Stochastic Stabilization and Destabilization of Nonlinear Hybrid Sdes System with Lévy Noise by Periodic Stochastic Controls

The aim of this paper is to determine whether or not a periodic stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system with Lévy noise. The time-inhomogeneous properties of the shift and diffusion coeffcients are used to provide some suffcient conditions on the almost surely exponential stability and instability of hybrid stochastic differential equations (SDEs) systems with Lévy noise. Based on these results of stability and instability, we use the Lévy jump and Brownian motion noise as the source of noises to stabilize (or destabilize) the given unstable (or stable) hybrid ordinary di_erential equations (ODEs) systems.