Prescribed exponential stabilization of scalar neutral differential equations: Application to neural control

TL;DR

This work develops a CRRID-based partial pole placement framework for scalar neutral functional differential equations to achieve exponential stabilization with a prescribed decay rate, and applies it to local stabilization of a one-layer Hopfield-type neural network with delayed feedback. By analyzing the spectrum of the associated quasipolynomial and leveraging real-root dominance, the authors derive conditions for three- and two-real-root configurations, extend beyond Frasson-Verduyn Lunel’s sufficient criteria, and provide explicit exponential estimates of solutions. The study then translates these spectral insights into practical delayed PD controller designs, computing controller gains and delays to stabilize various neural regimes (stable, asymptotically stable, and unstable) with prescribed decay rates, and comparing performance to delayed P control. The results offer a principled approach to neural stabilization with delays and open avenues for extending CRRID-based methods to more complex networks and combined Lyapunov-LMI analyses for global stability, while noting sensitivity issues associated with non-semi-simple spectral values.

Abstract

This paper presents a control-oriented delay-based modeling approach for the exponential stabilization of a scalar neutral functional differential equation, which is then applied to the local exponential stabilization of a one-layer neural network of Hopfield type with delayed feedback. The proposed approach utilizes a recently developed partial pole placement method for linear functional differential equations, leveraging the coexistence of real spectral values to explicitly prescribe the exponential decay of the closed-loop solution. While a delayed proportional (P) feedback control may achieve stabilization, it requires higher gains and only allows for a shorter maximum delay compared to the proportional-derivative (PD) feedback control presented in this work. The framework provides a practical illustration of the stabilization strategy, improving upon previous literature results that characterize the solution's exponential decay for simple real spectral values. This approach enhances neural stability in cases where the inherent dynamics are stable and offers a method to achieve local exponential stabilization with a prescribed decay rate when the inherent dynamics are unstable.

Figures (6)

• Figure 1: Spectrum of the quasipolynomial $\Delta$ for various parameters when $\tau=1$. This is obtained by assigning three real spectral values $s_3<s_2<s_1$ to $\Delta$ and computing each case's system parameters $a$, $\alpha$, and $\beta$.
• Figure 2: Plot of $\ln(\theta)$ and $\xi\frac{(\theta-1)}{2\theta}$ as functions of $d:=s_1-s_2$ and $\delta:=s_1-s_3$. Here $\theta:=\alpha e^{-\tau s_2}$ with $0<\alpha<e^{\tau s_1}$ given by \ref{['eq:alpha']} and $\xi$ is defined by \ref{['eq:xi']}. The red dots characterize values of $d$ and $\delta$ where \ref{['eq:xi identity 3rr']} is satisfied.
• Figure 3: Complete characterization of the regions with respect to the values of $(a+s_1)/\delta$ that are necessary and sufficient to ensure the strict dominance or not of the real spectral value $s_1$ when only two real spectral values $s_2<s_1$ are assigned to $\Delta$. The figure is depicted when the delay $\tau=1$ while $\delta:=s_1-s_2$ ranges between $0.01$ and $2$. The functions used are $\phi_1(\delta)=(1-e^{\delta})/(e^{\delta}(\delta-1)+1)$, $\phi_2(\delta) = -\delta/(e^{\delta}-(1+\delta))$, $\phi_3(\delta) = -1/(e^{\delta}-1)$, and $\phi_4(\delta) = 1$. We define the regions $R_1$ as the range of $\delta$ where $\frac{a+s_1}{\delta}\le\phi_1(\delta)$, $R_2$ as the range where $\phi_1(\delta)<\frac{a+s_1}{\delta}\le\phi_2(\delta)$, $R_3$ as the range where $\phi_2(\delta)<\frac{a+s_1}{\delta}\le\phi_3(\delta)$, $R_4$ as the range where $\phi_3(\delta)<\frac{a+s_1}{\delta}<\phi_4(\delta)$, and $R_5$ as the range where $\frac{a+s_1}{\delta}\ge\phi_4(\delta)$.
• Figure 4: Depiction of the functions $W(\tau d, \tau s_1)$ and $Z(2, \tau\delta,\tau s_1)$ defined respectively in \ref{['eq:function V-1']} and \ref{['eq:function Z']}. The red-colored region in the parameters space $(\tau d, \tau s_1)$ corresponds to the region where Frasson-Verduyn Lunel sufficient condition for the dominance of a simple real spectral value is satisfied. The CRRID property extends the conditions by the blue region.
• Figure 5: Solutions to \ref{['eq:nu grand que mu']} (left) and \ref{['eq:nu egal mu']} (right) when $I=0$ and when $I$ is the delayed PD controller given by \ref{['eq:feedback control']} where $\tau$, $k_d$ and $k_p$ are given by \ref{['eq:coeffs when nu grand que mu-2']} (left) and \ref{['eq:coeffs when nu egal mu']} (right) respectively. The initial condition is taken as $y_0 = 1$ for the solution with $I=0$, and $y_0(t)= 1+sin(t)$ for the solution with the delayed PD controller.

Theorems & Definitions (46)

• Lemma 2.1
• Lemma 2.2: Descartes' rule of signs
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 3.1: Dominance of a real root