Boundary Conditions

We consider two steady-state heat conduction problems $P$ and ${P}_{\alpha \,}$ (for each $\alpha >0$) with mixed boundary conditions for the same Poisson equation. The difference between both problems is that on the boundary portion $\Gamma _{1}$ a Dirichlet condition is verified for $P$ and a Newton condition with transfer coefficient $\alpha $ is verified for ${P}% _{\alpha \,}.$ We formulate distributed optimal control problems, for a suitable cost function, over the internal energy $g\,$ in the material. We make a new proof with respect to the one given in C.M. Gariboldi - D.A. Tarzia, App. Math. Optim. 47 (2003), 213-230 on the strongly convergence when $\alpha \rightarrow \infty $ of the optimal control $g_{op_{\alpha }}$, the system state $u_{g_{op_{\alpha }}\alpha }\,$and the adjoint state $% p_{g_{op_{\alpha }}\alpha }$ to the optimal control $g_{op},$ system state $% u_{g_{op}}$ and adjoint state ${p}_{g_{op}}$ corresponding to ${P}_{\alpha \,}$ and $P$ respectively. F...

Computational Optimization and Applications, 2012

First, let u g be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between u 3 (μ) = μu g 1 + (1 − μ)u g 2 and u 4 (μ) = u μg 1 +(1−μ)g 2 for μ ∈ [0, 1]. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy g for each positive heat transfer coefficient h given on a part of the boundary of the domain. For a given cost functional and using some monotony property between u 3 (μ) and u 4 (μ) given in Mignot (J. Funct. Anal. 22:130-185, 1976), we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter h goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot's conical differentiability) which is a great advantage with respect to the proof given in Gariboldi and Tarzia (Appl. Math. Optim. 47:213-230, 2003), for optimal control problems governed by elliptic variational equalities.

Communications in Nonlinear Science and Numerical Simulation, 2021

In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been 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 optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.

MAT Serie A, 2004

We consider two steady-state heat conduction problems P and P ® (for each ® > 0) with mixed boundary conditions for the same Poisson equation. The di®erence between both problems is that on the boundary portion ¡ 1 a Dirichlet condition is veri¯ed for P and a Newton condition with transfer coe±cient ® is veri¯ed for P ® : We formulate distributed optimal control problems, for a suitable cost function, over the internal energy g in the material. We make a new proof with respect to the one given in C.M. Gariboldi-D.A. Tarzia, App. Math. Optim. 47 (2003), 213-230 on the strongly convergence when ® ! 1 of the optimal control g op® , the system state u gop ® ® and the adjoint state p gop ® ® to the optimal control g op ; system state u gop and adjoint state p gop corresponding to P ® and P respectively. For this proof we eliminate the restriction on the constant of coerciveness of the bilinear form, and we use properties of the cost function and the theory of variational equalities instead of the¯xed point theorem.

2021

We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. 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 Γ2 of the boundary of Ω. 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 a explicit form for the optimal controls, we prove monotony properties and we obtain...

Proceedings in applied mathematics & mechanics, 2007

We consider a steady-state heat conduction problem Pα with mixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ1 of the boundary of Ω. We consider, for each α > 0, a cost function Jα and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ2 of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ2 in terms of the adjoint state. We prove that the optimal control qop α and its corresponding system state uq opα α and adjoint state pq opα α for each α are strongly convergent to qop, uq op and pq op in L 2 (Γ2), H 1 (Ω), and H 1 (Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ1. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ2 and the parameter α (which goes to infinity) is defined on Γ1.

IFIP Advances in Information and Communication Technology, 2013

I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term g through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over g for each parameter h > 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot's inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter h. We also prove, when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition. II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems. III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter h > 0, but in this case the convergence when h → +∞ is still an open problem.

CONVERGENCE, 2008

We consider a steady-state heat conduction problem P. with mixed boundary conditions for the Poisson equation in a bounded multidimensional domain n depending on a positive parameter a which represents the heat transfer coefficient on a portion 1-1 of the boundary of 0. We consider, for each a > 0, a cost function Ja and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion í2 of the boundary of S2. We obtain that the optimality conditions are given by a complementary free boundary problem in F2 in terms of the adjoint state. We prove that the optimal control gola and its corresponding system state uqopa a and adjoint state pgopaa for each a are strongly convergent to qop" ugop and pgop in L2(r2), H1(S2), and H1(S2) respectively when a -> m. We also 2000 Mathematics Subject Classification: 49J20, 35J85, 35R35.

Nonlinear Analysis: Real World Applications, 2011

Parabolic variational inequalities of the second kind Convex combination of solutions Monotony property Regularization method Dependency of the solutions on the data Strict convexity of cost functional Optimal control problems a b s t r a c t

arXiv (Cornell University), 2023

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 hα) 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 hα) 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 hα) 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 hα) 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 hα) , (P α) , (P h) and (P) by taking the limits h → 0 + and α → +∞ respectively. We also study the double convergence of (P hα) to (P) when (h, α) → (0 + , +∞) which represents the diagonal convergence in the above commutative diagram.