Stability Analysis of Impulsive Control Systems with Finite and Infinite Delays

This paper studies impulsive control systems with finite and infinite delays. Several stability criteria are established by employing the largest and smallest eigenvalue of matrix. Our sufficient conditions are less restrictive than the ones in the earlier literature. Moreover, it is shown that by using impulsive control, the delay systems can be stabilized even if it contains no stable matrix. Finally, some numerical examples are discussed to illustrate the theoretical results.


Introduction
As is known, a well-developed theory of impulsive control systems has come into existence (see [1][2][3][4][5][6][7][8] and the references therein). To our knowledge, such systems are now recognized as an excellent source of models to simulate processes and phenomena observed in control fields, such as orbital transfer of satellite [5,7], dosage supply in pharmacokinetics [9], stabilization, and synchronization in chaotic secure communication systems and other chaos systems [10,11]. Moreover, in some cases, continuous control is impossible and only impulsive control can be used. For example, a central bank cannot change its interest rate every day in order to regulate the money supply in a financial market.
On the other hand, time delay [12][13][14][15][16][17][18][19][20] occurs in many physical systems. So it is natural to study impulsive control systems with delays. Also, significant progress has been made in the theory of impulsive control systems with finite or infinite delays. Yang and Xu [21] have presented several interesting criteria on robust stability of uncertain impulsive control systems with time-varying delay by employing a formula for the variation of parameters and estimating the Cauchy matrix. Liu [22] has established several criteria on asymptotic stability of impulsive control systems with time delay by using the method of Lyapunov functions and Lyapunov functionals. By using the Razumikhin technique and Lyapunov functions, a new criterion on the uniform asymptotic stability and global stability of impulsive infinite delay differential systems has been obtained in [23]. However, to the best of our knowledge, there is no criterion on stability for impulsive control systems with delays by employing the largest and smallest eigenvalue of matrix. Hence, techniques and methods for impulsive control systems should be further developed and explored. The aim of this paper is to establish stability criteria for impulsive control systems with finite and infinite delays by employing the largest and smallest eigenvalue matrix.
Inspired by the idea in [22] dealing with impulsive control systems, some stability criteria are established. The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions. Then in Section 3, several stability criteria of impulsive control systems with finite and infinite delays by employing the largest and smallest eigenvalue of matrix are obtained. Finally, in Section 4, the advantages of the theoretical results are illustrated by two numerical examples.
Then we obtain the impulsive control system with finite delays as follows: where : [− , 0] → R is continuous. Furthermore, we can investigate the following impulsive control system with infinite delays: where : [ , 0] → R is continuous and ∫ +∞ 0 |ℎ( )| < ∞. For any matrix ∈ R × , max ( ) denotes the largest eigenvalue of and similarly min ( ) denotes the smallest eigenvalue of . The norm of matrix is ‖ ‖ = √ max ( ).
For a positive definite (symmetric) matrix ∈ R × , √ denotes the square root of , which is defined to be the unique positive definite matrix satisfying √ ⋅ √ = (see [24]).
Let us define the following class of functions for later use: is nondecreasing in } .
In order to prove our main results, we need the following definitions and lemmas.
Definition 1 (see [22]). A matrix ∈ R × is stable (or Hurwitz) if the eigenvalues of the matrix all have negative real parts and there is a unique positive definite matrix ∈ R × that solves the Lyapunov equation where is any × positive definite matrix.
Definition 2 (see [22]). A matrix ∈ R × is unstable if the eigenvalues of the matrix all have positive real parts and there is a unique positive definite matrix ∈ R × that solves the Lyapunov equation where is any × positive definite matrix.
Lemma 4 (see [25]). Assume that there exist functions , , ∈ exists. Moreover, , restricted to R + × ( ), is locally Lipschitz in and the following conditions are satisfied: Then the trivial solution of (3) is uniformly asymptotically stable.
Lemma 5 (see [25]). Assume that there exist functions , , ∈ exists. Moreover, , restricted to R + × ( ), is locally Lipschitz in and the following conditions are satisfied: Then the trivial solution of (3) is uniformly asymptotically stable.

Main Results
In this section, we will establish some stability criteria for system (3) and (4). Our first theorem provides conditions for uniform asymptotic stability of the trivial solution of (3), when matrix is Hurwitz.
Proof. Define the Lyapunov function Then, satisfies for all ∈ R . Furthermore, Since satisfies the strict inequality (8), then both equalities are satisfied by any chosen sufficiently close to . So let > satisfy The Scientific World Journal Let 1 and 2 be given by Then we have 1 > 2 , by equality (13) on . Thus, by Lemma 4, it follows that the trivial solution of (3) is uniformly asymptotically stable.
Our next theorem establishes conditions for uniform asymptotic stability of trivial solution of (3) when matrix is unstable.
Let 1 and 2 be given by Then, we have 2 > 1 , by inequality (16) on . Thus, by Lemma 5, it follows that the trivial solution of (3) is uniformly asymptotically stable.
Remark 9. Note that in Theorems 7 and 8, the criteria require the assumption that the time delay is finite, and so they may not be user-friendly in applications. Therefore, it is necessary to investigate the stability of impulsive control systems with infinite delays.

Theorem 10.
Let − −1 ≤ and assume that all the eigenvalues of have positive real parts. Suppose that there exist a constant 0 < < 1 and two symmetrical positive definite matrices , ∈ R × such that where N = ∫ +∞ 0 |ℎ( )| . Then the trivial solution of (4) is uniformly stable.
Let 1 and 2 be given by Then, we have 2 > 1 , by inequality (22) on . Thus, by Lemma 6, it follows that the trivial solution of (4) is uniformly stable.
Remark 11. In this paper, a new technique is offered to establish the stability criteria for finite and infinite delay impulsive control systems. In the above theories, rather than employing Razumikhin techniques and Lyapunov functions, we adopt the largest and smallest eigenvalue so that not only can they be easier constructed, but also the conditions ensuring the required stability are less restrictive. Furthermore, in Theorem 10, the obtained stability criterion is for impulsive control systems with both finite and infinite delays. Therefore, the result is rather general and has great power in applications.

Applications
where ℎ( ) = − , > 0 and with the following parameter matrices: Suppose in addition that the impulsive times occur with a frequency of = 0.04. Also, a simple check shows that both of these conditions are satisfied by choosing = 0.5. Therefore, uniformly stable of the trivial solution of (28) can be obtained by Theorem 10. .
Suppose in addition that the impulsive times occur with a frequency of = 0.009. Also, a simple check shows that both of these conditions are satisfied by choosing = 0.6. Therefore, robust exponential stability of the trivial solution of (28) can be obtained by Theorem 10.