Research On Stability And Applications For Nonlinear Systems With Time Delays

Time delays and nonlinearites are frequently encountered in various areas such asengineering, biology, mechanics and economics, which are often a source of instability.In view of this, stability analysis for nonlinear systems with time delays has receivedconsiderable attention by many researchers. Based on Lyapunov stability theory, thisdissertation is concerned with the stability and multiperiodicity problem of nonlinearsystems with time delays by using free weighing matrices, delay-dividing approach, M-matrix theory and nonsmooth analysis theory.The main work of this dissertation can be generalized as follows:1. The robust stability problem for uncertain neutral system and neutral systemwith nonlinear perturbations is investigated by utilizing free weighing matrices combinedwith Lyapunov-Krasovskii functional. The method employs free weighting matrices toexpress the relationships between the terms in the Leibniz-Newton formula, and to obtaindelay-dependent stability criteria. Numerical examples are given to show less conservatismcompared with some existing ones. Meanwhile, this dissertation ?rstly derives the neutral-delay dependent and discrete-delay dependent stability criterion for neutral system withmixed time-varying delays and nonlinear perturbations.2. The stability problems for uncertain neutral system, neutral system with nonlinearperturbations and Lurie control system are discussed by utilizing delay-dividing approachcombined with Lyapunov-Krasovskii functional. By dividing the whole delay intervalinto multiple segments such that each segment has a di?erent Lyapunov functional, someimproved delay-dependent stability conditions are derived. Numerical simulations aregiven to demonstrate the e?ectiveness and less conservativeness of the proposed methods.3. The stability problem for Cohen-Grossberg neural networks with time-varyingdiscrete delays and distributed delays is investigated based on M-matrix theory. Someglobal asymptotic stability criteria and the estimated location of the equilibrium points arederived without demanding the boundedness and the globally Lipschitz condition of theactivation functions. Numerical simulations are shown to demonstrate the e?ectivenessof the obtained results.4. The globally exponential stability problem for continuous-time high-order recur-rent neural network is investigated by considering variational slope of activation functions.The stability criteria are ?rst obtained for high order neural network involving externalinputs and the estimated locations of the equilibrium points. Meanwhile, the multiperi-odicity and attractivity of discrete-time high-order recurrent neural networks are studied,and the derived criteria guarantee locally and globally exponential attractiveness.