Homogenization of optimal control problems in perforated domains via periodic unfolding method

References

• Durante T, Faella L, Perugia C. Homogenization and behavior of optimal controls for the wave equation in domains with oscillating boundary. Nonlinear Differ. Equ. Appl. 2007;14:455–489.
   Web of Science ®Google Scholar
• Durante T, Mel’nyk TA. Homogenization of quasilinear optimal control problem involving a thick multilevel junction of type 3:2:1. ESAIM Control Optim. Calc. Var. 2012;18:583–610.
   Web of Science ®Google Scholar
• Buttazzo G, Dal Maso G. Γ-convergence and optimal control problems. J. Optim. Theory Appl. 1982;38:385–407.
   Web of Science ®Google Scholar
• Buttazzo G. Γ-convergence and its applications to some problem in the calculus of variations. Vol. 1993, School on homogenization. Trieste: ICTP; 1993. p. 38–61.
   Google Scholar
• Kesavan S, Muthukumar T. Homogenization of an optimal control problem with state constraints. Differ. Equ. Dyn. Syst. 2011;19:361–374.
   Google Scholar
• Kesavan S, Saint Jean Paulin J. Optimal control on perforated domains. J. Math. Anal. Appl. 1999;229:563–586.
   Web of Science ®Google Scholar
• Muthukumar T, Nandakumaran AK. Homogenization of low-cost control problems on perforated domains. J. Math. Anal. Appl. 2009;351:29–42.
   Web of Science ®Google Scholar
• Briane M, Damlamian A, Donato P. H-convergence in perforated domains. In: Cioranescu D, Lions JL, editors. Non-linear partial differential equations and their applications, Collége de France seminar. Vol. XIII, Pitman research notes in mathematics series. New York (NY): Longman; 1998. p. 62–100.
   Google Scholar
• Kesavan S, Rajesh M. Homogenization of periodic optimal control problems via multi-scale convergence. Proc. Indian Acad. Sci. (Math. Sci.) 1998;108:189–207.
   Web of Science ®Google Scholar
• Nandakumaran AK, Prakash R. Homogenization of boundary optimal control problems with oscillating boundary. Nonlinear Stud. Int. J. 2013;20:1–25.
   Google Scholar
• Nandakumaran AK, Prakash R, Raymond JP. Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries. Ann. Univ. Ferrara. 2012;58:143–166.
   Google Scholar
• Nandakumaran AK, Prakash R, Raymond J-P. Stokes’ system in a domain with oscillating boundary: homogenization and error analysis of an interior optimal control problem. Numer. Funct. Anal. Optim. 2013;35:323–355.
   Web of Science ®Google Scholar
• Prakash R. Optimal control problem for the time-dependent Kirchhoff–Love plate in a domain with rough boundary. Asymptotic Anal. 2013;81:337–355.
   Web of Science ®Google Scholar
• Zoubairi H. Optimal control and homogenization in a mixture of fluids separated by a rapidly oscillating interface. Electron. J. Differ. Equ. 2002;1–32.
   Google Scholar
• Allaire G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 1992;23:1482–1518.
   Web of Science ®Google Scholar
• Cabarrubias B, Donato P. Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Applicable Anal. 2011;91:1111–1127. doi:10.1080/00036811.2011.619982.
   Web of Science ®Google Scholar
• Cioranescu D, Saint Jean Paulin J. Homogenization of reticulated structures. Vol. 136, Applied mathematical sciences. New York (NY): Springer; 1999.
   Google Scholar
• Conca C, Diaz JI, Linan A, et al. Homogenization in chemical reactive flows. Electron. J. Differ. Equ. 2004;40:1–22.
   Google Scholar
• Conca C, Vanninathan M. Homogenization of periodic structures via Bloch decomposition. SIAM. J. Appl. Math. 1997;57:1639–1659.
   Web of Science ®Google Scholar
• Prakash R, Sili A. Asymptotic behavior of the solutions of a degenerating elliptic equation in a thin multidomain. Asymptotic Anal. 2014;90:345–365.
   Web of Science ®Google Scholar
• Cioranescu D, Donato P. An introduction to homogenization. New York: Oxford University Press; 1999.
   Google Scholar
• Hocking L. Optimal control, an introduction to the theory and applications. Oxford: Clarendon Press; 1991.
   Google Scholar
• Kogut P, Leugering G. Optimal control problems for partial differential equations on reticulated domains. New York: Birkhauser; 2011; Principle of the fictitious homogenization. Asymptotic Anal. 2001;26:37–72.
   Google Scholar
• Lions J-L. Optimal control of systems governed by partial differential equations. Berlin: Springer-Verlag; 1971.
   Google Scholar
• Macki J, Strauss A. Introduction to optimal control theory. New York (NY): Springer-Verlag; 1982.
   Google Scholar
• Abergel F, Temam R. On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1990;1:303–325.
   Google Scholar
• Bewley T, Moin P. Optimal control of turbulent channel flows, in active control of vibration and noise. ASME DE. DE-Vol. 75;1994. p.221–227. (Wang KW, Von Flotow AH, Shoureshi R, Hendricks EW, Farabee TW, editors).
   Google Scholar
• Blum J. Numerical simulation and optimal control in plasma physics, with applications to tokamaks. Chichester: Wiley; 1989.
   Google Scholar
• Christensen GS, El-Hawary ME, Soliman SA. Optimal control applications in electric power systems. New York (NY): Plenum; 1987.
   Google Scholar
• Ito K, Ravindran SS. Optimal control of thermally convected fluid flows. SIAM J. Sci. Comput. 1998;19:1847–1849.
   Web of Science ®Google Scholar
• Ramirez WF. Application of optimal control theory to enhanced oil recovery. Amsterdam: Elsevier; 1987.
   Google Scholar
• Cioranescu D, Damlamian A, Griso G. Periodic unfolding and homogenization. C.R. Acad. Sci. Paris, Ser. I. 2002;335:99–104.
   Web of Science ®Google Scholar
• Cioranescu D, Damlamian A, Griso G. The periodic unfolding and homogenization. SIAM J. Math. Anal. 2008;40:1585–1620.
   Web of Science ®Google Scholar
• Cioranescu D, Donato P, Zaki R. The periodic unfolding and Robin problems in perforated domains. C.R.A.S. Paris, Série 1. 2006;342:467–474.
   Google Scholar
• Cioranescu D, Donato P, Zaki R. The periodic unfolding method in perforated domains. Portugal Math. 2006;63:476–496.
   Google Scholar
• Cioranescu D, Donato P, Zaki R. Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions. Asymptotic Anal. 2007;53:209–235.
   Web of Science ®Google Scholar
• Cioranescu D, Damlamian A, Donato P, et al. The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 2012;44:718–760.
   Web of Science ®Google Scholar
• Donato P, Yang Z. The periodic unfolding method for the wave equation in domains with holes. Adv. Math. Sci. Appl. 2012;22:521–551.
   Google Scholar
• Nandakumaran AK, Prakash R, Sardar BC. Homogenization of an optimal control problem in a domain with highly oscillating boundary using periodic unfolding method. MESA. 2013;4:281–303.
   Google Scholar
• Raymond JP. Optimal control of partial differential equations. Universitè Paul Sabatier. Available from: www.math.univ-toulouse.fr.
   Google Scholar
• Cabarrubias B. Existence, uniqueness and homogenization results for a class of nonlinear PDE in perforated domains [Doctoral dissertation]. University of the Philippines Diliman and University of Rouen; 2012.
   Google Scholar
• Bensoussan A, Lions J-L, Papanicolaou G. Asymptotic analysis for periodic structures. Amsterdam: North-Holland; 1978.
   Google Scholar