Time-dependent mean-field games in the superquadratic case

Abstract

We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as the key techniques used in the subquadratic case do not extend to the superquadratic setting. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton-Jacobi equation are obtained in the present paper through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton-Jacobi equation with polynomial estimates for solutions of the Fokker-Planck equation. Existence of classical solutions can is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to the current understanding of superquadratic Hamilton-Jacobi equations.

Crowd-Averse Robust Mean-Field Games: Approximation via State Space Extension

“…The significance of the above result is that to find the optimal control input we need to solve the two coupled PDEs in (12) in vt and mt with given boundary conditions. This is usually done by iteratively solving the HJI equation for fixed mt and by entering the optimal u * t and ζ * t obtained from (13) in the FPK equation in (12), until a fixed point in vt and mt is reached.…”

“…The reduced mean-field system associated to the robust mean-field game for the problem under study is (12) complemented with the following additional equations…”

“…We now prove (12). Substituting the above expression for ζ * t back in the expression for the robust Hamiltonian we then havẽ H(xt, p,mt) = H(xt, p,mt) +…”

“…We are then in the position to establish the following result, which provides a lower bound for the value function in (12) when σ = 0.…”

“…In this sense, a variety of solution schemes has been recently proposed based on discretization and/or numerical approximations [1]. There has been substantial progress in the theory of weak solutions [9] and classical solutions [11], [12] for mean-field games. Robust mean-field games have been formulated in [5], [6].…”