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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:
34D30Structural stability of ODE and analogous concepts
34D20Stability of ODE
93D05Lyapunov and other classical stabilities of control systems
34A36Discontinuous equations
93B35Sensitivity (robustness) of control systems
37C20Generic properties, structural stability