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, cal K∞-exact detectability (resp. cal KWFT-exact detectability, cal KFT-exact detectability, cal KN-exact detectability) and cal K∞-exact observability (resp. cal KWFT-exact observability, cal KFT-exact observability, cal KN-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 KN-exact observability and cal KN-exact detectability, respectively.