Exact internal controllability in performed domains
type: Item
instance of: scholarly article
title: Exact internal controllability in performed domains
zbMATH Open document ID: 0627.35057
author: D. Cioranescu
author: P. Donato
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1989-01-01T00:00:00Z
review text: We consider a system described by a generalized wave equation in the cylinder \(\Omega _{\epsilon}\times (0,T)\) where \(\Omega _{\epsilon}\) is obtained by removing a number \(N_{\epsilon}\) of holes periodically distributed of size \(\epsilon\), from a fixed domain \(\Omega\). Hilbert uniqueness method (HUM) gives an exact control \(v_{\epsilon}\) for every fixed \(\epsilon\). This means we can act on the system to drive it to rest after a finite time T, arbitrarily chosen. If \(\epsilon\) \(\to 0\) (i.e. \(N_{\epsilon}\to \infty)\) the main questions are: does the control \(v_{\epsilon}\) converge to a limit v, and if so, does v control a limit system? In this work we give a positive answer to these questions in the case where \(v_{\epsilon}\) is an internal control, i.e., \(v_{\epsilon}\) is applied on the whole \(\Omega _{\epsilon}\). Let \(\epsilon\) \(\to 0\) be a homogenization process: the initial state equation with rapidly oscillating coefficients defined on a domain depending on \(\epsilon\), is replaced by a homogenized equation with constant coefficients, given on the whole domain \(\Omega\). We prove that \(v_{\epsilon}\) converges to a function f which is an exact control for the homogenized equation. In order to prove this result we give preliminary homogenization theorems for the wave equation in perforated domain.
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 4019603
zbMATH Keywords: generalized wave equation
zbMATH Keywords: Hilbert uniqueness method
zbMATH Keywords: exact control
zbMATH Keywords: limit system
zbMATH Keywords: internal control
zbMATH Keywords: homogenization process
zbMATH Keywords: initial state equation
zbMATH Keywords: rapidly oscillating coefficients
zbMATH Keywords: constant coefficients
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4019603