Geometric Carleman estimates for wave equations and its applications in control theory.

PhD Theses.The main question one considers in control theory is the following: Given a physical system, is it possible to drive the system from any given initial state to any prescribed nal state, using a suitable control? In this thesis, we consider this problem in the context of wave equations. In particular, we study some control problems for (n + 1)- dimensional wave equations with time dependent non-analytic lower order coe cients. For this purpose, we use suitable (geometric) Carleman estimates, which are weighted estimates used to prove certain unique continuation results. First, we obtain a Carleman estimate for ultrahyperbolic operators in Rm Rn. A special case of this estimate is then used to obtain an improved interior control result for waves. Second, we obtain a novel interior control result for waves on domains with time dependent boundaries. Finally, we prove a novel Carleman estimate for the wave operator on Lorentzian manifolds with some curvature bounds. This estimate is then used to solve a suitable control problem for general waves on Lorentzian manifolds with time dependent geometry