Criteria for stabilizing a multi-delay stochastic system with multiplicative control-dependent noises

In this paper, we investigate the mean-square stabilization for discrete-time stochastic systems that endure both multiple input delays and multiplicative control-dependent noises. For such multi-delay stochastic systems, we for the first time put forward two stabilization criteria: Riccati type and Lyapunov type. On the one hand, we adopt a reduction method to reformulate the original multi-delay stochastic system to a delay-free auxiliary system and present their equivalent proposition for stabilization. Then, by introducing a delay-dependent algebraic Riccati equation (DDARE), we prove that the system under consideration is stabilizable if and only if the developed DDARE has a unique positive definite solution. On the other hand, we characterize the delay-dependent Lyapunov equation (DDLE)-based criterion, which can be verified by linear matrix inequality (LMI) feasibility test. Besides, under some restricted structure, we propose an existence theorem of delay margin and more importantly, derive an explicit formula for computing its exact value.