Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 2, pp. 471-498.

We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.

DOI: 10.1051/cocv:2008037
Classification: 35B27, 35J25, 49J20, 93C20
Keywords: optimal control, homogenization, elliptic equation, periodic graph
@article{COCV_2009__15_2_471_0,
     author = {Kogut, Peter I. and Leugering, G\"unter},
     title = {Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {471--498},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {2},
     year = {2009},
     doi = {10.1051/cocv:2008037},
     mrnumber = {2513095},
     zbl = {1173.35015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008037/}
}
TY  - JOUR
AU  - Kogut, Peter I.
AU  - Leugering, Günter
TI  - Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 471
EP  - 498
VL  - 15
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008037/
DO  - 10.1051/cocv:2008037
LA  - en
ID  - COCV_2009__15_2_471_0
ER  - 
%0 Journal Article
%A Kogut, Peter I.
%A Leugering, Günter
%T Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 471-498
%V 15
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008037/
%R 10.1051/cocv:2008037
%G en
%F COCV_2009__15_2_471_0
Kogut, Peter I.; Leugering, Günter. Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 2, pp. 471-498. doi : 10.1051/cocv:2008037. http://www.numdam.org/articles/10.1051/cocv:2008037/

[1] H. Attouch, Variational Convergence for Functional and Operators, Applicable Mathematics Series. Pitman, Boston-London (1984). | MR | Zbl

[2] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). | MR | Zbl

[3] G. Bouchitte and I. Fragala, Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32 (2001) 1198-1226. | MR | Zbl

[4] A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002). | MR

[5] 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.

[6] G. Buttazzo and G. Dal Maso, Γ-convergence and optimal control problems. J. Optim. Theory Appl. 32 (1982) 385-407. | MR | Zbl

[7] J. Casado-Diaz, M. Luna-Laynez and J.D. Marin, 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. | MR | Zbl

[8] G. Chechkin, V. Zhikov, D. Lukkassen and A. Piatnitski, On homogenization of networks and junctions. J. Asymp. Anal. 30 (2000) 61-80. | MR | Zbl

[9] 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. | MR | Zbl

[10] D. Cioranescu, P. Donato and E. Zuazua, Exact boundary controllability for the wave equation in domains with small holes. J. Math. Pures Appl. 69 (1990) 1-31. | MR | Zbl

[11] C. Conca, A. Osses and J. Saint Jean Paulin, A semilinear control problem involving in homogenization. Electr. J. Diff. Equ. (2001) 109-122. | MR | Zbl

[12] G. Dal Maso, An Introduction of Γ-Convergence. Birkhäuser, Boston (1993). | MR | Zbl

[13] A. Haraux and F. Murat, Perturbations singulières et problèmes de contrôle optimal : deux cas bien posés. C. R. Acad. Sci. Paris Sér. I 297 (1983) 21-24. | MR | Zbl

[14] A. Haraux and F. Murat, Perturbations singulières et problèmes de contrôle optimal : un cas mal posé. C. R. Acad. Sci. Paris Sér. I 297 (1983) 93-96. | MR | Zbl

[15] S. Kesavan and M. Vanninathan, L'homogénéisation d'un problème de contrôle optimal. C. R. Acad. Sci. Paris Sér. A-B 285 (1977) 441-444. | MR | Zbl

[16] S. Kesavan and J. Saint Jean Paulin, Optimal control on perforated domains. J. Math. Anal. Appl. 229 (1999) 563-586. | MR | Zbl

[17] P.I. Kogut, S-convergence in homogenization theory of optimal control problems. Ukrain. Matemat. Zhurnal 49 (1997) 1488-1498 (in Russian). | MR | Zbl

[18] P.I. Kogut and G. Leugering, Homogenization of optimal control problems in variable domains. Principle of the fictitious homogenization. Asymptotic Anal. 26 (2001) 37-72. | MR | Zbl

[19] P.I. Kogut and G. Leugering, Asymptotic analysis of state constrained semilinear optimal control problems. J. Optim. Theory Appl. 135 (2007) 301-321. | MR | Zbl

[20] P.I. Kogut and G. Leugering, Homogenization of Dirichlet optimal control problems with exact partial controllability constraints. Asymptotic Anal. 57 (2008) 229-249. | MR | Zbl

[21] 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.

[22] 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). | MR | Zbl

[23] M. Lenczner and G. Senouci-Bereski, Homogenization of electrical networks including voltage to voltage amplifiers. Math. Meth. Appl. Sci. 9 (1999) 899-932. | MR | Zbl

[24] G. Leugering and E.J.P.G. Schmidt, On the modelling and stabilization of flows in networks of open canals. SIAM J. Contr. Opt. 41 (2002) 164-180. | MR | Zbl

[25] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Berlin, Springer-Verlag (1971). | MR | Zbl

[26] V. Mazja and A. Slutsckij, Averaging of a differential operator on thick periodic grid. Math. Nachr. 133 (1987) 107-133. | Zbl

[27] R. Orive and E. Zuazua, Finite difference approximation of homogenization for elliptic equation. Multiscale Model. Simul. 4 (2005) 36-87. | MR | Zbl

[28] G.P. Panasenko, Asymptotic solutions of the elasticity theory system of equations for lattice and skeletal structures. Russian Academy Sci. Sbornik Math. 75 (1993) 85-110. | MR | Zbl

[29] G.P. Panasenko, Homogenization of lattice-like domains. L-convergence. Reprint No. 178, Analyse numérique, Lyon Saint-Étienne (1994). | Zbl

[30] T. Roubiček, Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin, New York (1997). | MR | Zbl

[31] J. Saint Jean Paulin and D. Cioranescu, Homogenization of Reticulated Structures, Applied Mathematical Sciences 136. Springer-Verlag, Berlin-New York (1999). | MR | Zbl

[32] J. Saint Jean Paulin and H. Zoubairi, Optimal control and “strange term” for the Stokes problem in perforated domains. Portugaliac Mathematica 59 (2002) 161-178. | MR | Zbl

[33] M. Vogelius, A homogenization result for planar, polygonal networks. RAIRO Modél. Math. Anal. Numér. 25 (1991) 483-514. | Numdam | MR | Zbl

[34] V.V. Zhikov, Weighted Sobolev spaces. Sbornik: Mathematics 189 (1998) 27-58. | MR | Zbl

[35] V.V. Zhikov, On an extension of the method of two-scale convergence and its applications. Sbornik: Mathematics 191 (2000) 973-1014. | MR | Zbl

[36] V.V. Zhikov, Homogenization of elastic problems on singular structures. Izvestija: Math. 66 (2002) 299-365. | Zbl

[37] V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin (1994). | MR | Zbl

Cited by Sources: