Identification of an unknown spatial load distribution in a vibrating cantilevered beam from final overdetermination

Abstract

The inverse problem of determining the unknown spatial load distribution F(x) in the cantilever beam equation m(x)u_tt = -(EI(x)u_xx)_xx + F(x)H(t), with arbitrary but separable source term, from the measured data u_T(x) := u(x,T), x ∈ (0,l), at the final time T > 0 is considered. Some a priori estimates of the weak solution u ∈ H°^2,1(Ω_T) of the forward problem are obtained. Introducing the input-output map, it is proved that this map is a compact operator. The adjoint problem approach is then used to derive an explicit gradient formula for the Fréchet derivative of the cost functional J(F) = ∥ u(·,T;F) - u_T(·) ∥_L²(0,l)². The Lipschitz continuity of the gradient is proved. The collocation algorithm combined with the truncated singular value decomposition (TSVD) is used to estimate the degree of ill-posedness of the considered inverse source problem. The conjugate gradient algorithm (CGA), based on the explicit gradient formula, is proposed for numerical solution of the inverse problem. The algorithm is examined through numerical examples related to reconstruction of various spatial loading distributions F(x). The numerical results illustrate bounds of applicability of proposed algorithm, also its efficiency and accuracy.