×

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case. (English) Zbl 1106.93010

Summary: 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.

MSC:

93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] H. Amann , Parabolic evolution equations and nonlinear boundary conditions . J. Diff. Equ. 72 ( 1988 ) 201 - 269 . Zbl 0658.34011 · Zbl 0658.34011
[2] J. Arrieta , A. Carvalho and A. Rodríguez-Bernal , Parabolic problems with nonlinear boundary conditions and critical nonlinearities . J. Diff. Equ. 156 ( 1999 ) 376 - 406 . Zbl 0938.35077 · Zbl 0938.35077
[3] J.P. Aubin , L’analyse non linéaire et ses motivations économiques . Masson, Paris ( 1984 ). Zbl 0551.90001 · Zbl 0551.90001
[4] O. Bodart , M. González-Burgos and R. Pŕez-García , Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity . C. R. Math. Acad. Sci. Paris 335 ( 2002 ) 677 - 682 . Zbl 1021.35049 · Zbl 1021.35049
[5] A. Doubova , E. Fernández-Cara and M. González-Burgos , On the controllability of the heat equation with nonlinear boundary Fourier conditions . J. Diff. Equ. 196 ( 2004 ) 385 - 417 . Zbl 1049.35042 · Zbl 1049.35042
[6] 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 . Zbl 1038.93041 · Zbl 1038.93041
[7] L. Evans , Regularity properties of the heat equation subject to nonlinear boundary constraints . Nonlinear Anal. 1 ( 1997 ) 593 - 602 . Zbl 0369.35034 · Zbl 0369.35034
[8] C. Fabre , J.P. Puel and E. Zuazua , Approximate controllability of the semilinear heat equation . Proc. Roy. Soc. Edinburgh 125A ( 1995 ) 31 - 61 . Zbl 0818.93032 · Zbl 0818.93032
[9] L.A. Fernández and E. Zuazua , Approximate controllability for the semi-linear heat equation involving gradient terms . J. Optim. Theory Appl. 101 ( 1999 ) 307 - 328 . Zbl 0952.49003 · Zbl 0952.49003
[10] 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 . Numdam | Zbl 1106.93009 · Zbl 1106.93009
[11] 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 . Numdam | Zbl 0970.93023 · Zbl 0970.93023
[12] A. Fursikov and O.Yu. Imanuvilov , Controllability of Evolution Equations . Lecture Notes #34, Seoul National University, Korea ( 1996 ). MR 1406566 | Zbl 0862.49004 · Zbl 0862.49004
[13] I. Lasiecka and R. Triggiani , Exact controllability of semilinear abstract systems with applications to waves and plates boundary control . Appl. Math. Optim. 23 ( 1991 ) 109 - 154 . Zbl 0729.93023 · Zbl 0729.93023
[14] I. Lasiecka and R. Triggiani , Control Theory for Partial Differential Equations: Continuous and Approximation Theories . Cambridge University Press, Cambridge ( 2000 ). Zbl 0961.93003 · Zbl 0961.93003
[15] 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 . Zbl 0731.93011 · Zbl 0731.93011
[16] E. Zuazua , Exact controllability for the semilinear wave equation in one space dimension . Ann. I.H.P., Analyse non Linéaire 10 ( 1993 ) 109 - 129 . Numdam | Zbl 0769.93017 · Zbl 0769.93017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.