Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

Franck Boyer (1) , Jérôme Le Rousseau (2, 3)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 IUF - Institut universitaire de France
3 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabilic equations. Sub-linear and super-linear cases are considered.

Mots clés: Parabolic operator; semi-discrete Carleman estimates; observability; null controllability; semilinear equations

Domaines: Equations aux dérivées partielles [math.AP]

Franck Boyer, Jérôme Le Rousseau. Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2014, 31 (5), pp.1035-1078. ⟨10.1016/j.anihpc.2013.07.011⟩. ⟨hal-00724766⟩