  This paper studies network LQR problems with system matrices being
spatially-exponential decaying (SED) between nodes in the network. The major
objective is to study whether the optimal controller also enjoys a SED
structure, which is an appealing property for ensuring the optimality of
decentralized control over the network. We start with studying the open-loop
asymptotically stable system and show that the optimal LQR state feedback gain
$K$ is `quasi'-SED in this setting, i.e. $\|[K]_{ij}\|\sim
O\left(e^{-\frac{c}{\mathrm{poly}\ln(N)}}\mathrm{dist}(i,j)\right)$. The
decaying rate $c$ depends on the decaying rate and norms of system matrices and
the open-loop exponential stability constants. Then the result is further
generalized to unstable systems under a SED stabilizability assumption.
Building upon the `quasi'-SED result on $K$, we give an upper-bound on the
performance of $\kappa$-truncated local controllers, suggesting that
distributed controllers can achieve near-optimal performance for SED systems.
We develop these results via studying the structure of another type of
controller, disturbance response control, which has been studied and used in
recent online control literature; thus as a side result, we also prove the
`quasi'-SED property of the optimal disturbance response control, which serves
as a contribution on its own merit.
