# Exact Minimum Number of Bits to Stabilize a Linear System

We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$, where $a \geq 1$ is the system gain, $Z_n$'s are random variables with bounded $\alpha$-th moments, and $U_n$'s are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as its only information about system state $X_i$. We show that $M = \lfloor a + 1 \rfloor$ is necessary and sufficient for $\beta$-moment stability, for any $\beta < \alpha$. The converse is shown using information-theoretic techniques. The matching achievability scheme is a uniform quantizer of zoom-in / zoom-out type whose performance is analyzed using probabilistic arguments.

This is joint work with Yuval Peres, Gireeja Ranade, Mark Sellke.