  In this paper, we consider a first-order deterministic mean field game model
inspired by crowd motion in which agents moving in a given domain aim to reach
a given target set in minimal time. To model interaction between agents, we
assume that the maximal speed of an agent is bounded as a function of their
position and the distribution of other agents. Moreover, we assume that the
state of each agent is subject to the constraint of remaining inside the domain
of movement at all times, a natural constraint to model walls, columns, fences,
hedges, or other kinds of physical barriers at the boundary of the domain.
After recalling results on the existence of Lagrangian equilibria for these
mean field games and the main difficulties in their analysis due to the
presence of state constraints, we show how recent techniques allow us to
characterize optimal controls and deduce that equilibria of the game satisfy a
system of partial differential equations, known as the mean field game system.
