A sufficient condition for k-contraction of the series connection of two systems

The paper’s Theorem 2 states that for the series interconnection with Jacobian J(t,x) = [[J11, 0]; [J21, J22]], if J21 is uniformly bounded and if, for each i in {max(0,k−n),…,min(m,k)}, there exist matrix measures μi (induced by possibly different Lp norms) with μi(J11[k−i]) + μi(J22[i]) ≤ −ηi < 0, then the interconnection is k‑contracting; the proof scales the state with T(ε) to make the off‑diagonal J21 arbitrarily small and then applies a block-diagonal compound formula (Theorem 3) and its corollary, yielding a uniform negative matrix-measure bound for J[k] in a scaled norm . The candidate solution proves the same statement by a different construction: it decomposes the exterior space ∧^k into blocks Wi = ∧^{k−i}R^n ⊗ ∧^{i}R^m, shows that J[k] is block lower‑bidiagonal with diagonal blocks A[k−i]⊕C[i] and subdiagonal terms linear in J21, then designs a weighted block‑∞ norm that suppresses the off‑diagonal contribution; the diagonal blocks satisfy the same additive-μ inequality by assumption, so one obtains μ(J[k]) ≤ −η < 0 and thus k‑contraction. This aligns with the paper’s conditions and conclusions and uses standard compound/exterior‑algebra facts (e.g., exp(A)(k) = exp(A[k])) that the paper reviews in its preliminaries . The paper’s approach uses a similarity scaling T(ε) to control J21, whereas the model builds weights directly on ∧^k; both yield the same negative logarithmic‑norm bound and hence k‑contraction. The sufficiency and the need for the additive j<k conditions are also consistent with the discussion and examples in the paper (showing the condition is generally unimprovable) .