Distributed Identification of Contracting and/or Monotone Network Dynamics

The paper’s Theorem 1 states (a) well-posedness under a uniform positive-definiteness condition on the symmetric part of E; (b) `l1`-type contraction under (a) plus inequalities (15)–(16); (c) monotonicity under (a) with F ≥ 0, K ≥ 0, and E ∈ M_n; (d) positivity from (c) and e(0)=f(0,0); (e) a combined contracting-and-monotone condition. Part (a) is substantively correct (strong monotonicity implies bijectivity of x ↦ e(x,u)), though the proof text in Appendix A contains a likely stray assertion (“Since E is a non-singular M-matrix...”), not implied by (14) alone . Part (b) as written omits an essential inverse-positivity assumption: the Appendix A derivation steps from 1⊤(αE−S) ≥ 0 to 1⊤(α−|F|E−1) ≥ 0 and then to 1⊤(α−|FE−1|) ≥ 0 implicitly require E−1 ≥ 0 (i.e., E ∈ M_n) to preserve inequality directions; without this, the chain (40)→(41)→(42) is not valid in general . Part (c) is false as stated: the paper’s proof drops the dependence on u_{t+1} and writes δx_{t+1} = E_{t+1}^{−1}F δx_t, omitting terms from K δu_t and −(∂e/∂u) δu_{t+1}; by the paper’s own monotonicity definition (inputs ordered for all t), the u_{t+1}-channel must have nonnegative effect, which requires −E^{-1}(∂e/∂u) ≥ 0 (i.e., ∂e/∂u ≤ 0) in addition to E ∈ M_n, F ≥ 0, K ≥ 0 . A scalar counterexample (e(x,u)=x+u, f(x,u)=x) satisfies the paper’s (a) and (c) hypotheses (E=1∈M_1, F=1≥0, K=0≥0) but is not monotone since increasing u_{t+1} decreases x_{t+1} (∂a/∂u_{t+1} = −1) — contradicting the claim. With the corrections proposed by the model (add E ∈ M_n to (b) and add ∂e/∂u ≤ 0 to (c)), the results become correct; (d) and (e) then follow correctly under the corrected hypotheses (and (e) inherits E ∈ M_n from (c)) .