A polynomial chaos approach to stochastic LQ optimal control: Error bounds and infinite-horizon results

Ruchuan Ou, Jonas Schießl, Michael Heinrich Baumann, Lars Grüne, Timm Faulwasser

Abstract: The stochastic linear–quadratic regulator problem subject to Gaussian disturbances is well known and usually addressed via a moment-based reformulation. Here, we leverage polynomial chaos expansions, which model random variables via series expansions in a suitable L2<math><msup is="true"><mrow is="true"><mi mathvariant="script" is="true">L</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math> probability space, to tackle the non-Gaussian case. We present the optimal solutions for finite and infinite horizons and we analyze the infinite-horizon asymptotics. We show that the limit of the optimal state-input trajectory is the unique solution to a corresponding stochastic stationary optimization problem in the sense of probability measures. Moreover, we provide a constructive error analysis for finite-dimensional polynomial chaos approximations of the optimal solutions and of the optimal stationary pair in non-Gaussian settings. A numerical example illustrates our findings.