A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems

Farid Ammar-Khodja1, Assia Benabdallah2, Cédric Dupaix1, Manuel González-Burgos3
1Laboratoire de Mathématiques (UMR 6623), Université de Franche-Comté, Besançon Cedex, France
2CMI-LATP, UMR 6632, Université de Provence, Technopôle Château-Gombert, Marseille cedex 13, France
3Dpto, E.D.A.N, Universidad de Sevilla, Aptdo. 1160, Sevilla, Spain

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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), no. 2, 928–943

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonzá lez-Burgos, Controllability for a class of reaction-diffusion systems: the generalized Kalman’s condition, C. R.Acad.Sci. Paris, Ser. I 345 (2007), no. 10, 543–548

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonzá lez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems in preparation.

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostine, Null controllability of some systems of parabolic type by one control force ESAIM Control Optim.Calc.Var.11 (2005), no. 3, 426–448.

A. Benabdallah, Y. Dermenjian, J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math.Anal.Appl. 336 (2007), no. 2, 865–887.

A. Doubova, A. Osses, and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim.Calc.Var. 8 (2002), 621–661.

E. Fernández-Cara, M. Gonzá lez-Burgos, L. de Teresa, Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations, Commun.Pure Appl.Anal. 5 (2006), no. 3, 637–656.

E. Fernández-Cara, M. Gonzá lez-Burgos, L. de Teresa, About boundary controllability of cascade heat equations, In preparation.

E. Fernández-Cara, L. de Teresa, Null controllability of a cascade system of parabolic- hyperbolic equations, Discrete Contin.Dyn. Syst. 11 (2004), no. 2–3, 699–714.

E. Fernández-Cara, E. Zuazua, The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations 5 (2000), no. 4–6, 465–514.

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

M. González-Burgos, R. Pérez-García, M. González-Burgos, R. Pérez-Garcia, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot.Anal.46 (2006), no. 2, 123–162.

M. González-Burgos, L. de Teresa Controllability results for cascade systems of m coupled parabolic PDEs by one control force, submitted to Systems & Control Letters.

S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim. 46, (2007), no. 2, 379–394.

E. L. Ince., Ordinary Differential Equations, Dover Publications, New York, 1944.

O. Yu. Imanuvilov, 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), no. 2, 227–274.

R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical Control Theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London 1969.

G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm.Partial Differential Equations 20 (1995), no. 1–2, 335–356.

H. Leiva, Controllability of a system of parabolic equations with non-diagonal diffusion matrix, IMA J. Math.Control Inform. 22 (2005), no. 2, 187–199.

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations 233 (2007), no. 2, 417–447.

A. Pazy, Semigroups of linear operators and application to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm.Partial Differential Equations 25 (2000), no. 1–2, 39–72.

R. Triggiani, Constructive steering control functions for linear systems and abstract rank conditions, J. Optim.Theory Appl. 74 (1992), no. 2, 347–367.

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