# 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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.  Google Scholar [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.  Google Scholar [13] S. Lang, Algebra, Springer New York, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.  Google Scholar [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.  Google Scholar [13] S. Lang, Algebra, Springer New York, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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
 [1] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [2] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [3] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [4] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [5] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [6] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [7] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [8] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 [9] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [10] Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 [11] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [12] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [13] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [14] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122 [15] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [16] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [17] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [18] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355 [19] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [20] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 0.857

## Metrics

• PDF downloads (167)
• HTML views (479)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]