  We consider the problems of tracking an ensemble of indistinguishable agents
with linear dynamics based only on output measurements. In this setting, the
dynamics of the agents can be modeled by distribution flows in the state space
and the measurements correspond to distributions in the output space. In this
paper we formulate the corresponding state estimation problem using optimal
mass transport theory with prior linear dynamics, and the optimal solution
gives an estimate of the state trajectories of the ensemble. For general
distributions of systems this can be formulated as a convex optimization
problem which is computationally feasible with when the number of state
dimensions is low. In the case where the marginal distributions are Gaussian,
the problem is reformulated as a semidefinite programming and can be
efficiently solved for tracking systems with a large number of states.
