The solution of PDE-constrained optimal control problems is a computationally challenging task, and it involves the solution of structured algebraic linear systems whose blocks stem from the discretized first-order optimality conditions. In this paper we analyze the numerical solution of this large-scale system: we first perform a natural order reduction, and then we solve the reduced system iteratively by exploiting specifically designed preconditioning techniques. The analysis is accompanied by numerical experiments on two application problems.
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