Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

The paper proves, under Assumption 7 (uniform ES of the fast block Σ_D), the chain λ(Σ̄) ≤ λ(Σ̌) ≤ lim inf_{ε→0+} λ(Σε) ≤ lim sup_{ε→0+} λ(Σε) ≤ λ(Σ̂), plus the transfer statements: Σ̌ EU ⇒ ε-EU and Σ̂ ES ⇒ ε-ES (Theorem 9) . The candidate solution establishes the same inequalities and transfer claims. For the lower bound and Σ̌ construction, both use an O(ε) one-microcycle expansion and show M̄ ⊂ M̌ via constant signals (paper’s Lemma 13 and Proposition 14; model’s Step 2) . For the upper bound, the candidate argues via an outer differential-inclusion envelope and upper semicontinuity of solution sets, whereas the paper constructs a converse Lyapunov function V=V1+χV2 and compares short-time evolutions to solutions of Σ̂ (Section 5), still yielding lim sup ≤ λ(Σ̂) and ε-ES when λ(Σ̂)<0 . Both rely on the same structural properties of K(x) (compactness, homogeneity, global Hausdorff-Lipschitz continuity) proved in the paper (Prop. 17 and Lemma 19) . Hence, results agree; the proof strategies differ mainly for the rightmost inequality.