American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On a final value problem for a class of nonlinear hyperbolic equations with damping term
  • EECT Home
  • This Issue
  • Next Article
    Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions
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  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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021  doi: 10.3934/fods.2021005
[2]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[3]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
[4]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[5]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[6]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015
[7]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[8]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[9]

Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2509-2535. doi: 10.3934/dcdsb.2020193
[10]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[11]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[12]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[13]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[14]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[15]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[16]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[17]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[18]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[19]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[20]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (151)
  • HTML views (343)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences