June  2020, 9(2): 447-468. doi: 10.3934/eect.2020013

Moving and oblique observations of beams and plates

1. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France, CNRS, IMB, UMR 5251, F-33400 Talence, France

2. 

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

3. 

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

* Corresponding author: Philippe Jaming

Received  March 2019 Revised  April 2019 Published  June 2020 Early access  August 2019

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.

Citation: Philippe Jaming, Vilmos Komornik. Moving and oblique observations of beams and plates. Evolution Equations and Control Theory, 2020, 9 (2) : 447-468. doi: 10.3934/eect.2020013
References:
[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.

show all references

References:
[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.

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