Constructive stability and asymptotic stability of dynamical systems

In an earlier paper, the authors presented an algorithm for constructing a Liapunov function for a dynamical system. In this paper, we present theorems which allow the algorithm to be used In proving the asymptotic stability of dynamical systems, both difference and differential equations. The notion of an asymptotically stable set of matrices is introduced, and is shown to be a sufficient condition for the algorithm's termination in a finite number of steps. The instability stopping criterion is strengthened and the efficiency of the algorithm is improved in a number of ways. We investigate the tightness of our method by applying it to two-dimensional systems for which necessary and sufficient conditions for stability are known.