Abstract
An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray–Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.
Highlights
The goal of this work is to prove the existence and uniqueness of a classical solution to the following system of partial differential equations: Applied Mathematics & Optimization (2021) 83:1431–1464 ⎧ ⎪⎪⎪⎪⎪⎪⎨ (i (i ) i)−∂t u − σ Δu + H (x, t, ∇u(x, t) + φ(x, t) P(t))= f (x, t, m(t)) (x, t) ∈ Q,∂t m − σ Δm + div(vm) = 0 (x, t) ∈ Q, ⎪⎪⎪⎪⎪⎪⎩(i i i ) (i v) (v)
A similar approach has been employed in [16], [17], and [18] for the analysis of a mean field game problem proposed by Chan and Sircar [9]
The application of the Leray–Schauder theorem relies on a priori bounds for fixed points. These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are solutions to an optimal control problem of the Fokker–Planck equation
Summary
The goal of this work is to prove the existence and uniqueness of a classical solution to the following system of partial differential equations:. A similar approach has been employed in [16], [17], and [18] for the analysis of a mean field game problem proposed by Chan and Sircar [9] In this model, each agent exploits an exhaustible resource and fixes its price. The application of the Leray–Schauder theorem relies on a priori bounds for fixed points These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are solutions to an optimal control problem of the Fokker–Planck equation. Some parabolic estimates, used all along the article, are provided and proved in the appendix
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