The Method of Lyapunov Function and Exponential Stability of Impulsive Delay Systems with Delayed Impulses

This paper investigates the exponential stability of general impulsive delay systems with delayed impulses. By using the Lyapunov functionmethod, some Lyapunov-based sufficient conditions for exponential stability are derived, which aremore convenient to be applied than those Razumikhin-type conditions in the literature. Their applications to linear impulsive systems with time-varying delays are also proposed, and a set of sufficient conditions for exponential stability is provided in terms of matrix inequalities. Meanwhile, two examples are discussed to illustrate the effectiveness and advantages of the results obtained.

In the previous works on stability and stabilization of IDSs, the impulses are assumed to take the form of Δ(  ) =   (  , ( −  )), which indicates that the state "jump" at impulse times   is only related to the present state variables (see, e.g., [18][19][20][21][22][23][24][25][26]).But, in most cases, it is more applicable that the state variables on the impulses are also related to the past states.For example, in the transmission of the impulse information, input delays are often encountered.So, compared with the nondelayed impulses described above, it is much more meaningful to model the impulses as In fact, there have been several attempts in the literature to study the stability and control problems of a particular class of delayed impulsive systems [27,28].For example, Lian et al. [27] investigated the optimal control problem of linear continuous-time systems possessing delayed discretetime controllers in networked control systems.For nonlinear impulsive systems, Khadra et al. [28] studied the impulsive synchronization problem coupled by linear delayed impulses.By using the Razumikhin techniques, some sufficient conditions for asymptotic stability and exponential stability of general IDS-DI were established in [29][30][31][32], and sufficient conditions for exponential stability of impulsive stochastic functional differential systems with delayed impulses were obtained in [33].
In this paper, we will further investigate the stability of IDS-DI.By using the Lyapunov functions, some sufficient conditions ensuring exponential stability of IDS-DI are derived, which are more convenient to be applied than those Razumikhin-type conditions in [31,32].Their applications to linear impulsive systems with time-varying delays are also proposed, and a set of sufficient conditions for exponential stability are derived in terms of matrix inequalities.

Preliminaries
Let R denote the set of real numbers, R + the set of nonnegative real numbers, Z + the set of positive integers, and R  the -dimensional real space equipped with the Euclidean norm for all but at most a finite number of points  ∈ (−, 0]} be with the norm ‖‖ = sup −⩽⩽0 |()|, where ( + ) and ( − ) denote the right-hand and left-hand limits of function () at  respectively.
At the end of this section, let us introduce the following definitions.
Definition 3. The trivial solution of system (2) or, simply, system (2) is said to be exponentially stable if there is a pair of positive constants ,  such that, for any initial data

Main Results
In this section, we will analyze the exponential stability of system (2) by employing the Lyapunov functions.
Remark 5.The parameters  1 and  2 in condition (ii) describe the influence of impulses on the stability of the underlying continuous systems.Conditions (iv) and (v) in Theorem 4 show that the system will be stable if the impulses frequency and amplitude are suitably related to the increase or decrease of the continuous flows.
Remark 6.It is well known that the Razumikhin techniques are very effective in the study of stability problems for ordinary and functional differential systems.However, when we use the Razumikhin techniques, we need to choose an appropriate minimal class of functionals relative to which the derivative of the Lyapunov function or Lyapunov functional is estimated, which is not entirely convenient.In this sense, Theorem 4 is more convenient to be applied than those Razumikhin-type theorems in [31,32].
Then, system (21) is exponentially stable for any time delay  ∈ (0, ∞), and the convergence rate should not be greater than /, where  is the unique positive solution of  +  +  1 +  2   = 0.
In particular, if one takes   ≡  for all  ∈ Z + , then suppose the impulsive instances   satisfy For system (21), Theorem 7 yields the following result.
Proof.We just need to apply Theorem 7 with  = ln / and  = ().
Remark 9. When  > 1, the Lyapunov function  may jump up along the state trajectories of system (21) at impulse times   .Thus, the impulses may be viewed as disturbances; that is, they potentially destroy the stability of continuous system.In this case, it is required that the impulses do not occur too frequently.
Remark 10.When  < 1, the Lyapunov function  may jump down along the state trajectories of system (21) at impulse times   .Thus, the impulses may be treated as a stabilizing factor; that is, they may be used to stabilize an unstable continuous system.In this case, the impulses must take place frequently enough, and their amplitude must be suitably related to the growth rate of .
Remark 11.When  = 1, both the continuous dynamics and the discrete dynamics are stable, so the system can preserve exponential stability regardless of how often or how seldom impulses occur.

Applications and Example
Consider the following linear impulsive systems with timevarying delays: where () ∈ R  is the system state vector, , ,   , and   are  ×  matrices, and  : R + → [0, ] with  < ∞ is the time-varying delay.
Consequently, the conclusion follows from Theorem 4 immediately, and the proof is complete.
For  ̸ =   , by simple calculation, we have Remark 15.Since the system without impulses is exponentially stable and the impulses are destabilizing, the existing results in [29,32] cannot be applied to (30).The Razumikhintype theorem in [31] is also not convenient to be applied to this system since it is not easy to find an appropriate constant  > 1 to satisfy the Razumikhin-type condition.
Solving the linear matrix inequalities (24) and (25) in Theorem 12, we obtain the following feasible solution  = [ 0.3507 0.0332 0.0332 0.3076 ].Then, by Theorem 12, we know the given system is exponentially stable.
Remark 17.In Example 16, the impulses are used to stabilize an unstable system.In this case, the impulses must be frequent enough, and their amplitude must be suitably related to the growth rate of the continuous flow.

Conclusions
This paper has studied the exponential stability of impulsive delay systems in which the state variables on the impulses are related to the time delay.By using the Lyapunov function method, some criteria on the exponential stability are established.Moreover, the stability criteria obtained are applied to linear impulsive systems with time-varying delays, and a set of sufficient conditions for exponential stability is provided in terms of matrix inequalities.The obtained results improve and complement some recent works.Two examples have been given to illustrate the effectiveness and advantages of the results obtained.