March  2020, 10(1): 89-112. doi: 10.3934/mcrf.2019031

Minimal time of null controllability of two parabolic equations

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

* Corresponding author

Received  May 2018 Revised  December 2018 Published  April 2019

We consider a one-dimensional 2 × 2 parabolic equations, simultaneously controllable by a localized function in their source term. We also consider a simultaneous boundary control. In each case, we prove the existence of minimal time T0(q) of null controllability, that is to say, the corresponding problem is null controllable at any time T > T0(q) and not null controllable for T < T0(q). We also prove that one can expect any minimal time associated to the boundary control problem.

Citation: Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031
References:
[1]

D. AllonsiusF. 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

[2]

F. Ammar-KhodjaA. BenabdallahC. 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.  Google Scholar

[3]

F. Ammar KhodjaA. BenabdallahC. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var, 11 (2005), 426-448.  doi: 10.1051/cocv:2005013.  Google Scholar

[4]

F. Ammar KhodjaA. BenabdallahM. 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

[5]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence, J. Math. Anal. Appl, 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[6]

F. Ammar KhodjaA. BenabdallahM. 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

[7]

V. Bernstein, Leçcons sur les Progrès Réscents de la Théorie des Séries de Dirichlet, Gauthier-Villars, Paris, 1933. Google Scholar

[8]

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

[9]

H. O. Fattorini and D. L. Russell., Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal, 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[10]

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

[11]

E. Fernández-CaraM. 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

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, , Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

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.  Google Scholar

[14]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, volume 120 of Applied Mathematical Sciences, second edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[15]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[16]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987.  Google Scholar

[17]

J. R. Shackell, Overconvergence of Dirichlet series with complex exponents, J. Analyse Math, 22 (1969), 135-170.  doi: 10.1007/BF02786787.  Google Scholar

[18]

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

[19]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

show all references

References:
[1]

D. AllonsiusF. 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

[2]

F. Ammar-KhodjaA. BenabdallahC. 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.  Google Scholar

[3]

F. Ammar KhodjaA. BenabdallahC. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var, 11 (2005), 426-448.  doi: 10.1051/cocv:2005013.  Google Scholar

[4]

F. Ammar KhodjaA. BenabdallahM. 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

[5]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence, J. Math. Anal. Appl, 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[6]

F. Ammar KhodjaA. BenabdallahM. 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

[7]

V. Bernstein, Leçcons sur les Progrès Réscents de la Théorie des Séries de Dirichlet, Gauthier-Villars, Paris, 1933. Google Scholar

[8]

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

[9]

H. O. Fattorini and D. L. Russell., Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal, 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[10]

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

[11]

E. Fernández-CaraM. 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

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, , Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

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.  Google Scholar

[14]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, volume 120 of Applied Mathematical Sciences, second edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[15]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[16]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987.  Google Scholar

[17]

J. R. Shackell, Overconvergence of Dirichlet series with complex exponents, J. Analyse Math, 22 (1969), 135-170.  doi: 10.1007/BF02786787.  Google Scholar

[18]

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

[19]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

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