Constructive approximate solutions to classes of state-feedback control problems for stochastic nonlinear systems

In this brief paper we propose a systematic method to construct approximate solutions to classes of state-feedback control problems involving stochastic nonlinear systems. The method avoids solving partial differential equations and instead requires only the solution of simpler algebraic equations. Exploiting a dynamic extension, we provide approximate solutions, with an exactly quantifiable level of approximation, for optimal control problems and min–max zero-sum games, and exact solutions for the $H_{\infty }$ control problem. Extensive numerical simulations demonstrate improved performances with respect to the control laws obtained via the corresponding linear quadratic approximations and deterministic nonlinear approaches.