Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C1). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.