Identification of unknown spatial load distributions in a vibrating Euler-Bernoulli beam from limited measured data

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler-Bernoulli beam equation rho(x) w(tt)+ mu(x) w(t)+ (EI (x) w(xx))(xx) - T(r)u(xx) = Sigma(M)(m=1) g(m)(t)f(m)(x), (x. t) is an element of Omega(T) :=(0, l) x (0, T), with hinged-clamped ends (w(0, t) = w(xx)(0, t) = 0, w(l, t) = w(x)(l, t) = 0, t is an element of (0, T)), are studied. Here g(m) (t) are linearly independent functions, describing an asynchronous temporal loading, and f(m) (x) are the spatial load distributions. In the first identification problem the values nu(k)(t), k= (1, K) over bar, of the deflection w(x, t), are assumed to be known, as measured output data, in a neighbourhood of the finite set of points P := {x(k) is an element of (0, l), k = (1, K) over bar} subset of ( 0, l), corresponding to the internal points of a continuous beam, for all t is an element of ]0, T[. In the second identification problem the values theta(k)(t), k = (1, K) over bar, of the slope w(x)(x, t), are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all t is an element of ]0, T[. These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data nu(k)(t) and theta k(t) contain random noise. In the first part of the study we prove that each measured output data nu(k)(t) and theta k(t), k = (1, K) over bar can uniquely determine the unknown functions f(m) is an element of H-1 (]0, l[), m= (1, M) over bar. In the second part of the study we will introduce the input-output operators K-d : L-2(0, T) -> L-2 (0, T),(K(d)f)(t) := w(x, t; f) x is an element of P, f(x) := (f(1)(x)), and K-s : L-2(0, T) -> L-2 (0, T), (K(s)f)(t) := w(x) (x, t; f), x is an element of P, corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: K(d)f = nu and K(s)f = theta, where nu(t) := (nu(1)(t),...,nu(K)(t)) and theta k(t) := (theta(1)(t),...,theta(K)(t)). Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals J(1 alpha)(f) := (1/2)vertical bar vertical bar K(d)f-nu vertical bar vertical bar(2)(L2(0,T)) + (1/2)alpha vertical bar vertical bar f vertical bar vertical bar(2)(L2(0,T)) and J(2 alpha)(f) := (1/2)vertical bar vertical bar K(s)f-theta vertical bar vertical bar(2)(L2(0,T)) + (1/2)alpha vertical bar vertical bar f vertical bar vertical bar(2)(L2(0,T)). Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Frechet gradients J(1)'( f) and J(2)'( f) of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads gm(t).