Dubovitskii-Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data

An optimal control problem with a convex cost functional subject to a (linear) non-well-posed problem (Dirichlet heat equation with a given final data) is considered. The control is distributed and a convex constraint on the control is imposed. For a globally distributed control and a convex constraint on the control with non-empty interior, we deduce first-order necessary (and sufficient) optimality conditions using the so-called Dubovitskii-Milyutin formalism, obtaining, in particular, the existence of the corresponding adjoint problem (which is again a non-well-posed problem). In other cases (either empty interior convex constraint on the control or partially distributed control), we arrive at the optimality conditions but admitting the existence of the adjoint problem. Finally, numerical results are also presented approximating the optimality conditions for 1D domains by finite differences in time and space.