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

In this paper, a novel mathematical model and new approach is proposed for identification of an unknown shear force g(t) in a system governed by the general form Euler Bernoulli beam equation rho(x)u(tt) + mu(x)u(t) +(r(x)u(xx))(xx) = T(r)u(xx) = 0, (x, t) is an element of (0,1) x (0, T) subject to the boundary conditions u(0,t) = u(x)(0,t)) = u(xx)(x,t)vertical bar(x=l) 0, -r(x)u(xx) (x,t))(x) = g(t), from available boundary observation (measured output data), namely, the measured deflection v(t) := u(1, t) at x = 1. 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 (1)11 : g C H2(0, T) H L2 (0,T) corresponding to the inverse problem is introduced, It is shown that this operator is infective, compact and Lipschitz continuous. The last property allows us to prove an existence of a quasi -solution of the inverse problem, Frechet differentiability of the Tikhonov functional is also proved. In the case when = 0, an implicit formula for the Frechet 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 Frechet 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.