Uniqueness and stability in the Cauchy problem

Here, following the classical Carleman idea, we apply Carleman estimates to derive uniqueness results and stability estimates of the continuation of solutions to partial differential equations. We give the counterexample of Fritz John which shows importance of pseudo-convexity and outline recent progress in increased stability for the Helmholtz equation.

Applications to inverse problems and optimal control

By studying an "adjoint" problem we show that uniqueness of the continuation implies the so-called approximate controllability by solutions of PDE. For hyperbolic equations we will derive from Carleman estimates a stronger property called an exact controllability and its dual which is a Lipschitz stability estimate of the initial data by the lateral boundary data. Finally we outline the method of Bukhgeim-Klibanov which under certain conditions transform Carleman estimates into uniqueness results for unknown source terms and coeffieints of hyperbolic PDE. In conclusion we discuss open problems and further possibilities of Carleman estimates.