On the Status of the Stability Theory of Discontinuous Dynamical Systems

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. The stability analyses of continuous dynamical systems and discrete-time dynamical systems constitute mature subjects. This is not the case for the DDS. Such systems arise in the modeling process of a variety of systems, including hybrid dynamical systems, discrete-event systems, switched systems, systems subjected to impulse effects, and the like. The qualitative analysis of such systems has been of great interest over the past decades. In this chapter we will give an overview of the stability results of the DDS with an emphasis on the authors’ work, along the lines indicated below. Due to space limitations we will present only sample results (concerning uniform asymptotic stability of invariant sets for dynamical systems defined on metric spaces). In addition to proving that our DDS stability results are in general less conservative than the corresponding classical Lyapunov stability results for continuous dynamical systems and discrete-time dynamical systems, we establish here a unifying framework for the stability analysis of continuous dynamical systems, discrete-time dynamical systems, and DDSs. Finally, we also point to several references describing applications of the results addressed herein.