The local turnpike property in Mean Field Control and Games with quadratic Hamiltonian

Dipartimento di Matematica “Tullio Levi-Civita” Università di Padova via Trieste 63, 35121 Padova, Italy

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 4 December 2025
Accepted: 2 May 2026

Abstract

We study the local stability properties of solutions to ergodic and discounted mean field games systems, as the time horizon T → +∞, around stationary equilibria, when the Hamiltonian is quadratic. We replace the usual monotonicity of the coupling term with a weaker, local assumption on the stationary equilibrium (that need not be unique), stemming from a second-order strict positivity condition. This new stability assumption, together with a symmetry property of the system, allows us to derive an exponential turnpike property for those solutions that are close to the stationary one, whenever the spatial domain Ω is either the flat torus Tⁿ or ℝⁿ. Finally, through a fixed-point argument, we establish the actual existence of stable solutions, both on the finite horizon [0, T] and on the infinite horizon, in the periodic setting Ω = Tⁿ, provided that the initial (and terminal) data are close enough to the stationary equilibrium.