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

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.