Minimal time of null controllability of two parabolic equations

Ouaili Lydia

doi:10.3934/mcrf.2019031

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2020, Volume 10, Issue 1: 89-112. Doi: 10.3934/mcrf.2019031

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Minimal time of null controllability of two parabolic equations

Ouaili Lydia^,

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

^* Corresponding author
^* Corresponding author

Received: May 2018

Revised: December 2018

Early access: April 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K10, 93B05, 34L10.

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[1]	D. Allonsius, F. 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.
[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 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.
[4]	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.
[5]	F. Ammar Khodja, A. Benabdallah, M. 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.
[6]	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.
[7]	V. Bernstein, Leçcons sur les Progrès Réscents de la Théorie des Séries de Dirichlet, Gauthier-Villars, Paris, 1933.
[8]	J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
[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.
[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.
[11]	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.
[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.
[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.
[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.
[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.
[16]	J. Pöschel and E. Trubowitz, Inverse Spectral Theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987.
[17]	J. R. Shackell, Overconvergence of Dirichlet series with complex exponents, J. Analyse Math, 22 (1969), 135-170. doi: 10.1007/BF02786787.
[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.
[19]	J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.