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The inverse problem of determining the unknown spatial load distribution ( ) in the cantilever beam equation ( ) = −( ( ) ) + ( ) ( ), with arbitrary but separable source term, from the measured data ( ) := ( , ), ∈ (0, ), at the nal time > 0 is considered. Some a priori estimates of the weak solution ∈ ∘ 2,1 ( ) 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 ( ) = ‖ ( ⋅ , ; ) − ( ⋅ )‖ 2 2 (0, ) . 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 ( ). The numerical results illustrate bounds of applicability of proposed algorithm, also its e ciency and accuracy.

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In this paper, a novel mathematical model and new approach is proposed for identification of an unknown shear force in a system governed by the general form Euler–Bernoulli beam equation , subject to the boundary conditions u(0,t)  =  ux(0,t)  =  0, , , from available boundary observation (measured output data), namely, the measured deflection at x  =  l. The approach is based on weak solution theory for PDEs, Tikhonov regularization combined with the adjoint method. A uniqueness result for the problem under consideration is proved. The Neumann-to-Dirichlet operator corresponding to the inverse problem is introduced. It is shown that this operator is injective, compact and Lipschitz continuous. The last property allows us to prove an existence of a quasi-solution of the inverse problem. Fréchet differentiability of the Tikhonov functional is also proved. In the case when Tr  =  0, an implicit formula for the Fréchet gradient of this functional is derived by making use of the unique solution to corresponding adjoint problem. Furthermore, a class of admissible shear forces in which the Fréchet gradient of the Tikhonov functional is Lipschitz continuous, is derived. Numerical examples with random noisy measured outputs are presented to illustrate the validity and effectiveness of the proposed approach.