Topological Entropy, Observability, Robustness, Stabilizability, and Optimal Control

Here, we study connections among observability, stabilizability, and optimal control via digital channels on the one hand, and topological entropy of the open-loop system on the other hand. The concept of entropy of dynamic systems was originated in the work of Kolmogorov and was inspired by the Shannon’s pioneering paper. Kolmogorov’s work started a whole new research direction in which entropy appears as a numerical invariant of a class of deterministic dynamic systems. Later, Adler and his co-authors introduced topological entropy of dynamic systems, which is a modification of Kolmogorov’s metric entropy. The paper imported the concept of topological entropy into the theory of networked control systems. The concept of feedback topological entropy was introduced, and the condition of a local stabilizability of nonlinear systems via a limited capacity channel was given. In this chapter, we extend the concept of topological entropy to the case of uncertain dynamic systems with noncompact state space. Unlike, we use a less common “metric”definition of topological entropy introduced by Bowen. The “metric definition” is, in our opinion, more suitable to the theory of networked control systems. The main results of the chapter are necessary and sufficient conditions of robust observability, stabilizability, and solvability of the optimal control problem that are given in terms of inequalities between the communication channel data rate and the topological entropy of the open- loop system. The main results of the chapter were originally published in. Notice that the results on stabilizability of linear plants via limited capacity communication channels were proved by Nair and Evans.