September  2011, 1(3): 267-306. doi: 10.3934/mcrf.2011.1.267

Recent results on the controllability of linear coupled parabolic problems: A survey

1. 

Laboratoire de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besancon cedex, France

2. 

Centre de Mathématiques et Informatique, Université Aix-Marseille 1, Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France

3. 

Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla

4. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

Received  April 2011 Revised  July 2011 Published  September 2011

This paper tries to summarize recent results on the controllability of systems of (several) parabolic equations. The emphasis is placed on the extension of the Kalman rank condition (for finite dimensional systems of differential equations) to parabolic systems. This question is itself tied with the proof of global Carleman estimates for systems and leads to a wide field of open problems.
Citation: Farid Ammar-Khodja, Assia Benabdallah, Manuel González-Burgos, Luz de Teresa. Recent results on the controllability of linear coupled parabolic problems: A survey. Mathematical Control & Related Fields, 2011, 1 (3) : 267-306. doi: 10.3934/mcrf.2011.1.267
References:
[1]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions,, C. R. Acad. Sci. Paris, 349 (2011), 395. Google Scholar

[2]

F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force,, J. Math. Anal. Appl., 320 (2006), 928. doi: 10.1016/j.jmaa.2005.07.060. Google Scholar

[3]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force,, SIAM J. Control Optim., 42 (2003), 1661. doi: 10.1137/S0363012902417826. Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null controllability of some systems of parabolic type by one control force,, ESAIM Control Optim. Calc. Var., 11 (2005), 426. doi: 10.1051/cocv:2005013. Google Scholar

[5]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems,, Differ. Equ. Appl., 1 (2009), 427. Google Scholar

[6]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems,, J. Evol. Equ., 9 (2009), 267. doi: 10.1007/s00028-009-0008-8. Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials,, to appear in J. Math. Pures Appl., (). doi: 10.1016/j.matpur.2011.06.005. Google Scholar

[8]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, A Kalman condition for the boundary approximate controllability of non scalar parabolic problems with diffusion matrices,, in preparation., (). Google Scholar

[9]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Condensation index and necessary and sufficient conditions for the null controllability of abstract systems. Application to the boundary null controllability of coupled parabolic systems,, in preparation., (). Google Scholar

[10]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. Google Scholar

[11]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications,, C. R. Math. Acad. Sci. Paris, 348 (2010), 25. Google Scholar

[12]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate,, Abstr. Appl. Anal., 7 (2002), 585. doi: 10.1155/S108533750220408X. Google Scholar

[13]

O. Bodart and C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation,, J. Math. Anal. Appl., 195 (1995), 658. doi: 10.1006/jmaa.1995.1382. Google Scholar

[14]

O. Bodart, M. González-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity,, Comm. Partial Differential Equations, 29 (2004), 1017. Google Scholar

[15]

O. Bodart, M. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient,, Nonlinear Anal., 57 (2004), 687. doi: 10.1016/j.na.2004.03.012. Google Scholar

[16]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125. doi: 10.1016/S0022-247X(03)00457-8. Google Scholar

[17]

T. Carleman, Sur une problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939). Google Scholar

[18]

M. Chapouly, "Contrôlabilité d'Équations Issues de la Mécanique des Fluides,", Thèse pour obtenir le grade de Docteur en Mathématiques, (2009). Google Scholar

[19]

J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions,, ESAIM Contrôle Optim. Calc. Var., 1 (): 35. Google Scholar

[20]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007). Google Scholar

[21]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls,, J. Differential Equations, 246 (2009), 2908. Google Scholar

[22]

J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term,, SIAM J. Control Optim., 48 (2010), 5629. doi: 10.1137/100784539. Google Scholar

[23]

R. Dáger, Insensitizing controls for the 1-D wave equation,, SIAM J. Control Optim., 45 (2006), 1758. doi: 10.1137/060654372. Google Scholar

[24]

S. Dolecki and D. L. Russell, A general theory of observation and control,, SIAM J. Control Optim., 15 (1977), 185. doi: 10.1137/0315015. Google Scholar

[25]

A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient,, SIAM J. Control Optim., 41 (2002), 798. doi: 10.1137/S0363012901386465. Google Scholar

[26]

P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models,", Nonlinear Science: Theory and Applications, (1989). Google Scholar

[27]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 43 (1971), 272. Google Scholar

[28]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45. Google Scholar

[29]

E. Fernández-Cara, Some controllability results in fluid mechanics, in "Partial Differential Equations and Fluid Mechanics," 64-80,, London Math. Soc. Lecture Note Ser., 364 (2009). Google Scholar

[30]

E. Fernández-Cara, G. García and A. Osses, Controls insensitizing the observation of a quasi-geostrophic ocean model,, SIAM J. Control and Optim., 43 (2005), 1616. Google Scholar

[31]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability,, SIAM J. Control Optim., 45 (2006), 1399. Google Scholar

[32]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct. Anal., 259 (2010), 1720. doi: 10.1016/j.jfa.2010.06.003. Google Scholar

[33]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Controllability of linear and semilinear non-diagonalizable parabolic systems,, in preparation., (). Google Scholar

[34]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system,, J. Math. Pures Appl. (9), 83 (2004), 1501. Google Scholar

[35]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Some controllability results for the $N$ -dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls,, SIAM J. Control Optim., 45 (2006), 146. doi: 10.1137/04061965X. Google Scholar

[36]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. Google Scholar

[37]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Adv. Differential Equations, 5 (2000), 465. Google Scholar

[38]

A. Fursikov and O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, (Russian), Uspekhi Mat. Nauk, 54 (1999), 93. Google Scholar

[39]

A. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series, 34 (1996). Google Scholar

[40]

M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force,, Asymptot. Anal., 46 (2006), 123. Google Scholar

[41]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force,, Port. Math., 67 (2010), 91. Google Scholar

[42]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135. Google Scholar

[43]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029. Google Scholar

[44]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[45]

L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963). Google Scholar

[46]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,", Corrected reprint of the 1985 original, 275 (1985). Google Scholar

[47]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations,, Publ. Res. Inst. Math. Sci., 39 (2003), 227. doi: 10.2977/prims/1145476103. Google Scholar

[48]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations,, ESAIM Control Optim. Calc. Var., 16 (2010), 247. doi: 10.1051/cocv/2008077. Google Scholar

[49]

D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth,, Biophysical Journal, 40 (1982), 209. doi: 10.1016/S0006-3495(82)84476-7. Google Scholar

[50]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967). Google Scholar

[51]

J. E. Lagnese, The reachability problem for thermoelastic plates,, Arch. Rational Mech. Anal., 112 (1990), 223. doi: 10.1007/BF00381235. Google Scholar

[52]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., (). Google Scholar

[53]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems,, J. Funct. Anal., 258 (2010), 2739. doi: 10.1016/j.jfa.2009.10.011. Google Scholar

[54]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. Google Scholar

[55]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. Google Scholar

[56]

J.-L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes,, in, (1989), 43. Google Scholar

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F. Luca and L. de Teresa, System controllability and Diophantine approximations,, preprint., (). Google Scholar

[58]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Appl. Math., 52 (1973), 189. Google Scholar

[59]

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. Google Scholar

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N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

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L. de Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39. Google Scholar

[64]

L. de Teresa and E. Zuazua, Identification of the class of initial data for the insensitizing control of the heat equation,, Commun. Pure Appl. Anal., 8 (2009), 457. doi: 10.3934/cpaa.2009.8.457. Google Scholar

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show all references

References:
[1]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions,, C. R. Acad. Sci. Paris, 349 (2011), 395. Google Scholar

[2]

F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force,, J. Math. Anal. Appl., 320 (2006), 928. doi: 10.1016/j.jmaa.2005.07.060. Google Scholar

[3]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force,, SIAM J. Control Optim., 42 (2003), 1661. doi: 10.1137/S0363012902417826. Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null controllability of some systems of parabolic type by one control force,, ESAIM Control Optim. Calc. Var., 11 (2005), 426. doi: 10.1051/cocv:2005013. Google Scholar

[5]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems,, Differ. Equ. Appl., 1 (2009), 427. Google Scholar

[6]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems,, J. Evol. Equ., 9 (2009), 267. doi: 10.1007/s00028-009-0008-8. Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials,, to appear in J. Math. Pures Appl., (). doi: 10.1016/j.matpur.2011.06.005. Google Scholar

[8]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, A Kalman condition for the boundary approximate controllability of non scalar parabolic problems with diffusion matrices,, in preparation., (). Google Scholar

[9]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Condensation index and necessary and sufficient conditions for the null controllability of abstract systems. Application to the boundary null controllability of coupled parabolic systems,, in preparation., (). Google Scholar

[10]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. Google Scholar

[11]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications,, C. R. Math. Acad. Sci. Paris, 348 (2010), 25. Google Scholar

[12]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate,, Abstr. Appl. Anal., 7 (2002), 585. doi: 10.1155/S108533750220408X. Google Scholar

[13]

O. Bodart and C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation,, J. Math. Anal. Appl., 195 (1995), 658. doi: 10.1006/jmaa.1995.1382. Google Scholar

[14]

O. Bodart, M. González-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity,, Comm. Partial Differential Equations, 29 (2004), 1017. Google Scholar

[15]

O. Bodart, M. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient,, Nonlinear Anal., 57 (2004), 687. doi: 10.1016/j.na.2004.03.012. Google Scholar

[16]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125. doi: 10.1016/S0022-247X(03)00457-8. Google Scholar

[17]

T. Carleman, Sur une problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939). Google Scholar

[18]

M. Chapouly, "Contrôlabilité d'Équations Issues de la Mécanique des Fluides,", Thèse pour obtenir le grade de Docteur en Mathématiques, (2009). Google Scholar

[19]

J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions,, ESAIM Contrôle Optim. Calc. Var., 1 (): 35. Google Scholar

[20]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007). Google Scholar

[21]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls,, J. Differential Equations, 246 (2009), 2908. Google Scholar

[22]

J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term,, SIAM J. Control Optim., 48 (2010), 5629. doi: 10.1137/100784539. Google Scholar

[23]

R. Dáger, Insensitizing controls for the 1-D wave equation,, SIAM J. Control Optim., 45 (2006), 1758. doi: 10.1137/060654372. Google Scholar

[24]

S. Dolecki and D. L. Russell, A general theory of observation and control,, SIAM J. Control Optim., 15 (1977), 185. doi: 10.1137/0315015. Google Scholar

[25]

A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient,, SIAM J. Control Optim., 41 (2002), 798. doi: 10.1137/S0363012901386465. Google Scholar

[26]

P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models,", Nonlinear Science: Theory and Applications, (1989). Google Scholar

[27]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 43 (1971), 272. Google Scholar

[28]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (): 45. Google Scholar

[29]

E. Fernández-Cara, Some controllability results in fluid mechanics, in "Partial Differential Equations and Fluid Mechanics," 64-80,, London Math. Soc. Lecture Note Ser., 364 (2009). Google Scholar

[30]

E. Fernández-Cara, G. García and A. Osses, Controls insensitizing the observation of a quasi-geostrophic ocean model,, SIAM J. Control and Optim., 43 (2005), 1616. Google Scholar

[31]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability,, SIAM J. Control Optim., 45 (2006), 1399. Google Scholar

[32]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct. Anal., 259 (2010), 1720. doi: 10.1016/j.jfa.2010.06.003. Google Scholar

[33]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Controllability of linear and semilinear non-diagonalizable parabolic systems,, in preparation., (). Google Scholar

[34]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system,, J. Math. Pures Appl. (9), 83 (2004), 1501. Google Scholar

[35]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Some controllability results for the $N$ -dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls,, SIAM J. Control Optim., 45 (2006), 146. doi: 10.1137/04061965X. Google Scholar

[36]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. Google Scholar

[37]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Adv. Differential Equations, 5 (2000), 465. Google Scholar

[38]

A. Fursikov and O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, (Russian), Uspekhi Mat. Nauk, 54 (1999), 93. Google Scholar

[39]

A. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series, 34 (1996). Google Scholar

[40]

M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force,, Asymptot. Anal., 46 (2006), 123. Google Scholar

[41]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force,, Port. Math., 67 (2010), 91. Google Scholar

[42]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135. Google Scholar

[43]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029. Google Scholar

[44]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[45]

L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963). Google Scholar

[46]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators,", Corrected reprint of the 1985 original, 275 (1985). Google Scholar

[47]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations,, Publ. Res. Inst. Math. Sci., 39 (2003), 227. doi: 10.2977/prims/1145476103. Google Scholar

[48]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations,, ESAIM Control Optim. Calc. Var., 16 (2010), 247. doi: 10.1051/cocv/2008077. Google Scholar

[49]

D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth,, Biophysical Journal, 40 (1982), 209. doi: 10.1016/S0006-3495(82)84476-7. Google Scholar

[50]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967). Google Scholar

[51]

J. E. Lagnese, The reachability problem for thermoelastic plates,, Arch. Rational Mech. Anal., 112 (1990), 223. doi: 10.1007/BF00381235. Google Scholar

[52]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations,, to appear in ESAIM Control Optim. Calc. Var., (). Google Scholar

[53]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems,, J. Funct. Anal., 258 (2010), 2739. doi: 10.1016/j.jfa.2009.10.011. Google Scholar

[54]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. Google Scholar

[55]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. Google Scholar

[56]

J.-L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes,, in, (1989), 43. Google Scholar

[57]

F. Luca and L. de Teresa, System controllability and Diophantine approximations,, preprint., (). Google Scholar

[58]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Appl. Math., 52 (1973), 189. Google Scholar

[59]

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. Google Scholar

[60]

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