Journal of Dynamical and Control Systems

, Volume 12, Issue 3, pp 405–418 | Cite as

Regularity of an Euler-Bernoulli Equation with Neumann Control and Collocated Observation

Article
First Online:
Received:
Revised:

Abstract.

This paper studies the regularity of an Euler-Bernoulli plate equation on a bounded domain of ℝn, n ≥ 2, with partial Neumann control and collocated observation. It is shown that the system is not only well posed in the sense of D. Salamon but also regular in the sense of G. Weiss. It is also shown that the corresponding feedthrough operator is zero.

Key words and phrases.

Euler-Bernoulli beam boundary control well-posedness regularity 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    1. K. Ammari, Dirichlet boundary stabilization of the wave equation. Asymptot. Anal. 30 (2002), 117–130.MATHMathSciNetGoogle Scholar
  2. 2.
    2. K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var. 6 (2001), 361–386.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    3. C. I. Byrnes, D. S. Gilliam, V. I. Shubov, and G. Weiss, Regular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Systems 8 (2002), 341–370.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    4. A. Cheng and K. Morris, Well-posedness of boundary control systems. SIAM J. Control Optim. 42 (2003), 1244–1265.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    5. R. F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math.Control Inform. 14 (1997), 207–223.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    6. ———, Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Systems 14 (2001), 299–337.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    7. R. F. Curtain and G. Weiss, Well posedness of triples of operators, in the sense of linear systems theory. In: Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, Eds.) Vol. 91, BirkhÄuser, Basel (1989), pp. 41–59.Google Scholar
  8. 8.
    8. R. F. Curtain and H. J. Zwart, An introduction to infinite-dimensional linear systems theory. Springer-Verlag, New York (1995).MATHGoogle Scholar
  9. 9.
    9. L. C. Evans, Partial differential equations. AMS, Providence, Rhode Island (1998).Google Scholar
  10. 10.
    10. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1977).Google Scholar
  11. 11.
    11. B. Z. Guo and X. Zhang, On the regularity of wave equation with partial Dirichlet control and observation. SIAM J. Control Optim. 44 (2005), 1598–1613.MathSciNetCrossRefGoogle Scholar
  12. 12.
    12. B. Z. Guo and Z. C. Shao, Regularity of a Schrodinger equation with Dirichlet control and colocated observation. Systems Control Lett. 54 (2005), 1135–1142.MathSciNetCrossRefGoogle Scholar
  13. 13.
    13. B. Z. Guo and Y. H. Luo, Controllability and stability of a second order hyperbolic system with colocated sensor/actuator. Systems Control Lett. 46 (2002), 45–65.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    14. V. Komornik, Exact controllability and stabilization: the multiplier method. John Wiley and Sons, Chichester (1994).Google Scholar
  15. 15.
    15. I. Lasiecka and R. Triggiani, L 2(Σ)-regularity of the boundary to boundary operator B*L for hyperbolic and Petrowski PDEs. Abstr. Appl. Anal. 19 (2003), 1061–1139.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    16. J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag, Berlin (1972).MATHGoogle Scholar
  17. 17.
    17. M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. II. Controllability and stability. SIAM J. Control Optim. 42 (2003), 907–935.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    18. D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987), 383–431.MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    19. O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. I. Well-posed systems. Math. Control Signals Systems. 15 (2002), 291–315.MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    20. G. Weiss, The representation of regular linear systems in Hilbert spaces. In: Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, and W. Schappacher, Eds.) BirkhÄuser, Basel (1989), pp. 401–416.Google Scholar
  21. 21.
    21. ———, Transfer functions of regular linear systems, I. Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994), 827–854.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    22. G. Weiss and R. F. Curtain, Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Control 42 (1997), 4–21.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    23. G. Weiss, O. J. Staffans, and M. Tucsnak, Well-posed linear systems—a survey with emphasis on conservative systems. Int. J. Appl. Math. Comput. Sci. 11 (2001), 7–33.MATHMathSciNetGoogle Scholar
  24. 24.
    24. G. Weiss and M. Tucsnak, How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance. ESAIM J.Control Optim. Calc. Var. 9 (2003), 247–274.MATHMathSciNetGoogle Scholar
  25. 25.
    25. G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2000), 1204–1232.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of Systems ScienceAcademy of Mathematics and System SciencesBeijingP.R. China
  2. 2.School of Computational and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  3. 3.Graduate School of the Chinese Academy of SciencesBeijingP.R. China

Personalised recommendations