Quadratic Optimal Control of Graphon Q-noise Linear Systems

TL;DR

This paper develops a framework for optimal control of large complex networks modeled by graphons with Q-noise disturbances, extending finite network solutions to infinite-dimensional limits.

Contribution

It introduces a novel approach to control large stochastic networks using graphon theory and Q-noise, providing a way to approximate finite network control problems by infinite-dimensional operators.

Findings

Optimal control solutions converge from finite networks to graphon limits.

Extended theory to low rank systems and special cases.

Computed worst-case control performance for standard graphon limits.

Abstract

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.