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

Abstract

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, c ε , and passing to the limit in the Carleman estimates associated to the operators defined with c ε . Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.