On the observability inequalities for exact controllability of wave equations with variable coefficients

This paper deals with boundary exact controllability for the dynamics gover ned by the wave equation with nonconstant coefficients in the principal par t, subject to Dirichlet or Neumann boundary controls. The observability ine qualities are established by the Riemannian geometry method under some geom etric condition for the Dirichlet problem and for the Neumann problem, resp ectively. Next, a number of nontrivial examples are presented to verify the observability inequality. In particular, a counterexample is given without boundary exact controllability, where the control is exerted on the whole boundary.