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

Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems... We consider a steady-state heat conduction problem P α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ 1 of the boundary of a given bounded domain in R n . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op α and its corresponding system u_ g_ op α α and adjoint p_ g_ op α α states for each α are strongly convergent to g op , u_ g op and p _ g op , respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ 1 . We use the fixed point and elliptic variational inequality theories. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 47 (3) – May 21, 2003
18 pages

/lp/springer_journal/convergence-of-distributed-optimal-controls-on-the-internal-energy-in-ETL50jfawv
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-003-0761-y
Publisher site
See Article on Publisher Site

### Abstract

We consider a steady-state heat conduction problem P α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ 1 of the boundary of a given bounded domain in R n . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op α and its corresponding system u_ g_ op α α and adjoint p_ g_ op α α states for each α are strongly convergent to g op , u_ g op and p _ g op , respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ 1 . We use the fixed point and elliptic variational inequality theories.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: May 21, 2003

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