  We consider the inverse problem of reconstructing an unknown function $u$
from a finite set of measurements, under the assumption that $u$ is the
trajectory of a transport-dominated problem with unknown input parameters. We
propose an algorithm based on the Parameterized Background Data-Weak method
(PBDW) where dynamical sensor placement is combined with approximation spaces
that evolve in time. We prove that the method ensures an accurate
reconstruction at all times and allows to incorporate relevant physical
properties in the reconstructed solutions by suitably evolving the dynamical
approximation space. As an application of this strategy we consider Hamiltonian
systems modeling wave-type phenomena, where preservation of the geometric
structure of the flow plays a crucial role in the accuracy and stability of the
reconstructed trajectory.
