Fundamental Theory: The Principal Stability and Boundedness Results on Metric Spaces

Abstract

We present the Principal Lyapunov and Lagrange Stability Results, including Converse Theorems for continuous dynamical systems, discrete-time dynamical systems and discontinuous dynamical systems defined on metric spaces. All results presented involve the existence of either monotonic Lyapunov functions or non-monotonic Lyapunov functions. We show that the results involving monotonic Lyapunov functions reduce to corresponding results involving non-monotonic Lyapunov functions. Furthermore, in most cases, the results involving monotonic Lyapunov functions are in general more conservative than the corresponding results involving non-monotonic Lyapunov functions.We present stability results (sufficient conditions) for uniform stability, local and global uniform asymptotic stability, local and global exponential stability, and instability of invariant sets. We also present Converse Theorems (necessary conditions) for most of the enumerated stability types. Furthermore, we present Lagrange stability results (sufficient conditions) for the uniform boundedness and the uniform ultimate boundedness of motions of dynamical systems, as well as corresponding Converse Theorems (necessary conditions).The results of this chapter constitute the fundamental theory for the entire book because most of the general results that we develop in the subsequent chapters concerning finite-dimensional systems and infinite-dimensional systems can be deduced as consequences of the results of the present chapter.