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This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\mathbb{N}$, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615--646; Anal. Appl., 9 (2011), pp. 1--37], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\geq 1}$ of the random inputs, and we prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the...

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