A rolled-off passivity theorem

The paper’s Theorem 1 states that if H1 is incrementally (µ1,γ1)-dissipative and H2 is ε-strongly incrementally (µ2,γ2)-dissipative, then the negative-feedback interconnection F has finite incremental gain when µ1µ2<1, µ1γ2<1, and µ2γ1<1, and it provides a proof via Scaled Relative Graphs (SRGs) plus a homotopy argument to ensure the closed loop maps L2→L2; this addresses well-posedness and yields an explicit gain bound 1/r1 with r1 = min{ε, 1/µ1−µ2, 1/µ1−γ2, 1/γ1−µ2} . The candidate solution reproduces the same pointwise inequalities and achieves the same bound by a direct four-case estimate, but it assumes the existence of y ∈ F(u) without proving the closed-loop relation maps L2 into L2 for all inputs; the paper explicitly solves this with the τ-homotopy continuity argument, which the model omits . Hence, while the algebraic estimates in the model are sound, the proof is incomplete relative to the theorem’s claim about the closed-loop operator.