This paper which is a variant [6] studies the Riesz basis property and the exponential stability of a flexible Euler-Bernoulli beam with variable coefficients, clamped at one end and submitted at its free end at two control forces. We begin by establishing the spectral properties of this dynamical system, which allows us to show that there exists a sequence of generalized eigenfunctions forming a Riesz basis for the energy space considered. Consequently, the exponential stability under some conditions is derived.

Euler-Bernoulli beam, variable coefficients, Riesz basis, exponential stability.

Received: March 4, 2023; Accepted: May 1, 2023; Published: May 23, 2023

How to cite this article: Bomisso Gossrin Jean-Marc, Koffi Claude Jean Joris, Touré Kidjégbo Augustin and Coulibaly Adama, Riesz basis property and exponential stability of a second order system in time with variable coefficients, Far East Journal of Dynamical Systems 36(1) (2023), 77-92. http://dx.doi.org/10.17654/0972111823003

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

References:

[1] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Academic Press, 65, New York-London, 1975.
[2] M. D. Aouragh and N. Yebari, Stabilisation exponentielle d’une quation des poutres d’Euler-Bernoulli à coefficients variables, Annales Mathématiques Blaise Pascal 16 (2009), 483-510.
[3] G. D. Birkhoff, On the Asymptotic Character of the Solutions of Certain Linear Differential Equations Containing a Parameter, Trans. Amer. Math. Soc. 9 (1908), 219-231.
[4] G. D. Birkhoff, Boundary Value and Expansion Problems of Ordinary Linear Differential Equations, Transactions of the American Mathematical Society 9 (1908), 373-395.
[5] B. Z. Guo, Riesz Basis Property and Exponential Stability of Controlled Euler-Bernoulli Beam Equations with Variable Coefficients, SIAM Journal on Control and Optimization 40 (2002), 1905-1923.
[6] Mensah E. Patrice, On the numerical approximation of the spectrum of a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity, Far East Journal of Mathematical Sciences (FJMS) 126(1) (2020), 13-31.
[7] M. A. Naimark, Linear Differential Operators, Vol. I Ungar, New York, 1967.
[8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 44 (1983).
[9] A. Shkalikov, Boundary Problem for Ordinary Differential Operators with Parameter in Boundary Conditions, J. Soviet Math. 33 (1986), 1311-1342.
[10] Koffi Claude Jean Joris, Bomisso Gossrin Jean-Marc, Touré Kidjégbo Augustin and Coulibaly Adama, Riesz basis property and exponential stability for a damped system and controlled dynamically, International Journal of Numerical Methods and Applications 22 (2022), 19-40.