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Unifying theory for stability of continuous, discontinuous, and discrete-time dynamical systems. (English) Zbl 1117.93051
Summary: 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 H. Ye, A. N. Michel and L. Hou, IEEE Trans. Autom. Control 43, No. 4, 461–474 (1998; Zbl 0905.93024)], 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 by H. Ye, A. N. Michel and L. Hou [loc. cit.] are much more general than was previously known, and that the quality of the DDS results therein 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 we have equivalent stability properties! Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C55 Discrete-time control/observation systems
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