[Paper Review] Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG

Samer Dweik, Guilherme Mazanti|Dec 20, 2018

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes sharp semi-concavity of the value function in a non-autonomous optimal control problem with only lower Lipschitz regularity in time, and applies this to prove uniform $L^p$ bounds on agent distributions in an optimal-exit mean field game (MFG), ensuring propagation of $L^p$ integrability from initial to all future times under mild regularity assumptions on the dynamics.

ABSTRACT

This paper studies a mean field game inspired by crowd motion in which agents evolve in a compact domain and want to reach its boundary minimizing the sum of their travel time and a given boundary cost. Interactions between agents occur through their dynamic, which depends on the distribution of all agents. We start by considering the associated optimal control problem, showing that semi-concavity in space of the corresponding value function can be obtained by requiring as time regularity only a lower Lipschitz bound on the dynamics. We also prove differentiability of the value function along optimal trajectories under extra regularity assumptions. We then provide a Lagrangian formulation for our mean field game and use classical techniques to prove existence of equilibria, which are shown to satisfy a MFG system. Our main result, which relies on the semi-concavity of the value function, states that an absolutely continuous initial distribution of agents with an $L^p$ density gives rise to an absolutely continuous distribution of agents at all positive times with a uniform bound on its $L^p$ norm. This is also used to prove existence of equilibria under fewer regularity assumptions on the dynamics thanks to a limit argument.

Motivation & Objective

• To establish semi-concavity of the value function in a non-autonomous optimal control problem with minimal time regularity assumptions on the dynamics.
• To prove differentiability of the value function along optimal trajectories under stronger regularity conditions.
• To formulate a Lagrangian approach for optimal-exit MFGs and prove existence of equilibria.
• To derive uniform $L^p$ bounds on the distribution of agents at all positive times when the initial distribution has an $L^p$ density.
• To extend existence of equilibria to less regular dynamics using a limit argument based on $L^p$ estimates.

Proposed method

• Prove sharp semi-concavity of the value function using only a lower Lipschitz bound on the dynamics in time, without requiring upper bounds.
• Apply the Pontryagin Maximum Principle to characterize optimal trajectories and derive necessary conditions.
• Use a Lagrangian formulation to describe the evolution of agents and derive the MFG system in terms of continuity and Hamilton–Jacobi equations.
• Establish uniform $L^p$ bounds on the distribution of agents at all times by leveraging the semi-concavity of the value function.
• Construct a sequence of regularized dynamics and use weak convergence and compactness to pass to the limit and prove existence of equilibria under minimal regularity.
• Use viscosity solution theory to verify that the limit satisfies the Hamilton–Jacobi equation in the viscosity sense.

Experimental results

Research questions

• RQ1 Can semi-concavity of the value function be established under only a lower Lipschitz regularity condition on the time-dependent dynamics in a non-autonomous optimal control problem?
• RQ2 Does an initial $L^p$-integrable distribution of agents lead to uniformly bounded $L^p$ norms of the agent distribution at all future times in the optimal-exit MFG framework?
• RQ3 Can equilibria be proven to exist in the MFG model under weaker regularity assumptions on the dynamics by using $L^p$ estimates and a limit argument?
• RQ4 How does the semi-concavity of the value function contribute to the propagation of integrability in the MFG system?
• RQ5 What is the role of the Lagrangian formulation in proving existence of equilibria and deriving the MFG system?

Key findings

• The value function is semi-concave in space with a sharp semi-concavity constant that depends only on the lower Lipschitz constant of the dynamics in time.
• Differentiability of the value function along optimal trajectories holds under additional regularity assumptions on the dynamics.
• An absolutely continuous initial distribution with an $L^p$ density leads to an absolutely continuous distribution at all positive times with a uniform bound on its $L^p$ norm.
• The $L^p$ estimate is used to prove existence of equilibria in a less regular model via a limit argument on regularized dynamics.
• The limiting distribution satisfies the MFG system in the sense of distributions, with the continuity equation and Hamilton–Jacobi equation holding in the weak sense.
• The convergence of gradients of regularized value functions to the gradient of the limit value function is established, ensuring consistency in the dynamics of the MFG system.