Markov Chains, Entropy, and Fundamental Limitations in Nonlinear Stabilization

In this paper, we propose a novel methodology for establishing fundamental limitations in nonlinear stabilization. To aid the analysis, we express the stabilization problem as control of Markov chains. Using Markov chains, we derive the limitations as certain maximum probability bounds or as positive conditional entropy of the certain signals in the feedback loop. The former is related to the infeasibility of the asymptotic stabilization in the presence of quantization and the latter to the Bode integral formula. In either cases, it is shown that uncertainty - associated here with the unstable eigenvalues of the linearization - leads to fundamental limitations.