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June  2016, 6(2): 185-216. doi: 10.3934/mcrf.2016001

Partial null controllability of parabolic linear systems

Farid Ammar Khodja 1, , Franz Chouly 1, and  Michel Duprez 1,

1. 

Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France, France, France

Received  February 2015 Revised  October 2015 Published  April 2016

This paper is devoted to the partial null controllability issue of parabolic linear systems with $n$ equations. Given a bounded domain $\Omega$ in $\mathbb{R}^N$ ($N\in \mathbb{N}^*$), we study the effect of $m$ localized controls in a nonempty open subset $\omega$ only controlling $p$ components of the solution ($p,m \le n$). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled $2\times2$ parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.
Keywords: Kalman condition, observability, Controllability, parabolic systems., moment method.
Mathematics Subject Classification: Primary: 93B05, 93B07; Secondary: 93C20, 93C05, 35K4.
Citation: Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls,, Adv. Differential Equations, 18 (2013), 1005.   Google Scholar

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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

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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

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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

[5]

F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains,, C. R. Math. Acad. Sci. Paris, 352 (2014), 391.  doi: 10.1016/j.crma.2014.03.004.  Google Scholar

[6]

F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence,, preprint, ().   Google Scholar

[7]

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.   Google Scholar

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F. Boyer, On the penalised {HUM} approach and its applications to the numerical approximation of null-controls for parabolic problems,, in CANUM 2012, 41 (2013), 15.  doi: 10.1051/proc/201341002.  Google Scholar

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F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients,, Math. Control Relat. Fields, 4 (2014), 263.  doi: 10.3934/mcrf.2014.4.263.  Google Scholar

[10]

S. Chakrabarty and F. B. Hanson, Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method,, Math. biosci., 219 (2009), 129.  doi: 10.1016/j.mbs.2009.03.005.  Google Scholar

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J.-M. Coron, Control and Nonlinearity,, American Mathematical Society, 136 (2007).   Google Scholar

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I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels,, Dunod, (1974).   Google Scholar

[13]

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

[14]

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

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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

[16]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations,, Seoul National University, (1996).   Google Scholar

[17]

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

[18]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems,, A numerical approach. Encyclopedia of Mathematics and its Applications, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar

[19]

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

[20]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory,, ESAIM Control Optim. Calc. Var., 19 (2013), 288.  doi: 10.1051/cocv/2012013.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications,, Dunod, 2 (1968).   Google Scholar

[22]

K. Mauffrey, On the null controllability of a $3 \times 3$ 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

[23]

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

[24]

G. Olive, Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary,, Math. Control Signals Systems, 23 (2012), 257.  doi: 10.1007/s00498-011-0071-x.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls,, Adv. Differential Equations, 18 (2013), 1005.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains,, C. R. Math. Acad. Sci. Paris, 352 (2014), 391.  doi: 10.1016/j.crma.2014.03.004.  Google Scholar

[6]

F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence,, preprint, ().   Google Scholar

[7]

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.   Google Scholar

[8]

F. Boyer, On the penalised {HUM} approach and its applications to the numerical approximation of null-controls for parabolic problems,, in CANUM 2012, 41 (2013), 15.  doi: 10.1051/proc/201341002.  Google Scholar

[9]

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients,, Math. Control Relat. Fields, 4 (2014), 263.  doi: 10.3934/mcrf.2014.4.263.  Google Scholar

[10]

S. Chakrabarty and F. B. Hanson, Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method,, Math. biosci., 219 (2009), 129.  doi: 10.1016/j.mbs.2009.03.005.  Google Scholar

[11]

J.-M. Coron, Control and Nonlinearity,, American Mathematical Society, 136 (2007).   Google Scholar

[12]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels,, Dunod, (1974).   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations,, Seoul National University, (1996).   Google Scholar

[17]

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

[18]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems,, A numerical approach. Encyclopedia of Mathematics and its Applications, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar

[19]

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

[20]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory,, ESAIM Control Optim. Calc. Var., 19 (2013), 288.  doi: 10.1051/cocv/2012013.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications,, Dunod, 2 (1968).   Google Scholar

[22]

K. Mauffrey, On the null controllability of a $3 \times 3$ 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

[23]

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

[24]

G. Olive, Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary,, Math. Control Signals Systems, 23 (2012), 257.  doi: 10.1007/s00498-011-0071-x.  Google Scholar

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