Abstract
We study the semigroup associated to the Euler-Bernoulli beam equation with localized (discontinuous) dissipation. We assume that the beam is composed of three components: elastic, viscoelastic of Kelvin-Voigt type, and thermoelastic parts. We prove that this model generates a semigroup of Gevrey class that in particular implies the exponential stability of the model. To our knowledge, this is the first positive result giving increased regularity for the Euler-Bernoulli beam with localized damping.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2174-2194 |
| Number of pages | 21 |
| Journal | SIAM Journal on Control and Optimization |
| Volume | 59 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2021 |
Bibliographical note
Funding Information:\ast Received by the editors January 15, 2020; accepted for publication (in revised form) February 28, 2021; published electronically June 17, 2021. https://doi.org/10.1137/20M1312800 Funding: The second author was supported by CNPq project 310249/2018-0. \dagger Departamento de Matem\a'ticas, Universidad Andres Bello, Autopista Concepci\o'n-Talcahuano 7100, Talcahuano, Chile (gegova@gmail.com). \ddagger Departamento de Matem\a'tica, Universidad del B\{\'i}o B\{\'i}o, Concepci\o'n, Chile, and National Laboratory for Scientific Computation, Brasil (jemunozrivera@gmail.com). \S Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812 USA, and Beijing Institute of Technology, China (zliu@d.umn.edu).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics
Keywords
- C-semigroup
- Differentiability
- Exponential stability
- Gevrey class
- Kelvin-Voigt damping
- Thermoviscoelasticity
- Viscoelasticity