Exact controllability of a hybrid system, membrane with strings, on general polygon domains
type: Item
instance of: scholarly article
title: Exact controllability of a hybrid system, membrane with strings, on general polygon domains
zbMATH Open document ID: 0942.93014
DOI: 10.1023/A:1022868815952
author: Bo Liu
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: The author uses very effectively the Hilbert uniqueness method (HUM) pioneered in the late 1980s by J. E. Lagnese, and J. L. Lions and used also by R. Triggiani, I. Lasiecka and their collaborators. The proof of Hilbert concerning uniqueness of solutions to PDEs extended to certain types of inhomogeneous equations with certain types of inner product, was cleverly applied to control theory problems, and in particular to inverse problems of boundary controllability. In this paper, the author considers a hybrid system of a convex polygonal domain occupied by a membrane, which is reinforced on its boundary by strings. The membrane equation \(u_{tt}-\Delta u=0\), valid in \(\Omega\), the string equation: \(\rho u_{tt}- \beta\partial^2 u/\partial\tau^2-\partial u/\partial\nu= 0\) valid on the boundary part \(\Gamma_1\) of \(\partial\Omega\), \(\partial u/\partial\nu= 0\) on the boundary part \(\Gamma_2\), and \(u= 0\) on \(\Gamma_3\), respectively, define the mathematical model for the time period \([0,T]\). A Neumann-type boundary control is considered. For \(T\) sufficiently large (as is usually the case for boundary input) the author derives an energy inequality for the inverse boundary control problem, subject to some geometric constraints. This result establishes exact boundary controllability. He also considers a Carleman-type interior energy inequality for Neumann controls, and shows that for certain types of geometric conditions, the unique solution vanishes at a given, but sufficiently large, time \(T\).
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 74M05
Mathematics Subject Classification ID: 74K05
zbMATH DE Number: 1431854
zbMATH Keywords: exact controllability
zbMATH Keywords: Hilbert uniqueness method
zbMATH Keywords: boundary controllability
zbMATH Keywords: hybrid system
zbMATH Keywords: membrane
zbMATH Keywords: strings
zbMATH Keywords: boundary control
MaRDI profile type: Publication
full work available at URL: a:1022868815952
OpenAlex ID: W2228516742
description: scientific article; zbMATH DE number 1431854