Fundamental Theory: Specialized Stability and Boundedness Results on Metric Spaces

Abstract

We present several important specialized stability and boundedness results for dynamical systems defined on metric spaces. It turns out that a number of the results that we will develop in the subsequent chapters concerning finite-dimensional and infinite-dimensional systems can be deduced as consequences of corresponding results of the present chapter.For autonomous and periodic dynamical systems we show that when an invariant set is stable (asymptotically stable) then it is also uniformly stable (uniformly asymptotically stable). For autonomous dynamical systems we also present necessary and sufficient conditions for the stability and the asymptotic stability of invariant sets.For dynamical systems determined by continuous-time and discrete-time semigroups defined on metric spaces we establish stability and boundedness results which comprise the LaSalle-Krasovskii invariance theory.We present for both continuous-time and discrete-time dynamical systems a comparison theory for the various Lyapunov and Lagrange stability types. This comparison theory enables us to deduce the qualitative properties of a complex dynamical system (the object of inquiry) from the qualitative properties of a simpler dynamical system (the comparison system).For general continuous-time dynamical systems we establish a Lyapunov-type result which ensures the uniqueness of motions of a dynamical system.