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Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs
Authors:LS Hou  H Manouzi
Institution:a Department of Mathematics, Iowa State University, Ames, IA 50011, United States
b Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, United States
c Dèpartement de Mathèmatiques et Statistique Universitè Laval, Quèbec, Canada, G1V 0A6
Abstract:In this paper 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-Itô chaos or the Karhunen-Loève 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 input data in their Wiener-Itô 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.
Keywords:Distributed control  Stochastic partial differential equation  Finite element methods
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