Stability And Stabilization Problems Of Nonlinear Systems With Discontinuous Right-hand Sides

Nonlinear systems with discontinuous right-hand sides have been paid attention in recent years. They arise from mechanics, a lot of mathematical models in automatic control and electronic engineering are differential equations with discontinuous right-hand sides. In particular differential equations have items which are discontinuous with respect to the state variable (such as friction, viscosity, and so on). Variable structure control systems and switched systems are also systems with discontinuous right-hand sides. The control laws derived from optimal control strategy are always discontinuous, hence the closed-loop systems are discontinuous, too. In the framework of Filippov solutions and on the basis of Lipschitz continuous and regular scalar (or vector) Lyapunov functions, this dissertation discusses the stability problems of nonlinear systems with discontinuous right-hand sides with respect to a given point (equilibrium point generally) or a closed invariant set, and have some related results.The main contributions of this dissertation are as follows:In Chapter 1, we introduce the research background and present results. Meanwhile fundamental mathematical concepts are given and main results of this dissertation are shown.In Chapter 2, we define appropriate set-valued derivative for discontinuous systems. What's more, the stability problem of nonautonomous discontinuous systems is discussed. In the sense of Filippov solutions, globally uniformly asymptotical stability of nonautonomous discontinuous systems is considered, Matrosov stability theororems are given. At last the results have been applied to the tracking problem of a class of mechanical systems with discontinuous friction term.Chapter 3 mainly deals with uniformly ultimate boundedness of a class of nonautonomous systems with discontinuous right-hand sides and corresponding perturbed systems (in the sense of Filippov solutions). The definitions of globally uniformly strongly ultimate boundedness and globally uniformly weakly ultimate boundedness of discontinuous systems and the definition of globally equiuniformly strongly ultimate boundedness of corresponding perturbed systems are presented firstly. Moreover Lyapunov theorems for globally uniformly (equiuniformly) strongly ultimate boundedness of discontinuous systems (corresponding perturbed systems) are shown respectively.In Chapter 4, it is mainly discussed the stability problem of a class of nonlinear autonomous systems with discontinuous right-hand side (in the sense of Filippov solutions) based on Lipschitz continuous and regular vector Lyapunov functions. The generalized comparison principles are shown on the discontinuous autonomous systems which extends the stability results of discontinuous autonomous systems essentially. Furthermore. Lasalle invariant priciple is provided based on Lipschitz continuous and regular vector Lyapunov functions in the sense of Filippov solutions. Finally, based on two comparison systems, stability theories on the discontinuous systems are established.In Chapter 5, it is mainly discussed some stability problems with respect to a closed invariant set of a class of nonlinear systems with discontinuous right-hand side (in the sense of Filippov solution). When scalar Lyapunov function is Lipschitz continuous and regular, Matrosov theorems with respect to a closed invariant set of a class of nonlinear discontinuous systems are proposed firstly and finally, the problem of finite-time stability with respect to a closed invariant set is considered.In Chapter 6, the singular H_∞control problems for a class of linear impulsive systems, singular linear impulsive systems, linear uncertain, linear uncertain impulsive systems for given decay factor and nonlinear impulsive systems are considered respectively. When the systems don't satisfy the regular condition, sufficient conditions for the solvablitity of singular H_∞control problem for various impulsive systems are established. The control law guarantees the closed-loop systems disturbance attenation with internal stability.