Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the number of parameters may be countable infinite, i.e., σj with j ∈ N, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters σj . Polynomial approximations of state and control in terms of the possibly countably many stochastic coordinates σj will be used to establish sparsity of polynomial “generalized polynomial chaos (gpc)” expansions of the state and the control with respect to the parameter sequence (σj)j≥1. These imply, in particular, convergence rates of best N -term truncations of these expansions. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes as in [SG11, G] for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K].