Stochastic stabilization and destabilization of nonlinear switched system by periodic stochastic controls based on Lévy jump noise

The aim of this paper is to determine whether or not periodic stochastic feedback controls can stabilize or destabilize a given nonlinear switched system described by Markovian switching stochastic differential equations (MS-SDEs) with Lévy jump noise. The time-inhomogeneous property of the drift, diffusion and Lévy jump coefficient functions are used to establish some sufficient conditions on almost surely exponential stability and instability of stochastic switched system described by MS-SDEs with Lévy jump noise. Based on the established results on stability and instability, we use Brownian motion and Lévy jump as noise sources to design stochastic feedback controls to stabilize or destabilize a given unstable (or stable) Markovian switching ordinary differential equations (MS-ODEs) system. Two examples and simulations are given to illustrate our theory.