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Finite-time stability of continuous autonomous systems. (English) Zbl 0945.34039

Summary: Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Hölder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

MSC:

34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34A36 Discontinuous ordinary differential equations
93B35 Sensitivity (robustness)
37C20 Generic properties, structural stability of dynamical systems
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