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

We derive global Carleman estimates for one-dimensional linear parabolic equations ∂t±∂x(c∂x) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.