In this paper, we use asymptotic techniques and the finite difference method to study the spectrum of differential operator arising in exponential stabilization of non-uniform Euler-Bernoulli beam with indefinite damping that is clamped at one end and is free at the other. We build a numerical scheme and investigate the eigenvalues locus as a function of the positive feedback parameters α, β and the damping coefficient γ.

beam equation, semigroup theory, asymptotic analysis, Riesz basis, exponential stability, finite difference method

Received: May 3, 2024; Revised: June 13, 2024; Accepted: June 26, 2024; Published: July 19, 2024

How to cite this article: Kouassi Ayo Ayébié Hermith, N. Diop Fatou and Touré K. Augustin, Numerical study of spectrum for non‑uniform Euler-Bernoulli beam with indefinite damping under a force control in position and velocity, International Journal of Numerical Methods and Applications 24(2) (2024), 145-163. https://doi.org/10.17654/0975045224010

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References
[1] M. D. Aouragh and N. Yebari, Stabilisation exponentielle d’une équation des poutres d’Euler-Bernoulli à coefficients variables, Annales Mathématiques Blaise Pascal 16(2) (2009), 483-510.
[2] B. Z. Guo, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equation with variable coefficients, SIAM J. Control Optim. 40(6) (2002), 1905-1923.
[3] P. G. Ciarlet, Introduction à l’analyse numérique matricielle et à l’optimisation, Ed. Masson, 1982.
[4] M. Cherkaoui, F. Conrad and N. Yebari, Optimal decay rate of energy for wave equation with boundary feedback, Advances in Mathematical Sciences and Applications 12 (2002), 549-568.
[5] S. Cox and E. Zuazua, The rate at which energy decays in a damping string, Comm. Partial Differential Equations 19 (1994), 213-243.
[6] R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear System Theory, Springer Verlag, New York, 1995.
[7] G. H. Golub and C. F. V. Loan, Matrix Computations, The Johns Hopkins University Press, 1989.
[8] H. A. A. Kouassi, C. Adama and A. K. Touré, Numerical approximation of spectrum for variable coefficients Euler-Bernoulli beams under a force control in position and velocity, International Journal of Applied Mathematics 30 (2017), 211-288.
[9] C. B. Moler and G. W. Stewart, An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal. 10 (1973), 241-256 (Collection of articles dedicated to the memory of George E. Forsythe).
[10] Ö. Morgül, A dynamic control law for the wave equation, Automatic 30(11) (1994), 1785-1792.
[11] J. Rappaz and M. Picasso, Introduction à l’analyse numérique, Press Polytechniques et Universitaires, Lausanne, 1998.
[12] P. Rideau, Controle d’assemblage de poutres flexibles par des capteurs actionneurs ponctuels: étude du spectre du système, Thèse, École Nationale Supérieure des Mines de Paris, Sophia-Antipolis, 1985.
[13] K. A. Touré, A. Coulibaly and A. A. H. Kouassi, Riesz basis, exponential stability of variable coefficients Euler-Bernoulli beams with a force control in position and velocity, Far East Journal of Applied Mathematics 88(3) (2014), 193-220.
[14] T. K. Augustin, C. Adama and K. A. A. Hermith, Riesz basis, exponential stability of variable coefficients Euler-Bernoulli beams and indefinite damping under a force control in position and velocity, Electron. J. Differential Equations 2015(54) (2015), 1-20.
[15] J. M. Wang, G. Q. Xu and S. P. Yung, Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping, IMA J. Appl. Math. 70 (2005), 459-477.