Hou, Ling (St. Cloud State Univ.), Michel, Anthony N. (Univ. of Notre Dame)

Stability of Continuous, Discontinuous and Discrete-Time Dynamical Systems: Unifying Results

Scheduled for presentation during the Regular Session "Stability II" (FrP07), Friday, July 28, 2006, 16:10−16:35, Room H

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 25, 2024

Abstract

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuoustime 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 in [4]), 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 in [4] are much more general than was previously known, and that the quality of the DDS results in [4] 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 stability results for DDS in [4] are much more general than previously known, having connections even with discrete-time dynamical systems. The stability results considered herein include uniform boundedness, uniform ultimate boundedness, uniform asymptotic stability in the large, exponential stability in the large, and instability. In a companion paper, we establish unifying results in the sense discussed herein for uniform stability, uniform asymptotic stability and exponential stability.