Fixed Time Stability of Discrete Autonomous Systems

Unlike finite time stability, wherein the upper bound of the settling-time function capturing the finite settling time behavior of the dynamical system depends on the system initial conditions, fixed time stability involves finite time stable systems for which the minimum bound of the settling-time function is guaranteed to be independent of the system initial conditions and can a priori be adjusted. In this paper, we develop several fixed time stability results for discrete autonomous systems including a fixed-time Lyapunov theorem that involves a Lyapunov difference that satisfies an exponential inequality of the Lyapunov function giving rise to a minimum bound on the settling-time function characterized by the principal and secondary branches of the Lambert W function.