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March  2017, 7(1): 51-76. doi: 10.3934/naco.2017003

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

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

* Corresponding author: Erik Kropat

Received  August 2016 Revised  February 2016 Published  February 2017

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.

Citation: Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
##### References:

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##### References:
Two-scale transform: Mapping from $\mathscr{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathscr{Y}$.
Function spaces and operators: Notation of abstract op-timal control problems.
 ${\cal U} = {L^2}(\Omega \times {\cal Y}),\;{\Phi _\varepsilon } = {{\hat \Phi }_\varepsilon }$ ${\cal V} = {L^2}(\Omega \times {\cal Y}),\;{{\cal C}_\varepsilon } = {\widehat {\cal C}_\varepsilon }$ ${\cal W} = {L^2}(\Omega \times {\cal Y}),\;{{\cal Z}^\varepsilon } = \widehat {{\zeta ^\varepsilon }} \in {L^2}(\Omega \times {\cal Y})$ ${{\cal G}^\varepsilon } = {\widehat {\cal G}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{L^2}(\Omega \times {\cal Y})),\;{{\cal G}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {g^\varepsilon }{\xi ^\varepsilon }$ ${{\cal P}^\varepsilon } = {\widehat {\cal P}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal P}^\varepsilon }\widehat {{\phi ^\varepsilon }}: = {p^\varepsilon }{\phi ^\varepsilon }$ ${{\cal Q}^\varepsilon } = {\widehat {\cal Q}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal Q}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {q^\varepsilon }{\xi ^\varepsilon }$
 ${\cal U} = {L^2}(\Omega \times {\cal Y}),\;{\Phi _\varepsilon } = {{\hat \Phi }_\varepsilon }$ ${\cal V} = {L^2}(\Omega \times {\cal Y}),\;{{\cal C}_\varepsilon } = {\widehat {\cal C}_\varepsilon }$ ${\cal W} = {L^2}(\Omega \times {\cal Y}),\;{{\cal Z}^\varepsilon } = \widehat {{\zeta ^\varepsilon }} \in {L^2}(\Omega \times {\cal Y})$ ${{\cal G}^\varepsilon } = {\widehat {\cal G}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{L^2}(\Omega \times {\cal Y})),\;{{\cal G}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {g^\varepsilon }{\xi ^\varepsilon }$ ${{\cal P}^\varepsilon } = {\widehat {\cal P}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal P}^\varepsilon }\widehat {{\phi ^\varepsilon }}: = {p^\varepsilon }{\phi ^\varepsilon }$ ${{\cal Q}^\varepsilon } = {\widehat {\cal Q}^\varepsilon } \in {\cal L}({L^2}(\Omega \times {\cal Y}),{\mkern 1mu} {L^2}(\Omega \times {\cal Y})),{{\cal Q}^\varepsilon }\widehat {{\xi ^\varepsilon }}: = {q^\varepsilon }{\xi ^\varepsilon }$
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