Numerical solution of an optimal control problem with probabilistic and almost sure state constraints

TL;DR

This paper develops numerical algorithms for solving PDE-based optimal control problems with probabilistic and almost sure constraints, providing new formulas and comparing five different solution methods.

Contribution

It offers an exact Clarke subdifferential formula for the probability function and compares five numerical approaches for handling probabilistic and almost sure constraints.

Findings

The Clarke subdifferential formula improves theoretical understanding.

The spherical radial decomposition effectively handles probabilistic constraints.

Monte Carlo sampling aids in solving almost sure constraints.

Abstract

We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated.