• Previous Article
    Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method
  • EECT Home
  • This Issue
  • Next Article
    Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains
2014, 3(1): 167-189. doi: 10.3934/eect.2014.3.167

Boundary approximate controllability of some linear parabolic systems

1. 

LATP, UMR 7353, Aix-Marseille université, Technopôle Château-Gombert, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France

Received  April 2013 Revised  December 2013 Published  February 2014

This paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of $n$ heat equations coupled through constant terms and a $2 \times 2$ cascade system coupled by means of a first order partial differential operator with space-dependent coefficients.
    For each system we prove a sufficient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we prove the distributed approximate controllability in any space dimension of a cascade system coupled by a constant first order term.
    The method relies on a general characterization due to H.O. Fattorini.
Citation: Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167
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. doi: 10.1016/j.matpur.2012.09.012.

[2]

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 parbolic systems,, J. Evol. Equ., 9 (2009), 267. doi: 10.1007/s00028-009-0008-8.

[3]

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. doi: 10.7153/dea-01-24.

[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. doi: 10.1016/j.matpur.2011.06.005.

[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. doi: 10.3934/mcrf.2011.1.267.

[6]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems,, to appear in ESAIM Control Optim. Calc. Var., (2014).

[7]

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,, to appear in SIAM J. Control Optim., (2014).

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications,, C. R. Math. Acad. Sci. Paris, 348 (2010), 25. doi: 10.1016/j.crma.2009.11.001.

[9]

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients,, to appear in Math. Control Relat. Fields, (2014).

[10]

N. Dunford and J. T. Schwartz, Linear Operators. Part III : Spectral Operators,, Wiley-Interscience, (1971).

[11]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000).

[12]

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

[13]

H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon,, SIAM J. Control, 13 (1975), 1. doi: 10.1137/0313001.

[14]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct.Anal., 259 (2010), 1720. doi: 10.1016/j.jfa.2010.06.003.

[15]

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. doi: 10.4171/PM/1859.

[16]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135.

[17]

M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems,, Ned. Akad. Wetenschappen, 31 (1969), 443.

[18]

L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).

[19]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations,, ESAIM Control Optim. Calc. Var., 16 (2010), 247. doi: 10.1051/cocv/2008077.

[20]

R. C. MacCamy, V. J. Mizel and T. I. Seidman, Approximate boundary controllability for the heat equation,, Jour. of Math. Anal. and Appl., 23 (1968), 699. doi: 10.1016/0022-247X(68)90148-0.

[21]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc. 71, (1988).

[22]

K. Mauffrey, On the null controllability of a $3\times3$ parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl., 99 (2013), 187. doi: 10.1016/j.matpur.2012.06.010.

[23]

L. Miller, On the null-controllability of the heat equation in unbounded domains,, Bull. Sci. Math., 129 (2005), 175. doi: 10.1016/j.bulsci.2004.04.003.

[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. doi: 10.1007/s00498-011-0071-x.

[25]

L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291. doi: 10.1016/j.crma.2011.01.014.

[26]

L. de Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39. doi: 10.1080/03605300008821507.

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. doi: 10.1016/j.matpur.2012.09.012.

[2]

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 parbolic systems,, J. Evol. Equ., 9 (2009), 267. doi: 10.1007/s00028-009-0008-8.

[3]

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. doi: 10.7153/dea-01-24.

[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. doi: 10.1016/j.matpur.2011.06.005.

[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. doi: 10.3934/mcrf.2011.1.267.

[6]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems,, to appear in ESAIM Control Optim. Calc. Var., (2014).

[7]

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,, to appear in SIAM J. Control Optim., (2014).

[8]

A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications,, C. R. Math. Acad. Sci. Paris, 348 (2010), 25. doi: 10.1016/j.crma.2009.11.001.

[9]

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients,, to appear in Math. Control Relat. Fields, (2014).

[10]

N. Dunford and J. T. Schwartz, Linear Operators. Part III : Spectral Operators,, Wiley-Interscience, (1971).

[11]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000).

[12]

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

[13]

H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon,, SIAM J. Control, 13 (1975), 1. doi: 10.1137/0313001.

[14]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct.Anal., 259 (2010), 1720. doi: 10.1016/j.jfa.2010.06.003.

[15]

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. doi: 10.4171/PM/1859.

[16]

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force,, SIAM J. Control Optim., 46 (2007), 379. doi: 10.1137/060653135.

[17]

M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems,, Ned. Akad. Wetenschappen, 31 (1969), 443.

[18]

L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).

[19]

O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations,, ESAIM Control Optim. Calc. Var., 16 (2010), 247. doi: 10.1051/cocv/2008077.

[20]

R. C. MacCamy, V. J. Mizel and T. I. Seidman, Approximate boundary controllability for the heat equation,, Jour. of Math. Anal. and Appl., 23 (1968), 699. doi: 10.1016/0022-247X(68)90148-0.

[21]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc. 71, (1988).

[22]

K. Mauffrey, On the null controllability of a $3\times3$ parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl., 99 (2013), 187. doi: 10.1016/j.matpur.2012.06.010.

[23]

L. Miller, On the null-controllability of the heat equation in unbounded domains,, Bull. Sci. Math., 129 (2005), 175. doi: 10.1016/j.bulsci.2004.04.003.

[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. doi: 10.1007/s00498-011-0071-x.

[25]

L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291. doi: 10.1016/j.crma.2011.01.014.

[26]

L. de Teresa, Insensitizing controls for a semilinear heat equation,, Comm. Partial Differential Equations, 25 (2000), 39. doi: 10.1080/03605300008821507.

[1]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[2]

Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465

[3]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[4]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037

[5]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014

[6]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations & Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557

[7]

Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613

[8]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[9]

Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419

[10]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405

[11]

John E. Lagnese. Controllability of systems of interconnected membranes. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 17-33. doi: 10.3934/dcds.1995.1.17

[12]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[13]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

[14]

Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485

[15]

Larbi Berrahmoune. Constrained controllability for lumped linear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 159-175. doi: 10.3934/eect.2015.4.159

[16]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243

[17]

Assia Benabdallah, Michel Cristofol, Patricia Gaitan, Luz de Teresa. Controllability to trajectories for some parabolic systems of three and two equations by one control force. Mathematical Control & Related Fields, 2014, 4 (1) : 17-44. doi: 10.3934/mcrf.2014.4.17

[18]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263

[19]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[20]

Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations & Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621

2017 Impact Factor: 1.049

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]