Analysis and finite element approximations of stochastic optimal control problems constrained by stochastic elliptic partial differential equations

Analysis and finite element approximations of stochastic optimal control problems constrained by stochastic elliptic partial differential equations

Jangwoon Lee, Iowa State University

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Abstract
In this thesis we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Ito chaos or the Karhunen-Loeve expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the stochastic functions in their Wiener-Ito chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations. Finally, we present some results of numerical experiments.