Sensing-Limited Control Under Non-Designable Observation Mechanisms

We study the information-theoretic limits of controlling unstable linear systems through non-designable observation mechanisms. Unlike classical communication-constrained control, the information bottleneck lies in the observation mechanism rather than in a designable encoder-channel interface. For noiseless linear dynamics, we derive necessary conditions for mean-square observability and stabilizability, showing that the directed information rate from the unstable state process to the observation process must dominate the open-loop expansion rate of the unstable modes. We further show that this lower bound persists under additive process disturbances. In the Linear-Gaussian setting, although the unstable-state directed information rate remains intractable in closed form, we obtain an exact characterization of the full-state directed information rate, which upper-bounds the unstable-state quantity and yields computable necessary conditions. Under suitable posterior regularity conditions, we also establish sufficient conditions for asymptotic mean-square observability and, via certainty-equivalence control, asymptotic mean-square stabilizability. The key step is an entropy-to-error bridge: a strict surplus in directed information over the expansion rate forces posterior uncertainty to collapse and thereby drives the estimation error covariance to zero. These results identify a fundamental feasibility boundary for sensing-limited control and clarify how classical communication-based limits must be reinterpreted when the sensing interface is non-designable.