Control of a clamped-free beam by a piezoelectric actuator
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 545-563.

We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.

DOI : https://doi.org/10.1051/cocv:2006008
Classification : 93C20,  35B75,  35B60
Mots clés : piezoelectric actuator, metallic beam, exact controlability
@article{COCV_2006__12_3_545_0,
     author = {Cr\'epeau, Emmanuelle and Prieur, Christophe},
     title = {Control of a clamped-free beam by a piezoelectric actuator},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {545--563},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {3},
     year = {2006},
     doi = {10.1051/cocv:2006008},
     zbl = {1106.93008},
     mrnumber = {2224825},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2006008/}
}
Crépeau, Emmanuelle; Prieur, Christophe. Control of a clamped-free beam by a piezoelectric actuator. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 545-563. doi : 10.1051/cocv:2006008. http://www.numdam.org/articles/10.1051/cocv:2006008/

[1] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. 2 (1999) 33-63. | Zbl 0924.42022

[2] V. Balamurugan and S. Narayanan, Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control. Finite Elem. Anal. Des. 37 (2001) 713-738. | Zbl 1094.74683

[3] J.W.S. Cassels, An introduction to diophantine approximation. Moskau: Verlag 213 S. (1961). | MR 120219 | Zbl 0098.26301

[4] E. Crépeau, Exact boundary controllability of the Boussinesq equation on a bounded domain. Diff. Int. Equ. 16 (2003) 303-326.

[5] E. Crépeau, Contrôlabilité exacte d'équations dispersives issues de la mécanique. Thèse de l'Université de Paris-Sud, avalaible at www.math.ursq.fr/~crepeau/PUBLI/these1.html.

[6] P. Destuynder, I. Legrain, L. Castel and N. Richard, Theorical, numerical and experimental discussion of the use of piezoelectric devices for control-structure interaction. Eur. J. Mech., A/Solids 11 (1992) 97-106.

[7] P. Destuynder, A mathematical analysis of a smart-beam which is equipped with piezoelectric actuators. Control Cybern. 28 (1999) 503-530. | Zbl 0962.93049

[8] M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, Linear systems over Mikusinski operators and control of a flexible beam. ESAIM: Proc. 2 (1997) 183-193. | Zbl 0898.93018

[9] P. Germain, Mecanique, Tome II. Ellipses (1996).

[10] D. Halim and S.O.R. Moheimani, Spatial Resonant Control of Flexible Structures - Applications to a Piezoelectric Laminate Beam. IEEE Trans. Control Syst. Tech. 9 (2001) 37-53.

[11] D. Halim and S.O.R. Moheimani, Spatial H 2 control of a piezoelectric laminate beam: experimental implementation. IEEE Trans. Control Syst. Tech. 10 (2002) 533-546.

[12] W.S. Hwang and H.C. Park, Finite element modeling of piezoelectric sensors and actuator. AIAA Journal 31 (1993) 930-937.

[13] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Zeitschr. 41 (1936) 367-379. | Zbl 0014.21503

[14] S. Lang, Introduction to diophantine approximations. Springer-Verlag (1991). | MR 1348400 | Zbl 0826.11030

[15] S. Leleu, Amortissement actif des vibrations d'une structure flexible de type plaque à l'aide de transducteurs piézoélectriques. Thèse, ENS de Cachan (2002).

[16] J.L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité exacte, Collection de recherche en mathématiques appliquées 8 (Masson, Paris), 1988. distribués, Masson, Paris (1988). | MR 953547 | Zbl 0653.93002

[17] L. Meirovitch, Elements of vibration analysis, Düsseldorf, McGraw-Hill (1975). | Zbl 0359.70039

[18] R. Rebarber, Spectral assignability for distribued parameter systems with unbounded scalar control. SIAM J. Control Optim. 27 (1989) 148-169. | Zbl 0681.93037

[19] L. Rosier, Exact boundary controllability for the linear KdV equation - a numerical study. ESAIM: Proc. 4 (1998) 255-267. | Zbl 0919.93039

[20] J. Rudolph and F. Woittennek, Flatness based boundary control of piezoelectric benders. Automatisierungstechnik 50 (2002) 412-421.

[21] R.S. Smith, C.C. Chu and J.L. Fanson, The design of H controllers for an experimental non-collocated flexible structure Problem. IEEE Trans. Control Syst. Tech. 2 (1994) 101-109.

[22] S. Tliba and H. Abou-Kandil, H controller design for active vibration damping of a smart flexible structure using piezoelectric transducers, in 4th Symp. IFAC on Robust Control Design (ROCOND 2003), Milan, Italy (2003).

[23] M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930. | Zbl 0853.73051