Finite-time stability of nonlinear systems with state-dependent delayed impulses
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ORIGINAL PAPER
Finite-time stability of nonlinear systems with state-dependent delayed impulses Xiaoyu Zhang · Chuandong Li
Received: 24 February 2020 / Accepted: 9 September 2020 © Springer Nature B.V. 2020
Abstract This paper addresses the issue of finitetime stability (FTS) and finite-time contractive stability (FTCS) of nonlinear systems involving statedependent delayed impulsive perturbation. Several sufficient conditions are obtained by using theories of impulsive control and Lyapunov stability. The relation between impulsive perturbation and state-dependent delay is established to achieve FTS and FTCS. For time-varying nonlinear system and nonlinear system with fixed parameters, we derive some sufficient conditions based on the main thought of this paper, respectively. Finally, three numerical examples are provided to illustrate the effectiveness and validity of achieved results. Keywords Finite-time stability · Finite-time contractive stability · State-dependent delay · Delayed impulses · Impulsive theory
1 Introduction In plenty cases of practical situations, there should consider the behavior of systems over a period of time, X. Zhang · C. Li (B) Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing 400715, People’s Republic of China e-mail: [email protected]; [email protected]
which is finite. For instance, in the process of some chemical experiments, will the pressure or humidity or other parameters be kept within suitable bounds in a fixed time interval? Furthermore, under the circumstance of launching a satellite from neighborhood of a place A to neighborhood of the destination D, will the satellite be placed into the appropriate orbit? Thus, Kamenkov [7] introduced the concept of FTS in 1953. Dorato pointed out that FTS is a much more natural concept of “stability” in contrast with classical Lyapunov stability [3]. Specifically, there are two main aspects of differences. First, FTS processes systems which operating time is limited to a finite-time interval. Second, a prescribed bound of variables is essentially required by FTS. It should be emphasized that there is another concept of FTS, which has been extensively considered in mountains of publications, e.g., [2,5,10,12,26,28,38]. The latter FTS refers to the case that states of a system converge to the equilibrium at setting time, which is finite. A great variety of research regarding FTS has been devoted to because of its wide range of applications over many practical areas. Authors in [16] investigated the problem of FTS for time-varying systems by applying the Lyapunov–Razumikhn technique to deal with the time delay. In particular, this paper deeply investigated FTS for linear time-varying systems with time-varying parameters by constructing an auxiliary function. In [26], the concept of FTS was developed into the interconnected impulsive switched system and the problem of FTS for interconnected switching sys-
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