Long time behavior and turnpike solutions in mildly non-monotone mean field games

Abstract

We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

Turnpike in optimal control of PDEs, ResNets, and beyond

“…Cardaliaguet et al 2012Cardaliaguet et al , 2013, and also more recently Cardaliaguet andPorretta 2019, 2020) preceded and motivated those on the turnpike property (first appearing in Porretta and Zuazua 2013). Regarding this exponential estimate, Cirant and Porretta (2021), for example, have recently shown, under the assumption that the Hamiltonian H(x, p) is C 2 and locally Lipschitz with respect to p, and locally convex with respect to p, with f 0 (x, α), f 1 (α, m) also satisfying Lipschitz assumptions, and being locally bounded, that any classical solution (u T , m T ) to (14.5) then satisfies This is an exponential turnpike-like property. Indeed, we can see system (14.7) as similar to the steady optimality system in turnpike theory.…”

Transport equations with nonlocal diffusion and applications to Hamilton-Jacobi equations

“…Note that the threshold = ( +2) ( −1)/ can be regarded as a parabolic analogue to the one in (51). We also refer to [40] for more recent maximal regularity results for viscous ( = 1) problems with quadratic gradient growth and right-hand sides in mixed Lebesgue scales, and to [58] (and the references therein) for some maximal regularity properties for fully nonlinear second order uniformly parabolic problems in the context of viscosity solutions.…”

“…The proof we are going to present can be made rigorous by regularization (cf [82, Lemma 2.3]), using Duhamel's formula for the regularized PDE and then passing to the limit. The approach is inspired by [19] and it has been also recently implemented in [40,Lemma A.3] to get estimates in mixed Lebesgue scales and in [35,Lemma A.3]. We claim that there exists * ∈ (0, ] independently of ∈ (T ) such that…”

Turnpike in optimal control of PDEs, ResNets, and beyond

“…Indeed, works on a double-arc exponential estimate for (14.5) (e.g., [25,26], and also more recently [27,28]) precede and have motivated those on the turnpike property (first appearing in [144]). Regarding this exponential estimate, in the recent paper [37] for instance, the authors roughly show under the assumption that the Hamiltonian H(x, p) is C 2 and locally Lipschitz with respect to p, and locally convex with respect to p, with f 0 (x, α), f 1 (α, m) also satisfying Lipschitz assumptions, and being locally bounded, then any classical solution (u T , m T ) to (14.5 T d u = 0. This is an exponential turnpike-like property.…”