Controllability of linear and semilinear non-diagonalizable parabolic systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1178-1204.

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.

Reçu le :
DOI : 10.1051/cocv/2014063
Classification : 93B05, 35K20
Mots clés : Null controllability, parabolic, non-diagonalizable
Fernández-Cara, Enrique 1 ; González-Burgos, Manuel 1 ; de Teresa, Luz 2

1 Dpto, E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 D.F. México, Mexico
@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/

F. Alabau-Boussouira, 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

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 | MR | Zbl

F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Null controllability of some reaction-diffusion systems with one control force. J. Math. Anal. Appl. 320 (2006) 928–943. | DOI | MR | Zbl

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. | MR | Zbl

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 parabolic systems. J. Evol. Equ. 9 (2009) 267–291. | DOI | MR | Zbl

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force. SIAM J. Control Optim. 42 (2003) 1661–1680. | DOI | MR | Zbl

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 | MR | Zbl

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. De Teresa, Recent results on the controllability of coupled parabolic problems: a survey. Math. Control Relat. Fields 1 (2011) 267–306. | DOI | MR | Zbl

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. De Teresa, 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

F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. De Teresa, 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

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 | MR | Zbl

J.-P. Aubin, L’analyse non linéaire et ses motivations économiques. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1984). | MR | Zbl

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. SIAM J. Control Optim. 52 (2014) 2970–3001. | DOI | MR | Zbl

O. Bodart, M. González-Burgos and R. Pérez-García, 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

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Math. Control Relat. Fields 4 (2014) 263–287. | DOI | MR | Zbl

J.-M. Coron and P. Lissy, 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

A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, 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

C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31–61. | DOI | MR | Zbl

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 | MR | Zbl

E. Fernández-Cara, M. González-Burgos and L. De Teresa, Boundary controllability of parabolic coupled equations. J. Funct. Anal. 259 (2010) 1720–1758. | DOI | MR | Zbl

E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.-P. Puel, 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

E. Fernández-Cara and E. Zuazua,The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5 (2000) 465–514. | MR | Zbl

E. Fernández-Cara and E. Zuazua, 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

E. Fernández-Cara, and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446. | DOI | MR | Zbl

X. Fu, 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

M. González-Burgos and R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptot. Anal. 46 (2006) 123–162. | MR | Zbl

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 | MR | Zbl

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force. SIAM J. Control Optim. 46 (2007) 379–394. | DOI | MR | Zbl

O. Yu. Imanuvilov and M. Yamamoto, 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

G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur.Comm. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. | DOI | MR | Zbl

I. Steinbach and F. Pezzolla, A generalized field method for multiphase transformations using interface fields, Physica. D 134 (1999) 385–393. | DOI | MR | Zbl

I. Steinbach, F. Pezzolla, B. Nestler, M. Seebelger, R. Prieler, G.J. Schimitz and J.L.L. Rezende, A phase field concept for multiphase systems. Physica D 94 (1996) 135–147. | DOI | Zbl

L. De Teresa, Insensitizing controls for a semilinear heat equation. Comm. Partial Differ. Eq. 25 (2000) 39–72. | DOI | MR | Zbl

Cité par Sources :