  We study the boundary exact controllability for the quasilinear wave equation
in the higher-dimensional case. Our main tool is the geometric analysis. We
derive the existence of long time solutions near an equilibrium, prove the
locally exact controllability around the equilibrium under some checkable
geometrical conditions. We then establish the globally exact controllability in
such a way that the state of the quasilinear wave equation moves from an
equilibrium in one location to an equilibrium in another location under some
geometrical condition. The Dirichlet action and the Neumann action are studied,
respectively. Our results show that exact controllability is geometrical
characters of a Riemannian metric, given by the coefficients and equilibria of
the quasilinear wave equation. A criterion of exact controllability is given,
which based on the sectional curvature of the Riemann metric. Some examples are
presented to verify the global exact controllability.
