Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T10:36:45.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact controllability to the trajectories of the heat equation withFourier boundary conditions: the semilinear case

Published online by Cambridge University Press:  20 June 2006

Enrique Fernández-Cara ,
Manuel González-Burgos ,
Sergio Guerrero  and
Jean-Pierre Puel
Enrique Fernández-Cara
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]; [email protected]
Manuel González-Burgos
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected]; [email protected]; [email protected]
Sergio Guerrero
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France; [email protected]
Jean-Pierre Puel
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles – St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; [email protected]
Get access

Abstract

This paper is concerned with the global exact controllability ofthe semilinear heat equation (with nonlinear terms involving the state andthe gradient) completed with boundary conditions of the form ${\partialy\over\partial n} + f(y) = 0$ . We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzedin a previous first part of this work. In this second part we show that, when the nonlinear terms arelocally Lipschitz-continuous and slightly superlinear, one has exactcontrollability to the trajectories.

Keywords

Controllabilityheat equationFourier boundary conditionssemilinear.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 466 - 483
Copyright
© EDP Sciences, SMAI, 2006

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

Amann, H., Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Equ. 72 (1988) 201–269.
Arrieta, J., Carvalho, A. and Rodríguez-Bernal, A., Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. 156 (1999) 376–406.
J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984).
Bodart, O., González-Burgos, M. and Pŕez-García, R., Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. C. R. Math. Acad. Sci. Paris 335 (2002) 677–682. CrossRef
Doubova, A., Fernández-Cara, E. and González-Burgos, M., On the controllability of the heat equation with nonlinear boundary Fourier conditions. J. Diff. Equ. 196 (2004) 385–417.
Doubova, A., Fernández-Cara, E., González-Burgos, M. and Zuazua, E., On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798–819. CrossRef
Evans, L., Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593–602. CrossRef
Fabre, C., Puel, J.P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31–61. CrossRef
Fernández, L.A. and Zuazua, E., Approximate controllability for the semi-linear heat equation involving gradient terms. J. Optim. Theory Appl. 101 (1999) 307–328. CrossRef
E. Fernández-Cara, M. González-Burgos, S. Guerrero and J.P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case. ESAIM: COCV 12 442–465.
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Anal. non Linéaire 17 (2000) 583–616.
A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996).
Lasiecka, I. and Triggiani, R., Exact controllability of semilinear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23 (1991) 109–154. CrossRef
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000).
E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, H. Brezis and J.L. Lions Eds. Pitman (1991) 357–391.
E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension. Ann. I.H.P., Analyse non Linéaire 10 (1993) 109–129.