A comnutative diagram among discrete and continuous Neumann boundary optimal control problems

Abstract

We consider a bounded domain nR⊂Ω whose regular boundary 21ΓΓ=Ω∂=Γ∪ consists of the union of two disjoint portions 1Γ and 2Γ with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (),αP governed by elliptic variational equalities, when the parameter α of the family (the heat transfer coefficient on the portion of the boundary )1Γ goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary .2Γ It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another Neumann boundary mixed elliptic optimal control problem ()P governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary .1ΓWe consider the discrete approximations ()αhP and ()hP of the optimal control problems ()αP and (),P respectively, for each 0>h and for each ,0>α through the finite element method with Lagrange’s triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems ()αP and ().P The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems ()αhP when the parameter α goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family ()αhP to the corresponding discrete Neumann boundary mixed elliptic optimal control problem ()hP when ∞→α for each ,0>h in adequate functional spaces. We also study the convergence when 0→h and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (),αhP(),αP()hP and ()P by taking the limits 0→h and ,+∞→α respectively.