Social Optima in Linear Quadratic Graphon Field Control: Analysis via Infinite Dimensional Approach

This paper is concerned with linear quadratic graphon field social control problem where the noises of individual agents are correlated. Compared with the well-studied mean field system, the graphon field system consists of a large number of agents coupled weakly via a weighted undirected graph where each node represents an individual agent. Another notable feature of this paper is that the dynamics of states of agents are driven by Brownian motions with a correlation matrix. The infinite dimensional approach is adopted to design the centralized and decentralized controls for our large population system. By graphon theory, we prove that the linear quadratic (LQ) social optimum control problem under the centralized information pattern is equivalent to an LQ optimal control problem concerned with a stochastic evolution equation, and the feedback-type optimal centralized control is obtained. Then, by designing an auxiliary infinite dimensional optimal control problem through agent number $N\rightarrow\infty$, a set of decentralized strategies are constructed, which are further shown to be asymptotically social optimal.