Advanced Search
Article Contents
Article Contents

Moving and oblique observations of beams and plates

  • * Corresponding author: Philippe Jaming

    * Corresponding author: Philippe Jaming 
Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • We study the observability of the one-dimensional Schrödinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at the end of the paper.

    Mathematics Subject Classification: Primary: 93B07; Secondary: 74K10, 74K20, 42A99.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure  .  Case (ⅰ)

    Figure  .  Case (ⅱ)

    Figure  .  A case where none of (ⅰ) and (ⅱ) is satisfied

  • [1] C. BaiocchiV. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. (8), 2 (1999), 33-63. 
    [2] C. BaiocchiV. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956.
    [3] J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587.  doi: 10.1002/cpa.3160320405.
    [4] A. Beurling, Interpolation for an interval in $\mathbb R^1$, in The collected works of Arne Beurling. Vol. 2. Harmonic analysi (eds. L. Carleson, P. Malliavin, J. Neuberger and J. Wermer) Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989.
    [5] H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.
    [6] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. 
    [7] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.
    [8] S. Jaffard, Contrôle interne exact des vibrations d'une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 759-762. 
    [9] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire, Port. Math., 47 (1990), 423-429. 
    [10] P. Jaming and K. Kellay, A dynamical system approach to Heisenberg Uniqueness Pairs, J. Analyse Math., 134 (2018), 273-301.  doi: 10.1007/s11854-018-0010-6.
    [11] J.-P. Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. Ecole Norm. Sup. (3), 79 (1962), 93-150.  doi: 10.24033/asens.1108.
    [12] A. Y. Khapalov, Exact observability of the time-varying hyperbolic equation with finitely many moving internal observations, SIAM J. Control Optim., 33 (1995), 1256-1269.  doi: 10.1137/S0363012992236218.
    [13] A. Y. Khapalov, Mobile Point Sensors and Actuators in the Controllability Theory of Partial Differential Equations, Springer, 2017. doi: 10.1007/978-3-319-60414-5.
    [14] V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl. (9), 71 (1992), 331-342. 
    [15] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Collection RMA, vol. 36. Masson-John Wiley, Paris-Chicester, 1994.
    [16] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.
    [17] V. Komornik and P. Loreti, Observability of rectangular membranes and plates on small sets, Evol. Equ. Control Theory, 3 (2014), 287-304.  doi: 10.3934/eect.2014.3.287.
    [18] V. Komornik and B. Miara, Cross-like internal observability of rectangular membranes, Evol. Equ. Control Theory, 3 (2014), 135-146.  doi: 10.3934/eect.2014.3.135.
    [19] W. Krabs, On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes, Lecture Notes in Control and Information Sciences, 173. Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0039513.
    [20] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.
    [21] A. Szijártó and J. Heged'ús, Observation problems posed for the Klein-Gordon equation, Electron. J. Qual. Theory Differ. Equ., (2012), 13 pp. doi: 10.14232/ejqtde.2012.1.7.
    [22] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), 951-977.  doi: 10.1090/S0002-9947-08-04584-4.
  • 加载中



Article Metrics

HTML views(602) PDF downloads(244) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint