Hybrid Gauss Pseudospectral and Generalized Polynomial Chaos Algorithm to Solve Stochastic Optimal Control Problems

Gerald C. Cottrill and Frederick G. Harmon


Optimal Control, Pseudospectral Methods, Stochastic Collocation, Generalized Polynomial Chaos


A numerical algorithm combining the Gauss Pseudospectral Method (GPM) with a Generalized Polynomial Chaos (gPC) method to solve nonlinear stochastic optimal control problems with state variable uncertainties is presented. The GPM and gPC have been shown to be spectrally accurate numerical methods for solving deterministic optimal control problems and stochastic differential equations, respectively. The gPC uses collocation nodes to sample the random space, which are then inserted into the differential equations and solved with a deterministic solver generating a set of deterministic solutions used to characterize the distribution of the solution by constructing a polynomial representation of the output as a function of uncertain inputs. Optimal control problems are generally challenging due to path constraints, bounded controls, boundary conditions, and requirements to minimize a cost functional. Adding random parameters can make these problems even more challenging. This paper proposes using a GPM software package in place of the deterministic differential equation solvers used in the traditional gPC, providing minimum cost solutions to the sampled deterministic problems that satisfy specified constraints. This combination of deterministic and stochastic spectral methods was applied to a challenging nonlinear optimal control problem with multiplicative uncertain elements and the results are reported in this paper.

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