September  2014, 4(3): 263-287. doi: 10.3934/mcrf.2014.4.263

Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients

1. 

Aix-Marseille Université, CNRS, Centrale Marseille, Laboratoire d'Analyse Topologie et Probabilités, UMR 7353, 13453 Marseille, France, France

Received  August 2013 Revised  December 2013 Published  April 2014

In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms.
    Our analysis is mainly based on a criterion given by Fattorini in [12] (and systematically used in [22] for instance), that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems. This work gives theoretical justifications of some numerical observations described in [9].
Citation: Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263
References:
[1]

F. Alabau-Boussouira, Controllability of cascade coupled systems of multi-dimensional evolution PDEs by a reduced number of controls,, C. R. Math. Acad. Sci. Paris, 350 (2012), 577.  doi: 10.1016/j.crma.2012.05.009.  Google Scholar

[2]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications,, J. Math. Pures Appl., 99 (2013), 544.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[3]

G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 284.  doi: 10.1051/cocv:2007055.  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. Dupai 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, 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

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences,, preprint, (2013).   Google Scholar

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, Controllability to trajectories for some parabolic systems of three and two equations by one control force,, Math. Control Relat. Fields, 4 (2014), 17.  doi: 10.3934/mcrf.2014.4.17.  Google Scholar

[9]

F. Boyer, On the penalized hum approach and its applications to the numerical approximation of null-controls for parabolic problems,, ESAIM Proceedings, 41 (2013), 15.  doi: 10.1051/proc/201341002.  Google Scholar

[10]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems,, preprint, (2012).   Google Scholar

[11]

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

[12]

H. O. Fattorini, Some remarks on complete controllability,, SIAM J. Control, 4 (1966), 686.  doi: 10.1137/0304048.  Google Scholar

[13]

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

[14]

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

[15]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic {PDE}s by one control force,, Port. Math., 67 (2010), 91.  doi: 10.4171/PM/1859.  Google Scholar

[16]

J.-M. Ghidaglia, Some backward uniqueness results,, Nonlinear Anal., 10 (1986), 777.  doi: 10.1016/0362-546X(86)90037-4.  Google Scholar

[17]

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

[18]

F. Luca and L. de Teresa, Control of coupled parabolic systems and Diophantine approximations,, S$\vec e$MA J., 61 (2013), 1.  doi: 10.1007/s40324-013-0004-3.  Google Scholar

[19]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils,, Translated from the Russian by H. H. McFaden, (1988).   Google Scholar

[20]

K. Mauffrey, On the null controllability of a $3\times3$ parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl., 99 (2013), 187.  doi: 10.1016/j.matpur.2012.06.010.  Google Scholar

[21]

S. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques,, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 31 (1958), 219.   Google Scholar

[22]

G. Olive, Boundary approximate controllability of some linear parabolic systems,, Evol. Equ. Control Theory, 3 (2014), 167.  doi: 10.3934/eect.2014.3.167.  Google Scholar

[23]

L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291.  doi: 10.1016/j.crma.2011.01.014.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, Controllability of cascade coupled systems of multi-dimensional evolution PDEs by a reduced number of controls,, C. R. Math. Acad. Sci. Paris, 350 (2012), 577.  doi: 10.1016/j.crma.2012.05.009.  Google Scholar

[2]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications,, J. Math. Pures Appl., 99 (2013), 544.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[3]

G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 284.  doi: 10.1051/cocv:2007055.  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. Dupai 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, 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

[7]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences,, preprint, (2013).   Google Scholar

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, Controllability to trajectories for some parabolic systems of three and two equations by one control force,, Math. Control Relat. Fields, 4 (2014), 17.  doi: 10.3934/mcrf.2014.4.17.  Google Scholar

[9]

F. Boyer, On the penalized hum approach and its applications to the numerical approximation of null-controls for parabolic problems,, ESAIM Proceedings, 41 (2013), 15.  doi: 10.1051/proc/201341002.  Google Scholar

[10]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems,, preprint, (2012).   Google Scholar

[11]

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

[12]

H. O. Fattorini, Some remarks on complete controllability,, SIAM J. Control, 4 (1966), 686.  doi: 10.1137/0304048.  Google Scholar

[13]

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

[14]

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

[15]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic {PDE}s by one control force,, Port. Math., 67 (2010), 91.  doi: 10.4171/PM/1859.  Google Scholar

[16]

J.-M. Ghidaglia, Some backward uniqueness results,, Nonlinear Anal., 10 (1986), 777.  doi: 10.1016/0362-546X(86)90037-4.  Google Scholar

[17]

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

[18]

F. Luca and L. de Teresa, Control of coupled parabolic systems and Diophantine approximations,, S$\vec e$MA J., 61 (2013), 1.  doi: 10.1007/s40324-013-0004-3.  Google Scholar

[19]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils,, Translated from the Russian by H. H. McFaden, (1988).   Google Scholar

[20]

K. Mauffrey, On the null controllability of a $3\times3$ parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl., 99 (2013), 187.  doi: 10.1016/j.matpur.2012.06.010.  Google Scholar

[21]

S. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques,, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 31 (1958), 219.   Google Scholar

[22]

G. Olive, Boundary approximate controllability of some linear parabolic systems,, Evol. Equ. Control Theory, 3 (2014), 167.  doi: 10.3934/eect.2014.3.167.  Google Scholar

[23]

L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291.  doi: 10.1016/j.crma.2011.01.014.  Google Scholar

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