Input-to-state stability of hybrid stochastic systems with unbounded delays and impulsive effects

In this paper, we investigate input-to-state stability (ISS) problem for a general model of hybrid stochastic systems with unbounded delays and impulsive effects, in which stochastic disturbances involve white noise and Markov chain. Firstly, we establish a novel differential inequality with unbounded delays and variable inputs, which extends and improves some existing results. Then, by using the obtained inequality, the sufficient conditions are derived to determine ISS properties on impulsive robustness and impulsive stabilization for the impulsive stochastic systems with unbounded delays. The results not only show that the ISS properties still remain under certain impulsive perturbations for some continuous stable systems, but also indicate that an unstable system can be successfully stabilized to be input-to-state stable by impulses even if the corresponding continuous system is unstable. The obtained criteria are applied to study the ISS problems for impulsive stochastic neural networks and the systems of energy-storing electrical circuit. Finally, two numerical examples and their simulations are given to illustrate the validity of theoretical results.