# Exponential Decay and Numerical Solution for a Timoshenko System with Delay Term in the Internal Feedback

## Main Article Content

### Abstract

In this work we study the asymptotic behavior as t â†’ ∞ of the solution for the Timoshenko system with delay term in the feedback. We use the semigroup theory for to prove the well-posedness of the system and for to establish the exponential stability. As far we know, there exist few results for problems with delay, where the asymptotic behavior is based on the Gearhart- Herbst-Pruss-Huang theorem to dissipative system. See [4], [5], [6]. Finally, we present numerical results of the solution of the system.

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### References

- A. Soufyane, A. Wehbe, Exponential stability for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Equ. 29 (2003), 1-14.
- J. E. M. Rivera, R. Racke, Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341 (2008), 1068-1083.
- F. Amar-Khodja. A. Benabdallah, J. E. M. Rivera, R. Racke, Energy decay for Timoshenko systems of memory type, J. Differ. Equ. 194 (2003), 82-115.
- L. Gearhart, Spectral Theory for the Contractions Semigroups on Hilbert Spaces. Trans. of the American Mathematical Society . 236 (1978), 385-349.
- F. Huang, Characteristic Conditions for Exponential Stability of the Linear Dynamical Systems in Hilbert Spaces. Annals of Differential Equations. 1 (1985), 43-56.
- J. Pr ¨uss, On the Spectrumm of C0-semigroups. Trans. of the American Mathematical Society 284 (1984) 847-857.
- C. A. Raposo, J. Ferreira, M. L. Santos, N. N. Castro. Exponential Stability for the Timoshenko System with two weak Dampings. Applied Mathematics Letters. 18 (2010), 535-541.
- B. Said-Houari, Y. Laskri. A stabilit result of a Timoshenko system with a delay term in the internal feedback. Applied Mathematics and Computation . 217 (2010), 2857-2869.
- C. Abdallah, P. Dorato, J. Benitez-Read, R. Byrne. Delayed Positive Feedback Can Stabilize Oscillatory System. ACC, San Francisco (1993) 3016-3107.
- L. H Suh, Z. Bien. Use of time delay action in the controller designe. IEEE Trans. Autom. Control. 25 (1980) 600-603.
- R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
- S. Nicaise, J. Valein. Stabilization of second order evolution eqaution with unbounded feedback with delay. ESAIM Control. Optim. Calc. Var. 16 (2010).
- Z. Liu, S. Zheng, Semigroups Associated with dissipative systems, Chapman, New Y & Hallo/CRC, New York, 1999.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1993.