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  December 2020 Early access  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 and 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. 

[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. Cassels, An 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.

show all references

References:
[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. Cassels, An 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.

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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