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We study sparse solutions of optimal control problems governed by PDEs with uncertain coefficients. We propose two formulations, one where the solution is a deterministic control optimizing the mean objective, and a formulation aiming at stochastic controls that share the same sparsity structure. In both formulations, regions where the controls do not vanish can be interpreted as optimal locations for placing control devices. In this paper, we focus on linear PDEs with linearly entering uncertain parameters. Under these assumptions, the deterministic formulation reduces to a problem with known structure, and thus we mainly focus on the stochastic control formulation. Here, shared sparsity is achieved by incorporating the $L^1$-norm of the mean of the pointwise squared controls in the objective. We reformulate the problem using a norm reweighting function that is defined over physical space only and thus helps to avoid approximation of the random space using samples or quadrature. We show that a fixed poin...

Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case