Determination of a spatial load in a damped Kirchhoff–Love plate equation from final time measured data

Abstract

Abstract In this paper, we study the inverse problem of determining an unknown spatial load F(x) in the damped non-homogeneous isotropic rectangular Kirchhoff–Love plate equation ρ h ( x ) u t t + μ ( x ) u t + D ( x ) ( u x 1 x 1 + ν u x 2 x 2 ) x 1 x 1 + D ( x ) ( u x 2 x 2 + ν u x 1 x 1 ) x 2 x 2 + 2 ( 1 − ν ) D ( x ) u x 1 x 2 x 1 x 2 = F ( x ) G ( t ) , ( x , t ) ∈ Ω × 0 , T from final time measurement data u T (x) = u(x, T). Using the quasi-solution approach, the inverse problem is posed as a least square minimization problem of the Tikhonov functional, and the existence of minimum is shown. We prove that this functional is Fréchet differentiable and the derivative is written in terms of an adjoint problem associated with the Kirchhoff–Love plate equation. We establish sufficient conditions on the final time T and a lower bound of the damping parameter μ(x) to derive stability estimates for the determination of F(x) by invoking a first-order necessary optimality condition of the minimization problem. By the method of singular value decomposition of the input–output operator, sufficient conditions on the temporal load G(t) and the singular values are obtained to express the source term as a Fourier series representation of the measured data. We establish a relationship between the representation formulas for the regularized solution F α ∈ L 2(Ω) obtained by Tikhonov regularization and singular value decomposition methods. A numerical example of reconstructing the spatial load by applying the conjugate gradient algorithm is also presented. In the end, we derive another stability estimate by using the spectral properties of the input–output operator and regularity assumption on G(t).