Controllability and optimal control of the transport equation with a localized vector field

Michel Duprez (1) , Francesco Rossi (2) , Morgan Morancey (1)

1 I2M - Institut de Mathématiques de Marseille
2 LSIS - Laboratoire des Sciences de l'Information et des Systèmes

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 ω. 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 ω. 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 ω to feed the desired final configuration.

Mots clés: Distributed systems; Agents and agent-based systems; Optimisation

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Automatique

Michel Duprez, Francesco Rossi, Morgan Morancey. Controllability and optimal control of the transport equation with a localized vector field. 2017. ⟨hal-01474551v2⟩