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

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 rho(h)(x)u(tt)+mu(x)u(t) +(D(u)(ux(1)x(1) + Vu(x2x2)))(x1x1) + (D(x)=(ux(2)x(2) + Vu(x1x1))(x2x2) + 2(1-v)D(x)ux(1)x(1))(x1x2) = F((x)G(t), (x,t) is an element of Omega x (0, T] from final time measurement data uT(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 Frechet 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 mu(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-alpha is an element of L-2(omega) 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).

Automated summary

In this paper, the inverse problem of determining an unknown spatial load in the damped non-homogeneous isotropic rectangular Kirchhoff-Love plate equation is studied. The quasi-solution approach is used to pose the inverse problem as a least square minimization problem. The existence of minimum is shown, and the derivative of the Tikhonov functional is expressed in terms of an adjoint problem. Sufficient conditions for stability estimates and the relationship between the representation formulas for the regularized solution obtained by Tikhonov regularization and singular value decomposition methods are established. A numerical example of reconstructing the spatial load by applying the conjugate gradient algorithm is also presented.