Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

Jérôme Le Rousseau (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We derive global Carleman estimates for one-dimensional linear parabolic equations $\d_t \pm \d_x(c \d_x)$ with a coefficient of bounded variations. These estimates are obtained by approximating $c$ by piecewise constant coefficients, $c_\eps$, and passing to the limit in the Carleman estimates associated to the operators defined with $c_\eps$. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semi-linear equations.

Mots clés: Carleman estimate; observability; non-smooth coefficients; parabolic equations; control

Domaines: Equations aux dérivées partielles [math.AP]

Jérôme Le Rousseau. Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. Journal of Differential Equations, 2007, 233, pp.417-447. ⟨10.1016/j.jde.2006.10.005⟩. ⟨hal-00105669v2⟩