Stability of continuous, discontinuous and discrete-time dynamical systems: unifying results

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors´ stability results for DDS (given by Ye et al., 1998), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS of Ye et al. are much more general than was previously known, and that the quality of the DDS results of Ye et al. is consistent with that of the classical Lyapunov stability results for continuous dynamical systems. By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS of Ye et al. are much more general than previously known, having connections even with discrete-dynamical systems! Most of the existing stability results for DDS have thus far been applied to finite-dimensional dynamical systems. We apply our results in the analysis of a class of infinite-dimensional dynamical systems to demonstrate their wide applicability