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Exponential Stability of Impulsive Delay Differential Equations
The main objective of this paper is to further investigate the exponential stability of a class of impulsive delay differential equations. Several new criteria for the exponential stability are analytically established based on Razumikhin techniques. Some sufficient conditions, under which a class of linear impulsive delay differential equations are exponentially stable, are also given. An Euler method is applied to this kind of equations and it is shown that the exponential stability is preserved by the numerical process.
Impulsive differential equations arise widely in the study of medicine, biology, economics, engineering, and so forth. In recent years, theory of impulsive differential delay equations (IDDEs) has been an object of active research (see [1–18] and references therein). The results about the existence and uniqueness of IDDEs have been studied in [2, 7]. The stability of IDDEs has attracted increasing interest in both theoretical research and practical applications (see [1, 3, 5–18] and references therein). In particular, special attention has been focused on exponential stability of IDDEs (see [1, 3, 8, 9, 15]) because it has played an important role in many areas.
There is a little work done on exponential stability for IDDEs by the Lyapunov-Razumikhin method. Wang and Liu  have extended Lyapunov-Razumikhin method to IDDEs and established some exponential stability criteria. In this paper we restrict the length of each impulsive interval instead of some conditions in . As a result, several new criteria on exponential stability are analytically derived.
There are a few papers on numerical methods of impulsive differential equations. In , Covachev et al. obtained a convergent difference approximation for a nonlinear impulsive ordinary system in a Banach space. In [20, 21], the authors studied the stability of Runge-Kutta methods for linear impulsive ordinary differential equations. In , Ding et al. studied the convergence property of an Euler method for IDDEs. In , asymptotic stability of numerical solutions and exact solutions of a class of linear IDDEs was studied by the property of DDEs without impulsive perturbations. The convergence of the numerical methods for this kind of equations was studied. In this paper, we study exponential stability of the numerical solutions of linear IDDEs.
The rest of the paper is organized as follows. In Section 2, we obtained two criteria on exponential stability for IDDEs by the Lyapunov-Razumikhin method. The results obtained are applied to a class of linear IDDEs. In the last section, we prove that the Euler method for the linear IDDEs preserves the analytic exponential stability.
2. Stability of Analytic Solutions
Consider the impulsive delay differential system where ; ; , with as ; is a set of piecewise continuous functions which have a finite number of points of discontinuity in a finite interval and for all . We assume that , , so that is a solution of (1) as , which we call the zero solution.
Definition 1 (see ). A function : is said to belong to the class if(i) is continuous in each of the sets and for each , , exists, ;(ii) is locally Lipschitzian in all , and for all .
Theorem 4. Assume that there exist a function ; constants , ; positive constants , , , ; and a function with , such that for any with the following conditions hold:(i) for all , ;(ii) for all , whenever for ;(iii) for ;(iv) for , and , where .
Then the zero solution of (1) is exponentially stable.
Remark 5. Theorem 3.1 in  requires that , and , which implies . In our Theorem 4, we require for , and instead. This means that the impulsive effects are bounded instead of tending to zero (see Example 13).
Theorem 6. Assume that there exist a function ; constants , ; positive constants , , , , ; and a function with , such that for any with , the following conditions hold:(i) for all , ;(ii) for all , , whenever for , where is a constant and , and is the smallest integer larger than or equal to ; (iii);(iv) for , and , where , .
Then the zero solution of (1) is exponentially stable.
Proof. Let be the solution of system (1) and . We will prove
Let We need to show that for all . It is clear that for , since by condition (i).
Next we shall show , for . Suppose this is not true. Then there is a such that , , , and Note that . Then for . By condition (ii), . So which contradicts (8). Hence , for all .
Assume that , for . We shall show that , for . Obviously, by condition (iii) Suppose that there exists a such that and . There is a such that , , , and
Since , then for any , we have Thus by condition (ii), , then which contradicts (11). Hence for all . By induction, for all . In view of for all , we obtain for , . Since , there exists such that and . By condition (i) where . Hence the zero solution of (1) is exponentially stable.
Remark 7. Theorem 6 says that the delay differential equation is unstable and the suitable impulse effects are given, then it will become stable (see Example 14). Compared with the Theorem 3.1 in , we do not require that . For example, by Theorem 6 we know that the zero solution of the following system is exponentially stable:
In the following we consider where , and are constants.
Theorem 8. Assume that , and there is a constant such that and , where . Then the zero solution of (17) is exponentially stable.
Proof. Assume that .(i)Obviously, there exist , such that .(ii) Assume that for all . For any , if
we have . For , we have
(iii) Suppose that . Hence
(iv) Obviously, and .
By Theorem 4, the zero solution of (11) is exponentially stable.
Similarly, by Theorem 6 we have the following theorem.
Theorem 9. Assume that , , and there are constants and such that , , and , where , . Then the zero solution of (17) is exponentially stable.
3. The Euler Method for Linear IDDEs
In this section, we consider the exponential stability of the Euler method for (17). The convergence property can be proved similarly to . The Euler method for (17) with initial function is given by where . Let , then is an approximation for the exact solution for , and is an approximation for .
Definition 10. The Euler method for (11) is said to be exponentially stable if there exist positive constants , and , for any , such that for , , and .
The following theorem indicates that the Euler method preserves the property of exact solutions which was obtained above.
Proof. If , then . (i) If , we want to prove that
for , . Obviously, . Firstly, we consider the case and . Because and , so . Hence we have
Because , we have . By the inequality holding for all , we get .
Assume that for . Then So (22) holds for , . Suppose that (22) holds for , . Next, we shall prove (22) holds, when , . Hence Assume that for . Then Hence (22) holds. Since , there exists such that and . Hence (ii)If , we can prove that Consequently, the theorem holds.
Example 13. Consider the system Obviously, satisfies the Theorem 8. Therefore the zero solution of (29) is exponentially stable. By Theorem 11, the Euler method for (29) is also exponentially stable (see Figure 1).
Example 14. Obviously, the zero solution of the system is unstable (see Figure 2) while the zero solution of the following system is exponentially stable by Theorem 9 with . By Theorem 12, the Euler method for (31) is also exponentially stable (see Figure 3).
The authors wish to thank referees for valuable comments. The research was supported by the NSF of China no. 11071050.
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