Exact controllability for problems of transmission of the plate equation with lower-order terms
Authors:
Weijiu Liu and Graham H. Williams
Journal:
Quart. Appl. Math. 58 (2000), 37-68
MSC:
Primary 93B05; Secondary 74K20, 74M05
DOI:
https://doi.org/10.1090/qam/1738557
MathSciNet review:
MR1738557
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider the exact controllability for the problem of transmission of the plate equation with lower-order terms. Using Lions’ Hilbert Uniqueness Method (HUM for short), we show that the system is exactly controllable in ${L^2}\left ( \Omega \right ) \times {H^{ - 2}}\left ( \Omega \right )$. We also obtain some uniqueness theorems for the problem of transmission of the plate equation and for the operator $a\left ( x \right ){\Delta ^2} + q$.
- Nicolas Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 55 (1993), 126 (French, with English and French summaries). MR 1254820
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075
- A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl. (9) 68 (1989), no. 4, 457–465 (1990) (French, with English summary). MR 1046761
- Stéphane Jaffard, Contrôle interne exact des vibrations d’une plaque carrée, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 14, 759–762 (French, with English summary). MR 972075
- V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
- I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a nonconservative case, SIAM J. Control Optim. 27 (1989), no. 2, 330–373. MR 984832, DOI https://doi.org/10.1137/0327018
- I. Lasiecka and R. Triggiani, Further results on exact controllability of the Euler-Bernoulli equation with controls on the Dirichlet and Neumann boundary conditions, Stabilization of flexible structures (Montpellier, 1989) Lect. Notes Control Inf. Sci., vol. 147, Springer, Berlin, 1990, pp. 226–234. MR 1179444, DOI https://doi.org/10.1007/BFb0005156
- G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9) 71 (1992), no. 3, 267–291 (French, with English summary). MR 1172452
- J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
- J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1–68. MR 931277, DOI https://doi.org/10.1137/1030001
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 0350178
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- E. Zuazua, Exact boundary controllability for the semilinear wave equation, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987–1988) Pitman Res. Notes Math. Ser., vol. 220, Longman Sci. Tech., Harlow, 1991, pp. 357–391 (English, with French summary). MR 1131832
N. Burq, Contrôle de l’équation des plaques en présence d’ obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 55, 1–126 (1993)
R. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1992
R. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 5, Evolution Problems I, Springer-Verlag, Berlin, 1992
A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures et Appl. 68, 457–465 (1989)
S. Jaffard, Contrôle interne exact des vibrations d’une plaque carrée, C. R. Acad. Sci. Paris 307, Série I, 759–762 (1988)
V. Komornik, Exact controllability and stabilization: The multiplier method, John Wiley and Sons, Masson, Paris, 1994
I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A nonconservative case, SIAM J. Control and Optimization 27, No. 2, 330–373 (1989)
I. Lasiecka and R. Triggiani, Further results on exact controllability of the Euler-Bernoulli equation with controls on the Dirichlet and Neumann boundary conditions, Stabilization of Flexible Structures (Montpellier, 1989), Lecture Notes in Control and Information Sciences, 147, Springer-Verlag, 1990, pp. 226–234
G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures et Appl. 71, 267–291 (1992)
J. L. Lions, Contrôlabilité exacte perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité Exacte, Masson, Paris, Milan, Barcelone, Mexico, 1988
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review 30, no. 1, 1–68 (1988)
J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vols. I and II, Springer-Verlag, Berlin, Heidelberg, New York, 1972
J. Simon, Compact sets in the space ${L^{p}}\left ( 0, T; B \right )$, Annali di Matematica Pura ed Applicata (IV), Vol. CXLVI, 1987, pp. 65–96
E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. X (Paris, 1987–1988) 357–391, Pitman Research Notes in Mathematics Series 220, Longman Science and Technology, Harlow, 1991 (H. O. Fattorini)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
93B05,
74K20,
74M05
Retrieve articles in all journals
with MSC:
93B05,
74K20,
74M05
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society
