On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

Abstract

This paper is concerned withpth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic systems with time delays. Razumikhin-type theorems ensuringp-ISS/SISS are established for the mentioned systems with external input affecting both the continuous and the discrete dynamics. It is shown that when the impulse-free delayed stochastic dynamics arep-ISS/SISS but the impulses are destabilizing, thep-ISS/SISS property of the impulsive stochastic systems can be preserved if the length of the impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics arep-ISS/SISS and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system isp-ISS/SISS regardless of how often or how seldom the impulses occur. To overcome the difficulties caused by the coexistence of time delays, impulses, and stochastic effects, Razumikhin techniques and piecewise continuous Lyapunov functions as well as stochastic analysis techniques are involved together. An example is provided to illustrate the effectiveness and advantages of our results.

On the pth moment integral input‐to‐state stability and input‐to‐state stability criteria for impulsive stochastic functional differential equations

“…Huang and Mao considered the ISS for stochastic retarded systems with Markovian switching. Wu et al and Yao et al investigated the ISS for impulsive stochastic differential systems, which extended the correspondent results in the work of Ning et al Wu et al and Yao et al studied the iISS and ISS for impulsive stochastic systems with delays by applying the Krasovskii approach and Razumikhin approach, respectively. Recently, Peng and Deng used the average dwell‐time (ADT) approach to investigate the ISS for impulsive stochastic delayed systems.…”

pth moment exponential input‐to‐state stability of nonlinear discrete‐time impulsive stochastic delay systems

“…ISS property analysis of nonlinear systems with disturbance inputs is one of active research topics in modern nonlinear control theory. Therefore, it is of great significance to study ISS for impulsive systems, see, eg, other works …”

“…Most of the existing results pertain to the study of ISS property for continuous‐time impulsive systems, see other works and the references therein, but fewer studies the ISS property for discrete‐time impulsive systems . In addition, time delays occur frequently in many real‐world phenomena.…”

“…Besides, many systems in the real world may be disturbed by stochastic factors. In the past decades, there has been an increasing interest in the ISS property analysis of continuous‐time stochastic impulsive systems, see other works and the references therein. In the work of Wu et al, by means of the Lyapunov‐Krasovskii functional and the average impulsive interval approach, the ISS, integral‐ISS, and stochastic‐ISS for continuous‐time impulsive stochastic delayed systems are considered under linear assumptions.…”

Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects

“…In the past decades, the problems of stability and impulsive stabilization for impulsive differential systems have received considerable research attention [1][2][3][4][5][6][7][8][9][10][11]. In [12][13][14][15][16][17][18][19]26,27], the exponential stability results for impulsive functional differential systems are established.…”