Approximate dynamic programming for optimal stationary control with control-dependent noise

This brief studies the stochastic optimal control problem via reinforcement learning and approximate/adaptive dynamic programming (ADP). A policy iteration algorithm is derived in the presence of both additive and multiplicative noise using It calculus. The expectation of the approximated cost matrix is guaranteed to converge to the solution of some algebraic Riccati equation that gives rise to the optimal cost value. Moreover, the covariance of the approximated cost matrix can be reduced by increasing the length of time interval between two consecutive iterations. Finally, a numerical example is given to illustrate the efficiency of the proposed ADP methodology.