SIMULTANEOUS IDENTIFICATION OF SPATIAL LOAD AND EXTERNAL HEAT SOURCE IN THERMOELASTIC PLATE FROM FINAL TIME MEASURED DISPLACEMENT

In this paper, we study the inverse problem of simultaneously iden-tifying the mechanical load F(x, t) and heat source G(x, t) in structurally damped thermoelastic plate equations describing a homogeneous and elas-tically as well as thermally isotropic plate, from the vertical displacement uT(x) = u(x,T) measured at the final time T > 0. We establish the well-posedness of the initial boundary value problem and corresponding adjoint problem by using Galerkin's approximation method. The inverse problem is reformulated as a minimization problem for the Tikhonov functional using the Tikhonov regularization method. We prove that the regularized Tikhonov functional admits a unique solution in the naturally defined set of admissible sources. Furthermore, the Frechet differentiability of this functional is proved, and an explicit gradient formula is derived through the weak solution of the corresponding adjoint problem. An upper bound for the final time T > 0 is established to derive the stability estimate for the inverse problem by invok-ing a first-order necessary optimality condition for the minimization problem. This stability result also gives rise to the uniqueness of the solution to the regularized inverse problem. The results obtained in this paper help to ana-lyze the influence of thermal and mechanical loading that results in materials deflection, which, in turn, is vital in terms of physical applications.