Finite Necessary and Sufficient Stability Conditions for Linear System with Pointwise and Distributed Delays

The paper establishes two finite necessary-and-sufficient stability criteria for linear systems with pointwise and distributed delays: (A) positivity of a block Toeplitz matrix Kr̂ built from the delay Lyapunov matrix U at a computable order r̂, and (B) positivity of Kr* − α0Pᵀr*Pr* at a (smaller) computable order r*, where Pr uses the fundamental matrix K. Both the statements and their proofs hinge on (i) the Lyapunov condition ensuring existence/uniqueness of U, (ii) a prescribed-derivative Lyapunov–Krasovskii functional v0 with dv0/dt = −x(t)ᵀWx(t), and its variant v1 with dv1/dt = −x(t−H)ᵀW x(t−H), (iii) approximation of functions in a compact set S by ψr built from K, giving v1(ψr) = γᵀKrγ, and (iv) explicit bounds on the interpolation error εr = H(M1+L)e^{LH}/(r−1+LH) that yield the r̂ and r* thresholds. These ingredients are clearly stated and used correctly in the paper’s Theorems 13 and 14 and their proofs . In contrast, the candidate solution asserts a different energy identity dV/dt = −∫_{−H}^0 x(t+θ)ᵀW x(t+θ)dθ for its canonical functional V and builds a decay estimate from it. This identity is not the one used or established in the paper (which uses dv0/dt = −x(t)ᵀWx(t) and dv1/dt = −x(t−H)ᵀWx(t−H)), and no justification is provided for replacing v1 by such a V in the distributed-delay setting. The candidate also collapses the lower bound to α1∥x_t∥² without the α0∥x(t)∥² term essential to the paper’s mixed criterion. While the model reproduces the correct threshold formulas and much of the surrounding structure, the core Lyapunov-derivative step it relies on is unsupported in this context, so its proof outline is not sound. The paper’s argument remains correct; the model’s is not. The paper itself already notes the conservatism of the r̂ estimate and motivates the mixed criterion to reduce r*, consistent with the numerical evidence provided .