Finite-time stability for time-varying nonlinear dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies non-uniqueness of system solutions in backward time, such systems possess non-Lipschitzian dynamics. In this paper, we address finite-time and uniform finite-time stability of time-varying systems. Specifically, we provide Lyapunov and converse Lyapunov conditions for finite-time stability of a time-varying system. Furthermore, we show that finite-time stability leads to uniqueness of solutions in forward time. In addition, we establish necessary and sufficient conditions for continuity of the settling-time function of a nonlinear time-varying system.