Time-domain Lyapunov/LMI stability for the neutral Hopfield-type equation when ν−μ ≤ 0

Determine whether time-domain methods based on Lyapunov functionals and linear matrix inequalities can establish asymptotic stability for the scalar neutral functional differential equation \dot{y}(t) = −(ν − μ) y(t) − k_p y(t − τ) − k_d \dot{y}(t − τ) when ν − μ ≤ 0, and, if possible, derive explicit conditions on the controller gains k_p, k_d and the delay τ that ensure such stability in this regime.

Background

The paper studies the scalar neutral functional differential equation \dot{y}(t) = −(ν − μ) y(t) − k_p y(t − τ) − k_d \dot{y}(t − τ), motivated by stabilization of Hopfield-type neural dynamics. Prior time-domain results using Lyapunov functionals and LMIs provide sufficient delay-independent conditions for stability when ν − μ > 0 (e.g., Lien et al. 2000), but these do not cover cases with ν − μ ≤ 0.

The authors adopt a frequency-domain approach to analyze the characteristic quasipolynomial and prescribe exponential stabilization via CRRID and related spectral placement methods. They explicitly note a gap: no known Lyapunov/LMI time-domain results apply to the asymptotic stability analysis of the neutral equation when ν − μ ≤ 0, framing an unresolved question about the existence of such methods and conditions.