Approximate Controllability of a Parabolic
System with Imperfect Interfaces
Patrizia Donato1, and Editha C. Jose2
1Normandie Universit´e, Universit´e de Rouen, Laboratoire de Math´ematiques Rapha¨el Salem, CNRS, UMR 6085, Avenue de l’Universit´e, BP 12, 76801 Saint-E´tienne du Rouvray cedex, France
2Institute of Mathematical Sciences and Physics, UP Los Baños,College, Los Baños Laguna, Philippines
In this paper, we complete the investigation of the asymptotic behavior of the approximate control for a parabolic equation with periodic rapidly oscillating coefficients depending on a parameter γ and modeling composites with interfacial resistance. The approximate control and its asymptotic behavior as ε → 0 for the case −1 < γ ≤ 1was done recently by the authors in DONATO P. and JOSE E. 2015. Asymptotic Behavior of the Approximate Controls for Parabolic Equations with Interfacial Contact resistance. ESAIM: COCV 21(1): 100-127]. We considered here the remaining case γ ≤ −1. The corrector results for the latter case given in [YANG Z. 2014. The Periodic Unfolding Method for a Class of Parabolic Problems with Imperfect Interfaces. ESAIM: M2AN 48: 1279-1302] play an important role when proving this result. Following an idea introduced by J.-L. Lions, the approximate control is constructed as the solutions of a related transposed problem having as final data the (unique) minimum point of a suitable functional. Then we showed that the control and the corresponding solution of the periodic problem converge respectively to the control and to the solution of the homogenized problem. Let us mention here that one of the main difficulties is to find the appropriate limit functionals in order to obtain the convergence results. This study addresses the problem of homogenization in the context of controllability and vice-versa, showing the interplay of two approaches in the study of partial differential equations.
In this paper, we study the asymptotic behavior, as , of an approximate control for a system of linear parabolic equations with rapidly oscillating coecients on an “-periodic two-component composite.
More precisely, let ! to be a given open non-empty . . . . . read more
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