Finite-time stability and stabilisation of time-varying stochastic systems with multiplicative noise and time-delays

This article investigates the problem of finite-time stability and stabilisation (FTSS) in time-varying stochastic systems with multiplicative noise and time-delays. To simplify research process, the method of dimension expansion is employed to transform systems characterised by constant delays into delay-free models, which enables the extraction of the finite-time stability (FTS) of the original systems by leveraging the FTS analysis conducted on the new model. Two distinct approaches are employed to establish criteria for determining FTSS in systems. The first approach leverages the state transition matrix (STM) method to acquire both necessary and sufficient stability conditions for the simplified models. Furthermore, sufficient conditions for the FTS of the original systems can be obtained. For ease of verification, our second approach integrates the Lyapunov equation with linear matrix inequalities (LMIs), offering a framework to obtain control gains. Finally, the proposed results are instantiated through three simulation examples to demonstrate their efficacy.