Finite horizon mean field games on networks
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Calculus of Variations
Finite horizon mean field games on networks Yves Achdou1 · Manh-Khang Dao2 · Olivier Ley3 · Nicoletta Tchou4 Received: 14 July 2019 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton–Jacobi–Bellman equation and a forward in time Fokker–Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy some compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence, uniqueness and regularity results are proven. Mathematics Subject Classification 91A16 · 35R02 · 49N70
1 Introduction and main results The theory of mean field games (MFGs for short) is more and more investigated since the pioneering works [15–17] of Lasry and Lions: it aims at studying the asymptotic behavior of differential games (Nash equilibria) as the number of agents tends to infinity. In the present work, we study stochastic MFGs on networks with finite time horizon; they are described by a system of PDEs coupling a Fokker–Planck (FP) equation for the density of the distribution of states (forward in time) and a Hamilton–Jacobi–Bellman (HJB) equation for the optimal
Communicated by T.Riviere.
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Yves Achdou [email protected] Olivier Ley [email protected] Nicoletta Tchou [email protected]
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Laboratoire Jacques-Louis Lions, (LJLL), Université de Paris and Sorbonne Université, CNRS, 75006 Paris, France
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Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
3
University of Rennes, INSA Rennes, CNRS, IRMAR-UMR 6625, 35000 Rennes, France
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University of Rennes, CNRS, IRMAR-UMR 6625, 35000 Rennes, France 0123456789().: V,-vol
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value of a representative agent (backward in time). This work is a continuation of [2], which was devoted to MFGs on networks in the case of an infinite time horizon. We refer to [2] for a more extended discussion on MFGs and for additional references on the analysis of the systems of PDEs that stem from the model when there is no common noise. A network (or a graph) is a set of items, referred to as vertices (or nodes or crosspoints), with connections between them referred to as edges. In the recent years, there has been an increasing interest in the investigation of dynamical systems and differential equations on networks, in particular in connection with problems of data transmission and traffic management. The literature on optimal control in which the state variable takes its values on a network is recent: deterministic control problems and related Hamilton–Jacobi equations were studi
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