##
**Controllability of the linear system of thermoelastic plates.**
*(English)*
Zbl 0853.93017

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 J. Lagnese and 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 decoupled by putting \(\Delta \theta = - \beta_3 \Delta w_t\). (This technique has been suggested in a preprint of a paper of Henry, Lopes and Perissinotto). Then the initial parabolic system is replaced by an elliptic system. A theorem showing that the decoupled system is associated with a continuous and compact semigroup is crucial to the remainder of the paper. The variable \(w\) that solves the decoupled system obeys a well known damped equation for dynamic behavior of a thin plate. The so-called multiplier technique of J. L. Lions is applied to its solution. Analyticity of the coefficients allows the authors to apply Holmgren’s uniqueness theorem. Some delicate estimates based on energy and compactness arguments lead to the proof of the crucial observability inequality.

This is a lengthy paper, one in a series of papers on thermoelastic plate theory by Zuazua, Lions, Lagnese, and their collaborators. It is packed with new results combined with clever application of established functional analytic and semigroup techniques. These papers must be read, or studied in a seminar by young mathematicians wishing to work in the field of elasticity, and particularly in thermoelasticity. Here the energy conserving, and symmetric in time equations of elastic plate or possibly (in the near future) of shell dynamics are coupled with the parabolic equations of heat transfer, which are most sensitive to the direction of time passage. While existence and uniqueness of the solutions to these equations can be proved, as the authors do in the paper under review, it is perhaps also necessary to establish some estimates of the type given by F. John, which take a second look at the usual \(\{\varepsilon, \delta\}\) definition of well-posedness, and the “practical” meaning of uniqueness as viewed by a “practically oriented” engineer, or scientist. In the F. John example the value of \(\delta\) becomes proportional to roughly \(e^{-400} \times \varepsilon\), making such uniqueness and \(\{\varepsilon, \delta\}\) definition of well-posedness of no practical value. Here some of the estimates given by the authors are potentially helpful.

\(w_{tt} - \gamma \Delta w_{tt} + \Delta^2 w + \beta_1 \Delta \theta = 0\) 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 J. Lagnese and 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 decoupled by putting \(\Delta \theta = - \beta_3 \Delta w_t\). (This technique has been suggested in a preprint of a paper of Henry, Lopes and Perissinotto). Then the initial parabolic system is replaced by an elliptic system. A theorem showing that the decoupled system is associated with a continuous and compact semigroup is crucial to the remainder of the paper. The variable \(w\) that solves the decoupled system obeys a well known damped equation for dynamic behavior of a thin plate. The so-called multiplier technique of J. L. Lions is applied to its solution. Analyticity of the coefficients allows the authors to apply Holmgren’s uniqueness theorem. Some delicate estimates based on energy and compactness arguments lead to the proof of the crucial observability inequality.

This is a lengthy paper, one in a series of papers on thermoelastic plate theory by Zuazua, Lions, Lagnese, and their collaborators. It is packed with new results combined with clever application of established functional analytic and semigroup techniques. These papers must be read, or studied in a seminar by young mathematicians wishing to work in the field of elasticity, and particularly in thermoelasticity. Here the energy conserving, and symmetric in time equations of elastic plate or possibly (in the near future) of shell dynamics are coupled with the parabolic equations of heat transfer, which are most sensitive to the direction of time passage. While existence and uniqueness of the solutions to these equations can be proved, as the authors do in the paper under review, it is perhaps also necessary to establish some estimates of the type given by F. John, which take a second look at the usual \(\{\varepsilon, \delta\}\) definition of well-posedness, and the “practical” meaning of uniqueness as viewed by a “practically oriented” engineer, or scientist. In the F. John example the value of \(\delta\) becomes proportional to roughly \(e^{-400} \times \varepsilon\), making such uniqueness and \(\{\varepsilon, \delta\}\) definition of well-posedness of no practical value. Here some of the estimates given by the authors are potentially helpful.

Reviewer: V.Komkov (Roswell)

### MSC:

93B05 | Controllability |

74B05 | Classical linear elasticity |

35B37 | PDE in connection with control problems (MSC2000) |