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Constrained Mean Field Games with Grushin type dynamics

Alessandra Cutrì, Paola Mannucci, Claudio Marchi, Nicoletta Tchou

TL;DR

This work analyzes finite-horizon deterministic mean field games with Grushin-type degenerate dynamics and state constraints, establishing robust single-agent well-posedness and a Lagrangian MFG framework under two complementary local conditions. It proves existence of optimal trajectories, a closed-graph property for the optimal-trajectory map, and continuity of the value function, even in settings with degenerate directions and irregular boundaries. Building on these results, the paper constructs relaxed MFG equilibria via Kakutani fixed-point arguments and derives mild solutions (value function and time-evolving measures), with uniqueness arising from monotonicity assumptions. The findings extend mean field game theory to degenerate, constrained dynamics with nonlocal coupling, offering a rigorous basis for existence (and potential uniqueness) of equilibria in these complex systems with practical modeling relevance.

Abstract

This paper is devoted to a class of finite horizon deterministic mean field games with Grushin type dynamics, state constraints and nonlocal coupling. First, we consider the optimal control problem that each agent aims to solve when the evolution of the population is given and we establish some properties as: the existence of an optimal trajectory for any starting point $(x,t)$, the closed graph property for the multivalued map which associates to each point $(x,t)$ the set of optimal trajectories starting from that point, endowed with a suitable notion of convergence, the continuity of the value function. The main issue to overcome is due to the local interplay at boundary points between the set of state constraints and the degenerate dynamics. To this end, we shall point out two different sets of assumptions which are both sufficient for these properties. Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state).

Constrained Mean Field Games with Grushin type dynamics

TL;DR

This work analyzes finite-horizon deterministic mean field games with Grushin-type degenerate dynamics and state constraints, establishing robust single-agent well-posedness and a Lagrangian MFG framework under two complementary local conditions. It proves existence of optimal trajectories, a closed-graph property for the optimal-trajectory map, and continuity of the value function, even in settings with degenerate directions and irregular boundaries. Building on these results, the paper constructs relaxed MFG equilibria via Kakutani fixed-point arguments and derives mild solutions (value function and time-evolving measures), with uniqueness arising from monotonicity assumptions. The findings extend mean field game theory to degenerate, constrained dynamics with nonlocal coupling, offering a rigorous basis for existence (and potential uniqueness) of equilibria in these complex systems with practical modeling relevance.

Abstract

This paper is devoted to a class of finite horizon deterministic mean field games with Grushin type dynamics, state constraints and nonlocal coupling. First, we consider the optimal control problem that each agent aims to solve when the evolution of the population is given and we establish some properties as: the existence of an optimal trajectory for any starting point , the closed graph property for the multivalued map which associates to each point the set of optimal trajectories starting from that point, endowed with a suitable notion of convergence, the continuity of the value function. The main issue to overcome is due to the local interplay at boundary points between the set of state constraints and the degenerate dynamics. To this end, we shall point out two different sets of assumptions which are both sufficient for these properties. Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state).
Paper Structure (17 sections, 12 theorems, 61 equations, 5 figures)

This paper contains 17 sections, 12 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Settings of the control problem
  3. Closed graph property
  4. A sufficient condition for the reachability property
  5. Assumptions
  6. The value function
  7. Examples and counter-examples
  8. A class of Mean Field Games
  9. Setting and notations
  10. Probability sets .
  11. Assumptions.
  12. The set $\Gamma_C$.
  13. The set ${\mathcal{P}}(\Gamma_C)$ and the associated costs.
  14. Optimal trajectories.
  15. The sets ${\mathcal{P}}_{m_0}(\Gamma_C)$.
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $x\in\Sigma$. Assume one of the following properties Then, $\Gamma^{\rm{opt}}[x]$ has the closed graph property.

Figures (5)

  • Figure 1: Reachability at interior points
  • Figure 2: Curved cone
  • Figure 3: Example \ref{['ex:5.8']}
  • Figure 4: Example \ref{['example1']}
  • Figure 5: Example \ref{['ex5.6']}

Theorems & Definitions (49)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Remark 2.2
  • Proposition 2.1
  • proof
  • Definition 2.4
  • Lemma 2.1
  • ...and 39 more