Markov Chains, Entropy, and Fundamental Limitations in Nonlinear Stabilization

This paper is concerned with entropy based fundamental limitation results for the nonlinear stabilization of a scalar dynamical system. Using methods based on ergodic theory, we pose the problem as control of Markov chains. It is shown that uncertainty, associated here with the unstable eigenvalue of the linearization, leads to fundamental limitations. These limitations arise as certain in-feasibility conditions for nonlinear stabilization in the presence of quantization or equivalently as positive conditional entropy of the output signal in the feedback loop. The former leads to a nonlinear stabilization result and latter to a fundamental limitation result