Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs

Abstract

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., $\sigma_j$ with $j\in N$, 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 $\sigma_j$. Polynomial approximations of state and control in terms of the possibly countably many stochastic coordinates $\sigma_j$ 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 $(\sigma_j)_{j\geq 1}$. 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,CJG11] for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK,GK,K].