Stabilization via localized controls in nonlocal models of crowd dynamics

We consider a control system driven by a nonlocal continuity equation. Admissible controls are Lipschitz vector fields acting inside a fixed open set. We demonstrate that small perturbations of the initial measure, traced along Wasserstein geodesics, may be neutralized by admissible controls. More specifically, initial perturbations of order $\epsilon$ can be reduced to order ${\epsilon }^{1+\kappa }$, where $\kappa$ is a positive constant.