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.


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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