Original articles
Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefinite derivative

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Abstract

This paper investigates the input-to-state stability (ISS) and integral-input-to-state stability (iISS) of nonlinear impulsive systems. By using Lyapunov method involving indefinite derivative and average dwell-time (ADT) method, some sufficient conditions for ISS are obtained, where both types of impulses, stabilizing impulses and destabilizing impulses, are considered. In our approach, the time-derivative of the Lyapunov function is not necessarily negative definite, that allows wider applications than existing results in the literature. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.

Introduction

Impulsive system combines continuous and discontinuous behavior of dynamical system. The continuous dynamics are typically described by differential equations, and the discontinuous behavior in instantaneous state jumps that occur at given times, also are referred to as impulses. Earlier works [[1], [37], [10]] have successfully built a theory with formal definitions and initial assumptions on impulsive systems. Impulsive control has attracted considerable interest in biology, medicine, engineering and economics. It is much more attractive because it is discontinuous and usually has a simple structure, and only discrete control is needed to achieve the desired performance. Impulsive control is quite different from the cases of impulsive perturbation, which is a type of robustness problem. Many researchers have studied impulsive control systems in recent years; see [[17], [18], [15], [11], [12]]. For instance, impulsive control allows the stabilization of a chaotic system using only small control impulses, which could reduce the transmission of information and increase the robustness against the disturbance, see [31]. In population models [5], it is natural to release enemies at proper instances but not in a continuous release to control the population of a type of insect by using its natural enemies. Some sufficient conditions were derived in [36] for impulsive control of a class of nonlinear systems. In [16], the authors presented some sufficient conditions for stability of impulsive systems with impulses at fixed times.

When a system is affected by external inputs, it is important to guarantee the system to be input-to-state stable. The concepts of input-to-state stability (ISS) introduced by Sontag in [30] and [9] have been proved useful in this regard. Roughly speaking, the ISS property means that no matter what the size of the initial state is, the state will eventually approach a neighborhood of the origin whose size is proportional to the magnitude of the input. The analysis of ISS aims to investigate the effects of external inputs on dynamic systems. Integral-input-to-state stability (iISS), which is a natural generalization of ISS introduced in [6], is weaker than ISS. Recently, some other ISS properties have been proposed for various of systems, such as discrete systems, switched systems, and hybrid systems; see [[2], [24], [28], [19], [7], [3], [4], [35], [22], [32], [29], [38], [8]]. For example, [22] presented converse Lyapunov theorems for ISS and iISS of switched nonlinear systems; [38] studied the ISS of nonlinear systems subject to delayed impulses; [8] dealt with the ISS of discrete-time nonlinear systems; In [20], the authors investigated the input-to-state exponents and the related ISS for delayed discrete-time systems by the method of variation of parameters and introducing notions of uniform and weak uniform M-matrix. [21] investigated input-to-state-KL-stability for hybrid dynamical systems with external inputs. However, one may observe that most of them, such as those in [[22], [34], [33], [38], [8]], require the derivative of Lyapunov functions to be negative definite in order to derive the desired ISS property. Recently, [25] proposed a new approach for ISS property of nonlinear systems. It presents a new comparison principle for estimating an upper bound on the state of the nonlinear system in which the derivative of the Lyapunov function may be indefinite, rather than negative definite, which improves the previous work on this topic greatly. Because of the strong restriction of the method developed in [25], the authors in [13] improved the method and developed it to switched systems by ADT method. The authors of [26] developed the idea to delayed systems and established a class of continuously differentiable Lyapunov–Krasovskii functionals involving indefinite derivative, which generalizes the classic Lyapunov–Krasovskii functional method. However, those studies exclude impulsive structures.

Motivated by the above discussions, in this paper we shall study the ISS/iISS property for impulsive systems via Lyapunov method involving indefinite derivative. Some sufficient conditions based on ADT method are derived. The rest of this paper is organized as follows. In Section 2 the problem is formulated and some notations and definitions are given. In Section 3, we present some new characterizations of ISS/iISS based on Lyapunov method involving indefinite derivative. Examples are given in Section 4. Finally, the paper is concluded in Section 5.

Section snippets

Preliminaries

Notations. Let Z+ denote the set of positive integer numbers, R the set of real numbers, R+ the set of all nonnegative real numbers, and Rn the n-dimensional real spaces equipped with the Euclidean norm ||. ab and ab are the minimum and maximum of a and b, respectively.

Consider the following impulsive systems ẋ(t)=f(t,x(t),u(t)),ttk,x(t)=g(t,x(t),u(t)),t=tk,where kZ+, xRn is the system state, uRm is a measurable locally bounded disturbance input, tk is a strictly

Main results

In this section, we shall present some ADT results for ISS and iISS of impulsive system (1) based on Lyapunov method involving indefinite derivative.

Theorem 1

Assume that there exist functions α1,α2K, ρK, a continuous function ϕ:R+R, a locally Lipschitz function V:R+×RnR+ and constants ηR,τ>0,c0 such that for all tR+, xRn, uRm α1(|x|)V(t,x)α2(|x|);D+V(t,x)ϕ(t)V(t,x)forVρ(|u|)exp(t0tlncτ+ϕ(s)d

Applications

In this section, we present two examples to illustrate our main results.

Example 1

Consider the impulsive system (1) with f(t,x,u)=(sint0.15)x+0.05exp(cost1)u,g(t,x,u)=2x+0.1exp(cost1)u.Choose V(t,x)=|x| as ISS-Lyapunov function. It is easy to see that condition (6) holds. Let ρ(s)=|s| and ϕ(t)=sint0.1, and then when |x||u|exp(cost1), it leads to V̇(t,x)(sint0.15)|x|+0.05|x|(sint0.1)|x|ϕ(t)V(t,x),and V(g(t,x,u))2|x|+0.1|x|2.1|x|.Thus conditions (7) and (8) are satisfied

Conclusion

In this paper, we presented some new ADT-based sufficient conditions for ISS and iISS of impulsive systems via Lyapunov method involving indefinite derivative. The ISS property of the impulsive system can be guaranteed under the designed ADT scheme. Our results improved some recent work in the literature. Two examples were given to show the effectiveness and advantage of the obtained results. Since complex factors such as nonlinearities and delays exist widely in various engineering systems,

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    This work was supported by National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Excellent Youth Scholars of Shandong Province (ZR2016JL024, JQ201719). The paper has not been presented at any conference.

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