Non-autonomous hybrid stochastic systems with delays

TL;DR

This paper investigates the behavior of non-autonomous stochastic hybrid systems with delays, establishing conditions for the existence and convergence of evolution systems of measures, with applications to control systems with time-varying delays.

Contribution

It introduces new sufficient conditions for the existence and convergence of evolution systems of measures in non-autonomous stochastic systems with delays.

Findings

Established conditions for existence of evolution systems of measures.

Proved convergence of measures as delays approach zero.

Applied theory to control systems with time-varying delays.

Abstract

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic hybrid systems with delays. By general Krylov-Bogolyubov's method, we first obtain the sufficient conditions for the existence of an evolution system of measures of the non-autonomous stochastic system and also give some easily verifiable conditions. We then prove a sufficient condition for convergence of evolution systems of measures as the delay approaches zero. As an application of the abstract theory, we first prove the existence of evolution systems of measures for stochastic system with time-vary delays, which comes from feedback control problem based on discrete-time state observations. Furthermore, when observation interval goes to zero, we show every limit point of a sequence of evolution system of measures of the non-autonomous stochastic system must be a evolution system of measures of the limiting system.