  The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition
is central in the study of the observability of the wave equation linking this
property to high-frequency propagation along geodesics that are therays of
geometric optics. This connection is best understood through the propagation
properties of microlocal defect measures that appear as solutions to the wave
equation concentrate. For a sufficiently smooth metric this propagation occurs
along the bicharacteristic flow. If one considers a merely
$\mathscr{C}^1$-metric this bicharacteristic flow may however not exist. The
Hamiltonian vector field is only continuous; bicharacteristics do exist (as
integral curves of this continuous vector field) but uniqueness is lost. Here,
on a compact manifold without boundary, we consider this low regularity
setting, revisit the geometric control condition, and address the question of
support propagation for a measure solution to an ODE with continuous
coefficients. This leads to a sufficient condition for the observability and
equivalently the exact controllability of the wave equation. Moreover, we
investigate the stabililty of the observability property and the sensitivity of
the control process under a perturbation of the metric of regularity as low as
Lipschitz.
