December  2020, 15(4): 633-652. doi: 10.3934/nhm.2020017

Simultaneous observability of infinitely many strings and beams

1. 

College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, People's Republic of China

2. 

Département de mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

3. 

Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Via A. Scarpa, 16, 00161, Roma, Italy

* Corresponding author: Paola Loreti

Received  June 2019 Revised  June 2020 Published  August 2020

Fund Project: The first author was supported by the Visiting Professor Programme, Sapienza Università di Roma

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.

Citation: Vilmos Komornik, Anna Chiara Lai, Paola Loreti. Simultaneous observability of infinitely many strings and beams. Networks & Heterogeneous Media, 2020, 15 (4) : 633-652. doi: 10.3934/nhm.2020017
References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395-1410.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. BaiocchiV. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B, 2 (1999), 33-63.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. BarhoumiV. Komornik and M. Mehrenberger, A vectorial Ingham–Beurling theorem, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 53 (2010), 17-32.   Google Scholar

[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.  Google Scholar

[10] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[13]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[14]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Collection RMA, vol. 36, Masson–John Wiley, Paris–Chicester, 1994.  Google Scholar

[19]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[23]

J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués I-II, Masson, Paris, 1988.  Google Scholar

[24]

P. Loreti, On some gap theorems, European Women in Mathematics–Marseille 2003, 39–45, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005.  Google Scholar

[25]

M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258.   Google Scholar

[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.  Google Scholar

[27]

Q. Wu, The smallest Perron numbers,, Mathematics of Computation, 79 (2010), 2387-2394.  doi: 10.1090/S0025-5718-10-02345-8.  Google Scholar

show all references

References:
[1]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395-1410.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. BaiocchiV. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B, 2 (1999), 33-63.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. BarhoumiV. Komornik and M. Mehrenberger, A vectorial Ingham–Beurling theorem, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 53 (2010), 17-32.   Google Scholar

[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.  Google Scholar

[10] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[13]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[14]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Collection RMA, vol. 36, Masson–John Wiley, Paris–Chicester, 1994.  Google Scholar

[19]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[23]

J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués I-II, Masson, Paris, 1988.  Google Scholar

[24]

P. Loreti, On some gap theorems, European Women in Mathematics–Marseille 2003, 39–45, CWI Tract, 135, Centrum Wisk. Inform., Amsterdam, 2005.  Google Scholar

[25]

M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258.   Google Scholar

[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.  Google Scholar

[27]

Q. Wu, The smallest Perron numbers,, Mathematics of Computation, 79 (2010), 2387-2394.  doi: 10.1090/S0025-5718-10-02345-8.  Google Scholar

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 $
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