence rate 1.25.By applying the 4-step, 2nd-order Runge-Kutta method with step size 0.01, the numerical simulation of the system of delay differential equations with delayed impulses (4.3) with the initial function φ(s) = (sin(s), e −s , 1 − 2s) T for s ∈ [−0.2, 0] is given in Figure 3, the graph of solution of the corresponding system without impulse is given in Figure 4.   We note that the linear part X ′ (t) = AX(t) in the above example is unstable since all eigenvalues of A have positive real parts (l 1 = 0.143, l 2 = 0.1035 + 0.3342i, l 3 = 0.1035 − 0.3342i).As shown in Figure 4, the corresponding nonlinear system without impulses is unstable, however Figure 3 shows that it can be exponentially stabilized by impulses.
Remark 4.2.The stability theorems in [10]- [12], [15]- [18] can not apply to the above examples because their proposed Lyapunov functions or functionals do not deal with time delays at the impulsive moments.