Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output. (22nd June 2021) - Hasanov, Alemdar; Romanov, Vladimir; Baysal, Onur
Abstract: In this paper we discuss the unique determination of unknown spatial load F ( x ) in the damped Euler–Bernoulli beam equation ρ ( x ) u t t + μ u t + ( r ( x ) u x x ) x x = 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 u t t − ( k ( x ) u x ) x = F ( x ) G ( t ) from final time output is not possible. This result is also valid for the undamped beam equation ρ ( x ) u t t + ( r ( x ) u x x ) x x = 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 ) = ∑ n = 1 ∞ u T, n ψ n ( x ) / σ n, 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 ) − u T ‖ L 2 ( 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.