ASYMPTOTIC EXPANSION AND CONVERGENCE THEOREM OF CONTROL AND OBSERVATION ON THE BOUNDARY FOR SECOND-ORDER ELLIPTIC EQUATION WITH HIGHLY OSCILLATORY COEFFICIENTS

In this paper, we shall study systems governed by the Neumann problem of second-order elliptic equation with rapidly oscillating coefficients and with control and observations on the boundary. The multiscale asymptotic expansions of the solution for considering problem in the case without any constraints, and homogenized equation in the case with constraints will be given, their rigorous proofs will also be proposed.

[1]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1966 .

[2]  P. Grisvard,et al.  BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN , 1976 .

[3]  J. L. Lions,et al.  INTRODUCTION TO “REMARKS ON THE THEORY OF OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS” , 1977 .

[4]  A. Aziz,et al.  Control theory of systems governed by partial differential equations , 1977 .

[5]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[6]  Jean-Pierre Puel,et al.  Approximate controllability of the semilinear heat equation , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  S. Kesavan,et al.  Homogenization of an Optimal Control Problem , 1997 .

[8]  Axel Osses,et al.  On the controllability of the Laplace equation observed on an interior curve , 1998 .

[9]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[10]  S. Kesavan,et al.  Optimal Control on Perforated Domains , 1999 .

[11]  Jun-zhi Cui,et al.  Multiscale Asymptotic Analysis and Numerical Simulation for the Second Order Helmholtz Equations with Rapidly Oscillating Coefficients Over General Convex Domains , 2002, SIAM J. Numer. Anal..

[12]  Jun-zhi Cui,et al.  Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains , 2004, Numerische Mathematik.