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Observability of heat processes by transmutation without geometric restrictions
1. | CNRS, Institut de Mathématiques de Toulouse, UMR 5219, F-31062 Toulouse, France |
2. | Basque Center for Applied Mathematics (BCAM), Bizkaia Technology Park, Building 500, E-48160 Derio - Basque Country, Spain |
References:
[1] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control and Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[2] |
M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping, J. Differential Equations, 211 (2005), 303-332.
doi: 10.1016/j.jde.2004.12.010. |
[3] |
N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, (French) [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.
doi: 10.1016/S0764-4442(97)80053-5. |
[4] |
S. Ervedoza and E. Zuazua, "Sharp Observability Estimates for the Heat Equation," preprint, 2011. |
[5] |
A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. |
[6] |
L. Hörmander, "Linear Partial Differential Operators," Springer Verlag, Berlin-New York, 1976. |
[7] |
G. Lebeau, Contrôle analytique. I. Estimations a priori, (French) [Analytic control. I. A priori estimates], Duke Math. J., 68 (1992), 1-30.
doi: 10.1215/S0012-7094-92-06801-3. |
[8] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, (French) [Exact control of the heat equation], Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[9] |
G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, (French) [Stabilization of the wave equations by the boundary], Duke Math. J., 86 (1997), 465-491.
doi: 10.1215/S0012-7094-97-08614-2. |
[10] |
G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.
doi: 10.1007/s002050050078. |
[11] |
W. Li and X. Zhang, Controllability of parabolic and hyperbolic equations: Toward a unified theory, in "Control Theory of Partial Differential Equations," Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, (2005), 157-174. |
[12] |
J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1," (French) [Exact Controllability, Perturbations and Stabilization of Distributed Systems.Vol. 1], Contrôlabilité Exacte, [Exact Controllability], RMA, 8, Masson, Paris, 1988. |
[13] |
L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.
doi: 10.3934/dcdsb.2010.14.1465. |
[14] |
L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim., 45 (2006), 762-772 (electronic).
doi: 10.1137/S0363012904440654. |
[15] |
K. D. Phung, Waves, damped wave and observation, in "Some Problems on Nonlinear Hyperbolic Equations and Applications" (eds. Ta-Tsien Li, Yue-Jun Peng and Bo-Peng Rao), Series in Contemporary Applied Mathematics CAM 15, 2010. |
[16] |
L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, (French) [Uniqueness theorem adapted to the control of the solutions of hyperbolic problems], in "Équations aux Dérivées Partielles," École Polytech., Palaiseau, 1991. |
[17] |
L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, (French) [Cost function and control of solutions of hyperbolic equations], Asymptotic Anal., 10 (1995), 95-115. |
[18] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[19] |
X. Zhang, A remark on null exact controllability of the heat equation, SIAM J. Control Optim., 40 (2001), 39-53 (electronic).
doi: 10.1137/S0363012900371691. |
show all references
References:
[1] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control and Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[2] |
M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping, J. Differential Equations, 211 (2005), 303-332.
doi: 10.1016/j.jde.2004.12.010. |
[3] |
N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, (French) [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749-752.
doi: 10.1016/S0764-4442(97)80053-5. |
[4] |
S. Ervedoza and E. Zuazua, "Sharp Observability Estimates for the Heat Equation," preprint, 2011. |
[5] |
A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. |
[6] |
L. Hörmander, "Linear Partial Differential Operators," Springer Verlag, Berlin-New York, 1976. |
[7] |
G. Lebeau, Contrôle analytique. I. Estimations a priori, (French) [Analytic control. I. A priori estimates], Duke Math. J., 68 (1992), 1-30.
doi: 10.1215/S0012-7094-92-06801-3. |
[8] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, (French) [Exact control of the heat equation], Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[9] |
G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, (French) [Stabilization of the wave equations by the boundary], Duke Math. J., 86 (1997), 465-491.
doi: 10.1215/S0012-7094-97-08614-2. |
[10] |
G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.
doi: 10.1007/s002050050078. |
[11] |
W. Li and X. Zhang, Controllability of parabolic and hyperbolic equations: Toward a unified theory, in "Control Theory of Partial Differential Equations," Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, (2005), 157-174. |
[12] |
J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1," (French) [Exact Controllability, Perturbations and Stabilization of Distributed Systems.Vol. 1], Contrôlabilité Exacte, [Exact Controllability], RMA, 8, Masson, Paris, 1988. |
[13] |
L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.
doi: 10.3934/dcdsb.2010.14.1465. |
[14] |
L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim., 45 (2006), 762-772 (electronic).
doi: 10.1137/S0363012904440654. |
[15] |
K. D. Phung, Waves, damped wave and observation, in "Some Problems on Nonlinear Hyperbolic Equations and Applications" (eds. Ta-Tsien Li, Yue-Jun Peng and Bo-Peng Rao), Series in Contemporary Applied Mathematics CAM 15, 2010. |
[16] |
L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, (French) [Uniqueness theorem adapted to the control of the solutions of hyperbolic problems], in "Équations aux Dérivées Partielles," École Polytech., Palaiseau, 1991. |
[17] |
L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, (French) [Cost function and control of solutions of hyperbolic equations], Asymptotic Anal., 10 (1995), 95-115. |
[18] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[19] |
X. Zhang, A remark on null exact controllability of the heat equation, SIAM J. Control Optim., 40 (2001), 39-53 (electronic).
doi: 10.1137/S0363012900371691. |
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