1,720,973 research outputs found
Time-Dependent Focusing Mean-Field Games: The Sub-critical Case
Get PDFWe consider time-dependent viscous mean-field games systems in the case of local, decreasing and unbounded couplings. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimizers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimizers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker–Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short
The master equation and the convergence problems in mean field games with several populations.
Get PDFopenThe purpose of this paper is to develop the mean field game theory with several populations. In particular we exploit the connection between the MFG system and master equation
and at the end of this paper we show that the solution of the multi-popultaion Nash system converges to the solution of the multi-population master equation
Wasserstein regularity in mean field control problems
Get PDFopenThis work deals with a class of mean field control problems that are obtained as limits of optimal control problems for large particle systems.
Developing on [Cardaliaguet, P. & Souganidis, P. E.(2023). Regularity of the value function and quantitative propagation of chaos for mean field control problems, Nonlinear Differ. Equ. Appl.], we analyse the value function U in Wasserstein metric and we prove its smoothness in an open and dense set of the space, time and probability measures using the strategy of the linearized system. The definition of this set exploits the concept of strong stability.
Then, we focus on chaos propagation: we study the properties of the optimal solutions of the interacting particle system starting from the aforementioned open and dense set.
We also show some classical results on flows of probability measures via simple analytical tools
Mean field optimal control with piecewise deterministic Markov processes
Get PDFopenThis work studies the theoretical resolution and numerical approximation of a mean field optimization problem. More precisely, a mean field optimal control problem of piecewise deterministic Markov processes is formulated, modeling the optimal charging of a large fleet of electric vehicles. Optimality conditions are obtained through a linearization procedure and are studied via a system of coupled partial differential equations, similar to those encountered in mean field games. The problem is numerically solved using the General Frank-Wolfe algorithm, a variant of the conditional gradient algorithm. This algorithm is analyzed, and it is proved that it has linear convergence and satisfies the mesh-independence property: its rate and the underlying convergence constants are independent of the discretization parameters
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