Uniqueness and regularization in the flexural stiffness coefficient identification problem for a statically determined Euler–Bernoulli beam

Abstract

In this contribution, we present a rigorous analysis for the identification of the flexural stiffness coefficient in the statically determined Euler–Bernoulli beam, from measurements of the beam’s transverse deflection, corresponding to an applied transverse load. Such analysis follows by defining the parameter-to-solution map and then proving its injectivity, continuity, compactness, Fréchet differentiable, and that it satisfies the so-called tangential cone condition in the L2[0,L]-topology. In particular, injectivity (which implies uniqueness identification) is proved with a less restrictive assumption on the smoothness flexural stiffness coefficient than the ones in Lesnic et al. (1999). On the other hand, compactness implies the ill-posedness of the coefficient identification in the sense that, small perturbations in the measurements might lead to large perturbations in the coefficient recovery. Therefore, some regularization approach is needed. It is worth mentioning that the aforementioned properties of the parameter-to-solution map imply the convergence and stability w.r.t. noise data (regularization properties) of approximated solutions obtained by some iterative methods. We discuss the particular cases of steepest-descent and Landewber iterations and, besides, we present the numerical implementation with distinct noise levels on the deflection measurements.