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

Graphon control has been proposed and developed in [1]-[3] to approximately solve control problems for very large-scale networks (VLSNs) of linear dynamical systems based on graphon limits. This paper provides a solution method based on invariant subspace decompositions for a class of graphon linear quadratic regulation (LQR) problems where the local dynamics share homogeneous parameters but the graphon couplings may be heterogeneous among the coupled agents. Graphon couplings in this paper appear in states, controls and costs, and these couplings may be represented by different graphons. 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 where each agent computes its part of the optimal solution locally, are established. The application of these solutions to finite network LQR problems may be via (i) the graphon control methodology [3], or (ii) the representation of finite LQR problems as special cases of graphon LQR problems. The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN X nN dimensional Riccati equation, where N is the size of the network. For situations where the graphon couplings do not admit exact low-rank 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.