In this Note we study the exact controllability of a three-dimensional body made of a material whose constitutive law introduces an elasticity-electricity coupling. We show that, without any geometrical assumption, two controls (the elastic and the electric controls) acting on the whole boundary drive the system to rest in finite time.
On considère un corps constitué d'un matériau dont la loi constitutive introduit un couplage élastique-électrique. On montre que, sans faire aucune hypothèse géométrique, l'application de deux contrôles agissant sur la totalité de la frontière (un contrôle élastique et un contrôle électrique) permet de contrôler le système en temps fini.
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Irena Lasiecka 1; Bernadette Miara 2
@article{CRMATH_2009__347_3-4_167_0, author = {Irena Lasiecka and Bernadette Miara}, title = {Exact controllability of a {3D} piezoelectric body}, journal = {Comptes Rendus. Math\'ematique}, pages = {167--172}, publisher = {Elsevier}, volume = {347}, number = {3-4}, year = {2009}, doi = {10.1016/j.crma.2008.12.007}, language = {en}, }
Irena Lasiecka; Bernadette Miara. Exact controllability of a 3D piezoelectric body. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 167-172. doi : 10.1016/j.crma.2008.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.007/
[1] Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2006
[2] Inverse Problems and Partial Differential Equations, Springer, 2005
[3] Mathematical Control Theory of Coupled PDEs, SIAM, 2002 (NSF-CMBS Lecture Notes)
[4] Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., Volume 19 (1989), pp. 243-290
[5] , Control Theory for Partial Differential Equations: Continuous and Approximations Theories, vols. I, II, Cambridge University Press, Cambridge, 2000
[6] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Masson, 1988
[7] Contrôlabilité d'un corps piézoélectrique, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2001), pp. 267-270
[8] B. Miara, Exact controllability of piezoelectric shells, in: Fourth Conference on Elliptic and Parabolic Problems, Gaeta, 2002, pp. 434–441
[9] Exact controllability of a piezoelectric body. Theory and numerical simulation, Appl. Math. Optim. (2009)
[10] B. Miara, M.L. Santos, Stabilization of piezoelectric body, in preparation
[11] Controllability and stabilizability theory for linear partial differential equations, SIAM Rev., Volume 28 (1978), pp. 639-739
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