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Homogenization of constrained optimal control problems forone-dimensional elliptic equations on periodic graphs

Published online by Cambridge University Press:  24 June 2008

Peter I. Kogut  and
Günter Leugering
Peter I. Kogut
Affiliation:
Department of Differential Equations, Dnipropetrovsk National University, Naukova str., 13, 49050 Dnipropetrovsk, Ukraine. [email protected]
Günter Leugering
Affiliation:
Institüt für Angewandte Mathematik Lehrstuhl II, Universität Erlangen-Nürnberg Martensstr.3, 91058 Erlangen, Germany. [email protected]
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Abstract

We are concerned with the asymptotic analysis of optimal controlproblems for 1-D partial differential equations defined on aperiodic planar graph, as the period of the graph tends to zero. Wefocus on optimal control problems for elliptic equations withdistributed and boundary controls. Using approaches of the theory ofhomogenization we show that the original problem on the periodicgraph tends to a standard linear quadratic optimal control problemfor a two-dimensional homogenized system, and its solution can beused as suboptimal controls for the original problem.

Keywords

Optimal controlhomogenizationelliptic equationperiodic graph
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 2 , April 2009 , pp. 471 - 498
Copyright
© EDP Sciences, SMAI, 2008

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References

H. Attouch, Variational Convergence for Functional and Operators, Applicable Mathematics Series. Pitman, Boston-London (1984).
A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
Bouchitte, G. and Fragala, I., Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32 (2001) 1198–1226. CrossRef
A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002).
G. Buttazzo, Γ-convergence and its applications to some problems in the calculus of variations, in School on Homogenization, ICTP, Trieste, September 6–17, 1993, SISSA (1994) 38–61.
Buttazzo, G. and Dal Maso, G., Γ-convergence and optimal control problems. J. Optim. Theory Appl. 32 (1982) 385–407. CrossRef
Casado-Diaz, J., Luna-Laynez, M. and Marin, J.D., An adaption of the multi-scale methods for the analysis of very thin reticulated structures. C. R. Acad. Sci. Paris Sér. I 332 (2001) 223–228. CrossRef
Chechkin, G., Zhikov, V., Lukkassen, D. and Piatnitski, A., On homogenization of networks and junctions. J. Asymp. Anal. 30 (2000) 61–80.
D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topic in the Math. Modelling of Composit Materials, Boston, Birkhäuser, Prog. Non-linear Diff. Equ. Appl. 31 (1997) 49–93.
Cioranescu, D., Donato, P. and Zuazua, E., Exact boundary controllability for the wave equation in domains with small holes. J. Math. Pures Appl. 69 (1990) 1–31.
C. Conca, A. Osses and J. Saint Jean Paulin, A semilinear control problem involving in homogenization. Electr. J. Diff. Equ. (2001) 109–122.
G. Dal Maso, An Introduction of Γ-Convergence. Birkhäuser, Boston (1993).
Haraux, A. and Murat, F., Perturbations singulières et problèmes de contrôle optimal : deux cas bien posés. C. R. Acad. Sci. Paris Sér. I 297 (1983) 21–24.
Haraux, A. and Murat, F., Perturbations singulières et problèmes de contrôle optimal : un cas mal posé. C. R. Acad. Sci. Paris Sér. I 297 (1983) 93–96.
Kesavan, S. and Vanninathan, M., L'homogénéisation d'un problème de contrôle optimal. C. R. Acad. Sci. Paris Sér. A-B 285 (1977) 441–444.
Kesavan, S. and Saint Jean Paulin, J., Optimal control on perforated domains. J. Math. Anal. Appl. 229 (1999) 563–586. CrossRef
Kogut, P.I., S-convergence in homogenization theory of optimal control problems. Ukrain. Matemat. Zhurnal 49 (1997) 1488–1498 (in Russian).
Kogut, P.I. and Leugering, G., Homogenization of optimal control problems in variable domains. Principle of the fictitious homogenization. Asymptotic Anal. 26 (2001) 37–72.
Kogut, P.I. and Leugering, G., Asymptotic analysis of state constrained semilinear optimal control problems. J. Optim. Theory Appl. 135 (2007) 301–321. CrossRef
Kogut, P.I. and Leugering, G., Homogenization of Dirichlet optimal control problems with exact partial controllability constraints. Asymptotic Anal. 57 (2008) 229–249.
P.I. Kogut and T.A. Mel'nyk, Asymptotic analysis of optimal control problems in thick multi-structures, in Generalized Solutions in Control Problems, Proceedings of the IFAC Workshop GSCP-2004, Pereslavl-Zalessky, Russia, September 21–29 (2004) 265–275.
J.E. Lagnese and G. Leugering, Domain decomposition methods in optimal control of partial differential equations, International Series of Numerical Mathematics 148. Birkhäuser Verlag, Basel (2004).
Lenczner, M. and Senouci-Bereski, G., Homogenization of electrical networks including voltage to voltage amplifiers. Math. Meth. Appl. Sci. 9 (1999) 899–932. CrossRef
Leugering, G. and Schmidt, E.J.P.G., On the modelling and stabilization of flows in networks of open canals. SIAM J. Contr. Opt. 41 (2002) 164–180. CrossRef
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Berlin, Springer-Verlag (1971).
Mazja, V. and Slutsckij, A., Averaging of a differential operator on thick periodic grid. Math. Nachr. 133 (1987) 107–133.
Orive, R. and Zuazua, E., Finite difference approximation of homogenization for elliptic equation. Multiscale Model. Simul. 4 (2005) 36–87. CrossRef
Panasenko, G.P., Asymptotic solutions of the elasticity theory system of equations for lattice and skeletal structures. Russian Academy Sci. Sbornik Math. 75 (1993) 85–110. CrossRef
G.P. Panasenko, Homogenization of lattice-like domains. L-convergence. Reprint No. 178, Analyse numérique, Lyon Saint-Étienne (1994).
T. Roubiček, Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin, New York (1997).
J. Saint Jean Paulin and D. Cioranescu, Homogenization of Reticulated Structures, Applied Mathematical Sciences 136. Springer-Verlag, Berlin-New York (1999).
Saint Jean Paulin, J. and Zoubairi, H., Optimal control and “strange term” for the Stokes problem in perforated domains. Portugaliac Mathematica 59 (2002) 161–178.
Vogelius, M., A homogenization result for planar, polygonal networks. RAIRO Modél. Math. Anal. Numér. 25 (1991) 483–514. CrossRef
Zhikov, V.V., Weighted Sobolev spaces. Sbornik: Mathematics 189 (1998) 27–58. CrossRef
Zhikov, V.V., On an extension of the method of two-scale convergence and its applications. Sbornik: Mathematics 191 (2000) 973–1014. CrossRef
Zhikov, V.V., Homogenization of elastic problems on singular structures. Izvestija: Math. 66 (2002) 299–365.
V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin (1994).