Fixed time stability and optimal stabilisation of discrete autonomous systems

Unlike finite time stability, wherein the upper bound of the settling-time function capturing the finite settling time behaviour 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 characterised by the primary and secondary branches of the Lambert W function. Using these results, we develop an optimal control framework by exploiting connections between Lyapunov theory for fixed time stability and Bellman optimal control theory. In particular, we show that fixed time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that serves as the solution to the steady state Bellman equation guaranteeing both fixed time stability and optimality.