Detectability, Observability and Lyapunov-Type Theorems of Linear Discrete Time-Varying Stochastic Systems with Multiplicative Noise

Abstract

The objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, ${\mathcal{K}}^{\infty }$-exact detectability (resp. ${\mathcal{K}}^{WFT}$-exact detectability, ${\mathcal{K}}^{FT}$-exact detectability, ${\mathcal{K}}^N$-exact detectability) and ${\mathcal{K}}^{\infty }$-exact observability (resp. ${\mathcal{K}}^{WFT}$-exact observability, ${\mathcal{K}}^{FT}$-exact observability, ${\mathcal{K}}^N$-exact observability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalized Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, ${\mathcal{K}}^N$-exact observability and ${\mathcal{K}}^N$-exact detectability, respectively.