Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection

Abstract

Abstract The problem of identifying an unknown flexural rigidity r ⁢ ( x ) {r(x)} of the cantilever Euler–Bernoulli beam from measured boundary deflection is studied. The problem leads to the inverse coefficient problem of determining the unknown principal coefficient r ⁢ ( x ) {r(x)} in the Euler–Bernoulli beam equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x - ( T r ⁢ ( x ) ⁢ u x ) x = 0 , \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}-(T_{r}(x)u_{x})_{x}=0, ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) {(x,t)\in(0,\ell)\times(0,T)} subject to the boundary conditions u ⁢ ( 0 , t ) = u x ⁢ ( 0 , t ) = 0 , \displaystyle u(0,t)=u_{x}(0,t)=0, r ⁢ ( ℓ ) ⁢ u x ⁢ x ⁢ ( ℓ , t ) = 0 , \displaystyle r(\ell)u_{xx}(\ell,t)=0, - ( r ⁢ ( ℓ ) ⁢ u x ⁢ x ⁢ ( ℓ , t ) ) x + T r ⁢ ( ℓ ) ⁢ u x ⁢ ( ℓ , t ) = g ⁢ ( t ) , \displaystyle{-}(r(\ell)u_{xx}(\ell,t))_{x}+T_{r}(\ell)u_{x}(\ell,t)=g(t), from the measured deflection w ⁡ ( t ) := u ⁢ ( ℓ , t ) {\operatorname{w}(t):=u(\ell,t)} , t ∈ [ 0 , T ] {t\in[0,T]} , at the free end x = ℓ {x=\ell} of the cantilever beam. Compactness and Lipschitz continuity of the Neumann-to-Dirichlet operator Φ [ ⋅ ] : ℛ 2 ⊂ H 2 ( 0 , ℓ ) ↦ L 2 ( 0 , T ) , ( Φ r ) ( t ) := u ( ℓ , t ; r ) , \Phi[\,\cdot\,]:\mathcal{R}^{2}\subset H^{2}(0,\ell)\mapsto L^{2}(0,T),\quad(% \Phi r)(t):=u(\ell,t;r), corresponding to the inverse problem is proved. These properties allow us to prove the existence of a quasi-solution of the inverse problem as a solution of the minimization problem for the Tikhonov functional J ⁢ ( r ) := ( 1 2 ) ⁢ ∥ Φ ⁢ r - w ∥ L 2 ⁢ ( 0 , T ) 2 . J(r):=\Bigl{(}\frac{1}{2}\Bigr{)}\lVert\Phi r-\operatorname{w}\rVert^{2}_{L^{2% }(0,T)}. It is proved that this functional is Fréchet differentiable. Furthermore, an explicit formula for the Fréchet gradient of this functional is derived by making use of the unique solution to the corresponding adjoint problem.