Near-Optimal Distributed Linear-Quadratic Regulator for Networked Systems

The paper proves exponential spatial decay of the optimal LQR gain (Theorem 3.3), derives an operator-norm truncation bound (Corollary 3.5), and then establishes stability and near-optimality for the κ-truncated controller with explicit constants β, Ω, κ, Γ (Theorems 4.1–4.2) using a Lyapunov argument built around the DARE solution P* and a small-gain condition on B(K*−Kκ) that follows from Corollary 3.5 (see the definitions of ρ, Υ in (3.1) and δ, Ψ in (3.2), and the choice of κ in (4.1b) ). The candidate solution reaches the same conclusions with the same constants, but replaces the paper’s Lyapunov-based Step (for Theorem 4.1) with a resolvent/Neumann-series argument and Cauchy’s integral formula. That step cites a resolvent bound for Φ* that is not established in the paper; however, the rest of the derivation mirrors the paper’s logic: (i) exponential decay of the inverse KKT operator on the time–graph product leading to decay of K* blocks and the truncation bound (as in Appendix A and Theorem 3.3) , and (ii) the κ-choice and performance bound consistent with (4.1)–(4.2) . With the minor fix of replacing the resolvent step by the paper’s Lyapunov step, the model’s proof aligns fully. Hence, both are correct; the approaches differ.