Minimal time problem for crowd models with localized vector fields

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

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

In this work, we study a minimal time problem for a Partial Differential Equation of transport type, that arises in crowd models. The control is a Lipschitz vector field localized on a fixed control set ω. We provide a complete answer for the minimal time problem. After considering the discrete case, 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. We also give a numerical method to compute the minimal time and to build the corresponding control. These results are illustrated by some numerical simulations.

Mots clés: Transport equation; Control; Minimal time problem

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

Michel Duprez, Morgan Morancey, Francesco Rossi. Minimal time problem for crowd models with localized vector fields. 2017. ⟨hal-01493143v2⟩