This paper deals with exponential stability of a flexible Euler-Bernoulli beam with variable coefficients and internal damping. We begin by providing the well-posedness of problem in the sense of C₀-semi-groups of contractions. Then, by making use of Naimark [7] and Wang et al. [11, 12] ideas, the spectral properties of the studied problem are established. This leads to the construction of a Riesz basis for the energy space and allows to show the exponential stability of the system. We conclude by showing that even when the internal damping is not of constant sign, the exponential stability of the system remains.

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

Received: August 4, 2022; Accepted: September 26, 2022; Published: October 20, 2022

How to cite this article: 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. http://dx.doi.org/10.17654/0975045222005

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

References:

[1] 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.
[2] G. D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (1908), 373-395.
[3] G. J. M. Bomisso, K. A. Touré and G. Yoro, Stabilization of variable coefficients Euler-Bernoulli beam with viscous damping under a force control in rotation and velocity rotation, J. Math. Res. 9 (2017), 1-13.
[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5] B. Z. Guo, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM J. Control Optim. 40 (2002), 1905-1923.
[6] B. Z. Guo and J. M. Wang, Riesz basis generation of abstract second order partial equation systems with general non-separated boundary conditions, Numer. Funct. Anal. Optim. 27 (2006), 291-328.
[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, Vol. 44, 1983.
[9] A. Shkalikov, Boundary problem for ordinary differential operators with parameter in boundary conditions, J. Soviet Math. 33 (1986), 1311-1342.
[10] A. Touré, A. Coulibaly and H. Kouassi, Riesz basis property and exponential stability of Euler-Bernoulli beams with variable coefficients and indefinite damping under a force control in position and velocity, Electron. J. Differential Equations 54 (2015), 1-20.
[11] J. M. Wang, Riesz basis property of some infinite-dimensional control problems and its applications, Ph.D. Thesis, The University of Hong Kong, 2004.
[12] 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.