Abstract
We introduce a general formalism for linear evolution equations with skew adjoint operators. We make explicit the controllability operator as an expansion with respect to eigenfunctions. Using the fact that the eigenvalues are purely imaginary, we give sufficient controllability conditions. This approach is convenient for studying the asymptotic behaviour of the optimal control.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bardos, C., Lebeau, G., and Rauch, J.: Contrôle et stabilisation dans les problèmes hyperboliques, Appendice II in [12], Vol. 1.
Bardos, C., Lebeau, G., and Rauch, J.: Micro lobal ideas in control and stabilization, Proc. Clermont Ferrand Colloquium, June 1988, Springer-Verlag Lecture Notes.
Bardos, C., Lebeau, G., and Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, Preprint.
Bardos, C., Lebeau, G., and Rauch, J.: Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, CMA, DMI, Ecole Normale Supérieure, 87.17, October 1987.
Brezis, H.: Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983.
Chin-Hsien, Li: Preprint.
Lagnese, J. E. and Lions, J. L.: Modelling, Analysis and Control of Thin Plates, Masson, RMA, Vol. 6, 1988.
Lagnese, J. E.: Exact controllability of Maxwell's equations in a general region. SIAM J. Control Optim. 27 (1989), 374–388.
Lasiecka, I. and Triggiani, R.: Exact controllability for the wave equation with Neumann boundary control, Appl. Math. Optim. 19 (1989), 243–290.
Lebeau, G.: Contrôle de l'équation de Schrödinger, Preprint.
Lions, J. L.: Preprint.
Lions, J. L.: Controlabilité exacte, perturbations et stabilisation de systèmes distribués, Vols. 1 and 2, Masson, Paris, 1988.
Lions, J. L.: Exact controllability, stabilization and perturbations. SIAM Rev. 30 (1988), 1–68.
Triggiani, R.: Exact boundary controllability on L 2(Ω)×H-1(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary ∂Ω and related problems. Appl. Math. Optim. 18 (1988), 241–277.
Zuazua, E.: Controlabilité exacte d'un modèle de plaques vibrantes enun temps arbitrairement petit, C.R. Acad Sci. Paris Ser. A. 304 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bensoussan, A. On the general theory of exact controllability for skew symmetric operators. Acta Appl Math 20, 197–229 (1990). https://doi.org/10.1007/BF00049568
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00049568