Optimal control of a functional equation associated with closed range selfadjoint operators

Necessary and sufficient conditions for the optimality of a pair $(y^{*}, u^{*})$ subject to $Ay^{*} = Bu^{*} + f$ are given. Here $A$ is a selfadjoint operator with closed range on a Hilbert space $\mathcal {H}$ and $B \in L(\cal{H})$. The case $B$- unbounded is also discussed, which leads to some open problems. This general functional scheme includes most of the previous results on the optimal control of the $T$-periodic wave equation for all $T$ in a dense subset of $\mathbb{R}$. It also includes optimal control problems for some elliptic equations.