Linear Stochastic Graphon Systems with Q-Space Noise

The modelling and control of systems on large complex networks is intractable in general. One approach is to use graphon theory which provides limit objects for infinite sequences of graphs permitting one to approximate arbitrarily large networks by infinite dimensional operators. Such a formulation was initiated in the work of Gao and Caines (2020, 2021) extending classical linear system control theory to the control of systems on large networks. This paper introduces infinite dimensional stochastic processes called Q-space noise into this framework. First, Brownian motions in Hilbert spaces are defined. Second, stochastic dynamical systems on large graphs using Q-space noise processes are shown to converge in the graph limit in expectation. Third, state-to-state and linear-quadratic control of these systems is formulated and the limit approximations are established. Finally, the behavior of these approximations is illustrated numerically.