InverserObservability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients

We consider a general second-order hyperbolic equation defined on an open bounded domain V;Rn with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions B.C..are imposed. Our main result Theorem 3.5.is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H1 V.=L2 V.-energy at time tsT, or at time ts0, is dominated by the L2 S.-norms of the boundary traces ­ wr­nA and wt , modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield}under a uniqueness theorem, needed to absorb the lower-order term}continuous observability estimates for both the Dirichlet and Neumann case, with an explicit, sharp observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neumann case. In contrast with existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in V;Rn to a problem on an appropriate Riemann manifold determined by the coefficients of the principal part., where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the variable first-order energy level. terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate, to remove artificial geometrical conditions on the controlled part of the boundary, in the Neumann case.