Lyapunov Theorems for Finite Time and Fixed Time Semistability of Discrete-Time Stochastic Systems

In this paper, we build on the theory of semistability in probability to provide sufficient Lyapunov conditions for finite time and fixed time semistability in probability for discrete-time nonlinear stochastic dynamical systems. Specifically, we construct Lyapunov sufficient conditions for finite time semistability in probability using a Lyapunov difference operator which is a discrete-time analog of the infinitesimal generator for continuous-time Ito dynamical systems. In addition, we provide an upper bound for the average stochastic settling-time that captures the almost sure finite convergence behavior of the stochastic dynamical system. Furthermore, we develop a Lyapunov theorem for fixed time semistability in probability that guarantees the stochastic settling-time is almost surely uniformly bounded regardless of the system initial conditions. In particular, our Lyapunov theorem involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum guaranteed bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert
$W$
function. These results are then used to develop finite time and fixed time consensus protocols for discrete-time multiagent dynamical systems for achieving coordination tasks in finite and fixed time.