Existence, uniqueness and convergence of simultaneous distributed-boundary optimal control problems

Abstract

We consider a steady-state heat conduction problem P for the Poisson equation with mixed boundary conditions in a bounded multidimensional domain Ω. We also consider a family of problems Pα for the same Poisson equation with mixed boundary conditions, α > 0 being the heat transfer coefficient defined on a portion Γ1 of the boundary. We formulate simultaneous distributed and Neumann boundary optimal control problems on the internal energy g within Ω and the heat flux q, defined on the complementary portion Γ2 of the boundary of Ω for quadratic cost functional. Here, the control variable is the vector (g, q). We prove existence and uniqueness of the optimal control (g, q) for the system state of P, and (gα, qα) for the system state of Pα, for each α > 0, and we give the corresponding optimality conditions. We prove strong convergence, in suitable Sobolev spaces, of the vectorial optimal controls, system and adjoint states governed by the problems Pα to the corresponding vectorial optimal control, system and adjoint states governed by the problem P, when the parameter α goes to infinity. We also obtain estimations between the solutions of these vectorial optimal control problems and the solution of two scalar optimal control problems characterized by fixed g (with boundary optimal control q) and fixed q (with distributed optimal control g), respectively, for cases both of α > 0 and α = ∞.