Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination

Inverse problems of determining the unknown source term F(x, t) in the cantilevered beam equation utt = (EI(x)uxx)xx + F(x, t) from the measured data mu(x) := u(x, T) or nu(x) := ut(x, T) at the final time t = T are considered. In view of weak solution approach, explicit formulae for the Frechet gradients of the cost functionals J1(F) = ||u(x, T; w) - mu(x)||20 and J2(F) = ||ut(x, T; w) - nu(x)||20 are derived via the solutions of corresponding adjoint (backward beam) problems. The Lipschitz continuity of the gradients is proved. Based on these results the gradient-type monotone iteration process is constructed. Uniqueness and ill-conditionedness of the considered inverse problems are analyzed.