Stability of Stochastic Delay Differential Systems With Variable Impulses Due to Logic Choice

Chunxiang Li

doi:10.1109/access.2021.3085757

Abstract

This paper is concerned with stability problems of stochastic delay differential systems with variable impulses due to logic choice. Firstly, a class of variable impulses due to logic choice is introduced in this paper which is more general than the logic impulses established by (Suo and Sun, 2015) and (Zhang et al., 2018). Then, by establishing a connection between the stochastic delay differential system with logic impulses and a corresponding stochastic delay differential system without logic impulses, some sufficient conditions for stability of the systems are obtained. Finally, the application in a class of linear stochastic delay differential systems with logic impulses is discussed, and several stability criteria together with two numerical examples are given.

Highlights

It’s well known that, impulses described the abrupt changes at certain instants well and a large number of results related to impulsive system have been published both in theoretical research and practical applications in recent years, see [1]–[17] and references therein
Dynamic walking of biped robots is well researched by impulsive system, some stable characteristics of biped robots can be achieved by imposing some logic choice on the impulses
An interesting recent development of the impulsive system is the logic impulses, that is the impulsive effects are suffered by logic choice which was firstly established in [20]

Summary

INTRODUCTION

It’s well known that, impulses described the abrupt changes at certain instants well and a large number of results related to impulsive system have been published both in theoretical research and practical applications in recent years, see [1]–[17] and references therein. C. Li: Stability of Stochastic Delay Differential Systems by establishing the equivalent relation between the solutions of SDLI and SD, it is able to overcome the Ito’s formula application difficulty mentioned above that the impulses lie in the integrating intervals; (iii) the application of theorem results in a class of linear stochastic delay differential systems with logic impulses is discussed. This paper is organized as follows: in section 2, the author briefly recalls some basic notations and construct a class of linear stochastic delay differential system with variable impulses due to logic choice; in section 3, several sufficient conditions ensuring various stabilities of the SDLI and SD are provided; in section 4, the application of theorem results in a class of linear stochastic delay differential systems with logic impulses is discussed and several stability criteria together with two numerical examples are given; in section 5, concluding remarks are given

PRELIMINARIES

STABILITY CRITERIA

APPLICATION TO A CLASS OF LINEAR SDLI

CASE I

CASE II φk t

CONCLUSION