  Mean field optimal control problems are a class of optimization problems that
arise from optimal control when applied to the many body setting. In the noisy
case one has a set of controllable stochastic processes and a cost function
that is a functional of their trajectories. The goal of the optimization is to
minimize this cost over the control variables. Here we consider the case in
which we have $N$ stochastic processes, or agents, with the associated control
variables, which interact in a disordered way so that the resulting cost
function is random. The goal is to find the average minimal cost for $N\to
\infty$, when a typical realization of the quenched random interactions is
considered. We introduce a simple model and show how to perform a dimensional
reduction from the infinite dimensional case to a set of one dimensional
stochastic partial differential equations of the Hamilton-Jacobi-Bellman and
Fokker-Planck type. The statistical properties of the corresponding stochastic
terms must be computed self-consistently, as we show explicitly.

Optimization and Control;Disordered Systems and Neural Networks