Sufficient and Necessary Condition for the Asymptotic Stability of Stochastic Systems With Discrete Time Feedbacks and Applications

This article is devoted to reveal some essential features of stochastic control systems with sampled data (SCSwSD). First, it is shown by a proposition that the moment asymptotic stability of the underlying system is equivalent to that of any regular accurate numerical scheme under simple conditions, which is convenient to be structured specially for SCSwSD. This kind of principle provides a way for inferring moment asymptotic stability of SCSwSD by numerical simulations logically. As the first application of the proposition, the accurate scheme construction procedure is introduced in a general framework and illustrated for the quasi-linear models, the mean square asymptotic stability of linear SCSwSD is investigated. The restriction to the upper bound of the sampling period is confirmed by the way. As the second application of the proposition, the almost sure stability of a kind of controlled system with sampled noise is analyzed via the discrete scheme approach. The concepts of accurate numerical computation and simulation (ANCS) are proposed. A distinctive character, sampled data based control (SDBC) only, for SDBC is reported and studied preliminarily based on ANCS and the equivalence proposition. Some important remarks are given as further analyses on related issues. Finally, numerical examples are given to illustrate the applications of the theoretical results.