Double Convergence of a Family of Discrete Distributed Mixed Elliptic Optimal Control Problems with a Parameter

Abstract

We consider a bounded domain n  whose regular boundary 12        consists of the union of two disjoint portions 1  and 2  with meas 1 ( ) 0 . The convergence of a family of continuous distributed mixed elliptic optimal control problems ( P  ), governed by elliptic variational equalities, when the parameter  of the family (the heat transfer coefficient on the portion of the boundary 1  ) goes to infinity was studied in Gariboldi -Tarzia, Appl. Math. Optim., 47 (2003), 213-230. 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 distributed mixed elliptic optimal control problem ( P ) governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary 1 We consider the discrete approximations ( h P  ) and ( h P ) of the optimal control problems ( P  ) and ( P ) respectively, for each  0 h and for each   0 , through the finite element method with Lagrange's triangles of type 1 with parameter h (the longest side of the triangles). We also discretize the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the distributed optimal control problems ( P  ) and ( P ). The goal of this paper is to study the double convergence of this family of discrete distributed mixed elliptic optimal control problems ( h P  ) when the parameters  goes to infinity and the parameter h goes to zero simultaneously. We prove the convergence of the discrete optimal controls, the discrete system and adjoint states of the family ( h P  ) to the corresponding to the discrete distributed mixed elliptic optimal control problem ( h P ) when  , for each  0 h , in adequate functional spaces. We study the convergence of the discrete distributed optimal control problems ( h P  ) and ( h P ) when  0 h obtaining a commutative diagram which relates the continuous and discrete distributed mixed elliptic optimal control problems       ,, hh P P P  and ( P ) by taking the limits  0 h and   respectively. We also study the double convergence of ( h P  ) to ( P ) when    ( , ) (0, ) h which represents the diagonal convergence in the above commutative diagram.

Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems

“…In Section 3 and Section 4, similar results are obtained in an annulus in R 2 and a spherical shell in R 3 , respectively. In all cases, we obtain, in agreement with theory, the convergence of the optimal controls and values when α → ∞ as it was obtained in [3,11,12,13] and for numerical analysis in [27]. Also, the corresponding rates of convergence are studied, obtaining, in Appendix A, that the order of convergence in each case is 1/α which is new for these elliptic optimal control problems.…”

Numerical Analysis of a Family of Optimal Distributed Control Problems Governed by an Elliptic Variational Inequality

“…1. We generalize recent results obtained for optimal control problems governed by elliptic variational equalities given in [37,38].…”

“…Finally, we see that: Then, by using (3.24), (3.25), we obtain (3.20). Now, following the idea given in [38] we have this final theorem:…”

Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

“…Proof. We use the Lax-Milgram Theorem, the variational equalities ( 17), ( 18), ( 21) and ( 22), the coerciveness ( 5) and ( 6) and following [16,26,34,35].…”

“…Proof. We use the definitions ( 15) and ( 16), the elliptic variational equalities ( 17) and ( 18) and the coerciveness ( 5) and ( 6), following [15,16,26,34,35].…”