Quadratic Optimal Control of Graphon Q-noise Linear Systems

TL;DR

The paper addresses scalable control of very large networked systems by formulating Linear Quadratic Gaussian control on graphon limits subject to Q-noise, providing a rigorous convergence bridge from finite graphs to infinite-dimensional operator-based problems. The main approach uses graphon limits to replace large adjacency matrices with operator-valued kernels, and introduces Q-noise as a spatially distributed stochastic disturbance that yields well-defined limits; the core results show convergence of finite-dimensional Riccati equations to their operator-valued graphon counterparts and establish both long-range average and discounted-horizon solutions. A key contribution is the finite-rank reduction: when the graphon and noise covariance share a finite invariant subspace, the LQG problem decomposes into a finite-dimensional Riccati system plus a one-dimensional orthogonal component, enabling efficient computation. Numerical examples illustrate convergence and demonstrate the effectiveness of low-rank graphon models in approximating large networks, highlighting practical scalability for network control and mean-field-like analysis. Collectively, the work provides a mathematically principled, scalable framework for designing optimal controllers for very large stochastic networks with 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.

Figures (6)

• Figure 1: Top: Trajectory of system (\ref{['eq:indep_noise']}) with independent noise at each node. Because the magnitude of the independent noise is so high, there is no clear system structure in the state trajectory. Bottom: the state trajectory of system (\ref{['eq:indep_noise']}) with a Q-noise disturbance. While there is clearly noise present in the system, there is an overall structure suggesting that the limit will be continuous in both space and time.
• Figure 2: Top row: An Erdős-Renyi graphon and sample graph. Middle row: A uniform attachment graphon and graph. Bottom row: A small world graphon.
• Figure 3: Left: a fifty node W-random graph. Right: the associated adjacency matrix to be used for the numerical simulations. Yellow squares represent an edge, blue squares represent a lack of an edge. The adjacency matrix is rank 49, despite the limit system being rank one.
• Figure 4: Top: the system generated with the finite graph using the piecewise constant graphon $\boldsymbol{A}^{[N]}$. Middle: The system trajectory generated when $\boldsymbol{A}^{[N]}$ is projected onto the eigenspace spanned by ${\boldsymbol{f}}$. Bottom: the root squared distance of the finite graph system trajectory and the projected graph system trajectory. The root squared distance has a maximum deviation of $0.023$, showing that the two trajectory surfaces are very similar.
• Figure 5: Top: the trajectory of the system under the rank one limit control. Bottom: the positive root of the squared distance between the finite graph system and the limit system over time.

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2: Operators on $Q$-Space Noise
• Lemma 2.1
• Theorem 2.2
• Definition 2.3: Q-noise Dynamical Systems
• Definition 2.4: Mild solution
• Theorem 2.3
• Theorem 3.1
• Corollary 3.1.1: ichikawa_dynamic_1979
• Lemma 3.2