Stochastic Control on Large Networks: A Q-noise Formulation

Solving linear quadratic Gaussian optimal control problems on large complex networks is computationally intractable and may be impossible due to data-collection costs or privacy concerns. 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. By building on the foundations of Dunyak and Caines (2022), linear quadratic problems on graphon systems with Q-noise disturbances are defined and shown to be the limit of the finite graph optimal control problem. The result is demonstrated with a numerical example showing that even relatively small networks (N = 500) show this emergent limit behavior.