Trajectory stabilization of nonlocal continuity equations by localized controls

We discuss stabilization around trajectories of the continuity equation with nonlocal 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, measured in Wasserstein distance, one can improve the final error to an order ^1+ with >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 being in a very small neighborhood of it.