Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence

Erik Kropat

doi:10.3934/naco.2017003

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2017, Volume 7, Issue 1: 51-76. Doi: 10.3934/naco.2017003

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Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence

Erik Kropat^,

University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

* Corresponding author: Erik Kropat
* Corresponding author: Erik Kropat

Received: August 2016

Revised: February 2016

Published: May 2017

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Abstract

The homogenization of optimal control problems on periodic networks is considered. Traditional approaches for a homogenization of uncontrolled problems on graphs often rely on an artificial extension of branches. The main result shows that such an extension to thin domains is not required. A two-scale transform for network functions leads to a representation of the microscopic optimal control problem on the graph in terms of a two-scale transformed minimization problem that allows for a further homogenization. Here, the concept of $S$-homogenization is applied in order to prove the existence of an absolutely $S$-homogenized optimal control problem with respect to the superior domain and the microscopic scale encoded in the reference graph of the network. In addition, results on the $Γ$-convergence of optimal control problems on periodic networks are discussed.

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Mathematics Subject Classification: Primary: 34B45, 34E13, 34E05, 34E10.

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Figure 1. Two-scale transform: Mapping from $\mathscr{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathscr{Y}$.

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Table 1. Function spaces and operators: Notation of abstract op-timal control problems.

${\cal U} = {L^2}(\Omega \times {\cal Y}),\;{\Phi _\varepsilon } = {{\hat \Phi }_\varepsilon }$
${\cal V} = {L^2}(\Omega \times {\cal Y}),\;{{\cal C}_\varepsilon } = {\widehat {\cal C}_\varepsilon }$
${\cal W} = {L^2}(\Omega \times {\cal Y}),\;{{\cal Z}^\varepsilon } = \widehat {{\zeta ^\varepsilon }} \in {L^2}(\Omega \times {\cal Y})$
${{\cal G}^\varepsilon } = {\widehat {\cal G}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{L^2}(\Omega \times {\cal Y})),\;{{\cal G}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {g^\varepsilon }{\xi ^\varepsilon }$
${{\cal P}^\varepsilon } = {\widehat {\cal P}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal P}^\varepsilon }\widehat {{\phi ^\varepsilon }}: = {p^\varepsilon }{\phi ^\varepsilon }$
${{\cal Q}^\varepsilon } = {\widehat {\cal Q}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal Q}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {q^\varepsilon }{\xi ^\varepsilon }$

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