  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.
