Unique recovery of unknown spatial load in damped Euler-Bernoulli beam equation from final time measured output

In this paper we discuss the unique determination of unknown spatial load F(x) in the damped Euler-Bernoulli beam equation ρ (x)utt + μ ut + (r (x) uxx)xx = F (x)G (t) from final time measured output (displacement, u T (x) ≔ u(x, T) or velocity, ν t,T (x) ≔ u t (x, T)). It is shown in [Hasanov Hasanoglu and Romanov 2017 Introduction to Inverse Problems for Differential Equations (New York: Springer)] that the unique determination of F(x) in the undamped wave equation utt-(k (x)ux)x = F (x)G (t) from final time output is not possible. This result is also valid for the undamped beam equation ρ (x) utt+ (r (x) uxx)xx = F (x)G (t). We prove that in the presence of damping term μu t , the spatial load can be uniquely determined by the final time output, in terms of the convergent singular value expansion (SVE), as, F(x) = ςn = 1∞ uT,n ψn (x)/ σn under some acceptable conditions with respect to the final time T > 0, the damping coefficient μ > 0 and the temporal load G(t) > 0. As an alternative method we propose the adjoint problem approach (APA) and derive an explicit gradient formula for the Fréchet derivative of the Tikhonov functional J(F) = ∥u (˙ ,T;F)- uT∥L2 (0,l)2. Comparative analysis of numerical algorithms based on SVE and APA methods are provided for the harmonic loading G(t) = cos(ωt), ω > 0, as a most common dynamic loading case. The results presented in this paper not only clearly demonstrate the key role of the damping term μu t in the inverse problems arising in vibration and wave phenomena, but also allows us, firstly, to find admissible values of the final time T > 0, at which a final time measured output can be extracted, and secondly, to reconstruct the unknown spatial load F(x) in the damped Euler-Bernoulli beam equation from this measured output.