  We consider the task of remote state estimation and stabilization of
disturbed linear plants via noisy communication channels. In 2007 Matveev and
Savkin established a surprising link between this problem and Shannon's theory
of zero-error communication. By applying very recent results of computability
of the channel reliability function and computability of the zero-error
capacity of noisy channels by Boche and Deppe, we analyze if, on the set of
linear time-invariant systems paired with a noisy communication channel, it is
uniformly decidable by means of a Turing machine whether remote state
estimation and stabilization is possible. The answer to this question largely
depends on whether the plant is disturbed by random noise or not. Our analysis
incorporates scenarios both with and without channel feedback, as well as a
weakened form of state estimation and stabilization. In the broadest sense, our
results yield a fundamental limit to the capabilities of computer-aided design
and autonomous systems, assuming they are based on real-world digital
computers.
