Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping

Abstract

Abstract In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x + ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}+(\kappa(x)u_{xxt})_{xx}=0,\quad(x,% t)\in(0,\ell)\times(0,T], subject to the boundary conditions u ⁢ ( 0 , t ) = 0 , u x ⁢ ( 0 , t ) = 0 , [ r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ ( t ) , u(0,t)=0,\quad u_{x}(0,t)=0,\quad[r(x)u_{xx}+\kappa(x)u_{xxt}]_{x=\ell}=0,% \quad-[(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}]_{x=\ell}=g(t), for t ∈ [ 0 , T ] {t\in[0,T]} , from the measured deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) {\nu(t):=u(\ell,t)} , t ∈ [ 0 , T ] {t\in[0,T]} , and from the bending moment ω ⁢ ( t ) := - ( r ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ ( 0 , t ) + κ ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ t ⁢ ( 0 , t ) ) , t ∈ [ 0 , T ] , \omega(t):=-(r(0)u_{xx}(0,t)+\kappa(0)u_{xxt}(0,t)),\quad t\in[0,T], where the terms ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x {(\kappa(x)u_{xxt})_{xx}} and μ ⁢ ( x ) ⁢ u t {\mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.