Exact controllability of semilinear heat equations in spaces of analytic functions

Laurent, Camille; Rosier, Lionel

doi:10.1016/j.anihpc.2020.03.001

Laurent, Camille ¹ ; Rosier, Lionel ^2,³

¹ CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions, UMR 7598, Boite courrier 187, 75252, Paris Cedex 05, France
² ULCO, LMPA J. Liouville, BP 699, F-62228 Calais, France
³ CNRS FR 2956, France

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 1047-1073.

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Résumé

It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. In this paper, we pursue the study of the reachable states of parabolic equations based on a direct approach using control inputs in Gevrey spaces by considering a semilinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable x, the unknown y, and its derivative $\partial_{x} y$ . By investigating carefully a nonlinear Cauchy problem in the spatial variable and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane.

MR Zbl

DOI : 10.1016/j.anihpc.2020.03.001

Classification : 35K40, 93B05
Mots-clés : Semilinear heat equation, Exact controllability, Ill-posed problems, Gevrey class, Reachable states

Affiliations des auteurs :

Laurent, Camille ¹ ; Rosier, Lionel ^{2, 3}

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     title = {Exact controllability of semilinear heat equations in spaces of analytic functions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1047--1073},
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Laurent, Camille; Rosier, Lionel. Exact controllability of semilinear heat equations in spaces of analytic functions. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 4, pp. 1047-1073. doi : 10.1016/j.anihpc.2020.03.001. http://www.numdam.org/articles/10.1016/j.anihpc.2020.03.001/

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