## Taiwanese Journal of Mathematics

### Optimal Energy Decay for a Transmission Problem of Waves Under a Nonlocal Boundary Control

#### Abstract

In this paper, we consider a transmission problem in the presence of a boundary control condition of nonlocal type. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 5 (2019), 1201-1225.

Dates
Revised: 11 January 2019
Accepted: 27 January 2019
First available in Project Euclid: 20 February 2019

https://projecteuclid.org/euclid.twjm/1550631631

Digital Object Identifier
doi:10.11650/tjm/190108

Mathematical Reviews number (MathSciNet)
MR4012376

Zentralblatt MATH identifier
07126945

#### Citation

Atoui, Halim; Benaissa, Abbes. Optimal Energy Decay for a Transmission Problem of Waves Under a Nonlocal Boundary Control. Taiwanese J. Math. 23 (2019), no. 5, 1201--1225. doi:10.11650/tjm/190108. https://projecteuclid.org/euclid.twjm/1550631631

#### References

• Z. Achouri, N. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci. 40 (2017), no. 11, 3837–3854.
• W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), no. 2, 837–852.
• R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. 27 (1983), no. 3, 201–210.
• ––––, Fractional calculus - A different approach to the analysis of viscoelastically damped structures, AIAA J. 21 (1983), no. 5, 741–748.
• E. Blanc, G. Chiavassa and B. Lombard, Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives, J. Comput. Phys. 237 (2013), 1–20.
• A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), no. 2, 455–478.
• U. J. Choi and R. C. MacCamy, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl. 139 (1989), no. 2, 448–464.
• F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1 (1985), no. 1, 43–55.
• V. Komornik, Exact Controllability and Stabilization: The multiplier method, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Chichester, 1994.
• Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series, Springer-Verlag London, London, 1999.
• Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), no. 1, 37–42.
• M. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity, in: Progress and Trends in Rheology II, 64–67, Steinkopff, Heidelberg, 1988.
• B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform. 23 (2006), no. 2, 237–257.
• B. Mbodje and G. Montseny, Boundary fractional derivative control of the wave equation, IEEE Trans. Automat. Control 40 (1995), no. 2, 378–382.
• I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering 198, Academic Press, San Diego, CA, 1999.
• J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847–857.
• A. J. A. Ramos and M. W. P. Souza, Equivalence between observability at the boundary and stabilization for transmission problem of the wave equation, Z. Angew. Math. Phys. 68 (2017), no. 2, Art. 48, 11 pp.
• S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.
• P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mech. 51 (1984), no. 2, 294–298.