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.

中文翻译：

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同时分布边界抛物线最优控制问题的收敛性

我们考虑在n维域$ \ Omega $中具有规则边界$ \ Gamma $和一族问题$ S _ {\ alpha} $的热传导问题$ S $，其中参数$ \ alpha> 0 $是边界$ \ Gamma_ {1} $的一部分上的传热系数。关于这些状态系统，我们制定了同时分布边界关于内部能量$ g $和边界$ \ Gamma_ {2} $的互补部分的热通量$ q $的最优控制问题。当传热系数$ \ alpha $变为无穷大时，我们获得了最优控制的存在性和唯一性，以及伴随状态的一阶最优条件以及最优控制，系统和伴随状态的收敛性。最后，我们证明了第一作者的前一篇论文中研究的同时分布边界最优控制和分布最优控制问题之间的估计。