Quasi-stability regions of nonlinear dynamical systems: theory

The concept of a stability region (region of attraction) of nonlinear dynamical systems is widely accepted in many fields such as engineering and the sciences. When utilizing this concept, the Lyapunov function approach has been found to give rather conservative estimations of the stability regions of many nonlinear systems. In this paper we study the notion of a quasi-stability region and develop a comprehensive theory for it. It is shown that, working on the concept of quasi-stability regions, one can greatly overcome the problem of conservative estimations of stability regions using the Lyapunov function approach. A complete characterization of quasi-stability regions is presented. Dynamical as well as topological properties of quasi-stability regions are also derived. The quasi-stability regions are shown to be robust relative to small perturbations of the underlying vector fields. The class of nonlinear dynamical systems whose stability regions equal their quasi-stability regions is characterized.