March  2014, 4(1): 17-44. doi: 10.3934/mcrf.2014.4.17

Controllability to trajectories for some parabolic systems of three and two equations by one control force

1. 

Aix-Marseille University, LATP UMR 7353, France, France

2. 

Aix-Marseille University, CPT UMR 7332, France

3. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

Received  May 2012 Revised  November 2012 Published  December 2013

We present a controllability result for a class of linear parabolic systems of $3$ equations. We establish a global Carleman estimate for the solutions of systems of $2$ parabolic equations coupled with first order terms. Stability results for inverse coefficients problems are deduced.
Citation: Assia Benabdallah, Michel Cristofol, Patricia Gaitan, Luz de Teresa. Controllability to trajectories for some parabolic systems of three and two equations by one control force. Mathematical Control & Related Fields, 2014, 4 (1) : 17-44. doi: 10.3934/mcrf.2014.4.17
References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostine, 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

[2]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostine, 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

[3]

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

[4]

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.  doi: 10.7153/dea-01-24.  Google Scholar

[5]

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

[6]

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,, J. Math. Pures Appl. (9), 96 (2011), 555.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component,, Appl. Anal., 88 (2009), 683.  doi: 10.1080/00036810802555490.  Google Scholar

[9]

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

[10]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Soviet Math. Dokl., 17 (1981), 244.   Google Scholar

[11]

H. Cartan, Calcul Différentiel,, Hermann, (1967).   Google Scholar

[12]

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

[13]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a 2X2 reaction-diffusion system using a Carleman estimate with one observation,, Inverse Problems, 22 (2006), 1561.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[14]

M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system,, Applicable Analysis, 91 (2012), 2073.  doi: 10.1080/00036811.2011.583240.  Google Scholar

[15]

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

[16]

E. Fernández-Cara, Null controllability of the semilinear heat equation,, ESAIM Control Optim. Calc. Var., 2 (1997), 87.  doi: 10.1051/cocv:1997104.  Google Scholar

[17]

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

[18]

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.  doi: 10.1016/j.matpur.2004.02.010.  Google Scholar

[19]

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

[20]

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

[21]

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.  doi: 10.4171/PM/1859.  Google Scholar

[22]

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

[23]

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

[24]

O. Yu. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for dynamical Lamé systems with two sets of boundary,, Comm. Pure Appl. Math., LVI (2003), 1366.   Google Scholar

[25]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I,, Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[26]

K. Mauffrey, On the null controllability of a parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl. (9), 99 (2013), 187.  doi: 10.1016/j.matpur.2012.06.010.  Google Scholar

[27]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075007.  Google Scholar

[28]

L. de Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39.  doi: 10.1080/03605300008821507.  Google Scholar

[29]

K. Vo-Khac, Distributions, Analyse de Fourier, Opérateurs aux Dérivées Partielles: Cours et Exercices Résolus, avec Une étude Introductive sur des Espaces Vectoriels Topologiques. Tome II,, Vuibert, (1972).   Google Scholar

[30]

M. Wang, Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion,, Math. Biosci., 212 (2008), 149.  doi: 10.1016/j.mbs.2007.08.008.  Google Scholar

show all references

References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostine, 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

[2]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostine, 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

[3]

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

[4]

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.  doi: 10.7153/dea-01-24.  Google Scholar

[5]

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

[6]

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,, J. Math. Pures Appl. (9), 96 (2011), 555.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component,, Appl. Anal., 88 (2009), 683.  doi: 10.1080/00036810802555490.  Google Scholar

[9]

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

[10]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Soviet Math. Dokl., 17 (1981), 244.   Google Scholar

[11]

H. Cartan, Calcul Différentiel,, Hermann, (1967).   Google Scholar

[12]

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

[13]

M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a 2X2 reaction-diffusion system using a Carleman estimate with one observation,, Inverse Problems, 22 (2006), 1561.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[14]

M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system,, Applicable Analysis, 91 (2012), 2073.  doi: 10.1080/00036811.2011.583240.  Google Scholar

[15]

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

[16]

E. Fernández-Cara, Null controllability of the semilinear heat equation,, ESAIM Control Optim. Calc. Var., 2 (1997), 87.  doi: 10.1051/cocv:1997104.  Google Scholar

[17]

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

[18]

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.  doi: 10.1016/j.matpur.2004.02.010.  Google Scholar

[19]

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

[20]

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

[21]

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.  doi: 10.4171/PM/1859.  Google Scholar

[22]

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

[23]

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

[24]

O. Yu. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for dynamical Lamé systems with two sets of boundary,, Comm. Pure Appl. Math., LVI (2003), 1366.   Google Scholar

[25]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I,, Die Grundlehren der mathematischen Wissenschaften, (1972).   Google Scholar

[26]

K. Mauffrey, On the null controllability of a parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl. (9), 99 (2013), 187.  doi: 10.1016/j.matpur.2012.06.010.  Google Scholar

[27]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075007.  Google Scholar

[28]

L. de Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39.  doi: 10.1080/03605300008821507.  Google Scholar

[29]

K. Vo-Khac, Distributions, Analyse de Fourier, Opérateurs aux Dérivées Partielles: Cours et Exercices Résolus, avec Une étude Introductive sur des Espaces Vectoriels Topologiques. Tome II,, Vuibert, (1972).   Google Scholar

[30]

M. Wang, Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion,, Math. Biosci., 212 (2008), 149.  doi: 10.1016/j.mbs.2007.08.008.  Google Scholar

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