A (Strongly) Connected Weighted Graph is Uniformly Detectable based on any Output Node

The paper’s Theorem 1 asserts (i) uniform detectability of (e^{-LΔt}, C) if C has a nonzero row sum and (ii) uniform stabilizability of (e^{-LΔt}, B) if B has a nonzero column sum, claiming (ii) follows from duality. Its detectability proof hinges on a “unique ∞-norm maximizer” corollary for e^{-L} and then argues only the 1-vector matters; this misuses the uniform detectability quantifier and the corollary is itself false (−1 also attains the ∞-norm), so the proof is flawed even though claim (i) is true for primitive stochastic A via a standard Perron–Frobenius argument. More seriously, claim (ii) is false for strongly connected directed graphs; nonzero column sum does not ensure stabilizability. The model gives the correct PBH-based condition π^T B ≠ 0, provides a concrete counterexample, and supplies a sound detectability proof and a reasonable parameter-varying extension. See Theorem 1 and its proof sketch in the PDF and the parameter-varying section for how the paper argues these points , and note the problematic “unique ∞-norm” claim and its use in the detectability proof .