  In this paper we develop a time reversal method for the radiative transport
equation to solve two problems: an inverse problem for the recovery of an
initial condition from boundary measurements, and the exact boundary
controllability of the transport field with finite steering time. Absorbing and
scattering effects, modeled by coefficients with low regularity, are
incorporated in the formulation of these problems. This time reversal approach
leads to a convergent iterative procedure to reconstruct the initial condition
provided that the scattering coefficient is sufficiently small in the
$L^{\infty}$ norm. Then, using duality arguments, we show that the solvability
of the inverse problem leads to exact controllability of the transport field
with minimum-norm control obtained constructively. The solution approach to
both of these problems may have medical applications in areas such as optical
imaging and optimization of radiation delivery for cancer therapy.
