Trajectory Stabilization of Nonlocal Continuity Equations by Localized Controls

We discuss stabilization around trajectories of the continuity equation with nonlo cal vector fields, where the control is localized, i.e., it acts on a fixed subset of the configuration space. We first show that the correct definition of stabilization is the following: given an initial error of order epsilon , measured in Wasserstein distance, one can improve the final error to an order epsilon (1+kappa) with kappa > 0. We then prove the main result: assuming that the trajectory crosses the subset of control action, stabilization can be achieved. The key problem lies in regularity issues: the reference trajectory needs to be absolutely continuous, while the initial state to be stabilized needs to be realized by a small Lipschitz perturbation or by being in a very small neighborhood of it.