Partial null controllability of parabolic linear systems

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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

Abstract
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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 and 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), 1005-1072.
[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), 427-457. doi: 10.7153/dea-01-24.
[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), 267-291. doi: 10.1007/s00028-009-0008-8.
[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), 267-306. doi: 10.3934/mcrf.2011.1.267.
[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), 391-396. doi: 10.1016/j.crma.2014.03.004.
[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, Available from: https://hal.archives-ouvertes.fr/hal-01165713.
[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), 17-44.
[8]	F. Boyer, On the penalised {HUM} approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012, 41e Congrès National d'Analyse Numérique, EDP Sci., Les Ulis, 41 (2013), 15-58. doi: 10.1051/proc/201341002.
[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), 263-287. doi: 10.3934/mcrf.2014.4.263.
[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), 129-141. doi: 10.1016/j.mbs.2009.03.005.
[11]	J.-M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 136 2007.
[12]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, Gauthier-Villars, Paris-Brussels-Montreal, 1974.
[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), 272-292.
[14]	E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758. doi: 10.1016/j.jfa.2010.06.003.
[15]	E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.
[16]	A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, 1996.
[17]	J.-M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal., 10 (1986), 777-790. doi: 10.1016/0362-546X(86)90037-4.
[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, 117. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.
[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), 91-113. doi: 10.4171/PM/1859.
[20]	S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300. doi: 10.1051/cocv/2012013.
[21]	J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris, 2 1968.
[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), 187-210. doi: 10.1016/j.matpur.2012.06.010.
[23]	S. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., 31 (1958), 219-239.
[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), 257-280. doi: 10.1007/s00498-011-0071-x.

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), 1005-1072.
[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), 427-457. doi: 10.7153/dea-01-24.
[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), 267-291. doi: 10.1007/s00028-009-0008-8.
[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), 267-306. doi: 10.3934/mcrf.2011.1.267.
[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), 391-396. doi: 10.1016/j.crma.2014.03.004.
[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, Available from: https://hal.archives-ouvertes.fr/hal-01165713.
[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), 17-44.
[8]	F. Boyer, On the penalised {HUM} approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012, 41e Congrès National d'Analyse Numérique, EDP Sci., Les Ulis, 41 (2013), 15-58. doi: 10.1051/proc/201341002.
[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), 263-287. doi: 10.3934/mcrf.2014.4.263.
[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), 129-141. doi: 10.1016/j.mbs.2009.03.005.
[11]	J.-M. Coron, Control and Nonlinearity, American Mathematical Society, Providence, RI, 136 2007.
[12]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, Gauthier-Villars, Paris-Brussels-Montreal, 1974.
[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), 272-292.
[14]	E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758. doi: 10.1016/j.jfa.2010.06.003.
[15]	E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514.
[16]	A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, 1996.
[17]	J.-M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal., 10 (1986), 777-790. doi: 10.1016/0362-546X(86)90037-4.
[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, 117. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.
[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), 91-113. doi: 10.4171/PM/1859.
[20]	S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300. doi: 10.1051/cocv/2012013.
[21]	J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris, 2 1968.
[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), 187-210. doi: 10.1016/j.matpur.2012.06.010.
[23]	S. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., 31 (1958), 219-239.
[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), 257-280. doi: 10.1007/s00498-011-0071-x.

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