Some problems of distributed parameter control with unilateral constraints (Q752456)

The paper deals with linear-quadratic control problems for parabolic equations with convex constraints on the control. These problems are reformulated in abstract geometric framework as problems of seeking a nearest point in (\({\mathcal S}\cap {\mathcal C})\subset {\mathcal H}\) where \({\mathcal H}\) is a Hilbert space, \({\mathcal S}\) an affine subspace of \({\mathcal H}\) and \({\mathcal C}\) a (possibly) unbounded closed convex set in \({\mathcal H}\), describing the control constraint. It is proved that the optimal control with constraints is the nearest point (in \({\mathcal C})\) to the optimal control in the case without constraints.