Probabilistic criterion-based optimal retention of trajectories of a discrete-time stochastic system in a given tube: bilateral estimation of the Bellman function

This paper examines an optimal control problem with a probabilistic criterion for retaining the trajectories of a discrete-time stochastic system in given sets. The dynamic programming method is employed for obtaining the isobells of levels 1 and 0 of the Bellman function, two-sided estimates for the right-hand side of the dynamic programming equation, two-sided estimates for the Bellman function, and the optimal-value function of the probabilistic criterion. These results are then used for deriving an approximate formula for the optimal control. As an illustrative example the problem of keeping an inverted pendulum in the neighborhood of an unstable equilibrium is considered.