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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 260236, 10 pages
Semistability of Nonlinear Impulsive Systems with Delays
1Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China
2College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
Received 22 June 2012; Revised 23 August 2012; Accepted 15 September 2012
Academic Editor: Bo Shen
Copyright © 2012 Xiaowu Mu and Yongliang Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the stability analysis and semistability theorems for delay impulsive systems having a continuum of equilibria. We relate stability and semistability to the classical concepts of system storage functions to impulsive systems providing a generalized hybrid system energy interpretation in terms of storage energy. We show a set of Lyapunov-based sufficient conditions for establishing these stability properties. These make it possible to deduce properties of the Lyapunov functional and thus lead to sufficient conditions for stability and semistability. Our proposed results are evaluated using an illustrative example to show their effectiveness.
Due to their numerous applications in various fields of sciences and engineering, impulsive differential systems have become a large and growing interdisciplinary area of research. In recent years, the issues of stability in impulsive differential equations with time delays have attracted increasing interest in both theoretical research and practical applications [1–9], while difficulties and challenges remain in the area of impulsive differential equations , especially those involving time delays . Various mathematical models in the study of biology, population dynamics, ecology and epidemic, and so forth can be expressed by impulsive delay differential equations. These processes and phenomena, for which the adequate mathematical models are impulsive delay differential equations, are characterized by the fact that there is sudden change of their state and that the processes under consideration depend on their prehistory at each moment of time. In the transmission of the impulse information, input delays are often encountered. Control and synchronization of chaotic systems are considered in [12, 13]. By utilizing impulsive feedback control, all the solutions of the Lorenz chaotic system will converge to an equilibrium point. The application of networked control systems is considered in [14–17], while in , when analyzing the asymptotic stability for discrete-time neural networks, the activation functions are not required to be differentiable or strictly monotonic. The existence of the equilibrium point is first proved under mild conditions. By constructing a new Lyapnuov-Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the discrete-time neural networks to be globally asymptotically stable. In , Razumikhin-type theorems are established which guarantee ISS/iISS for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics. It is shown that when the delayed continuous dynamics are ISS/iISS but the discrete dynamics governing the impulses are not, the ISS/iISS property of the impulsive system can be retained if the length of the impulsive interval is large enough. Conversely, when the delayed continuous dynamics are not ISS/iISS but the discrete dynamics governing the impulses are, the impulsive system can achieve ISS/iISS. In [19, 20], the authors consider linear time invariant uncertain sampled-data systems in which there are two sources of uncertainty: the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable. They model such systems as linear impulsive systems and they apply their theorem to the analysis and state-feedback stabilization. They find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed. Population growth and biological systems are considered in [21, 22]. Stochastic systems are considered in [23–25], and so forth. However, the corresponding theory for impulsive systems with time delays having a continuum of equilibria has been relatively less developed.
The purpose of this paper is to study the stability and semistability properties for nonlinear delayed impulsive systems with continuum of equilibria. Examples of such systems include mechanical systems having rigid-body modes and isospectral matrix dynamical systems . Such systems also arise in chemical kinetics, compartmental modeling, and adaptive control. Since every neighborhood of a nonisolated equilibrium contains another equilibrium, a nonisolated equilibrium cannot be asymptotically stable. Thus asymptotic stability is not the appropriate notion of stability for systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. More precisely, an equilibrium is semistable if it is Lyapunov stable, and every trajectory starting in a neighborhood of the equilibrium converges to a (possibly different) Lyapunov stable equilibrium. It can be seen that, for an equilibrium, asymptotic stability implies semistability, while semistability implies Lyapunov stability. We will employ the method of Lyapunov function for the study of stability and semistability of impulsive systems with time delays. Several stability criteria are established. A set of Lyapunov-based sufficient conditions is provided for stability criteria, then we extend the notion of stability to develop the concept of semistability for delay impulsive systems. Finally, an example illustrates the effectiveness of our approach.
Let denote the set of positive integer numbers. Let denote the set of piecewise right continuous functions with the norm defined by . For simplicity, define , for . For given , if , then for each , we define by and , respectively. A function is of class , if is continuous, strictly increasing, and . For a given scalar , let .
Let be an open set and for some . Given functionals , satisfying . Considering the following nonlinear time-delay impulsive system described by the state equation where is the system state, denotes the right-hand derivative of , and denote the limit from the right and the limit from the left at point , respectively. is the initial time. Here we assume that the solutions of system are right continuous, that is, . is a strictly increasing sequence of impulse times in where .
Definition 2.1. The function is said to be composite-PC, if for each and , and is continuous at each in , then the composite function .
Definition 2.2. The function is said to be quasi-bounded, if for each , , and for each compact set , there exists some , such that for all .
Definition 2.3. The function with is said to be a solution of if(i) is continuous at each in ;(ii)the derivative of exists and is continuous at all but at most a finite number of points in ;(iii)the right-hand derivative of exists and satisfies (2.1) in , while for each , (2.2) holds;(iv)Equation (2.3) holds, that is, .
We denote by (or , if in not confusing) the solution of . is said to be a solution defined on if all above conditions hold for any .
We make the following assumptions on system .(A1) is composite-PC, quasi-bounded and locally Lipschitzian in .(A2) For each fixed , is a continuous function of on .
Under the assumptions above, it was shown in  that for any , system admits a solution that exists in a maximal interval ) and the zero solution of the system exists.
Definition 2.4. An equilibrium point of is a point satisfying for all where is the solution of . Let denote the set of equilibrium points of .
Definition 2.5. Consider the delay impulsive system .(i)An equilibrium point of is Lyapunov stable if for any there exists , such that implies for all , where is the initial function for . An equilibrium point is uniformly Lyapunov stable, if, in addition, the number is independent of .(ii)An equilibrium point of is semistable if it is Lyapunov stable and there exists an open subset of containing such that for all initial conditions in the trajectory of converges to a Lyapunov stable equilibrium point, that is, , where is a Lyapunov stable equilibrium point.(iii)System is said to be uniformly asymptotically stable in the sense of Lyapunov with respect to the zero solution, if it is uniformly stable and .
Definition 2.6. The 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 , and for all , .
Definition 2.7. Let . For any , the upper right-hand derivative of with respect to system is defined by
3. Main Results
In the following, we will establish several sufficient conditions for Lyapunov stability and semistability for impulsive differential system with time delays.
Theorem 3.1. System is uniformly stable, and the zero solution of is asymptotically stable if there exists a Lyapunov function which satisfies the following.(i) such that (ii)For any , and , there exists , such that (iii)There exist a and a subsequence of the impulsive moments such that (iv)For any , there exists a function , such that (v)For any , there exists a function such that
Proof. Let be an equilibrium point of the system . We first prove that is uniformly stable, that is, for , there exists such that implies for all .
For , let such that For any , by condition (3.4), we get
By (3.3), it is clear that is nonincreasing along the subsequence , so we have
For any , by (3.5), we get Combining (3.7), (3.8), and (3.9), we conclude that By condition (3.2), for any , we have and then, by (3.10), for any we derive that . Hence, by (3.1) we obtain that . Since , we get which implies that system is uniformly Lyapunov stable.
Next, we will prove that the zero solution of is asymptotically stable.
Since system is uniformly stable, from (3.1), there must exist a real number such that . Hence, there exists a such that
In the following, we will show that . Without loss of generality, we can suppose that there exists a sequence , such that From (3.3) we get Since , we obtain If the sequence is the same as the sequence , then it is obvious that . If , it follows from the assumptions above that (3.16) holds. Otherwise, we assume that ; there exists a such that . Then from condition (3.5) we get So which implies .
Hence, we derive that . Finally, by (3.1), we have which implies that the zero solution of the system is asymptotically stable. The proof is completed.
Next, we present a sufficient condition for semistability for system .
Theorem 3.2. Consider the system ; assume that there exists nonnegative-definite continuous function such that Let . If every equilibrium point of system is Lyapunov stable, then every point in is semistable.
It follows from (3.19) and (3.3) that
Since is nonnegative, it follows that . Next, we show that as .
If it is not true, then there exists and an infinite sequence of times such that . By definition of we have that does not belong to the set of impulsive times .
Note that from (3.19), it follows from Proposition 3.1 of  that is bounded for all . Hence, it follows from the Lipschitz continuity of that is bounded for all ; thus, is uniformly continuous on. So, there exists such that every is contained in some interval of of length on which . This contradicts . Hence as . It follows that as . Since is bounded, we get (as ).
Next, let . For every open neighborhood and , , it follows from Proposition 5.1 of  that there exists such that . Since every point in is Lyapunov stable, and hence is a Lyapunov stable equilibrium of , it follows that is semistable. Finally, since is arbitrary, this implies every point in is semistable. The proof is completed.
4. Numerical Example
In this section, we give an example about compartmental systems to illustrate the effectiveness of the proposed method. Compartmental systems involve dynamical models that are characterized by conservation laws (e.g., mass and energy) capturing the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous, that is, any material entering the compartment is instantaneously mixed with the material of the compartment.
Example 4.1. Consider the nonlinear two-compartment time-delay impulsive systems given by
where . Let Lyapunov function , then for any we have
Let , , , and , then the conditions of Theorem 3.1 are satisfied, which means the equilibrium points of the system are Lyapunov stable, and
Let then we derive that ; it follows from Theorem 3.2 that every point in is semistable.
The simulation result is depicted in Figure 1, where the length of the impulsive intervals is second and the time delay second.
The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions that improved the overall quality of this paper. This work was supported by Grants of NSFC-60874006.
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