In this paper, the regularity and stability of the semigroup associated with a system of coupled plate equations is considered. Indirect structural or Kelvin-Voigt damping is imposed, i.e., only one equation is directly damped by one of these two damping. By the frequency domain method, we show that the associated semigroup of the system with indirect structural damping is analytic and exponentially stable. However, with the much stronger indirect Kelvin-Voigt damping, we prove that, by the asymptotic spectral analysis, the semigroup is even not differentiable. The exponential stability is still maintained. Finally, some numerical simulations of eigenvalues of the corresponding one-dimensional systems are also given.
Accepté le :
DOI : 10.1051/cocv/2018060
Mots-clés : Analyticity, structural damping, Kelvin-Voigt damping, exponential stability, semigroup, spectrum
@article{COCV_2019__25__A51_0,
author = {Han, Zhong-Jie and Liu, Zhuangyi},
title = {Regularity and stability of coupled plate equations with indirect structural or {Kelvin-Voigt} damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018060},
zbl = {1437.35655},
mrnumber = {4019757},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018060/}
}
TY - JOUR AU - Han, Zhong-Jie AU - Liu, Zhuangyi TI - Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018060/ DO - 10.1051/cocv/2018060 LA - en ID - COCV_2019__25__A51_0 ER -
%0 Journal Article %A Han, Zhong-Jie %A Liu, Zhuangyi %T Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018060/ %R 10.1051/cocv/2018060 %G en %F COCV_2019__25__A51_0
Han, Zhong-Jie; Liu, Zhuangyi. Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 51. doi : 10.1051/cocv/2018060. http://www.numdam.org/articles/10.1051/cocv/2018060/
[1] , Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM J. Control Optim. 41 (2002), 511–541. | Zbl
[2] and , Indirect stabilization of locally coupled wave-type systems. ESAIM: COCV 18 (2012) 548–582. | Numdam | MR | Zbl
[3] , and , Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2 (2002) 127–150. | DOI | MR | Zbl
[4] , and , Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Math. Control Relat. Fields 1 (2011) 413–436. | Zbl
[5] , , A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39 (1982) 433–454. | DOI | MR | Zbl
[6] , and , Energy decay of type III linear thermoelastic plates with memory. J. Math. Anal. Appl. 401 (2013) 357–366. | Zbl
[7] and , Asymptotic stability of wave equations coupled by velocities. Math. Control Relat. Fields 6 (2016) 429–446. | DOI | MR | Zbl
[8] and , One-parameter semigroups for linear evolution equations. Springer Science & Business Media, New York (1999). | MR | Zbl
[9] , Sharp decay rates for the weakly coupled hyperbolic system with one internal damping. SIAM J. Control Optim. 50 (2012) 1643–1660. | Zbl
[10] , Spectral theory for contraction semigroups on Hilbert space. Trans. Am. Math. Soc. 236 (1978) 385–394. | Zbl
[11] , Indirect stabilization of hyperbolic systems through resolvent estimates. Evol. Equ. Control Theory 6 (2017) 59–75. | Zbl
[12] , Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56. | MR | Zbl
[13] , On the mathematical model for linear elastic systems with analytic damping. SIAM J. Control Optim. 26 (1988) 714–724. | DOI | MR | Zbl
[14] and , Holomorphic property and exponential stability of the semigroup associated with linear elastic systems with damping. Ann. Differ. Equ. 4 (1988) 411–424. | Zbl
[15] and , Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C. Abstr. Appl. Anal. 3 (1998) 153–169. | Zbl
[16] and , Analyticity of solutions in type III thermoelastic plates. IMA J. Appl. Math. 75 (2010) 637–646. | DOI | MR | Zbl
[17] and , A note on the equations of a thermoelastic plate. Appl. Math. Lett. 8 (1995) 1–6. | DOI | MR | Zbl
[18] and , Qualitative properties of certain C0 semigroups arising in elastic systems with various damping. Adv. Differ. Equ. 3 (1998) 643–686. | MR | Zbl
[19] and , Semigroups Associated with Dissipative Systems. Chapman and Hall/, Boca Raton (1999). | Zbl
[20] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | Zbl
[21] , On the spectrum of C0-semigroups. Trans. Am. Math. Soc. 284 (1984) 847–857. | MR | Zbl
[22] , A general framework for the study of indirect damping mechanisms in elastic systems. J. Math. Anal. Appl. 173 (1993) 339–358. | DOI | MR | Zbl
[23] , Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Math. Control Relat. Fields 2 (2012) 45–60. | DOI | MR | Zbl
[24] , Spectral methods in Matlab. SIAM, Philadelphia, PA (2000). | DOI | MR | Zbl
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