  We propose a neural network approach for solving high-dimensional optimal
control problems. In particular, we focus on multi-agent control problems with
obstacle and collision avoidance. These problems immediately become
high-dimensional, even for moderate phase-space dimensions per agent. Our
approach fuses the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman
(HJB) approaches and parameterizes the value function with a neural network.
Our approach yields controls in a feedback form for quick calculation and
robustness to moderate disturbances to the system. We train our model using the
objective function and optimality conditions of the control problem. Therefore,
our training algorithm neither involves a data generation phase nor solutions
from another algorithm. Our model uses empirically effective HJB penalizers for
efficient training. By training on a distribution of initial states, we ensure
the controls' optimality is achieved on a large portion of the state-space. Our
approach is grid-free and scales efficiently to dimensions where grids become
impractical or infeasible. We demonstrate our approach's effectiveness on a
150-dimensional multi-agent problem with obstacles.
