Some Problems About Stability And Stabilization For Nonlinear Systems

During the past decades, there have been significant advances in the area of non-linear control systems. The stability and stabilization problems play an important role in the development of nonlinear control systems analysis. In this dissertation, we discuss several problems of stability and stabilization. One is uniformly ultimate boundedness and adaptive control of discontinuous time-delay systems. Second is the new problems of uniformly ultimate boundednes and uniformly asymptotic sta-bility for nonlinear systems. Third, W-stability of multi-stable discrete nonlinear systems are focused. Finally, dissipativity problems of time-delay systems are dis-cussed.Uniformly ultimate boundedness of discontinuous systems with time-delays are discussed in the sense of Filippov solutions in Chapter Two. Based on Lyapunov-Krasovskii functional, Lyapunov theorem for globally strongly uniformly ultimate boundedness of discontinuous time-delay systems is shown. Furthermore, the result is applied to a class of mechanical systems with retarded discontinuous friction item.In the third chapter, the design problems of state-feedback adaptive control are presented for discontinuous nonlinear systems with time-delay in the sense of Filip-pov solutions. Based on nonsmooth Lyapunov-Krasovskii functional and chain rule, discontinuous state feedback controllers are showed under certain conditions. By using state feedback controllers, adaptive controller and adaptive L2 controller are designed which guarantee the state of the controlled closed-loop system convergence to the origin.In Chapter Four, based on Ayels-Peuteman condition, we relax the conditions of Lyapunov function and introduce new criterions of uniformly ultimate boundedness and Matrosov theorems of uniformly asymptotic stability for nonlinear time-varying systems. Unlike classical Lyapunov theory, Lyapunov function in this chapter is not assumed to be differentiable. The given theorems can be considered as the extension of the corresponding results of classical Lyapunov theory.In the fifth chapter, based on a new type stability notion of W-stability, a new Lyapunov characterization which is an extension of conventional Lyapunov characterization is presented for multi-stable discrete-time nonlinear systems. And the estimates of solution and the Lyapunov stability theorem and converse theorem are proposed.Dissipativity problem for time-delay systems is discussed in Chapter Six. Us-ing a new quadratic supply rate, dissipativity problem of linear state/input delay systems is discussed. And passivity and finite gain are explored, correspondingly. Finally, nonlinear time-delay dissipative systems are considered.