Homogenization of optimal control problems on curvilinear networks with a periodic microstructure – results on \(S\)-homogenization and \(\Gamma\)-convergence. (English) Zbl 1361.49004
Summary: 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 \(\Gamma\)-convergence of optimal control problems on periodic networks are discussed.
MSC:
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
| 34E13 | Multiple scale methods for ordinary differential equations |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
Keywords:
optimal control problems; \(S\)-homogenization; \(\Gamma\)-convergence; two-scale convergence; networks; microstructuresReferences:
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