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

Abstract

Necessary and sufficient conditions for the optimality of a pair ( y ∗ , u ∗ ) (y^{*}, u^{*}) subject to A y ∗ = B u ∗ + f Ay^{*} = Bu^{*} + f are given. Here A A is a selfadjoint operator with closed range on a Hilbert space H \mathcal {H} and B ∈ L ( H ) B \in L(\mathcal {H}) . The case B 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 T –periodic wave equation for all T T in a dense subset of R \mathbb {R} . It also includes optimal control problems for some elliptic equations.