  In this paper we consider the mathematical model of thermo- and
photo-acoustic tomography for the recovery of the initial condition of a wave
field from knowledge of its boundary values. Unlike the free-space setting, we
consider the wave problem in a region enclosed by a surface where an impedance
boundary condition is imposed. This condition models the presence of physical
boundaries such as interfaces or acoustic mirrors which reflect some of the
wave energy back into the enclosed domain. By recognizing that the inverse
problem is equivalent to a statement of boundary observability, we use control
operators to prove the unique and stable recovery of the initial wave profile
from knowledge of boundary measurements. Since our proof is constructive, we
explicitly derive a solvable equation for the unknown initial condition. This
equation can be solved numerically using the conjugate gradient method. We also
propose an alternative approach based on the stabilization of waves. This leads
to an exponentially and uniformly convergent Neumann series reconstruction when
the impedance coefficient is not identically zero. In both cases, if well-known
geometrical conditions are satisfied, our approaches are naturally suited for
variable wave speed and for measurements on a subset of the boundary.
