Optimal control with stochastic PDE constraints and uncertain controls

TL;DR

This paper develops a framework for optimal control problems constrained by stochastic PDEs with uncertain controls, exploring the use of stochastic finite element methods to control statistical system responses.

Contribution

It introduces a novel approach combining stochastic finite element methods with optimal control under uncertainty, including analysis of collocation and Galerkin methods.

Findings

Coupling between stochastic collocation points reduces method efficiency.

Stochastic control can influence statistical properties of the system.

Numerical examples demonstrate the framework's applicability to inverse problems.

Abstract

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented.