Stabilization of linear systems with multiple unknown time-varying input delays by linear time-varying feedback

The paper proves asymptotic stabilization for linear systems with multiple unknown time‑varying input delays using a memoryless, linear time‑varying gain u(t)=−B^T P(γ(t))x(t), where P(γ) solves the parametric Lyapunov equation ATP+PA−PBB^TP=−γP, under the crucial Assumption 1 that all eigenvalues of A are zero and (A,B) is controllable, and Assumption 3 that the delays are bounded and satisfy τ̇_i(t)≤d<1 (both τ̄ and d are unknown to the controller). The analysis constructs a specific Lyapunov–Krasovskii functional and derives a differential inequality V̇≤−(γ/4)V0−(γ/4)(...) leading to V(t)≤f(t1)/f(t) V(t1) with f(t)=exp(γ0(ωt+1)^{1−μ}/(4ω(1−μ))), and, via a lower bound P(γ)≽μ1γ^δ I (δ≥1), concludes x(t)→0 (Theorem 1 and its proof, including (21)–(23), (31)–(35), and (37)–(39) ). The model’s write‑up omits the key structural assumption on A (all eigenvalues at zero) that underpins property (15) and the boundedness of δ_c, and it does not confront the lack of a uniform lower bound for V0(t)=x^T P(t)x(t) (explicitly noted by the paper’s Remark 3), which is essential to deduce x(t)→0 from V→0; instead it asserts x→0 without establishing a lower bound coupling V0 and ∥x∥^2 as t→∞. It also relies on a non‑substantiated bound on ∥u̇∥^2 and contains a contradictory parameter choice (κ=qτ̄/η with “η small”), making the proposed proof incomplete and logically flawed relative to the paper’s precise argument (Assumptions and Lemma 3; Remark 3 and the construction around (33)–(35) and f(t) ).