Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T14:43:41.763Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact controllability to trajectories for semilinear heatequations with discontinuous diffusion coefficients

Published online by Cambridge University Press:  15 August 2002

Anna Doubova ,
A. Osses  and
J.-P. Puel
Anna Doubova
Affiliation:
Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain and École Polytechnique, 91128 Palaiseau Cedex, France; dubova@numer.us.es. doubova@cmapx.polytechnique.fr. This work has been partially supported by D.G.E.S., Spain, Grants PB98–1134.
A. Osses
Affiliation:
Departamento de Ingenería Matemática, Facultad de Ciencias de Físicas y Matemáticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile and Centro de Modelamiento Matemático, UMR 2071 CNRS-Uchile; axosses@dim.uchile.cl. This work has been partially supported by FONDECYT grants No. 1000955 and 7000955.
J.-P. Puel
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles Saint-Quentin, 45 avenue des États Unis, 78035 Versailles Cedex, France and École Polytechnique, 91128 Palaiseau Cedex, France; jppuel@cmapx.polytechnique.fr.
Get access

Abstract

The results of this paper concern exact controllability to thetrajectories for a coupled system of semilinear heat equations. Wehave transmission conditions on the interface and Dirichlet boundaryconditions at the external part of the boundary so that the system can beviewed as a single equation with discontinuous coefficients in theprincipal part. Exact controllability to the trajectories is proved when weconsider distributed controls supported in the part of the domain where thediffusion coefficient is the smaller and if the nonlinear term f(y) growsslower than |y|log3/2(1+|y|) at infinity. In the proof we use nullcontrollability results for the associate linear system and globalCarleman estimates with explicit bounds or combinations of several ofthese estimates. In order to treat the terms appearing on theinterface, we have to construct specific weight functions depending ongeometry.

Keywords

Carleman inequalitiescontrollabilitytransmission problems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 621 - 661
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anita, S. and Barbu, V., Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157-173. CrossRef
Aronson, D.G. and Serrin, J., Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25 (1967) 81-122. CrossRef
Aronson, D.G. and Serrin, J., A maximum principle for nonlinear parabolic equations. Ann. Scuola Norm. Sup. Pisa 3 (1967) 291-305
J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson (1984).
Barbu, V., Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. CrossRef
T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires. Ellipses, Paris, Mathématiques & Applications (1990).
Cox, S. and Zuazua, E., The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545-573. CrossRef
A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM: SICON (to appear).
Fabre, C., Puel, J.-P. and Zuazua, E., (a) Approximate controllability for the semilinear heat equation. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) 807-812; (b) Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61.
Fabre, C., Puel, J.-P. and Zuazua, E., Approximate controllability for the linear heat equation with controls of minimal L ∞ norm. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 679-684.
Fernández-Cara, E., Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-107. CrossRef
Fernández-Cara, E. and Zuazua, E., The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5 (2000) 465-514.
Fernández-Cara, E. and Zuazua, E., Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583-616. CrossRef
E. Fernández-Cara and E. Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients (to appear).
A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Seoul National University, Korea, Lecture Notes 34 (1996).
Imanuvilov, O.Yu., Controllability of parabolic equations. Mat. Sb. 186 (1995) 102-132.
Imanuvilov, O.Yu. and Yamamoto, M., Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. Lecture Notes in Pure Appl. Math. 218 (2001) 113-137.
O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskow (1967).
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983).
Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. CrossRef
Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in L p . Indiana Univ. Math. J. 29 (1980) 79-102. CrossRef
Weissler, F.B., Semilinear evolution equations in Banach spaces. J. Funct. Anal. 32 (1979) 277-296. CrossRef
E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, edited by H. Brezis and J.-L. Lions. Pitman (1991) 357-391.
Zuazua, E., Finite dimensional controllability for the semilinear heat equations. J. Math. Pures 76 (1997) 570-594.
Zuazua, E., Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control and Cybernetics 28 (1999) 665-683.