Exact Controllability of nonlinear Heat equations in spaces of analytic functions

Camille Laurent (1, 2) , Lionel Rosier (3, 4)

1 CNRS - Centre National de la Recherche Scientifique
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 CAS - Centre Automatique et Systèmes
4 CAOR - Centre de Robotique

It is by now well known that the use of Carleman estimates allows to establish the control-lability 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 nonlinear 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 ∂ 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. 2010 Mathematics Subject Classification: 35K40, 93B05

Mots clés: Nonlinear heat equation; viscous Burgers' equation; Allen-Cahn equation; exact controllability; ill-posed problems; Gevrey class; reachable states

Domaines: Equations aux dérivées partielles [math.AP]

Camille Laurent, Lionel Rosier. Exact Controllability of nonlinear Heat equations in spaces of analytic functions. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2020, ⟨10.1016/j.anihpc.2020.03.001⟩. ⟨hal-01955484⟩