Controllability of the linear system of thermoelasticity. (English) Zbl 0846.93008

The author considers a classical thermoelastic system modelled by the equations: \[ \begin{cases} u_{tt}- \mu \Delta u- (\lambda+ \mu) \nabla\text{ div } u+ \alpha V\theta= 0 & \text{in } \Omega\times (0, \infty),\\ \theta_t- \Delta \theta+ \beta\text{ div } u_t= 0 & \text{in } \Omega\times (0, \infty),\\ u= \theta= 0\quad & \text{on } \Gamma\times (0, \infty),\text{ where } \Gamma= \partial\Omega) \text{(of class } C^2)\end{cases}\tag{1} \] and \(u(x, 0)= u^0(x)\), \(u_t(x, 0)= u^1(x)\), \(\theta(x, 0)= \theta^0(x)\). \(\lambda\), \(\mu\) are the usual Lammé constants, while \(\alpha\), \(\beta\) are coupling constants. It is well-known that this system is well posed in the space: \(H= (H^1(\Omega))^n\times (L^2(\Theta))^n\times (L^2(\Omega))\). Moreover, it is known that there exists a unique continuous solution on some interval of time \((0, T)\). Let \(S(t)= \{u, u_t, \theta\}\) denote that solution. Then for \(t> 0\), \(S(t): H\to H\) is a strongly semigroup generated by the original system of partial differential equations (1). A control \(f_{\chi^{\overline\omega}}\) is introduced restricted to the region \(\overline\omega\). An exact controllability problem is posed. However, the author points out that the irreversibility of the system and the regularizing effect of the temperature imply that in general the exact controllability is not possible to attain. This approximate-exact controllability is sought. Let \(\overline\omega\) be a neighborhood of \(\Gamma\). It is generally true that exact controllability of the heat equation demands a time period proportional to the square root of the constant \(\mu\).
The author’s first theorem states that the system is approximately-exact controllable in time \(T\) if \(T> \text{diam}(\Omega \backslash \overline\omega)/ \sqrt\mu\). A simple result of this type is obtained for the one-dimensional case. Next, the author states and proves the continuity and compactness of the operator \(S(t)\) for a decoupled system. The proofs are based on the use of the multiplier technique, compactness arguments, Holmgren’s uniqueness theorem, and a result of Henry, Lopes and Perissinotto (HLP) [to be published]. For the sake of completeness, the author offers a proof of the HLP theorem. As in previous recent works of the same author, this paper contains many interesting details, and deserves a serious study of scientists interested in the control of thermoelastic systems.
Reviewer: V.Komkov (Roswell)


93B05 Controllability
74B10 Linear elasticity with initial stresses
35B37 PDE in connection with control problems (MSC2000)