Estimates for normal solutions of problems with zone controls in L2 for the wave equation

For the wave equation with variable coefficients and homogeneous boundary conditions of the first kind, we consider problems with regular zone controls and dual zone observation problems. For weak generalized solutions of the observation problem on sufficiently large time intervals, we obtain constructive estimates that imply the well-posed solvability of the observation operator. These estimates contain information that permits one to construct stable approximate solutions of both problems with the use of a variational method suggested earlier by the author for linear equations with nonuniformly perturbed operators.