Finite-Time Stability of Discrete Autonomous Systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. In this paper, we address finite time stability of discrete-time dynamical systems. Specifically, we show that finite time stability leads to uniqueness of solutions in forward time. Furthermore, we provide Lyapunov and converse Lyapunov theorems for finite-time stability of discrete autonomous systems involving scalar difference fractional inequalities and minimum operators. In addition, lower semicontinuity of the settling-time function capturing the finite settling time behavior of the dynamical system is studied and illustrated through several examples. In particular, it is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov theorems for finite time stability of discrete-time systems can only assure the existence of lower semicontinuous Lyapunov functions.