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

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, ${\mathcal{K}}^{\mathrm{\infty }}$-exact detectability (resp. ${\cal K}^{WFT}$-exactdetectability,${\cal K}^{FT}$-exactdetectability,${\cal K}^{N}$-exactdetectability)and${\cal K}^{\infty}$-exactobservability(resp\mathrm{.}${\cal K}^{WFT}$-exactobservability,${\cal K}^{FT}$-exactobservability,${\cal K}^{N}$-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, ${\cal K}^{N}$-exactobservabilityand${\cal K}^{N}$-exact detectability, respectively.