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, K∞-exact detectability (resp. KWFT-exact detectability, KFT-exact detectability, KN-exact detectability) and K∞-exact observability (resp. KWFT-exact detectability, KFT-exact detectability, KN-exact detectability) 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 generalised Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, KN-exact observability and KN-exact detectability, respectively.