A COMMUTATIVE DIAGRAM AMONG DISCRETE AND CONTINUOUS NEUMANN BOUNDARY OPTIMAL CONTROL PROBLEMS
We consider a bounded domain whose regular boundary consists of the union of two disjoint portions and with positive measures. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems governed by elliptic variational equalities, when the parameter aof the family heat transfer coefficient on the portion of the boundary goes to infinity was studied in Gariboldi-Tarzia [15], being the control variable the heat flux on the boundary 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 governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary
We consider the discrete approximations and of the optimal control problems and respectively, for each and for each 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 and The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems when the parameter agoes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family to the corresponding discrete Neumann boundary mixed elliptic optimal control problem when for each in adequate functional spaces. We also study the convergence when and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems and by taking the limits and respectively.
Neumann boundary optimal control problems, elliptic variational equalities, mixed boundary conditions, numerical analysis, finite element method, fixed points, optimality conditions, convergence with respect to a parameter, error estimations.