Stability criteria of linear delay differential systems based on fundamental matrix

The paper proves that for the retarded linear delay system x'(t)=A0 x(t)+∑ Aj x(t−τj) with fundamental matrix X(t), the following are equivalent: (i) asymptotic (hence exponential) stability of X(t); (ii) A0+∑Aj invertible and ∫0^∞ X=−(A0+∑Aj)^{-1} (Theorem 3.1); (iii) ∫0^∞ ||X|| < ∞ (Theorem 3.2); (iv) ∫0^∞ ||X||_F^2 < ∞ (Theorem 3.3). These claims and their proofs appear explicitly in the manuscript (Theorem 3.1: statement and proof steps around Eqs. (3.1)–(3.5) and the characteristic equation; Theorem 3.2: proof via boundedness, uniform continuity and Barbalat’s lemma; Theorem 3.3: proof via entrywise energy estimates and Barbalat’s lemma) . The candidate solution asserts the same equivalences but relies on an incorrect step: from (2) it concludes (3), i.e., that existence of the (entrywise) improper matrix integral ∫0^∞ X implies absolute integrability of ||X||. This does not follow in general (matrix/vector integrals can converge conditionally without ||X|| being L1). Because the candidate’s equivalence chain uses (2)⇒(3)⇒(1), its proof is logically flawed even though the final equivalence is true per the paper. Hence, paper correct, model wrong.