# American Institute of Mathematical Sciences

June  2020, 10(2): 217-256. doi: 10.3934/mcrf.2019037

## Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries

 1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France 2 Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France

* Corresponding author: Franck Boyer

Received  July 2018 Published  June 2020 Early access  August 2019

The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than $1$. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general.

The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls.

To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.

Citation: Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037
##### References:
 [1] F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, Journal de Mathématiques Pures et Appliquées, 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012. [2] D. Allonsius, F. Boyer and M. Morancey, Spectral analysis of discrete elliptic operators and applications in control theory, Numerische Mathematik, 140 (2018), 857-911.  doi: 10.1007/s00211-018-0983-1. [3] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267. [4] F. Ammar Khodja, F. Chouly and M. Duprez, Partial null controllability of parabolic linear systems, Math. Control Relat. Fields, 6 (2016), 185-216.  doi: 10.3934/mcrf.2016001. [5] A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the n-dimensional boundary null controllability in cylindrical domains, SIAM Journal on Control and Optimization, 52 (2014), 2970–3001. doi: 10.1137/130929680. [6] K. Bhandari and F. Boyer, Boundary null-controllability of coupled parabolic systems with Robin conditions, Submitted, (2019), https://hal.archives-ouvertes.fr/hal-02091091. [7] 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, ESAIM Proc., EDP Sci., Les Ulis, 41 (2013), 15–58. doi: 10.1051/proc/201341002. [8] F. Boyer, F. Hubert and J. L. Rousseau, Discrete carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, Journal de Mathématiques Pures et Appliquées, 93 (2010), 240-276.  doi: 10.1016/j.matpur.2009.11.003. [9] F. Boyer, F. Hubert and J. L. Rousseau, Discrete carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM Journal on Control and Optimization, 48 (2010), 5357-5397.  doi: 10.1137/100784278. [10] F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations, Annales de l'Institut Henri Poincaré Non Linear Analysis, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011. [11] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. [12] S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett., 55 (2006), 597-609.  doi: 10.1016/j.sysconle.2006.01.004. [13] S. Lang, Algebra, Springer New York, 2002. doi: 10.1007/978-1-4613-0041-0. [14] J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM: COCV, 18 (2012), 712-747.  doi: 10.1051/cocv/2011168. [15] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, (1995), 13 pp. [16] L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465. [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis, 2nd edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [18] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. [19] E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, Eur. Math. Soc., Zürich, 3 (2006), 1389–1417.

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##### References:
 [1] F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, Journal de Mathématiques Pures et Appliquées, 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012. [2] D. Allonsius, F. Boyer and M. Morancey, Spectral analysis of discrete elliptic operators and applications in control theory, Numerische Mathematik, 140 (2018), 857-911.  doi: 10.1007/s00211-018-0983-1. [3] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267. [4] F. Ammar Khodja, F. Chouly and M. Duprez, Partial null controllability of parabolic linear systems, Math. Control Relat. Fields, 6 (2016), 185-216.  doi: 10.3934/mcrf.2016001. [5] A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the n-dimensional boundary null controllability in cylindrical domains, SIAM Journal on Control and Optimization, 52 (2014), 2970–3001. doi: 10.1137/130929680. [6] K. Bhandari and F. Boyer, Boundary null-controllability of coupled parabolic systems with Robin conditions, Submitted, (2019), https://hal.archives-ouvertes.fr/hal-02091091. [7] 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, ESAIM Proc., EDP Sci., Les Ulis, 41 (2013), 15–58. doi: 10.1051/proc/201341002. [8] F. Boyer, F. Hubert and J. L. Rousseau, Discrete carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, Journal de Mathématiques Pures et Appliquées, 93 (2010), 240-276.  doi: 10.1016/j.matpur.2009.11.003. [9] F. Boyer, F. Hubert and J. L. Rousseau, Discrete carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM Journal on Control and Optimization, 48 (2010), 5357-5397.  doi: 10.1137/100784278. [10] F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations, Annales de l'Institut Henri Poincaré Non Linear Analysis, 31 (2014), 1035-1078.  doi: 10.1016/j.anihpc.2013.07.011. [11] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. [12] S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett., 55 (2006), 597-609.  doi: 10.1016/j.sysconle.2006.01.004. [13] S. Lang, Algebra, Springer New York, 2002. doi: 10.1007/978-1-4613-0041-0. [14] J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM: COCV, 18 (2012), 712-747.  doi: 10.1051/cocv/2011168. [15] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, (1995), 13 pp. [16] L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465. [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis, 2nd edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [18] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9. [19] E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, Eur. Math. Soc., Zürich, 3 (2006), 1389–1417.
Typical geometric situation
Grid geometry
Component ${\alpha}$ (left) and ${\beta}$ (right) of system (4) with no control
Component ${\alpha}$ (left) and ${\beta}$ (right) of system (4) with a boundary control
Norms of the components ${\alpha}$, ${\beta}$ of system (4) with and without control
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