Stochastic stabilization and destabilization of nonlinear and time‐varying hybrid systems by noise

Summary The aim of this article is to design a suitable strength function g ( t , x , r ( t )) such that the Wiener noise g ( t , x ( t ), r ( t )) dw ( t ) either stabilizes or destabilizes a given nonlinear and time‐varying hybrid system . To this end, the basic properties, including the existence and uniqueness of the local and global solutions and the nonzero property of solutions of the nonlinear and time‐varying hybrid stochastic systems, are first investigated as the theoretical basis of the article. Second, two theorems and the corresponding corollaries on the stability and instability of the hybrid stochastic systems are established. Third, the design method for the noise strength g ( t , x , r ( t )) is then proposed based on the established theorems. We also point out that the Markov jump r ( t ) may have a stabilizing (respectively, destabilizing) effect when we design the noise strength g ( t , x , r ( t )) so that the introduced noise g ( t , x ( t ), r ( t )) dw ( t ) stabilizes (respectively, destabilizes) the corresponding hybrid system. Finally, we illustrate our method using two examples. Compared with the existing literature, our method is suitable for a wider class of nonlinear and time‐varying systems with weaker conditions than quasi‐linear systems.