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March  2021, 10(1): 61-102. doi: 10.3934/eect.2020052

Boundary null-controllability of coupled parabolic systems with Robin conditions

Kuntal Bhandari 1, and  Franck Boyer 1,2,

1. 

Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier
2. 

Institut Universitaire de France, 31062 Toulouse Cedex 09, France

Received  July 2019 Revised  January 2020 Published  March 2021 Early access  May 2020

Fund Project: The work of the first author was partially supported by the Labex CIMI (Centre International de Mathématiques et d'Informatique), ANR-11-LABX-0040-CIMI

The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems.

Keywords: Parabolic systems, boundary controllability, moments method, spectral estimates, penalisation approach.
Mathematics Subject Classification: 35B30, 35K20, 93B05.
Citation: Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052
References:
[1]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[2]

D. Allonsius and F. Boyer, Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, Mathematical Control & Related Fields. doi: 10.3934/mcrf.2019037.  Google Scholar

[3]

D. Allonsius, F. Boyer and M. Morancey, Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., 140 (2018), 857-911.  doi: 10.1007/s00211-018-0983-1.  Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[5]

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.  Google Scholar

[6]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.  doi: 10.1016/j.jfa.2014.07.024.  Google Scholar

[7]

F. V. Atkinson, Discrete and continuous boundary problems, in Mathematics in Science and Engineering, 8, Academic Press, New York-London, 1964.  Google Scholar

[8]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[9]

F. B. Belgacem, H. E. Fekih and H. Metoui, Singular perturbation for the dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal., 37 (2003), 833-850.  doi: 10.1051/m2an:2003057.  Google Scholar

[10]

F. B. Belgacem, H. E. Fekih and J. P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136.   Google Scholar

[11]

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 J. Control Optim., 52 (2014), 2970–3001. doi: 10.1137/130929680.  Google Scholar

[12]

A. Benabdallah, F. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Annales Henri Lebesgue, preprint, URL https://hal.archives-ouvertes.fr/hal-01949391. Google Scholar

[13]

F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[14]

E. Casas, M. Mateos and J.-P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM Control Optim. Calc. Var., 15 (2009), 782–809. doi: 10.1051/cocv:2008049.  Google Scholar

[15]

J.-M. Coron, Control and nonlinearity, in Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[16]

K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[17]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.  Google Scholar

[18]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[19]

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.  Google Scholar

[20]

H. Hochstadt, Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math., 14 (1961), 749-764.  doi: 10.1002/cpa.3160140408.  Google Scholar

[21]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.  doi: 10.1006/jdeq.1996.0154.  Google Scholar

[22]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[23]

R. Nittka, Inhomogeneous parabolic Neumann problems, Czechoslovak Math. J., 64 (2014), 703-742.  doi: 10.1007/s10587-014-0127-4.  Google Scholar

[24]

G. Olive, Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, 3 (2014), 167-189.  doi: 10.3934/eect.2014.3.167.  Google Scholar

[25]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[2]

D. Allonsius and F. Boyer, Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, Mathematical Control & Related Fields. doi: 10.3934/mcrf.2019037.  Google Scholar

[3]

D. Allonsius, F. Boyer and M. Morancey, Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., 140 (2018), 857-911.  doi: 10.1007/s00211-018-0983-1.  Google Scholar

[4]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[5]

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.  Google Scholar

[6]

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.  doi: 10.1016/j.jfa.2014.07.024.  Google Scholar

[7]

F. V. Atkinson, Discrete and continuous boundary problems, in Mathematics in Science and Engineering, 8, Academic Press, New York-London, 1964.  Google Scholar

[8]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[9]

F. B. Belgacem, H. E. Fekih and H. Metoui, Singular perturbation for the dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal., 37 (2003), 833-850.  doi: 10.1051/m2an:2003057.  Google Scholar

[10]

F. B. Belgacem, H. E. Fekih and J. P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136.   Google Scholar

[11]

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 J. Control Optim., 52 (2014), 2970–3001. doi: 10.1137/130929680.  Google Scholar

[12]

A. Benabdallah, F. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Annales Henri Lebesgue, preprint, URL https://hal.archives-ouvertes.fr/hal-01949391. Google Scholar

[13]

F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[14]

E. Casas, M. Mateos and J.-P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM Control Optim. Calc. Var., 15 (2009), 782–809. doi: 10.1051/cocv:2008049.  Google Scholar

[15]

J.-M. Coron, Control and nonlinearity, in Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[16]

K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[17]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.  Google Scholar

[18]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[19]

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.  Google Scholar

[20]

H. Hochstadt, Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math., 14 (1961), 749-764.  doi: 10.1002/cpa.3160140408.  Google Scholar

[21]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.  doi: 10.1006/jdeq.1996.0154.  Google Scholar

[22]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[23]

R. Nittka, Inhomogeneous parabolic Neumann problems, Czechoslovak Math. J., 64 (2014), 703-742.  doi: 10.1007/s10587-014-0127-4.  Google Scholar

[24]

G. Olive, Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, 3 (2014), 167-189.  doi: 10.3934/eect.2014.3.167.  Google Scholar

[25]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

Figure 1.  The cylindrical geometry
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