Stochastic input-to-state stability of switched stochastic nonlinear systems

In this paper, global asymptotic stability in probability (GASiP) and stochastic input-to-state stability (SISS) for nonswitched stochastic nonlinear (nSSNL) systems and switched stochastic nonlinear (SSNL) systems are investigated. For the study of GASiP, the definition which we considered is not the usual notion of asymptotic stability in probability (stability in probability plus attractivity in probability); it can depict the properties of the system quantitatively. Correspondingly, based on this definition, some sufficient conditions are provided for nSSNL systems and SSNL systems. Furthermore, the definition of SISS is introduced and corresponding criteria are provided for nSSNL systems and SSNL systems. In the proof of the above results, to overcome the difficulties coming with the appearance of switching and the stochastic property at the same time, we generalize the past comparison principle and fully use the properties of the functions which we constructed. In terms of the average dwell-time of the switching laws, a sufficient SISS condition is obtained for SSNL systems. Finally, some examples are provided to demonstrate the applicability of our results.