Optimal Finite Time Control for Discrete-Time Stochastic Dynamical Systems

In this paper, we address finite time stabilization in probability of discrete-time stochastic dynamical systems. Specifically, a stochastic finite-time optimal control framework is developed by exploiting connections between stochastic Lyapunov theory for finite time stability in probability and stochastic Bellman theory. In particular, we show that finite time stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic finite time stability and optimality.