  The modelling of linear quadratic Gaussian optimal control problems on large
complex networks is intractable computationally. Graphon theory provides an
approach to overcome these issues by defining limit objects for infinite
sequences of graphs permitting one to approximate arbitrarily large networks by
infinite dimensional operators. This is extended to stochastic systems by the
use of Q-noise, a generalization of Wiener processes in finite dimensional
spaces to processes in function spaces. The optimal control of linear quadratic
problems on graphon systems with Q-noise disturbances are defined and shown to
be the limit of the corresponding finite graph optimal control problem. The
theory is extended to low rank systems, and a fully worked special case is
presented. In addition, the worst-case long-range average and infinite horizon
discounted optimal control performance with respect to Q-noise distribution are
computed for a small set of standard graphon limits.
