Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect

Ramos, A. J. A.; Freitas, M. M.; Almeida, de; Jesus, S. S.; Moura, Thiago Rafael da Silva

doi:10.1007/s00033-019-1106-2

Cited by 44 publications

(21 citation statements)

References 20 publications

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“…They proved that the dissipation produced by damping δv t acting in the transverse displacement of the beam is strong enough to stabilize exponentially the system (7) for whatever the physical parameters of the model and reproduced a numerical counterpart in a totally discrete domain, which preserve the important decay property of the numerical energy. In the same way, in [17] Ramos, A.J.A., Freitas, M.M., Almeida, D.S. et al proved the system's exponential stability independent of any relation between the coefficients using terms of feedback at the boundary and consequently proved their equivalence with the exact observability at the boundary.…”

mentioning

confidence: 88%

Piezoelectric beams with magnetic effect and localized damping

Afilal¹,

Soufyane²,

Santos³

2023

MCRF

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mentioning

confidence: 88%

Piezoelectric beams with magnetic effect and localized damping

Afilal¹,

Soufyane²,

Santos³

2023

MCRF

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“…In [34], Ramos et al studied a one-dimensional system of piezoelectric beams with magnetic effects, showed by ρv tt − αv xx + γβp xx = 0, (x, t) ∈ (0, L) × (0, T ), µp tt − βp xx + γβv xx = 0, (x, t) ∈ (0, L) × (0, T ), (1.3) with boundary conditions v(0, t) = αv x (L, t) − γβp x (L, t) + ξ 1 h v t (L, t) = 0, t ∈ (0, T ), p(0, t) = βp x (L, t) − γβv x (L, t) + ξ 2 h p t (L, t) = 0, t ∈ (0, T ), (1.4) where ξ i , i = 1, 2 are positive constant feedback gains. The authors first divided the original system into a conservative system and an auxiliary system.…”

Section: Introductionmentioning

confidence: 99%

Stability of piezoelectric beams with magnetic effects of fractional derivative type and with/without thermal effects

An¹,

Liu

Kong³

2021

Preprint

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In this paper, we first consider the well-posedness and asymptotic behavior of a onedimensional piezoelectric beam system with control boundary conditions of fractional derivative type, which represent magnetic effects on the system. By introducing two new equations to deal with control boundary conditions of fractional derivative type, we obtain a new equivalent system, so as to show the well-posedness of the system by using Lumer-Philips theorem. We then prove the lack of exponential stability by a spectral analysis, and obtain the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455-478). To find a more stable system, we then consider the stability of the above system with additional thermal effects described by Fourier's law, and achieve the exponential stability for the new model by using the perturbed functional method.

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“…In [34], Ramos et al studied a one-dimensional system of piezoelectric beams with magnetic effect, the system is shown as…”

Section: Introductionmentioning

confidence: 99%

Polynomial stability of piezoelectric beams with magnetic effect and tip body

An¹,

Liu²,

Kong³

2021

Preprint

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In this paper, we consider a dissipative system of one-dimensional piezoelectric beam with magnetic effect and a tip load at the free end of the beam, which is modeled as a special form of double boundary dissipation. Our main aim is to study the well-posedness and asymptotic behavior of this system. By introducing two functions defined on the right boundary, we first transform the original problem into a new abstract form, so as to show the well-posedness of the system by using Lumer-Philips theorem. We then divide the original system into a conservative system and an auxiliary system, and show that the auxiliary problem generates a compact operator. With the help of Wely's theorem, we obtain that the system is not exponentially stable. Moreover, we prove the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455-478).

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Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect

Cited by 44 publications

References 20 publications

Piezoelectric beams with magnetic effect and localized damping

Piezoelectric beams with magnetic effect and localized damping

Stability of piezoelectric beams with magnetic effects of fractional derivative type and with/without thermal effects

Polynomial stability of piezoelectric beams with magnetic effect and tip body

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