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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:
[1]

A. Braides, Γ-Convergence for Beginners Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar

[2]

G. Buttazzo, Gamma-convergence and its applications to some problem in the calculus of variations, Gamma-convergence and its applications to some problem in the calculus of variations, In School on homogenization, ICTP, Trieste, September 6-17,1993, 1993 (1994), 303-325. Google Scholar

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D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. Google Scholar

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S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259-2283. doi: 10.1016/j.jmps.2005.04.010. Google Scholar

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B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7. Google Scholar

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V. V. Jikov and S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar

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P. Kogut and G. Leugering, S-Homogenization of Optimal Control Problems in Banach Spaces, S-Homogenization of Optimal Control Problems in Banach Spaces, Mathematische Nachrichten, 1 (2002), 141-169. doi: 10.1002/1522-2616(200201)233:1<141::AID-MANA141>3.0.CO;2-I. Google Scholar

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P. Kogut, Homogenization of Optimal Control Problems for Distributed Systems Cybernetics Institute of Ukrainian National Academic Science, Kyiv, Glushkov, 40,1998 (in Russian).Google Scholar

[12]

E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen Wissenschaftlicher Verlag Berlin, Berlin, 2007.Google Scholar

[13]

E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83-88. Google Scholar

[14]

E. KropatS. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. Google Scholar

[15]

E. KropatS. Meyer-Nieberg and G.-W. Weber, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Numerical Algebra, Control and Optimization, 9 (2016), 183-219. doi: 10.3934/naco.2016008. Google Scholar

[16]

E. KropatS. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. Google Scholar

[17]

M. Lenczner, Homogénéisation d'un circuit électrique, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Academie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. Google Scholar

[18]

M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397. doi: 10.1137/S1540345903423919. Google Scholar

[19]

M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932. doi: 10.1142/S0218202599000415. Google Scholar

[20]

C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231-2258. doi: 10.1016/j.jmps.2005.04.006. Google Scholar

[21]

C. MieheS. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617-2660. doi: 10.1016/j.jmps.2004.03.011. Google Scholar

[22]

E. NuhnE. KropatW. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii(eds. Ralph H. Sprague, Jr.), IEEE Computer Society, (2012), 1089-1096. Google Scholar

[23]

G. A. Pavliotis and Andrew Stuart, Multiscale Methods. Averaging and Homogenization Springer, Texts in Applied Mathematics, 53, New York, 2008. Google Scholar

[24]

L. Tartar, The General Theory of Homogenization: A Personalized Introduction Springer, Berlin, Heidelberg, 2010.Google Scholar

[25]

M. Vogelius, A homogenization result for planar, polygonal networks, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514. Google Scholar

show all references

References:
[1]

A. Braides, Γ-Convergence for Beginners Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar

[2]

G. Buttazzo, Gamma-convergence and its applications to some problem in the calculus of variations, Gamma-convergence and its applications to some problem in the calculus of variations, In School on homogenization, ICTP, Trieste, September 6-17,1993, 1993 (1994), 303-325. Google Scholar

[3]

G. Buttazzo and G. Dal Maso, Gamma-convergence and optimal control problems, Gamma-convergence and optimal control problems, Journal of Optimization Theory and Applications, 38 (1982), 385-407. doi: 10.1007/BF00935345. Google Scholar

[4]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. Google Scholar

[5]

G. Dal Maso, An Introduction to Γ-Convergence Birkhäuser, Progress in Nonlinear Differential Equations and Their Applications, Basel, 1993. doi: 10.1007/978-1-4612-0327-8. Google Scholar

[6]

S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259-2283. doi: 10.1016/j.jmps.2005.04.010. Google Scholar

[7]

B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7. Google Scholar

[8]

V. V. Jikov and S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. Google Scholar

[9]

P. Kogut and G. Leugering, S-Homogenization of Optimal Control Problems in Banach Spaces, S-Homogenization of Optimal Control Problems in Banach Spaces, Mathematische Nachrichten, 1 (2002), 141-169. doi: 10.1002/1522-2616(200201)233:1<141::AID-MANA141>3.0.CO;2-I. Google Scholar

[10]

P. Kogut and G. Leugering, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs in Optimal Control Problems for Partial Differential Equations on Reticulated Domains (eds. P. Kogut and G. Leugering), Birkhäuser Boston, (2011), 409-440. doi: 10.1007/978-0-8176-8149-4. Google Scholar

[11]

P. Kogut, Homogenization of Optimal Control Problems for Distributed Systems Cybernetics Institute of Ukrainian National Academic Science, Kyiv, Glushkov, 40,1998 (in Russian).Google Scholar

[12]

E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen Wissenschaftlicher Verlag Berlin, Berlin, 2007.Google Scholar

[13]

E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, Homogenization of singularly perturbed diffusion-advection-reaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83-88. Google Scholar

[14]

E. KropatS. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. Google Scholar

[15]

E. KropatS. Meyer-Nieberg and G.-W. Weber, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -efficient solution strategies based on homogenization theory, Numerical Algebra, Control and Optimization, 9 (2016), 183-219. doi: 10.3934/naco.2016008. Google Scholar

[16]

E. KropatS. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. Google Scholar

[17]

M. Lenczner, Homogénéisation d'un circuit électrique, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Academie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. Google Scholar

[18]

M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397. doi: 10.1137/S1540345903423919. Google Scholar

[19]

M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932. doi: 10.1142/S0218202599000415. Google Scholar

[20]

C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231-2258. doi: 10.1016/j.jmps.2005.04.006. Google Scholar

[21]

C. MieheS. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, A micro-macro approach to rubber-like materials -Part I: the non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617-2660. doi: 10.1016/j.jmps.2004.03.011. Google Scholar

[22]

E. NuhnE. KropatW. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii(eds. Ralph H. Sprague, Jr.), IEEE Computer Society, (2012), 1089-1096. Google Scholar

[23]

G. A. Pavliotis and Andrew Stuart, Multiscale Methods. Averaging and Homogenization Springer, Texts in Applied Mathematics, 53, New York, 2008. Google Scholar

[24]

L. Tartar, The General Theory of Homogenization: A Personalized Introduction Springer, Berlin, Heidelberg, 2010.Google Scholar

[25]

M. Vogelius, A homogenization result for planar, polygonal networks, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514. Google Scholar

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