Unifying theory for stability of continuous, discontinuous, and discrete-time dynamical systems

We show that if the hypotheses for the classical Lyapunov stability results for continuous dynamical systems (Zubov, 1964; Hahn, 1967) (continuous-time dynamical systems whose motions are continuous with respect to time) are satisfied, then the hypotheses of the corresponding stability results for discontinuous dynamical systems (continuous-time dynamical systems whose motions are not necessarily continuous with respect to time, abbreviated DDS), reported by Ye et al. (1998), are also satisfied. We then embed discrete-time dynamical systems into a class of DDS with equivalent stability properties and we show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding stability results for DDS are also satisfied. This shows that the stability results for DDS given by Ye et al. (1998) are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems and discrete-time dynamical systems. This is demonstrated further by means of two specific examples. The results summarized above, establish a unifying stability theory for continuous dynamical systems, discrete-time dynamical systems and discontinuous dynamical systems