Abstract
In this paper, we study the boundary stabilizing feedback control problem of well-known Scole model that has nonhomogeneous spatial parameters. By using an abstract result of Riesz basis, we show that the closed-loop system is a Riesz spectral system. The asymptotic distribution of eigenvalues, the spectrum-determinded growth condition and the exponential stability are concluded.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation in Operator Methods for Optimal Control Problems, 2nd edn. (S. J. Lee, ed., Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York), 108 pp. 67–96
B. Z. Guo, R. Yu, On Riesz basis property of discrete operators with application to an Euler-Bernoulli beam equation with boundary linear feedback control. IMA J. Math. Control Inform. 18 pp. 241–251 (2001)
B. Z. Guo, R. Yu, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J. Control Optim. 40 pp. 1905–1923 (2002)
S. W. R. Lee, H. L. Li, Development and characterization of a rotary motor driven by anisotropic piezoelectric composite. SIAM J. Control Optim. Smart Materials Structures 7 pp. 327–336 (1998)
W. Littman, L. Markus, Exact boundary controllability of a hybrid system of elasticity. Arch. Rational Mech. Anal. Structures 103 pp. 193–236 (1988)
V. Komornik, Exact controllability and stabilization (The Multiplier Method) Masson, Paris:Wiley (1995)
M. A. Naimark, Linear Differential Operators, Part 1: Elementary Theory of Linear Differential Operators Ungar Publishing Co., New York (1967)
A. Pazy, Semigroups of linear operators and applications to partial differential equations Springer-Verlag, New York (1983)
B. Rao, Recent results in non-uniform and uniform stabilization of the Scole model by boundary feedbacks Lecture Notes in Pure and Applied Mathematics, J.-E Zolesio, ed., Marcel Dekker, New York163(1983) pp. 357–365 (1994)
J. M. Wang, G. Q. Xu, S. P. Yung, Riesz basis property, exponential stability of variable coefficient Euler Bernoulli beams with indefinite damping. IMA J. Appl. Math. 163 pp. 459–477 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Aouragh, M.D., El Boukili, A. (2020). On the Energy Decay of a Nonhomogeneous Hybrid System of Elasticity. In: Dos Santos, S., Maslouhi, M., Okoudjou, K. (eds) Recent Advances in Mathematics and Technology. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-35202-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-35202-8_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-35201-1
Online ISBN: 978-3-030-35202-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)