Carleman estimates and controllability results for the one-dimensional heat equation with {\em BV} coefficients

International audienceWe derive global Carleman estimates for one-dimensional linear parabolic operators $\d_t \pm \d_x(c \d_x)$ with a coefficient $c$ with bounded variations. These estimates are obtained by approximating $c$ by piecewise regular coefficients, $c_\eps$, and passing to the limit in the Carleman estimates associated to the operators defined with $c_\eps$. Such estimates yield results of controllability to the trajectories for a classe of {\em semilinear} parabolic equations