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

Abstract : The convergence of a family of continuous distributed mixed elliptic optimal control problems ($$ P_{\alpha } $$), governed by elliptic variational equalities, when the parameter $$ \alpha \to \infty $$ was studied in Gariboldi - Tarzia, Appl. Math. Optim., 47 (2003), 213-230 and it has been proved that it is convergent to a distributed mixed elliptic optimal control problem ($$ P $$). We consider the discrete approximations ($$ P_{h\alpha } $$) and ($$ P_{h} $$) of the optimal control problems ($$ P_{\alpha } $$) and ($$ P $$) respectively, for each $$ h > 0 $$ and $$ \alpha > 0 $$. We study the convergence of the discrete distributed optimal control problems ($$ P_{h\alpha } $$) and ($$ P_{h} $$) when $$ h \to 0 $$, $$ \alpha \to \infty $$ and $$ (h,\alpha ) \to (0, +\infty ) $$ obtaining a complete commutative diagram, including the diagonal convergence, which relates the continuous and discrete distributed mixed elliptic optimal control problems $$ \left( {P_{h\alpha } } \right),\;\left( {P_{\alpha } } \right),\;\left( {P_{h} } \right) $$ and ($$ P $$) by taking the corresponding limits. The convergent corresponds to the optimal control, and the system and adjoint system states in adequate functional spaces.
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Domingo Tarzia. Double Convergence of a Family of Discrete Distributed Mixed Elliptic Optimal Control Problems with a Parameter. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.493-504, ⟨10.1007/978-3-319-55795-3_47⟩. ⟨hal-01626894⟩

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