On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output

Abstract

Abstract In this paper, we discuss the role of the damping term μ ⁢ u t {\mu u_{t}} in unique determination of unknown spatial load F ⁢ ( x ) {F(x)} in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+\mu u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t)} from the measured final time displacement u T ⁢ ( x ) := u ⁢ ( x , T ) {u_{T}(x):=u(x,T)} or velocity ν T ⁢ ( x ) := u t ⁢ ( x , T ) {\nu_{T}(x):=u_{t}(x,T)} . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ ( x ) {F(x)} in the undamped wave equation u t ⁢ t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}-(k(x)u_{x})_{x}=F(x)G(t)} from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+(r(x)u_{xx})_{xx}=F(x)G(t)} . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 {T>0} and the damping coefficient μ > 0 {\mu>0} . The cases of pure spatial load G ⁢ ( t ) ≡ 1 {G(t)\equiv 1} , and the exponentially decaying load G ⁢ ( t ) = e - η ⁢ t {G(t)=e^{-\eta t}} are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ ( x ) {F(x)} in the damped Euler–Bernoulli beam equation.