Asymptotic stability and boundedness of stochastic functional differential equations with Markovian switching

Abstract

This paper is concerned with the boundedness, exponential stability and almost sure asymptotic stability of stochastic functional differential equations (SFDEs) with Markovian switching. The key technique used is the method of multiple Lyapunov functions. We use two auxiliary functions to dominate the corresponding different Lyapunov function in different mode while the diffusion operator in different model is controlled by other multiple auxiliary functions. Our conditions on the diffusion operator are weaker than those in the related existing works.

Stability of a class of neutral stochastic functional differential equations with Markovian switching

“…One contribution of this paper is that some novel boundedness and stability criteria are first established, which extend the results in [23] from time delay to more general functional condition. The other contribution is that the results of this paper generalise the results of [21] to the neutral case. Specifically, under neutral term and Markovian switching, we discuss the stability of the system in three progressive situations, that is, the diffusion operators are controlled by two auxiliary functions at first, then the diffusion operators are controlled by multiple auxiliary functions with constant coefficients and further time‐varying coefficients.…”

“…The conditions in Theorem 2 are relaxed such that the system can be better to describe the situation in which the operators are influenced by more complicated states in the interval . Second, Theorem 2 generalises Theorem 1 in [21] by adding the neutral term which is significant to the systems depending on the changing rate of past states.…”

“…Therefore, many works have attached much importance to the stability of SFDEwMS, such as [19–24]. Specifically, Feng et al [21] used the method of multiple Lyapunov functions and investigated the boundedness, exponential stability and almost sure asymptotic stability of SFDEwMS. Feng et al [21] applied two auxiliary functions to dominate the corresponding different Lyapunov functions in different modes while the diffusion operators in different models were controlled by other multiple auxiliary functions.…”

“…Specifically, Feng et al [21] used the method of multiple Lyapunov functions and investigated the boundedness, exponential stability and almost sure asymptotic stability of SFDEwMS. Feng et al [21] applied two auxiliary functions to dominate the corresponding different Lyapunov functions in different modes while the diffusion operators in different models were controlled by other multiple auxiliary functions. In [24], robust control for a class of uncertain stochastic Markovian jump systems with interval and distributed time‐varying delays was investigated.…”

“…Specifically, under neutral term and Markovian switching, we discuss the stability of the system in three progressive situations, that is, the diffusion operators are controlled by two auxiliary functions at first, then the diffusion operators are controlled by multiple auxiliary functions with constant coefficients and further time‐varying coefficients. We use multiple Lyapunov functions, generalised Itô formula and non‐negative semi‐martingale convergence theorem to obtain the desired results.The main challenge and difficulty in this paper are not only improving the system in [21] to neutral case, but also discussing the stability results of the system in three progressive situations. Moreover, to extend Theorem 3 in [23] to more general condition, some techniques are proposed to cope with the difficulties caused by the neutral term u in the proof of Corollary 1 in this paper.…”