Journal of Systems Science and Complexity

, Volume 27, Issue 3, pp 463–475 | Cite as

Spectral analysis and stabilization of a coupled wave-ODE system

  • Dongxia ZhaoEmail author
  • Junmin Wang


In this paper, an interconnected wave-ODE system with K-V damping in the wave equation and unknown parameters in the ODE is considered. It is found that the spectrum of the system operator is composed of two parts: Point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point \(- \tfrac{1} {d}\), and there are two branches of the asymptotic eigenvalues: The first branch is accumulating towards \(- \tfrac{1} {d}\), and the other branch tends to −∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum-determined growth condition and exponential stability of the system are concluded.


Exponential stability Kelvin-Voigt damping Riesz basis spectrum wave equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Zhou Z C and Tang S X, Boundary stabilization of a coupled wave-ODE system with internal anti-damping, International Journal of Control, 2012, 85: 1683–1693.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    Tang S X and Xie C K, State and output feedback boundary control for a coupled PDE-ODE system, Systems & Control Letters, 2011, 60: 540–545.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Kristic M and Smyshlyaev A, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Philadelphia, 2009.Google Scholar
  4. [4]
    Xu G Q and Feng D X, On the spectrum determined growth assumption and the perturbation of C 0 semigroups, Integral Equations and Operator Theory, 2001, 39: 363–376.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Guo B Z, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM Journal on Control and Optimization, 2001, 39: 1736–1747.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Guo B Z and Wang J M, Remarks on the application of the Keldysh theorem to the completeness of root subspace of non-self-adjoint operators and comments on “Spectral operators generated by Timoshenko beam model”, Systems & Control Letters, 2006, 55: 1029–1032.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Zhang X and Zuazua E, Polynomial decay and control of a hyperbolic-parabolic coupled system, Journal of Differential Equations, 2004, 204: 380–438.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Kristic M, Delay Compensation for Nonlinear, Adaptive, and PDE Systems, Birkhäuser, Boston, 2009.CrossRefGoogle Scholar
  9. [9]
    Kristic M, Guo B Z, and Smyshlyaev A, Boundary controllers and observers for the linearized Schrödinger equation, SIAM Journal on Control and Optimization, 2011, 49: 1479–1497.CrossRefMathSciNetGoogle Scholar
  10. [10]
    Wang J M, Guo B Z, and Krstic M, Wave equation stabilization by delays equal to even multiples of the wave propagation time, SIAM Journal on Control and Optimization, 2011, 49(2): 517–554.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Wang J M, Ren B B, and Krstic M, Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation, IEEE Transactions on Automatic Control, 2012, 57(1): 179–185.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Krstic M, Compensating a string PDE in the actuation or in sensing path of an unstable ODE, IEEE Transactions on Automatic Control, 2009, 54: 1362–1368.CrossRefMathSciNetGoogle Scholar
  13. [13]
    Wang J M, Lü X W, and Zhao D X, Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks, International Journal of Control, 2011, 84(5): 904–915.CrossRefzbMATHGoogle Scholar
  14. [14]
    Atay F M, Balancing the inverted pendulum using position feedback, Applied Mathemetics Letters, 1999, 12: 51–56.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    Liu B and Hu H Y, Stabilization of linear undamped systems via position and delayed position feedbacks, Journal of Sound and Vibration, 2008, 312: 509–528.CrossRefGoogle Scholar
  16. [16]
    Susto G A and Krstic M, Control of PDE-ODE cascades with Neumann interconnections, Journal of the Franklin Institute, 2010, 347: 284–314.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Pazy A, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.CrossRefGoogle Scholar
  18. [18]
    Guo B Z, Wang J M, and Zhang G D, Spectral analysis of a wave equation with Kelvin-Voigt damping, Z. Angew. Math. Mech., 2010, 90: 323–342.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Guo B Z and Zhang G D, On spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping, ESAIM: Control, Optimization and Calculus of Variations, 2012, 18: 889–913.CrossRefzbMATHGoogle Scholar
  20. [20]
    Opmeer M R, Nuclearity of Hankel operators for ultradifferentiable control systems, Systems & Control Letters, 2008, 57: 913–918.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsNorth University of ChinaTaiyuanChina
  2. 2.School of MathematicsBeijing Institute of TechnologyBeijingChina

Personalised recommendations