Stability and stabilization using discrete-time feedback control for hybrid stochastic delay systems with general delay

This paper develops stability and stabilization for hybrid stochastic differential delay equations (SDDEs) with general time-variable delays using the Halanay-type inequalities. First, a right-continuous version of the Halanay inequality is proved using methods different from the usual approach of proof by contradiction method. Because the Halanay inequality does not require much on the delay, boundedness and stability in mean square of hybrid SDDEs with general time delays are obtained. For stability not focusing on the equilibrium points, asymptotic stability in distribution is an appropriate criterion and has been studied extensively. For this type of stability, it is crucial to use time-homogeneous Markov processes. In this work, the problem is examined by treating delays that behave periodically. The proof is conducted using probabilistic argument, segment processes, and weaker conditions than that of the moment conditions. For a given hybrid SDDE being unstable in distribution, a feedback control based on discrete-time observations is constructed to stabilize the underlying system. For the controlled systems (hybrid SDDEs with non-constant delays), time homogeneous Markov processes are also identified. The use of Halanay inequality enables us to obtain the upper bound of observation duration using linear equations rather than the cumbersome exponential equations. Finally, two examples are given to demonstrate the effectiveness of our theory. It is shown that a better bound on observation duration can be achieved compared with the existing results.