Invariance Entropy for Uncertain Control Systems

The paper proves, for any quasi-invariant-partition (A,G), that hinv(A,G) equals the maximum mean cycle weight w*(A,G) and provides the equivalent path-sum limit formula. It does so by showing that Wm(A,G) is the unique minimal (m,Q)-spanning set and that rinv equals |A| times the maximal product of outdegrees along admissible words, yielding the limit of average vertex weights (Theorem 3.1), followed by a cycle decomposition argument to identify the limit with the maximum cycle mean (Theorem 3.2). The candidate solution mirrors this structure: it (i) encodes (A,G) as a directed graph with weights w(A)=log2|D(A)|, (ii) proves coverage and one-step forcing properties from (3.1)–(3.2), (iii) shows S must contain all admissible words, giving the exact rinv formula, and (iv) concludes via a cycle-mean lemma. The logic and end results align with the paper’s claims and proofs, with only stylistic differences in the cycle decomposition argument. Base-2 logarithms are consistent with the paper’s convention. Key paper steps referenced: definitions of quasi-invariant-partitions and D(·), P(·) (, ), the reduction to Wm(A,G) and the product/sum formulas (), and the maximum mean cycle characterization (, , ).