Identification of thermal expansion coefficient in a thermoelastic plate from final time-measured displacement
We investigate a coupled thermoelastic plate system consisting of a fourth-order displacement equation and a heat evolution equation linked through a spatially varying coupling factor $\alpha(x)$. The model accounts for thermoelastic interactions through the operators $\operatorname{div}(\alpha(x)\nabla \theta)$ and $\operatorname{div}(\alpha(x)\nabla u_t)$. We establish the well-posedness of the direct problem under homogeneous Neumann conditions for $u$ and Dirichlet conditions for $\theta$, deriving optimal energy estimates and demonstrating continuous dependence of solutions on the given data. We further introduce an input-output operator corresponding to the considered inverse problem and show that it is compact and Lipschitz continuous, confirming the ill-posed nature of the associated inverse problem. Using these properties, the inverse problem is formulated as a minimization problem for the Tikhonov functional, and we establish the existence of a minimizer.