This paper is concerned with the controllability of some (linear and semilinear) non-diagonalizable parabolic systems of PDEs. We will show that the well known null controllability properties of the classical heat equation are also satisfied by these systems at least when there are as many scalar controls as equations and some (maybe technical) conditions are satisfied. We will also show that, in some particular situations, the number of controls can be reduced. The minimal amount is then determined by a Kalman rank condition.
DOI : 10.1051/cocv/2014063
Mots-clés : Null controllability, parabolic, non-diagonalizable
@article{COCV_2015__21_4_1178_0,
author = {Fern\'andez-Cara, Enrique and Gonz\'alez-Burgos, Manuel and de Teresa, Luz},
title = {Controllability of linear and semilinear non-diagonalizable parabolic systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1178--1204},
publisher = {EDP-Sciences},
volume = {21},
number = {4},
year = {2015},
doi = {10.1051/cocv/2014063},
mrnumber = {3395760},
zbl = {1320.93017},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2014063/}
}
TY - JOUR AU - Fernández-Cara, Enrique AU - González-Burgos, Manuel AU - de Teresa, Luz TI - Controllability of linear and semilinear non-diagonalizable parabolic systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1178 EP - 1204 VL - 21 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014063/ DO - 10.1051/cocv/2014063 LA - en ID - COCV_2015__21_4_1178_0 ER -
%0 Journal Article %A Fernández-Cara, Enrique %A González-Burgos, Manuel %A de Teresa, Luz %T Controllability of linear and semilinear non-diagonalizable parabolic systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1178-1204 %V 21 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014063/ %R 10.1051/cocv/2014063 %G en %F COCV_2015__21_4_1178_0
Fernández-Cara, Enrique; González-Burgos, Manuel; de Teresa, Luz. Controllability of linear and semilinear non-diagonalizable parabolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1178-1204. doi : 10.1051/cocv/2014063. http://archive.numdam.org/articles/10.1051/cocv/2014063/
, Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE’s by a single control. Math. Control Signals Systems 26 (2014) 1–46. | DOI | MR | Zbl
and , Indirect controllability of locally coupled wave-type systems and applications. J. Math. Pures Appl. 99 (2013) 544–576. | DOI | MR | Zbl
, and , Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928–943. | DOI | MR | Zbl
, , and , A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems. Differ. Equ. Appl. 1 (2009) 427–457. | MR | Zbl
, , and , A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems. J. Evol. Equ. 9 (2009) 267–291. | DOI | MR | Zbl
, , and , Controllability to the trajectories of phase-field models by one control force. SIAM J. Control Optim. 42 (2003) 1661–1680. | DOI | MR | Zbl
, , and , 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 | MR | Zbl
, , and , Recent results on the controllability of coupled parabolic problems: a survey. Math. Control Relat. Fields 1 (2011) 267–306. | DOI | MR | Zbl
, , and , A new relation between the condensation index of complex sequences and the null controllability of parabolic systems. C. R. Math. Acad. Sci. Paris 351 (2013) 19-20, 743–746. | DOI | MR | Zbl
, , and , Minimal time of controllability of two parabolic equations with disjoint control and coupling domains. C. R. Math. Acad. Sci. Paris 352 (2014) 391–396. | DOI | MR | Zbl
, , and , 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 | MR | Zbl
J.-P. Aubin, L’analyse non linéaire et ses motivations économiques. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1984). | MR | Zbl
, , and , Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the -dimensional boundary null-controllability in cylindrical domains. SIAM J. Control Optim. 52 (2014) 2970–3001. | DOI | MR | Zbl
, and , Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient. Nonlin. Anal. 57 (2004) 687–711. | DOI | MR | Zbl
and , Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Math. Control Relat. Fields 4 (2014) 263–287. | DOI | MR | Zbl
and , Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components. Invent. Math. 198 (2014), no. 3, 833–880. | DOI | MR | Zbl
, , and , On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798–819. | DOI | MR | Zbl
, and , Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31–61. | DOI | MR | Zbl
and , Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. | DOI | MR | Zbl
, and , Boundary controllability of parabolic coupled equations. J. Funct. Anal. 259 (2010) 1720–1758. | DOI | MR | Zbl
, , and , Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case. ESAIM: COCV 12 (2006) 466–483. | Numdam | MR | Zbl
and ,The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5 (2000) 465–514. | MR | Zbl
and , Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 17 (2000) 583–616. | DOI | Numdam | MR | Zbl
, and , Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446. | DOI | MR | Zbl
, Null controllability for the parabolic equation with a complex principal part. J. Funct. Anal. 257 (2009) 1333–1354. | DOI | MR | Zbl
A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Vol. 34 of Lect. Notes Ser. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl
and , Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptot. Anal. 46 (2006) 123–162. | MR | Zbl
and , Controllability results for cascade systems of coupled parabolic PDEs by one control force. Port. Math. 67 (2010) 91–113. | DOI | MR | Zbl
, Null controllability of some systems of two parabolic equations with one control force. SIAM J. Control Optim. 46 (2007) 379–394. | DOI | MR | Zbl
and , Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (2003) 227–274. | DOI | MR | Zbl
and , Contrôle exact de l’équation de la chaleur.Comm. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl
, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. | DOI | MR | Zbl
and , A generalized field method for multiphase transformations using interface fields, Physica. D 134 (1999) 385–393. | DOI | MR | Zbl
, , , , , and , A phase field concept for multiphase systems. Physica D 94 (1996) 135–147. | DOI | Zbl
, Insensitizing controls for a semilinear heat equation. Comm. Partial Differ. Eq. 25 (2000) 39–72. | DOI | MR | Zbl
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