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In this paper, we study boundary optimal control problems on the heat flux and simultaneous distributed-boundary optimal control problems on the internal energy and the heat flux for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system was originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the boundary optimal control problem and simultaneous distributed-boundary optimal control problems. We show an asymptotic behavior result for the optimal controls and the system states for both optimal control problems, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary. [ABSTRACT FROM AUTHOR]

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain with regular boundary and a family of problems Sα with also mixed boundary conditions in, where α > 0 is the heat transfer coefficient on the portion of the boundary T1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion T2 of the boundary of. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα. We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity. [ABSTRACT FROM AUTHOR]

We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary Γ and a family of problems Sα, where the parameter α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy g and the heat flux q on the complementary portion of the boundary Γ2. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author. [ABSTRACT FROM AUTHOR]

In this paper, we consider a family of simultaneous distributed-boundary optimal control problems (P α) on the internal energy and the heat flux for a system governed by a mixed elliptic variational equality with a parameter α > 0 (the heat transfer coefficient on a portion of the boundary of the domain) and a simultaneous distributed-boundary optimal control problem (P) governed also by an elliptic variational equality with a Dirichlet boundary condition on the same portion of the boundary. We formulate discrete approximations P h α and P h of the optimal control problems P α and (P) respectively, for each h > 0 and for each α > 0 , through the finite element method with Lagrange's triangles of type 1 with parameter h (the longest side of the triangles). The goal of this paper is to study the convergence of this family of discrete simultaneous distributed-boundary mixed elliptic optimal control problems P h α when the parameters α goes to infinity and the parameter h goes to zero simultaneously. We prove the convergence of the family of discrete problems P h α to the discrete problem P h when α → + ∞ , for each h > 0 , in adequate functional spaces. We study the convergence of the discrete problems P h α and P h , for each α > 0 , when h → 0 + obtaining a commutative diagram which relates the continuous and discrete simultaneous distributed-boundary mixed elliptic optimal control problems P h α , P α , P h and (P) by taking the limits h → 0 + and α → + ∞ respectively. We also study the double convergence of P h α to (P) when (h , α) → (0 + , + ∞) which represents the diagonal convergence in the above commutative diagram. [ABSTRACT FROM AUTHOR]