Inverse/observability estimates for second-order hyperbolic equations withvariable coefficients

We consider a general second-order hyperbolic equation defined on an open b ounded domain Omega subset of R-n with variable coefficients in both the el liptic principal part and in the first-order terms as well. At first, no bo undary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a re construction, or inverse, estimate for solutions w: under checkable conditi ons on the coefficients of the principal part, the H-1(Omega) x L-2(Omega)- energy at time t = T, or at time t = 0, is dominated by the L-2(Sigma)-norm s of the boundary traces partial derivative w/partial derivative v(A) and w (t), module 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 Ne umann case, with an explicit, sharp observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditio ns 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 pri ncipal part problem in Omega subset of R-n to a problem on an appropriate R iemann manifold (determined by the coefficients of the principal part), whe re the principal part is the Laplacian. In our second step, we employ expli cit Carleman estimates at the differential level to take care of the variab le first-order (energy level) terms. In our third step, we employ micro-loc al analysis yielding a sharp trace estimate, to remove artificial geometric al conditions on the controlled part of the boundary, in the Neumann case. (C) 1999 Academic Press.