A Carleman approximation scheme for a stochastic optimal nonlinear control problem

The paper investigates the optimal linear quadratic control problem in the discrete-time framework, for stochastic systems affected by disturbances generated by a nonlinear stochastic exosystem. By applying the maximum principle to nonlinear optimal control problems, it does not admit, in general, implementable solutions. Therefore, it is worthwhile to look for finite-dimensional approximation schemes. The approach followed in this paper is based on the υ-degree Carleman approximation of a stochastic nonlinear system applied to the nonlinear exosystem and provides a real-time algorithm to design an implementable control law. Simulations support theoretical results and show the improvements when the approximation index υ is increased.