Neumann boundary optimal control problems governed by parabolic variational equalities

Abstract We consider a heat conduction problem S with mixed boundary conditions in an n -dimensional domain Ω regular and family of problems α also Ω, where > 0 is the transfer coefficient on portion Γ 1 . In relation to these state systems, we formulate Neumann optimal control flux q which definite complementary 2 Ω. obtain existence uniqueness controls, first order optimality terms adjoint convergence system when goes infinity. Furthermore, particular real parameter λ , parabolic elliptic P find explicit form for prove monotony properties results time