Title: Controllability and optimal control of the transport equation with a localized vector field,
Abstract: We study controllability of a Partial Differential Equation of transport
type, that arises in crowd models. We are interested in controlling such system
with a control being a Lipschitz vector field on a fixed control set $\omega$.
We prove that, for each initial and final configuration, one can steer one to
another with such class of controls only if the uncontrolled dynamics allows to
cross the control set $\omega$. We also prove a minimal time result for such
systems. We show that the minimal time to steer one initial configuration to
another is related to the condition of having enough mass in $\omega$ to feed
the desired final configuration.