Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

Gariboldi,; Tarzia, Domingo A.

doi:10.1007/s00245-003-0761-y

Cited by 21 publications

(47 citation statements)

References 10 publications

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“…Proof. We use the Lax-Milgram Theorem, the variational equalities (19), (20), (22) and (23), the coerciveness (11) and (12) and following [10,18,25].…”

Section: Discretization By Finite Element Methods and Propertiesmentioning

confidence: 99%

“…The purpose of this paper is to do the numerical analysis, by using the finite element method, of the convergence of the continuous distributed mixed optimal control problems with respect to a parameter (the heat transfer coefficient) given in [10,11] obtaining a double convergence when the parameter of the finite element method goes to zero and the heat transfer coefficient goes to infinity.…”

Section: Introductionmentioning

confidence: 99%

“…and   1 0 is the coercive constant for the bilinear form 1 a [16,21]. The unique continuous distributed optimal energies op g and op g  have been characterized in [10] as a fixed point on H for a suitable operators W and W  over their optimal adjoint system states…”

Section: Introductionmentioning

confidence: 99%

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

Tarzia

2016

IFIP Advances in Information and Communication Technology

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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.

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“…Proof. We use the Lax-Milgram Theorem, the variational equalities (19), (20), (22) and (23), the coerciveness (11) and (12) and following [10,18,25].…”

Section: Discretization By Finite Element Methods and Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Tarzia

2016

IFIP Advances in Information and Communication Technology

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“…We denote with u q and u qα the unique solutions of the mixed elliptic problems (1) and (2) respectively for each q ∈ Q and α > 0 whose variational formulations are given as in [3], [4] and [7].…”

Section: Introductionmentioning

confidence: 99%

Convergence of boundary optimal controls in mixed elliptic problems

Gariboldi

Tarzia

2007

Proc Appl Math and Mech

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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.

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“…Several optimal control problems are governed by elliptic variational inequalities ( [1], [2], [3], [5], [6], [13], [14], [27], [31], [32], [39]) and there exists an abundant literature about continuous and numerical analysis of optimal control problems governed by elliptic variational equalities or inequalities ( [4], [10], [11], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [26], [29], [30], [35], [36], [40]) and by parabolic variational equalities or inequalities ( [7], [28]).…”

Section: Introductionmentioning

confidence: 99%

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

Olguín¹,

Tarzia²

2016

ADECP

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The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter α > 0) is obtained through the finite element method when its parameter h → 0. We also obtain the limit of the discrete optimal control and the associated state system solutions when α → ∞ (for each h > 0) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when h → 0 and α → ∞. Moreover, the double convergence is also obtained when (h, α) → (0, ∞).

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Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

Cited by 21 publications

References 10 publications

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

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

Convergence of boundary optimal controls in mixed elliptic problems

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

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