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Convergence of simultaneous distributed-boundary parabolic optimal control problems

  • * Corresponding author: DTarzia@austral.edu.ar

    * Corresponding author: DTarzia@austral.edu.ar 

The first and third authors is supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH

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  • We consider a heat conduction problem $ S $ with mixed boundary conditions in a n-dimensional domain $ \Omega $ with regular boundary $ \Gamma $ and a family of problems $ S_{\alpha} $, where the parameter $ \alpha>0 $ is the heat transfer coefficient on the portion of the boundary $ \Gamma_{1} $. In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy $ g $ and the heat flux $ q $ on the complementary portion of the boundary $ \Gamma_{2} $. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $ \alpha $ goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.

    Mathematics Subject Classification: Primary: 49J20; Secondary: 35K05, 49K20.


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