Finite-Time Stability of Continuous Autonomous Systems

TLDR

Finite‑time stability is defined for equilibria of continuous but non‑Lipschitzian autonomous systems. The study investigates how finite‑time‑stable systems respond to perturbations. The authors analyze continuity, Lipschitz, and Hölder properties of the settling‑time function, present Lyapunov and converse Lyapunov results via scalar differential inequalities, and examine perturbation sensitivity. They demonstrate that the regularity of the Lyapunov function is linked to that of the settling‑time function, and that converse Lyapunov theorems can only guarantee continuous Lyapunov functions.

Abstract

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.