Robustness Of Exponential For Dissipativity Nonautonomous Dynamical Systems

As we know, the stability of nonautonomous systems plays an important role in the applications to the theory of control, computer, and the living creature network, etc. Correspondingly, more and more attention has been paid to the study of dynamical behavior for nonautonomous system in recent years. In comparison with autonomous case, the situation in nonautonomous systems seems to be more complicated. In fact, even if some basic concepts (such as the concepts for attractors) in the dynamical theory for nonautonomous systems are still undergoing investigations and need to be further developed.This thesis is concerned with the robustness of asymptotic stability of nonautonomous systems with respect to small time delays and perturbations. First, we give a Lyapunov characterization of exponential dissipativity for nonautonomous systems, which will be used in the proof of the later theorems. Second, the asymptotical stability, the exponentially asymptotical stability and globally exponential dissipativity of the nonautonomous systems with small time delays are studied via Lyapunov's method. It turns out that the nonautonomous systems with delays will remain exponential dissipativity provided the time lag is small enough. Finally, robustness of global exponential dissipativity of nonlinear system under perturbations is considered by using Lyapunov's method and some techniques in Uniform Gronwall Lemma. It is shown that the nonautonomous system under perturbations will remain exponential dissipativity under some types of bounded perturbations.