Graphon Field Tracking Games with Discrete Time Q-noise

Linear quadratic games on very large dense networks can be modelled with discrete time linear quadratic graphon field games with Q-noise. In such a game, each node in the graph corresponds to an agent weakly connected via an undirected network, with a correlated Brownian disturbance affecting each agent. The limit of the finite-sized linear quadratic network tracking game in discrete time is formulated, and it is shown that under the proper assumptions, the game has a graphon limit system with Q-noise. Then, the optimal control of the discrete time system is found in closed-form and the Nash equilibrium behavior of the game is numerically demonstrated.