Abstract

An optimal control problem in a two-dimensional domain with a rapidly oscillating boundary is considered. The main features of this article are on two points, namely, we consider periodic controls in the thin periodic slabs of period $\epsilon >0$, a small parameter, and height $O(1)$ in the oscillatory part, and the controls are characterized using unfolding operators. We then do a homogenization analysis of the optimal control problems as $\epsilon \rightarrow 0$ with $L^2$ as well as Dirichlet (gradient-type) cost functionals.

Keywords

  1. optimal control and optimal solution
  2. homogenization
  3. oscillating boundary
  4. internal periodic control
  5. adjoint system
  6. unfolding operator
  7. boundary unfolding

MSC codes

  1. 35B27
  2. 35B40
  3. 35B37
  4. 49J20
  5. 49K20

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References

1.
Y. Achdou, O. Pironneau, and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys., 147 (1998), pp. 187--218.
2.
G. Allaire and M. Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var., 4 (1999), pp. 209--243.
3.
Y. Amirat and O. Bodart, Boundary layer correctors for the solution of Laplace equation in a domain with oscillating boundary, Z. Anal. Anwend., 20 (2001), pp. 929--940.
4.
Y. Amirat, O. Bodart, U. De Maio, and A. Gaudiello, Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), pp. 1598--1616.
5.
J. M. Arrieta and S. M. Bruschi, Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation, Math. Models Methods Appl. Sci., 17 (2007), pp. 1555--1585.
6.
V. Barbu, Mathematical Methods in Optimization of Differential Systems, Math. Appl. 310, Kluwer Academic, Dordrecht, 1994.
7.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
8.
A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), pp. 53--157.
9.
J. F. Bonder, R. Orive, and J. D. Rossi, The best Sobolev trace constant in a domain with oscillating boundary, Nonlinear Anal., 67 (2007), pp. 1173--1180.
10.
R. Brizzi and J.-P. Chalot, Boundary homogenization and Neumann boundary value problem, Ric. Mat., 46 (1997), pp. 341--387.
11.
D. Bucur, E. Feireisl, S̆. Nečasová, and J. Wolf, On the asymptotic limit of the Navier--Stokes system on domains with rough boundaries, J. Differential Equations, 244 (2008), pp. 2890--2908.
12.
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math. Appl. 17, Oxford University Press, New York, 1999.
13.
D. Cioranescu, A. Damlamian, and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 99--104.
14.
D. Cioranescu, A. Damlamian, and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), pp. 1585--1620.
15.
A. Damlamian, An elementary introduction to periodic unfolding, in Multi Scale Problems and Asymptotic Analysis, GAKUTO Internat. Ser. Math. Sci. Appl. 24, 2006, pp. 119--136.
16.
A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete Contin. Dyn. Syst., 23 (2009), pp. 197--219.
17.
T. Durante, L. Faella, and C. Perugia, Homogenization and behavior of optimal controls for the wave equation in domains with oscillating boundary, NoDEA Nonlinear Differential Equations Appl., 14 (2007), pp. 455--489.
18.
A. C. Esposito, P. Donato, A. Gaudiello, and C. Picard, Homogenization of the p-Laplacian in a domain with oscillating boundary, Comm. Appl. Nonlinear Anal., 4 (1997), pp. 1--23.
19.
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier--Stokes Equations, Linearized Steady Problems, Vol. I, Springer Tracts Nat. Philos. 38, Springer-Verlag, Berlin, 1994.
20.
A. Gaudiello, Asymptotic behaviour of non-homogeneous Neumann problems in domains with oscillating boundary, Ric. Mat., 43 (1994), pp. 239--292.
21.
A. Gaudiello, R. Hadiji, and C. Picard, Homogenization of the Ginzburg--Landau equation in a domain with oscillating boundary, Commun. Appl. Anal., 7 (2003), pp. 209--223.
22.
V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
23.
S. Kesavan and J. Saint Jean Paulin, Homogenization of an optimal control problem, SIAM J. Control Optim., 35 (1997), pp. 1557--1573.
24.
S. Kesavan, Optimal control on perforated domains, J. Math. Anal. Appl., 229 (1999), pp. 563--586.
25.
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer-Verlag, Berlin, 1971.
26.
J.-L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Gordon and Breach Science, New York, 1981.
27.
J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1--68.
28.
J.-L. Lions, Exact Controllability, Perturbations and Stabilization of Distributed Systems, Vol. 1, Res. Appl. Math. 8, Masson, Paris, 1988.
29.
J.-L. Lions, Exact Controllability, Perturbations and Stabilization of Distributed Systems, Vol. 2, Res. Appl. Math. 9, Masson, Paris, 1988.
30.
U. De Maio, A. Gaudiello, and C. Lefter, Optimal control for a parabolic problem in a domain with highly oscillating boundary, Appl. Anal., 83 (2004), pp. 1245--1264.
31.
U. De Maio and A. K. Nandakumaran, Exact internal controllablity for a hyperbolic problem in a domain with highly oscillating boundary, Asymptot. Anal., 83 (2013), pp. 189--206.
32.
T. Muthukumar and A. K. Nandakumaran, Darcy-type law associated to an optimal control problem, Electron. J. Differential Equations, 2008 (2008), 12.
33.
T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problem on perforated domains, J. Math. Anal. Appl., 351 (2009), pp. 29--42.
34.
A. K. Nandakumaran and R. Prakash, Homogenization of boundary optimal control problems with oscillating boundaries, Nonlinear Stud., 20 (2013), pp. 401--425.
35.
A. K. Nandakumaran, R. Prakash, and J.-P. Raymond, Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries, Ann. Univ. Ferrara Sez. VII Sci. Mat., 58 (2012), pp. 143--166.
36.
A. K. Nandakumaran, R. Prakash, and J.-P. Raymond, Stokes system in a domain with oscillating boundary: Homogenization and error analysis of an interior optimal control problem, Numer. Funct. Anal. Optim., 35 (2014), pp. 323--355.
37.
R. Prakash, Optimal control problem for the time-dependent Kirchhoff--Love plate in a domain with rough boundary, Asymptot. Anal., 81 (2013), pp. 337--355.
38.
R. Prakash and A. Sili, Asymptotic behavior of the solutions of a degenerating elliptic equation in a domain with oscillating boundary, Asymptot. Anal., 90 (2014), pp. 345--365.
39.
J.-P. Raymond, Optimal Control of Partial Differential Equations, Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France; also available online from http://www.math.univ-toulouse.fr/\string raymond/book-ficus.pdf.
40.
L. Tartar, The general theory of homogenization---A personalized introduction, Lect. Notes Unione Mat. Ital. 7, Springer-Verlag, Berlin, 2009.

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 3245 - 3269
ISSN (online): 1095-7138

History

Submitted: 5 November 2014
Accepted: 10 July 2015
Published online: 27 October 2015

Keywords

  1. optimal control and optimal solution
  2. homogenization
  3. oscillating boundary
  4. internal periodic control
  5. adjoint system
  6. unfolding operator
  7. boundary unfolding

MSC codes

  1. 35B27
  2. 35B40
  3. 35B37
  4. 49J20
  5. 49K20

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