Randomly switching systems: Models, analysis, and applications

This work provides a survey on some of the recent progress of switching diffusion systems. In recent years, switching diffusion systems have gained much popularity owing to their flexibility in modeling and their nature to conveniently depict the coexistence of continuous dynamics and discrete events. In this paper, we begin with a number of motivating examples to display a variety of applications that can be covered by switching diffusions. Then we study several important properties of the underlying systems. First weak stability is treated, and then ergodicity is considered, which provides us with a useful tool to replace the time-varying system measures by an ergodic or limit measure. Stability for equilibria is also examined. For the totally degenerated diffusions (i.e., no Gaussian noise case), we are dealing with switched ordinary differential equations. A somewhat surprising discovery is an insight different from the well-know Hartman-Grobman theorem regarding linearization. Numerical approximation for the solution of controlled switching diffusions are also considered.