Publication Cover
Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 9
91
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Gevrey regularity for the Euler–Bernoulli beam equation with localized structural damping

&
Pages 1587-1603 | Received 09 Jan 2023, Accepted 02 Sep 2023, Published online: 08 Sep 2023

References

  • Belinskiy B, Lasiecka I. Gevrey's and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions. J Math Anal Appl. 2007;332(1):137–154. doi: 10.1016/j.jmaa.2006.10.025
  • Komornik V. Exact controllability and stabilization. The multiplier method. Paris: Masson; 1994.
  • Lagnese J. Boundary stabilization of thin plates. Philadelphia: SIAM; 1989.
  • Lasiecka I. Exponential decay rates for the solutions of Euler–Bernoulli equations with boundary dissipation occurring in the moments only. J Differ Eqs. 1992;95(1):169–182. doi: 10.1016/0022-0396(92)90048-R
  • Lasiecka I. Mathematical control theory of coupled PDEs. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2002.
  • Lasiecka I, Triggiani R. Exact controllability of the Euler–Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a nonconservative case. SIAM J Control Optim. 1989;27(2):330–373. doi: 10.1137/0327018
  • Lasiecka I, Triggiani R. Exact controllability of the Euler–Bernoulli equation with boundary controls for displacement and moment. J Math Anal Appl. 1990;146(1):1–33. doi: 10.1016/0022-247X(90)90330-I
  • Denk R, Kammerlander F. Exponential stability for a coupled system of damped-undamped plate equations. IMA J Appl Math. 2018;83(2):302–322. doi: 10.1093/imamat/hxy002
  • Gearhart L. Spectral theory for contraction semigroups on Hilbert space. Trans Amer Math Soc. 1978;236:385–394. doi: 10.1090/tran/1978-236-00
  • Prüss J. On the spectrum of C0-semigroups. Trans Amer Math Soc. 1984;284:847–857.
  • Ammari K, Hassine F, Robbiano L. Stabilization for vibrating plate with singular structural damping. Discrete Contin Dyn Syst Ser S. 2023;16(6):1168–1180.
  • Burq N. Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 1998;180(1):1–29. doi: 10.1007/BF02392877
  • Martinez BB, Denk R, Monzón JH, et al. Regularity and asymptotic behavior for a damped plate-membrane transmission problem. J Math Anal Appl. 2019;474(2):1082–1103. doi: 10.1016/j.jmaa.2019.02.005
  • Tebou L. Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping. Nonlinear Anal. 2009;71(12):e2288–e2297. doi: 10.1016/j.na.2009.05.026
  • Liu K, Liu Z. Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J Control Optim. 1998;36(3):1086–1098. doi: 10.1137/S0363012996310703
  • Taylor S. Chapter ‘Gevrey semigroups’ [Ph.D. Thesis]. School of Mathematics, University of Minnesota; 1989.
  • Sozzo B, Muñoz Rivera JE. The Gevrey class of the Euler–Bernoulli beam model. J Math Anal Appl. 2022;505(1):Article ID 125619, 13 pp. doi: 10.1016/j.jmaa.2021.125619
  • Adams RA, Fournier J. Sobolev spaces. 2nd ed. Amsterdam: Elsevier/Academic Press; 2003. xiv+305 pp.
  • Engel K-J, Nagel R. One-parameter semigroups for linear evolution equations. New York: Springer-Verlag; 2000.
  • Kato T. Perturbation theory for linear operators. New York: Springer-Verlag; 1980.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag; 1983.
  • Chen S, Triggiani R. Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<α<12. Proc Amer Math Soc. 1990;110:401–415.
  • Arendt W, Batty CJK. Tauberian theorems and stability of one-parameter semigroups. Trans Amer Math Soc. 1988;306(2):837–852. doi: 10.1090/tran/1988-306-02
  • Gómez Ávalos G, Muñoz Rivera JE, Liu Z. Gevrey class of locally dissipative Euler–Bernoulli beam equation. SIAM J Control Optim. 2021;59(3):2174–2194. doi: 10.1137/20M1312800

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.