Observability of heat processes by transmutation without geometric restrictions

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June 2011, 1(2): 177-187. doi: 10.3934/mcrf.2011.1.177

Observability of heat processes by transmutation without geometric restrictions

Sylvain Ervedoza ^1, and Enrique Zuazua ^2,

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

Received December 2010 Revised March 2011 Published June 2011

Abstract
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The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.

Keywords: Heat equation, observability, transmutation..

Mathematics Subject Classification: Primary: 35C15, 93B07; Secondary: 35K0.

Citation: Sylvain Ervedoza, Enrique Zuazua. Observability of heat processes by transmutation without geometric restrictions. Mathematical Control and Related Fields, 2011, 1 (2) : 177-187. doi: 10.3934/mcrf.2011.1.177

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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