Controllability of the linear system of thermoelastic plates
type: Item
instance of: scholarly article
title: Controllability of the linear system of thermoelastic plates
zbMATH Open document ID: 0853.93017
author: Luz de Teresa
author: Enrique Zuazua
published in: Advances in Differential Equations
publication date: 1997-01-12T00:00:00Z
review text: The paper under review is a continuation of studies by Professor Zuazua and his collaborators of control theory for a linear model of thermoelastic plates given by equations: \(w_{tt} - \gamma \Delta w_{tt} + \Delta^2 w + \beta_1 \Delta \theta = 0\) in \(Q\) (or \(w_{tt} - \gamma \Delta w_{tt} + \Delta^2 w + \beta_1 \Delta \theta = f\) in \(Q\), if control \(f(.,t)\) is applied) \(\beta_2 \theta_t - \Delta \theta - \beta_3 \Delta w_t = 0\) in \(Q\), with boundary conditions: \(w = \partial w/ \partial \nu = 0\) on \(\Sigma\), and initial conditions \(w(0) = w^0\), \(w_t (0) = w^1 \); \(\theta (0) = \theta^0\) in \(\Omega\). Here \(Q = \Omega \times (0,T)\), and \(\Sigma = \partial \Omega \times (0,T)\). \(\Omega \in \mathbb{R}^2\) is a bounded, open, connected set. A partially heuristic justification of the mathematical model presented above was given by \textit{J. Lagnese} and \textit{J. L. Lions} in their book [Modelling analysis and control of thin plates, Paris (1988; Zbl 0662.73039)]. It is known that for every initial data, as given above, there exists a unique solution in \(C([0,T],V)\), where \(V = H^2_0 (\Omega) \times H^1_0 (\Omega) \times L^2 (\Omega)\). Let \(\omega\) denote a control region. Then, the following theorem is proved: Let \(\omega\) be a neighborhood of \(\partial \Omega\). Suppose that \(T > \sqrt \gamma \text{diam} (\Omega \backslash \omega)\). Then the system is exact-approximately controllable (meaning exact in displacement, approximate in time). The key to the argument is the observability inequality \(\delta \leq \int_{[0,T]} \{\int_\Omega |\nabla \varphi |^2 dx\}dt\), where \(\{\varphi, \psi\}\) is a solution to the adjoint system \[ \varphi_{tt} - \gamma \Delta \varphi_{tt} + \Delta^2 \varphi + \beta_3 \Delta \psi_t = 0 \quad \text{in }Q \] \[ -\beta_2 \psi_t - \Delta \psi + \beta_1 \Delta \varphi = 0 \] in \(Q\), with similar final time conditions, and zero boundary conditions. The authors observe that the original equations can be …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 35B37
zbMATH DE Number: 889014
zbMATH Keywords: decoupling
zbMATH Keywords: thermoelastic plates
zbMATH Keywords: controllable
zbMATH Keywords: observability
zbMATH Keywords: adjoint system
zbMATH Keywords: multiplier technique
zbMATH Keywords: Holmgren's uniqueness theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 889014