  We consider scalar decentralized average-cost infinite-horizon LQG problems
with two controllers, focusing on the fast dynamics case when the (scalar)
eigenvalue of the system is large. It is shown that the best linear
controllers' performance can be an arbitrary factor worse than the optimal
performance. We propose a set of finite-dimensional nonlinear controllers, and
prove that the proposed set contains an easy-to-find approximately optimal
solution that achieves within a constant ratio of the optimal quadratic cost.
The insight for nonlinear strategies comes from revealing the relationship
between information flow in control and wireless information flow. More
precisely, we discuss a close relationship between the high-SNR limit in
wireless communication and fast-dynamics case in decentralized control, and
justify how the proposed nonlinear control strategy can be understood as
exploiting the generalized degree-of-freedom gain in wireless communication
theory. For a rigorous justification of this argument, we develop new
mathematical tools and ideas. To reveal the relationship between
infinite-horizon problems and generalized MIMO Witsenhausen's counterexamples,
we introduce the idea of geometric slicing. To analyze the nonlinear strategy
performance, we introduce an approximate-comb-lattice model for the relevant
random variables.
