Identification of an unknown shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection

Abstract

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.