Advertisement

On the Energy Decay of a Nonhomogeneous Hybrid System of Elasticity

  • Moulay Driss Aouragh
  • Abderrahman El Boukili
Chapter
  • 49 Downloads
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

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.

Keywords

Euler-Bernoulli beam Boundary control Stabilization Riesz basis 

References

  1. 1.
    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. 67–96Google Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    W. Littman, L. Markus, Exact boundary controllability of a hybrid system of elasticity. Arch. Rational Mech. Anal. Structures 103 pp. 193–236 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Komornik, Exact controllability and stabilization (The Multiplier Method) Masson, Paris:Wiley (1995)Google Scholar
  7. 7.
    M. A. Naimark, Linear Differential Operators, Part 1: Elementary Theory of Linear Differential Operators Ungar Publishing Co., New York (1967)Google Scholar
  8. 8.
    A. Pazy, Semigroups of linear operators and applications to partial differential equations Springer-Verlag, New York (1983)CrossRefGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Moulay Driss Aouragh
    • 1
  • Abderrahman El Boukili
    • 2
  1. 1.MAMCS Group, Department of Maths, M2I Laboratory, FSTMoulay Ismaïl UniversityErrachidiaMorocco
  2. 2.Department of Physics, FSTMoulay Ismaïl UniversityErrachidiaMorocco

Personalised recommendations