5,800 research outputs found
Long time average of first order mean field games and weak KAM theory
We show that the long time average of solutions of first order mean field
game systems in finite horizon is governed by an ergodic system of mean field
game type. The well-posedness of this later system and the uniqueness of the
ergodic constant rely on weak KAM theory
Long Time Behavior of First Order Mean Field Games on Euclidean Space
The aim of this paper is to study the long time behavior of solutions to
deterministic mean field games systems on Euclidean space. This problem was
addressed on the torus in [P. Cardaliaguet, {\it Long time
average of first order mean field games and weak KAM theory}, Dyn. Games Appl.
3 (2013), 473-488], where solutions are shown to converge to the solution of a
certain ergodic mean field games system on . By adapting the
approach in [A. Fathi, E. Maderna, {\it Weak KAM theorem on non compact
manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we
identify structural conditions on the Lagrangian, under which the corresponding
ergodic system can be solved in . Then we show that time
dependent solutions converge to the solution of such a stationary system on all
compact subsets of the whole space
Obstacle Mean-Field Game Problem
In this paper, we introduce and study a first-order mean-field game obstacle
problem. We examine the case of local dependence on the measure under
assumptions that include both the logarithmic case and power-like
nonlinearities. Since the obstacle operator is not differentiable, the
equations for first-order mean field game problems have to be discussed
carefully. Hence, we begin by considering a penalized problem. We prove this
problem admits a unique solution satisfying uniform bounds. These bounds serve
to pass to the limit in the penalized problem and to characterize the limiting
equations. Finally, we prove uniqueness of solutions
Mean field games systems of first order
We consider a system of mean field games with local coupling in the
deterministic limit. Under general structure conditions on the Hamiltonian and
coupling, we prove existence and uniqueness of the weak solution,
characterizing this solution as the minimizer of some optimal control of
Hamilton-Jacobi and continuity equations. We also prove that this solution
converges in the long time average to the solution of the associated ergodic
problem
Weak KAM approach to first-order Mean Field Games with state constraints
We study the asymptotic behavior of solutions to the constrained MFG system
as the time horizon goes to infinity. For this purpose, we analyze first
Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM
theory, constructing a Mather measure for the associated variational problem.
Using these results, we show that a solution to the constrained ergodic mean
field games system exists and the ergodic constant is unique. Finally, we prove
that any solution of the first-order constrained MFG problem on
converges to the solution of the ergodic system as
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