Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping
In this paper, we study the inverse problems of determining the unknown transverse shear force in a system governed by the damped Euler–Bernoulli equation subject to the boundary conditions for , from the measured deflection , , and from the bending moment where the terms and account for the Kelvin–...
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| Published in | Journal of inverse and ill-posed problems Vol. 32; no. 1; pp. 75 - 106 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin
De Gruyter
01.02.2024
Walter de Gruyter GmbH |
| Subjects | |
| Online Access | Get full text |
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| Abstract | In this paper, we study the inverse problems of determining the unknown transverse shear force
in a system governed by the damped Euler–Bernoulli equation
subject to the boundary conditions
for
,
from the measured deflection
,
, and from the bending moment
where the terms
and
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
, 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
, 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. |
|---|---|
| AbstractList | In this paper, we study the inverse problems of determining the unknown transverse shear force
in a system governed by the damped Euler–Bernoulli equation
subject to the boundary conditions
for
,
from the measured deflection
,
, and from the bending moment
where the terms
and
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
, 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
, 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. 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. In this paper, we study the inverse problems of determining the unknown transverse shear force [Image omitted] 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 ] , [Image omitted] \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 ) , [Image omitted] 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 [Image omitted], from the measured deflection [Image omitted], [Image omitted], 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 ] , [Image omitted] \omega(t):=-(r(0)u_{xx}(0,t)+\kappa(0)u_{xxt}(0,t)),\quad t\in[0,T], where the terms [Image omitted] and [Image omitted] 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 [Image omitted], 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 [Image omitted], 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. |
| Author | Hasanov, Alemdar Dileep, Anjuna Kumarasamy, Sakthivel |
| Author_xml | – sequence: 1 givenname: Sakthivel surname: Kumarasamy fullname: Kumarasamy, Sakthivel email: sakthivel@iist.ac.in organization: Department of Mathematics, Indian Institute of Space Science and Technology, Trivandrum 695 547, India – sequence: 2 givenname: Alemdar surname: Hasanov fullname: Hasanov, Alemdar email: alemdar.hasanoglu@gmail.com organization: Department of Mathematics, Kocaeli University; and Şehit Ekrem Mah., Altinşehir Sk., Ayazma Villalari, 22, Bahčecik, Kocaeli - 41030, Türkiye – sequence: 3 givenname: Anjuna surname: Dileep fullname: Dileep, Anjuna email: dileepanjuna10@gmail.com organization: Department of Mathematics, Indian Institute of Space Science and Technology, Trivandrum 695 547, India |
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| Snippet | In this paper, we study the inverse problems of determining the unknown transverse shear force
in a system governed by the damped Euler–Bernoulli equation... 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... In this paper, we study the inverse problems of determining the unknown transverse shear force [Image omitted] in a system governed by the damped... |
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| SubjectTerms | 35A01 35G05 35R30 49J20 bending moment Bending moments Boundary conditions Damping Euler-Bernoulli beams Euler-Bernoulli equation Euler–Bernoulli beam Inverse problems Kelvin–Voigt damping Lipschitz stability Regularity shear force identification Shear forces Transverse shear |
| Title | Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping |
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