A stochastic Galerkin method for optimal Dirichlet boundary control problems with uncertain data

Abstract The paper deals with a stochastic Galerkin approximation of elliptic Dirichlet boundary control problems with random input data. The expectation of a tracking-type cost functional with deterministic constrained control is minimized. Error estimates are derived for the control variable in the $$L^2(\partial {\mathcal {D}})$$ L 2 ( ∂ D ) -norm and for the state variable in the $${L^2(\Gamma ;L^2({\mathcal {D}}))}$$ L 2 ( Γ ; L 2 ( D ) ) -norm. To solve large linear systems, appropriate preconditioners are proposed for both unconstrained and constrained scenarios. To illustrate the validity and efficiency of the proposed approaches, some numerical experiments are performed.