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

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

Received  December 2010 Revised  March 2011 Published  June 2011

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.
Citation: Sylvain Ervedoza, Enrique Zuazua. Observability of heat processes by transmutation without geometric restrictions. Mathematical Control & 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. 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. 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. 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, (1996).

[6]

L. Hörmander, "Linear Partial Differential Operators,", Springer Verlag, (1976).

[7]

G. Lebeau, Contrôle analytique. I. Estimations a priori, (French) [Analytic control. I. A priori estimates],, Duke Math. J., 68 (1992), 1. 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. 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. 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. doi: 10.1007/s002050050078.

[11]

W. Li and X. Zhang, Controllability of parabolic and hyperbolic equations: Toward a unified theory,, in, 242 (2005), 157.

[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, (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. 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. doi: 10.1137/S0363012904440654.

[15]

K. D. Phung, Waves, damped wave and observation,, in, (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, (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.

[18]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095.

[19]

X. Zhang, A remark on null exact controllability of the heat equation,, SIAM J. Control Optim., 40 (2001), 39. 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. 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. 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. 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, (1996).

[6]

L. Hörmander, "Linear Partial Differential Operators,", Springer Verlag, (1976).

[7]

G. Lebeau, Contrôle analytique. I. Estimations a priori, (French) [Analytic control. I. A priori estimates],, Duke Math. J., 68 (1992), 1. 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. 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. 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. doi: 10.1007/s002050050078.

[11]

W. Li and X. Zhang, Controllability of parabolic and hyperbolic equations: Toward a unified theory,, in, 242 (2005), 157.

[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, (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. 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. doi: 10.1137/S0363012904440654.

[15]

K. D. Phung, Waves, damped wave and observation,, in, (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, (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.

[18]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095.

[19]

X. Zhang, A remark on null exact controllability of the heat equation,, SIAM J. Control Optim., 40 (2001), 39. doi: 10.1137/S0363012900371691.

[1]

Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1465-1485. doi: 10.3934/dcdsb.2010.14.1465

[2]

C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663

[3]

Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849

[4]

Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55

[5]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[6]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105

[7]

Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35

[8]

Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307

[9]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969

[10]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[11]

Filippo Gazzola, Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in RN. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 83-87. doi: 10.3934/dcdss.2008.1.83

[12]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[13]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[14]

Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109

[15]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012

[16]

Susana Merchán, Luigi Montoro, I. Peral. Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 245-268. doi: 10.3934/cpaa.2015.14.245

[17]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193

[18]

Chin-Chin Wu, Zhengce Zhang. Dead-core rates for the heat equation with a spatially dependent strong absorption. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2203-2210. doi: 10.3934/dcdsb.2013.18.2203

[19]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[20]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277

2017 Impact Factor: 0.542

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]