Stochastic Hybrid Systems: Stabilization And Optimal Control

The term "hybrid dynamical system" (HDS) has many meanings, the most common of which is a dynamical system that involves the interaction of discrete and continuous dynamic. Such dynamical systems typically contain variables that take values from a continuous set and also variables that take values from a discrete set. However, the field of stochastic hybrid systems (SHS) is rather young. In 2000, Hu,Lygeros and Sastry proposed the concept of SHS. The systems involve a hybrid state space, with both continuous and discrete states. The continuous state obeys a stochastic differential equation (SDE) that depends on the hybrid state. Transitions occur when the continuous state hits the boundary of the state space. Since the random factors are intrinsic in real world, the SHS have wide application. Such as traffic management, computer management, high-level flexible manufacturing systems, internet management, finance decision ,society management and so on.The main contributions of this paper are as follows:In chapter 1, we introduce the main results of this paper. Meanwhile, we introduce the preliminaries which include the basic concepts and results with stochastic systems and the viscosity solution theory of partially differential equation.In chapter 2, we studied the stability and stabilization of stochastic switching systems. First we extend the Peuteman-Aeyels asymptotic stability theorem to stochastic settings; as an application, we study the mean square stabilizability of linear stochastic systems via a synchronous switched controller or via asynchronous switched controller. A synchronous switching controller consists of a class of basic controllers. The real controller is a choice within basic controllers depending on time and state of the system. The stabilization problem for the system is to design a suitable rule for switching from one basic controller to another. By S-procedure we obtain a sufficient and necessary condition for the stabilizability via synchronous switched controller for linear stochastic systems.In Chapter 3, a robust optimal control is considered for a general stochastic nonlinear hybrid systems with finitely many admissible control settings and with switching cost. With dynamic programming principle and viscosity solution theory, two basic results have been given. One is the switching lower-value function is a viscosity solution of the appropriate system of quasi-...