Solving Stochastic LQR Problems by Polynomial Chaos

We consider the infinite dimensional stochastic linear quadratic optimal control problem for the infinite horizon case. We provide a numerical framework for solving this problem using a polynomial chaos expansion approach. By applying the method of chaos expansions to the state equation, we obtain a system of deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of infinite horizon deterministic linear quadratic regulator problems. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to the direct approach. We perform numerical experiments which validate our approach and compare the finite and infinite horizon case.