[Paper Review] Mild and weak solutions of Mean Field Games problem for linear control systems

Piermarco Cannarsa, Cristian Mendico|arXiv (Cornell University)|Jul 4, 2019

Guidance and Control Systems3 references4 citations

TL;DR

This paper establishes the existence and uniqueness of mild and weak solutions for first-order Mean Field Games with linear control dynamics on $\mathbb{R}^d$. It introduces a Lagrangian framework using measures on path space to define Nash equilibria, proves Hölder continuity of the equilibrium distribution and fractional semiconcavity of the value function, and shows equivalence between mild and weak solutions via a fixed-point argument and dynamic programming principles.

ABSTRACT

The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on $\\mathbb R^{d}$. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria), as Borel probability measures on the space of admissible trajectories, and mild solutions as solutions associated with such equilibria. Moreover, we prove the existence and uniqueness of mild solutions and we study their regularity: we prove H\\"older regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. Finally, we address the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincides with a suitable class of weak solutions.

Motivation & Objective

• To define and analyze Mean Field Games equilibria for first-order problems with linear controlled dynamics using a Lagrangian formulation on path space.
• To establish the existence and uniqueness of mild solutions via a fixed-point argument on the space of Borel probability measures over trajectories.
• To investigate the regularity of the value function and the evolution of the agent distribution, particularly Hölder continuity and fractional semiconcavity.
• To prove the equivalence between mild solutions (defined via equilibria on path space) and weak solutions of the associated PDE system.
• To extend regularity results under additional assumptions on the Lagrangian, including Lipschitz continuity of the equilibrium distribution and linear semiconcavity of the value function.

Proposed method

• Define the state dynamics as $\dot{\gamma}(t) = A\gamma(t) + Bu(t)$, with $A, B$ real matrices and $u$ an admissible control in $L^2$.
• Introduce the space $\Gamma_T$ of absolutely continuous paths equipped with the uniform norm, and consider Borel probability measures $\eta$ on $\Gamma_T$ with finite first moment.
• Define the flow of measures $m_t = e_t \sharp \eta$, where $e_t$ is the evaluation map at time $t$, representing the distribution of agents.
• Construct Nash equilibria as measures $\eta$ supported on minimizing trajectories of the cost functional $\int_0^T L(\gamma(s), u(s), m_s)\,ds + G(\gamma(T), m_T)$.
• Apply a fixed-point argument in the space of probability measures to prove existence and uniqueness of equilibria under suitable assumptions on $L$, $G$, and the control structure.
• Use dynamic programming and estimates on the flow of trajectories to derive Lipschitz and semiconcave regularity of the value function $V(t,x)$, and relate these to the regularity of $m_t$.

Experimental results

Research questions

• RQ1 Under what conditions does a Mean Field Games equilibrium exist for linear control systems with first-order dynamics?
• RQ2 What regularity properties does the value function $V(t,x)$ of the underlying optimal control problem possess, particularly in time and space?
• RQ3 How regular is the time-evolution of the agent distribution $m_t$ in the equilibrium, and under what conditions is it Lipschitz continuous?
• RQ4 What is the relationship between mild solutions (defined via path-space equilibria) and weak solutions of the PDE system associated with the Mean Field Games problem?
• RQ5 Can fractional semiconcavity estimates be established for the value function under minimal assumptions on the Lagrangian and terminal cost?

Key findings

• The value function $V(t,x)$ is locally Lipschitz continuous on $[0,T] \times \mathbb{R}^d$, with a Lipschitz constant depending on the bounds of $D_xL$, $u^*$, and $G$.
• The distribution $m_t$ is $\frac{1}{2}$-Hölder continuous in time, i.e., $\|m_t - m_s\|_{\text{TV}} \leq C|t-s|^{1/2}$, under standard assumptions.
• The value function $V(t,x)$ is locally semiconcave in space with a fractional modulus of semiconcavity in time, specifically $V(t,x) + C|t-s|^{\alpha}$ is concave in $t$ for some $\alpha \in (0,1)$.
• Under additional assumptions on the Lagrangian, the equilibrium distribution $m_t$ is Lipschitz continuous in time, leading to linear semiconcavity of $V$ in time.
• The class of mild solutions (defined via equilibria on path space) coincides with the class of weak solutions of the PDE system, as shown via duality and weak formulation techniques.
• The proof of Lipschitz regularity of $V$ relies on trajectory sensitivity estimates and bounds on $D_xL$, with explicit dependence on $\|A\|$, $T$, and $\|u^* angle_{L^2}$.