Exact boundary controllability for the wave equation with time-dependent and variable coefficients

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part, subject to Neumann boundary controls with time-dependent and variable coefficients. The observability inequalities are established by the Riemannian geometry method under some geometric condition for the Neumann problem. The result is obtained by the multiplier method and compactness-uniqueness argument.