Arbitrarily Fast Switched Distributed Stabilization of Partially Unknown Interconnected Multiagent Systems: A Proactive Cyber Defense Perspective

The paper’s Theorem 1 proves ISS for the two-layer MAS under arbitrarily fast switching using a common quadratic Lyapunov function V̄(x)=x^T P̄ x, Riccati-based local gains Ki=−μmin R_i^{-1} B_{ui}^T P_i, and the validation matrix Q̄vσ := Q̄ + K̄^T (R̄ + 2 R̄ Ēcσ) K̄ − (1/af) P̄ B̄f B̄f^T P̄ ≻ 0; the proof shows V̄̇ ≤ −x^T Q̄v x + γd ||d||^2 uniformly in σ, establishing a common ISS Lyapunov function and hence ISS under arbitrary switching . The candidate solution reproduces this structure and the same cancellations, including the choice Rf = (af γf γcy ||Aa||^2 + ad) I to absorb the sector-bounded nonlinearity and to handle disturbances, and it derives an explicit KL–K∞ bound, which is consistent with the paper’s ISS claim and definitions . Minor notation/sign issues aside (notably the sign on Ēcσ in the text), the arguments align step-by-step with the paper’s derivation (Design Properties 1 and the decomposition leading to −v^T(R̄+2R̄Ēcσ)v) . Therefore, both are correct and substantially the same proof.