Abstract: in the damped Euler–Bernoulli beam equation ρ(x)utt + μut + (r(x)uxx)xx = F(x)G(t) from final time measured output (displacement, uT (x) :=u(x, T) or velocity, νt,T (x) :=ut(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 undampedwave 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 μut, the spatial load can be uniquely determined by the final time output, in terms of the convergent singular value expansion (SVE)under some acceptable conditions with respect to the final time T > 0, the damping coefficient μ > 0 and the temporal load G(t) > 0.

Cite: Hasanov A. , Romanov V. , Baysal O.
Unique recovery of an unknown spatial load in damped beam equation from final time output
Inverse Problems. 2021. V.37. N7. P.37 07500. DOI: 10.1088/1361-6420/ ac01fb

Dates:

Submitted:	Mar 24, 2021
Accepted:	May 12, 2021

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