Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators

Abstract

Graphon control has been proposed and developed in [1]-[4] to approximately solve control problems for very largescale networks of linear dynamical systems. In this paper, linear quadratic regulation (LQR) problems for graphon dynamical systems are studied. Graphon couplings appear in states, controls and cost, and these couplings may be represented by different graphons. Based on invariant subspace decompositions, this work provides a solution method for a class of such problems where the local dynamics is homogeneous but the network couplings are heterogeneous among the coupled subsystems. By exploring a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal solution and a nodal collaborative optimal control solution are established. The complexity of these solutions involves solving one nd × nd dimensional Riccati equation and one n × n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the (nontrivial) invariant subspace shared by the coupling operators. For situations where the graphon couplings do not admit exact lowrank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.