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March  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 and 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), 544-576. 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), 267-291. 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), 427-457. 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), 555-590. 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), 267-306. 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). Available from: http://hal.archives-ouvertes.fr/hal-00743899. [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). Available from: http://hal.archives-ouvertes.fr/hal-00845994. [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), 25-29. 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). Available from: http://hal.archives-ouvertes.fr/hal-00848709. [10] N. Dunford and J. T. Schwartz, Linear Operators. Part III : Spectral Operators, Wiley-Interscience, New York, 1971. [11] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [12] H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694. doi: 10.1137/0304048. [13] H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13. 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), 1720-1758. 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), 91-113. 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), 379-394. doi: 10.1137/060653135. [17] M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems, Ned. Akad. Wetenschappen, Proc. Ser. A, 31 (1969), 443-448. [18] L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. [19] O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., 16 (2010), 247-274. 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), 699-703. 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, Providence (R.I.), 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), 187-210. 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), 175-185. 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), 257-280. 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), 291-296. 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), 39-72. 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), 544-576. 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), 267-291. 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), 427-457. 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), 555-590. 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), 267-306. 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). Available from: http://hal.archives-ouvertes.fr/hal-00743899. [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). Available from: http://hal.archives-ouvertes.fr/hal-00845994. [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), 25-29. 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). Available from: http://hal.archives-ouvertes.fr/hal-00848709. [10] N. Dunford and J. T. Schwartz, Linear Operators. Part III : Spectral Operators, Wiley-Interscience, New York, 1971. [11] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [12] H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694. doi: 10.1137/0304048. [13] H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13. 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), 1720-1758. 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), 91-113. 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), 379-394. doi: 10.1137/060653135. [17] M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems, Ned. Akad. Wetenschappen, Proc. Ser. A, 31 (1969), 443-448. [18] L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. [19] O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., 16 (2010), 247-274. 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), 699-703. 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, Providence (R.I.), 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), 187-210. 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), 175-185. 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), 257-280. 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), 291-296. 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), 39-72. doi: 10.1080/03605300008821507.
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