Uniform stabilization and simultaneous exact boundary controllability for a pair of hyperbolic systems
type: Item
instance of: scholarly article
title: Uniform stabilization and simultaneous exact boundary controllability for a pair of hyperbolic systems
zbMATH Open document ID: 0855.93063
DOI: 10.1007/BF02106615
published in: Siberian Mathematical Journal
publication date: 1996-10-15T00:00:00Z
review text: The following mixed problem in the cylinder \(\Omega\times (0, T)\) is considered: \[ {\partial^2 u\over \partial t^2}- {\partial\over \partial x_i} \Biggl(A_{ij} {\partial u\over \partial x_j}\Biggr)= 0,\quad {\partial^2 v\over \partial t^2}- {\partial\over \partial x_i} \Biggl(A_{ij} {\partial v\over \partial x_j}\Biggr)= 0, \] \[ u|_{t= 0}= \varphi(x),\;v|_{t= 0}= \psi(x),\;{\partial u\over \partial t}\Biggl|_{t= 0}= \varphi_1(x),\;{\partial v\over \partial t}\Biggr|_{t= 0}= \psi_1(x), \] \[ A_{ij} {\partial u\over \partial x_j} \nu_i+ au+ bu_t+ \xi v_t|_{S_0}= 0,\quad u|_{S_1}= 0, \] \[ A_{ij} {\partial v\over \partial x_j} \nu_i+ cv+ dv_t+ \xi u_t|_{S_0}= 0,\quad v|_{S_1}= 0, \] where the smooth boundary \(S\) of \(\Omega\) consists of two disjoint closed surfaces \(S_0\) and \(S_1\), \(A_{ij}= A^*_{ji}\) are square matrices of order \(m\), \(\nu= (\nu_1, \nu_2,\dots, \nu_n)\) is the unit outward normal to the boundary \(S\) and \(a\), \(b\), \(c\), \(d\), \(\xi\) are constants. Using a basic energy identity, the author constructs some stabilizing operators for the above pair of second-order hyperbolic systems; namely, for \(b> 0\), \(d> 0\) the stabilizing boundary operator represents \(2n\) scalar conditions and for \(b> 0\), \(d= 0\), the stabilizing operator is determined by only \(n\) scalar conditions. In this way, he proves the uniform simultaneous stabilization of solutions to the corresponding mixed problems. According to the uniform stabilization with a stabilizing operator having \(n\) scalar conditions, one obtains the exact controllability of the corresponding evolution system with control in only \(n\) boundary conditions on \(S_0\). Finally, the author considers the case of a pair of systems from elasticity theory.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35L99
zbMATH DE Number: 917092
zbMATH Keywords: mixed problem
zbMATH Keywords: energy identity
zbMATH Keywords: second-order hyperbolic systems
zbMATH Keywords: stabilizing boundary operator
zbMATH Keywords: uniform simultaneous stabilization
zbMATH Keywords: exact controllability
author: Boris V. Kapitonov
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 917092