  In this paper we study the linear quadratic regulation (LQR) problem for
dynamical systems coupled over large-scale networks and obtain locally
computable low-complexity solutions. The underlying large or even infinite
networks are represented by graphons and the couplings appear in both the
dynamics and the quadratic cost. The optimal solution is obtained first for
graphon dynamical systems for the special case where the graphons are exactly
characterized by finite spectral summands. The complexity of generating these
control solutions involves solving d+1 scalar Riccati equations where d is the
number of non-zero eigenvalues in the spectral representation. Based on this,
we provide a suboptimal low-complexity solution for problems with general
graphon couplings via spectral approximations and analyze the performance under
the approximate control. Finally, a numerical example is given to illustrate
the explicit solution and demonstrate the simplicity of the solution.
