We investigate a first-order mean field planning problem of the form-partial derivative(t)u + H(x, Du) = f(x,m) in (0,T) x R-d ,partial derivative(t)m - del.(mH(p) (x, Du)) = 0 in (0,T) x R-d ,m(0,center dot) = m(0) , m(T, center dot) = m(T) in R-d,associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m, u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form -partial derivative(t)u + H(x, Du) <= alpha, under minimal summability conditions on alpha, and to a measure theoretic description of the optimality via a suitable contact defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players. (C) 2019 Elsevier Inc. All rights reserved.

Orrieri, C., Porretta, A., Savare, G. (2019). A variational approach to the mean field planning problem. JOURNAL OF FUNCTIONAL ANALYSIS, 277(6), 1868-1957 [10.1016/j.jfa.2019.04.011].

A variational approach to the mean field planning problem

Porretta A.;
2019-01-01

Abstract

We investigate a first-order mean field planning problem of the form-partial derivative(t)u + H(x, Du) = f(x,m) in (0,T) x R-d ,partial derivative(t)m - del.(mH(p) (x, Du)) = 0 in (0,T) x R-d ,m(0,center dot) = m(0) , m(T, center dot) = m(T) in R-d,associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m, u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form -partial derivative(t)u + H(x, Du) <= alpha, under minimal summability conditions on alpha, and to a measure theoretic description of the optimality via a suitable contact defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players. (C) 2019 Elsevier Inc. All rights reserved.
2019
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Mean field planning; Optimal transport; Kantorovich duality; Superposition principle
Orrieri, C., Porretta, A., Savare, G. (2019). A variational approach to the mean field planning problem. JOURNAL OF FUNCTIONAL ANALYSIS, 277(6), 1868-1957 [10.1016/j.jfa.2019.04.011].
Orrieri, C; Porretta, A; Savare, G
Articolo su rivista
File in questo prodotto:
File Dimensione Formato  
OPS-JFA.pdf

solo utenti autorizzati

Descrizione: articolo di ricerca
Licenza: Copyright dell'editore
Dimensione 3.2 MB
Formato Adobe PDF
3.2 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/231065
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 28
  • ???jsp.display-item.citation.isi??? 28
social impact