Boundary Feedback Stabilization of the Undamped Euler--Bernoulli Beam with Both Ends Free

Guo, Faming; Huang, Fang‐Yang

doi:10.1137/s0363012901380961

Cited by 22 publications

(11 citation statements)

References 27 publications

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“…On the other hand, the boundary stabilization and boundary control of Euler‐Bernoulli beam without rotating (nonrotating case) by using different boundary dampers were considered by several authors, and many results have been obtained in this regard. We mention, among others, the work of Krall, Morgül, Conrad and Morgül, Canbolat et al, de Querioz et al, Li et al, Andrews et al, Guo and Huang, Guo and Guo, Yan et al, and the references therein. Also, the stabilization of viscoelastic Euler‐Bernoulli beam was considered earlier by Park et al, where the authors studied the following problem

y_{t t} false(x, t false) + y_{x x x x} false(x, t false) - true \int_{0}^{t} q false(t - τ false) y_{x x x x} false(τ false) d τ + g false(y_{t} false(x, t false) false) = 0, false(x, t false) \in false[0, L false] \times false(0, \infty false),

with the following boundary conditions and initial data

\{\begin{matrix} y false(0, t false) = y_{x} false(0, t false) = y_{x x} false(L, t false) = y_{x x x} false(0, t false) = 0, t \geq 0, \\ y_{x x x} false(L, t false) - true \int_{0}^{t} q false(t - τ false) y_{x x x} false(L, t false) d τ = f false(y false(L, t false) false), t \geq 0, \\ y false(x, 0 false) = y_{0} false(x false), \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

“…[10][11][12][13][14] On the other hand, the boundary stabilization and boundary control of Euler-Bernoulli beam without rotating (nonrotating case) by using different boundary dampers were considered by several authors, and many results have been obtained in this regard. We mention, among others, the work of Krall, 15 Morgül, 16 Conrad and Morgül, 17 Canbolat et al, 18 de Querioz et al, 19 Li et al, 20 Andrews et al, 21 Guo and Huang, 22 Guo and Guo, 23 Yan et al, 24 and the references therein. Also, the stabilization of viscoelastic Euler-Bernoulli beam was considered earlier by Park et al, 25 where the authors studied the following problem…”

Section: Introductionmentioning

confidence: 97%

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Stabilization of a viscoelastic rotating Euler‐Bernoulli beam

Berkani

2018

Math Methods in App Sciences

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y_{t t} false(x, t false) + y_{x x x x} false(x, t false) - true \int_{0}^{t} q false(t - τ false) y_{x x x x} false(τ false) d τ + g false(y_{t} false(x, t false) false) = 0, false(x, t false) \in false[0, L false] \times false(0, \infty false),

with the following boundary conditions and initial data

\{\begin{matrix} y false(0, t false) = y_{x} false(0, t false) = y_{x x} false(L, t false) = y_{x x x} false(0, t false) = 0, t \geq 0, \\ y_{x x x} false(L, t false) - true \int_{0}^{t} q false(t - τ false) y_{x x x} false(L, t false) d τ = f false(y false(L, t false) false), t \geq 0, \\ y false(x, 0 false) = y_{0} false(x false), \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

Stabilization of a viscoelastic rotating Euler‐Bernoulli beam

Berkani

2018

Math Methods in App Sciences

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“…Z. Guo & Jin, 2013;F. Guo & Huang, 2001;Harland, Mace, & Jones, 2001;He, Huang, & Li, 2017;He, Nie, Meng, & Liu, 2017;Jin & Guo, 2015;Li, Xu, & Han, 2016;Liu & Liu, 1998, 2000Lu, Chen, Yao, & Wang, 2013;Luo, Guo, & Morgul, 1999;Meurer, Thull, & Kugi, 2008;Miletíc, Stürzer, Arnold, & Kugi, 2016;Nguyen, Do, & Pan, 2013;Özer, 2017;Queiroz, Dawson, Nagarkatti, & Zhang, 2000) based on the Lyapunov direct and flatness methods and (Bohm, Krstic, Kuchler, & Sawodny, 2014;Krstic & Smyshlyaev, 2008;Paranjape, Chung, & Krstic, 2013) based on the backstepping method on single beams, and (Henikl, Kemmetmüller, Meurer, & Kugi, 2016;Kater & Meurer, 2016;Lagnese, Leugering, & Schmidt, 1994) on multiple beams.…”

Section: Introductionmentioning

confidence: 99%

Stabilisation of large motions of Euler–Bernoulli beams by boundary controls

Duc

2018

International Journal of Systems Science

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This paper proposes a design of boundary controls for stabilization of Euler-Bernoulli beams with large motions and non-neglectable moment of inertia under external loads. Common types of boundary conditions in practice are considered. The designed boundary controllers guarantee globally practically K∞-exponentially stability of the closed-loop system. The control design, well-posedness, and stability analysis are based on two Lyapunov-type theorems developed for a class of evolution systems in Hilbert space.

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“…where β(t) and Γ(t) are boundary control terms applied to the free end of the beam. The boundary feedback stabilization problem of this model, that is the problem of finding boundary controls capable to guarantee the exponential stability, has been studied at length (see [1,2,3,4,5] and references therein).…”

Section: Introductionmentioning

confidence: 99%