Stability

The stability problem is an essential and important problem in control theory and dynamic equations. In this chapter, the fundamental problem of stability for SMJSs with general TRMs is considered, whose TRM may be exactly, uncertain, partially unknown and designed. The conditions guaranteeing a given SMJS stochastically admissible are expressed in terms of LMIs or LMIs with equation constraints, which can be efficiently solved by using standard numerical algorithms. Especially, when TRM is given exactly, necessary and sufficient conditions with different forms are developed. Then, the robust stability of Markovian jump singularly perturbed systems with uncertain switchings and nonlinear perturbations for any perturbation parameter ? ? (0, ?¯] is solved by an LMI approach. Instead of containing ?, such conditions guaranteeing the existence and uniqueness of a solution in addition to stochastic admissibility are established by choosing an ?-dependent Lyapunov function and only depends on stability bound ?¯. It is worth mentioning that the stability results proposed in this chapter will play important roles in dealing with other problems.