Optimal retention of the trajectories of a discrete-time stochastic system in a tube: one problem statement

This paper considers an optimal control problem for a time-invariant linear stochastic system with discrete time, scalar unbounded control, additive noise, and a probabilistic criterion for retaining its trajectories in a given neighborhood of zero. We use dynamic programming and two-sided Bellman function estimates to derive analytical expressions for the optimal control at two time steps and a suboptimal control on any control horizon. The effectiveness of these controls is illustrated on a numerical example.