CONTROLLABILITY OF NETWORKED SYSTEM WITH HETEROGENEOUS DYNAMICS

The paper’s main theorem (Theorem 4.1) states exactly the controllability conditions the model proves, namely: (i) ei^T D ≠ 0 for all i, (ii) (Ai + λi H, B) is controllable for each i, and (iii) an eigenvalue-sharing linear-independence condition across blocks; see the theorem statement and proof structure in the PDF . However, the proof relies critically on Proposition 1, which asserts that for F̃ = A + (J ⊗ H) obtained via the similarity (T ⊗ I), the left eigenvectors of F̃ are of the simple separated form ei ⊗ ξ for ξ a left eigenvector of Ai + λi H; this is then transported back to F via Lemma 2.3 to conclude that ei^T ⊗ ξ are left eigenvectors of F . That claim is generally false when C has nontrivial Jordan blocks: for J with superdiagonal ones, (ei ⊗ ξ)(J ⊗ H) produces unavoidable contributions in later blocks unless ξH = 0, so ei ⊗ ξ is not a left eigenvector in general. The paper’s own examples use a nondiagonal J with a superdiagonal one (e.g., J with a 1 in the (2,3) position in Example 1) , contradicting the proposition’s eigenvector form. Because the necessity and sufficiency arguments in Theorem 4.1 repeatedly expand arbitrary left eigenvectors of F using that (incorrect) separated basis (e.g., the sums ∑αkl(eik^T ⊗ ξlik)), those steps are incomplete/incorrect as written . In contrast, the model’s solution performs the correct PBH analysis after the similarity transform, uses block-upper-triangular back-substitution to construct the left eigenspace of F̃ from the diagonal-block eigenspaces, and derives the PBH conditions via the induced block-upper-triangular map to the outputs; this fixes the gap without requiring C to be diagonalizable. Therefore, while the theorem statement matches the model’s result, the paper’s proof is incomplete in general, and the model’s solution is correct.