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Regularity of plate equations with control concentrated in interior curves

Published online by Cambridge University Press:  14 November 2011

Stéphane Jaffard  and
Marius Tucsnak
Stéphane Jaffard
Affiliation:
Centre de Mathématiques et de Leurs Applications (CMLA), ENS de Cachan, 61 Av. du Président Wilson, 94235 Cachanand Université Paris 12 Créteil, France
Marius Tucsnak
Affiliation:
Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique, 91128 Palaiseauand Université de Versailles, France
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Synopsis

We consider initial and boundary-value problems modelling the vibration of a plate with piezoelectric actuator. The usual models lead to the Bernoulli–Euler and Kirchhoff plate equations with right-hand side given by a distribution concentrated in an interior curve. We obtain regularity results which are stronger than those obtained by simply using the Sobolev regularity of the right-hand side. By duality, we obtain new trace regularity properties for the solutions of plate equations. Our results provide appropriate function spaces for the control of plates provided with piezoelectric actuators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 1005 - 1025
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Avalos, G. and Lasiecka, I.. A differential Riccati equation for the active control of a problem in structural acoustics (IMA Preprint 1345, 1995).Google Scholar
2Ball, J. M. and Slemrod, M.. Nonharmonic Fourier series and the stabilization of semilinear control systems. Comm. Pure Appl. Math. 32 (1979), 555–87.CrossRefGoogle Scholar
3Banks, H. T. and Smith, R. C.. The modelling of piezoceramic patch interactions with shells, plates and beams (ICASE Report No. 92–66).Google Scholar
4Banks, H. T., Fang, W., Silcox, R. J. and Smith, R. C.. Approximation methods for control of the acoustic/structure interaction with piezoceramic actuators (Preprint).Google Scholar
5Crawley, E. F. and Anderson, E. H.. Detailed models for piezoceramic actuation of beams. J. Intell. Mat. Systems 1 (1990), 4–25.CrossRefGoogle Scholar
6Destuynder, Ph., Legrain, I., Castel, L. and Richard, N.. Theoretical, numerical and experimental discussion of the use of piezoelectric devices for control-structure interaction. European J. Mech. A Solids 11 (1992), 181–213.Google Scholar
7Doetsch, G.. Introduction to the Theory and Applications of the Laplace Transformation (New York: Springer, 1974).CrossRefGoogle Scholar
8, Hardy and , Wright. An Introduction to the Theory of Numbers (Oxford: Clarendon Press, 1975).Google Scholar
9Herz, C. S.. Fourier transforms related to convex sets. Ann. of Math. 75 (1962), 81–92.CrossRefGoogle Scholar
10Kendall, D. G.. On the number of lattice-points inside a random oval. Quart. J. Math. Oxford Ser. 19 (1948), 1–26.CrossRefGoogle Scholar
11Lagnese, J. E.. Boundary stabilization of thin plates (Philadelphia: SIAM, 1989).CrossRefGoogle Scholar
12Lagnese, J. E. and Lions, J. L.. Modelling, Analysis and Control of Thin Elastic Plates (Paris: Masson, 1988).Google Scholar
13Lasiecka, I. and Triggiani, R.. A cosine operator approach to modelling L 2 boundary input hyperbolic equations. Appl. Math. Optim. 7 (1981), 35–93.CrossRefGoogle Scholar
14Lasiecka, I. and Triggiani, R.. A lifting theorem for time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations. Proc. Amer. Math. Soc. 10 (1988), 745–55.CrossRefGoogle Scholar
15Lions, J. L.. Contrôlabilité exacte des Systèmes distribués (Paris: Masson, 1988).Google Scholar
16Lions, J. L.. Pointwise control for distributed systems. In Control and Estimation in Distributed Parameter Systems, ed. Banks, H. T., 1–41 (Philadelphia, SIAM, 1992).Google Scholar
17Lions, J. L. and Magenes, E.. Nonhomogenous Boundary value Problems (Berlin: Springer, 1972).Google Scholar
18Meyer, Y.. Etude d'un modele mathématique issu du contrôle des structures spatiales déformables. In Nonlinear Partial Differential Equations and their Applications, vol. 2, Research Notes in Mathematics, 234–342 (Boston: Pitman, 1985).Google Scholar
19Randol, B.. On the Fourier transform of the indicator function of a planar set. Trans. Amer. Math. Soc. 139 (1969), 271–8.Google Scholar
20Triggiani, R.. Regularity with point control. Part 2. Kirchhoff equations. J. Differential Equations 103 (1993), 394–420.CrossRefGoogle Scholar
21Triggiani, R.. Regularity with point control. Part 1. Wave equations and Bernoulli-Euler equations (Berlin: Springer, 19xx).Google Scholar
22Triggiani, R.. Interior and boundary regularity of the wave equation with interior point control. Differential Integral Equations 6 (1993), 111–29.CrossRefGoogle Scholar
23Tucsnak, M.. Contrôle d'une poutre avec actionneur piézoélectrique. C. R. Acad. Sci. Paris Sér. 1 319 (1994), 697–702.Google Scholar
24Tucsnak, M.. Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. (to appear).Google Scholar