# American Institute of Mathematical Sciences

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

## Partial null controllability of parabolic linear systems

 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.
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.

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##### 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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