On the Status of the Stability Theory of Discontinuous Dynamical Systems

Abstract

Dynamical systems can be classified in a variety of ways. Thus, when the time set \(T=\mathbb{R}^+=[0, \infty)\) we speak of a continuous-time dynamical system and when \(T=\mathbb{N}=\{0, 1, 2, \cdots\}\) we speak of a discrete-time dynamical system. When the state space \(X\) is a finite dimensional linear space, we speak of a finite dimensional dynamical system, and otherwise, of an infinite dimensional dynamical system. When all the motions in a continuous-time dynamical system are continuous with respect to time, we speak of a continuous dynamical system and when at least one of the motions in a continuous-time dynamical system is not continuous with respect to time, we speak of a discontinuous dynamical system (DDS). Continuous dynamical systems may be viewed as special cases of DDSs.