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We study the controllability of a Partial Differential Equation of transporttype, that arises in crowd models. We are interested in controlling it with acontrol being a vector field, representing a perturbation of the velocity,localized on a fixed control set. We prove that, for each initial and finalconfiguration, one can steer approximately one to another with Lipschitzcontrols when the uncontrolled dynamics allows to cross the control set. Wealso show that the exact controllability only holds for controls with lessregularity, for which one may lose uniqueness of the associated solution.

Approximate and exact controllability of the continuity equation with a localized vector field