\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Simultaneous observability of infinitely many strings and beams

  • * Corresponding author: Paola Loreti

    * Corresponding author: Paola Loreti
The first author was supported by the Visiting Professor Programme, Sapienza Università di Roma
Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • We investigate the simultaneous observability of infinite systems of vibrating strings or beams having a common endpoint where the observation is taking place. Our results are new even for finite systems because we allow the vibrations to take place in independent directions. Our main tool is a vectorial generalization of some classical theorems of Ingham, Beurling and Kahane in nonharmonic analysis.

    Mathematics Subject Classification: Primary: 93B07; Secondary: 35L05, 74K10, 42A99.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A system of three strings with vibration planes $ \sf{p_j} $ spanned by $ d_j: = (\ell_j,\phi_j,\theta_j) $ and $ v_j\perp d_j $, $ j = 1,2,3 $. In (i) $ \ell_1 = \ell_2 = \ell_3 = 1 $, the $ v_j $'s are pairwise orthogonal, and $ v_1 = d_3 $, $ v_2 = d_1 $, $ v_3 = d_2 $. We have $ T_0 = 2\max\left\lbrace {\ell_1,\ell_2,\ell_3}\right\rbrace = 2 $. In (ii), we have $ \ell_1 = \ell_3 = 1 $, $ \ell_2 = 2/(2+\sqrt{2}) $ and $ v_1 = d_3\perp d_1 = v_2 = v_3 $. Then $ T_0 = 2\max\left\lbrace {\ell_1+\ell_2,\ell_3}\right\rbrace\approx 3.1715 $. In the planar case (iii) we have $ \ell_1 = 1 $, $ \ell_2 = 2/(2+\sqrt{2}) $ and $ \ell_3 = 2/(4+\sqrt{2}) $, so that $ T_0 = 2\max\left\lbrace {\ell_1+\ell_2+\ell_3}\right\rbrace\approx 3.9103 $

  • [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395-1410. 
    [2] K. Ammari and S. Farhat, Stability of a tree-shaped network of strings and beams,, Math. Methods Appl. Sci., 41 (2018), 7915-7935.  doi: 10.1002/mma.5255.
    [3] K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback,, Lecture Notes in Mathematics, 2124. Springer, Cham, 2015. doi: 10.1007/978-3-319-10900-8.
    [4] C. BaiocchiV. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B, 2 (1999), 33-63. 
    [5] C. BaiocchiV. Komornik and P. Loreti, Généralisation d'un théorème de Beurling et application à la théorie du contrôle, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 281-286.  doi: 10.1016/S0764-4442(00)00116-6.
    [6] 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.
    [7] J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems,, Comm. Pure Appl. Math., 32 (1979), 555-587.  doi: 10.1002/cpa.3160320405.
    [8] A. BarhoumiV. Komornik and M. Mehrenberger, A vectorial Ingham–Beurling theorem, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 53 (2010), 17-32. 
    [9] A. Beurling, Interpolation for an Interval in ${\mathbb R}^1$, in The Collected Works of Arne Beurling, Vol. 2. Harmonic Analysis (eds. L. Carleson, P. Malliavin, J. Neuberger and J. Wermer) Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989.
    [10] J. W. S. CasselsAn Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957. 
    [11] R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 621-626.  doi: 10.1016/S0764-4442(01)01876-6.
    [12] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, , Springer Science & Business Media, Vol. 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.
    [13] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. 
    [14] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.
    [15] S. JaffardM. Tucsnak and E. Zuazua, On a theorem of Ingham. Dedicated to the memory of Richard J. Duffin, J. Fourier Anal. Appl., 3 (1997), 577-582.  doi: 10.1007/BF02648885.
    [16] S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, J. Differential Equations, 145 (1998), 184-215.  doi: 10.1006/jdeq.1997.3385.
    [17] J.-P. Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. de l'E.N.S., 79 (1962), 93-150.  doi: 10.24033/asens.1108.
    [18] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Collection RMA, vol. 36, Masson–John Wiley, Paris–Chicester, 1994.
    [19] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.
    [20] V. Komornik and P. Loreti, Multiple-point internal observability of membranes and plates, Appl. Anal., 90 (2011), 1545-1555.  doi: 10.1080/00036811.2011.569497.
    [21] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8.
    [22] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.
    [23] J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués I-II, Masson, Paris, 1988.
    [24] P. Loreti, On some gap theorems, European Women in Mathematics–Marseille 2003, 39–45, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005.
    [25] M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258. 
    [26] E. Sikolya, Simultaneous observability of networks of beams and strings, Bol. Soc. Paran. Mat., 21 (2003), 31–41. doi: 10.5269/bspm.v21i1-2.7505.
    [27] Q. Wu, The smallest Perron numbers,, Mathematics of Computation, 79 (2010), 2387-2394.  doi: 10.1090/S0025-5718-10-02345-8.
  • 加载中

Figures(1)

SHARE

Article Metrics

HTML views(329) PDF downloads(384) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return