Continuing our effort of studying positive recurrence and ergodicity of switching diffusion processes in Chapters 3 and 4, this chapter focuses on stability of the dynamic systems described by switching diffusions. For some of the recent progress in stability analysis, we refer the reader to [48, 116, 136, 182, 183] and references therein. For treating dynamic systems in science and engineering, linearization techniques are used most often. Nevertheless, the nonlinear systems and their linearizations may or may not share similar asymptotic behavior. A problem of great interest is: If a linear system is stable, what can we say about the associated nonlinear systems? This chapter provides a systematic approach for treating such problems for switching diffusions. We solve these problems using Liapunov function methods.
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