Mean-Field Pontryagin Maximum Principle

Mattia Bongini (1) , Massimo Fornasier (1) , Francesco Rossi (2) , Francesco Solombrino (3)

1 Fakultät Mathematik
2 LSIS - Laboratoire des Sciences de l'Information et des Systèmes
3 University of Naples Federico II = Università degli studi di Napoli Federico II

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov-type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables.

Mots clés: mean-field limit; Γ -limit; optimal con-; trol with ODE-PDE constraints; subdifferential calculus; Hamiltonian flows; Sparse optimal control

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Automatique / Robotique

Mattia Bongini, Massimo Fornasier, Francesco Rossi, Francesco Solombrino. Mean-Field Pontryagin Maximum Principle. Journal of Optimization Theory and Applications, 2017, 211, pp.1-38. ⟨10.1007/s10957-017-1149-5⟩. ⟨hal-01577930⟩