Skip to main content Accessibility help
×
Home
Hostname: page-component-6c8bd87754-lkb8j Total loading time: 0.13 Render date: 2022-01-20T17:48:07.600Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; cara@us.es; manoloburgos@us.es; sguerrero@us.es
Manuel González-Burgos
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; cara@us.es; manoloburgos@us.es; sguerrero@us.es
Sergio Guerrero
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France; guerrero@ann.jussieu.fr
Jean-Pierre Puel
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Versailles – St. Quentin, 45 avenue des États-Unis, 78035 Versailles, France; jppuel@cmapx.polytechnique.fr
Get access

Abstract

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

Type
Research Article
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) 201269.
Arrieta, J., Carvalho, A. and Rodríguez-Bernal, A., Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. 156 (1999) 376406.
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) 677682. 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) 385417.
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) 798819. CrossRef
Evans, L., Regularity properties of the heat equation subject to nonlinear boundary constraints. Nonlinear Anal. 1 (1997) 593602. CrossRef
Fabre, C., Puel, J.P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 3161. CrossRef
Fernández, L.A. and Zuazua, E., Approximate controllability for the semi-linear heat equation involving gradient terms. J. Optim. Theory Appl. 101 (1999) 307328. 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) 109154. 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.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *