Quantum-Inspired Impulsive Continuous Hopfield Networks for Robust and Resilient Control

Abstract

This paper introduces the Quantum-Inspired Impulsive Continuous Hopfield Network (Q-ICHN), a novel hybrid control framework designed to handle non-smooth, high-energy perturbations in nonlinear dynamical systems. Standard Continuous Hopfield Networks (CHNs) rely on sigmoidal activation functions that are prone to gradient saturation, which leads to an insufficient corrective response when the system undergoes large deviations from equilibrium. To overcome this shortcoming, the proposed Q-ICHN adopts a wave-packet-based activation function grounded in the stationary Schrödinger equation, yielding a non-monotonic and oscillatory activation profile that sustains effective compensatory dynamics across a broad range of states. Furthermore, the proposed framework incorporates Madelung’s quantum potential into the control architecture, thereby enabling a fundamental reshaping of the system’s energy landscape. Specifically, this induces a tunneling-like mechanism that allows the system to circumvent local minima and rapidly recover from impulsive disturbances, manifested as a sharpened attractor structure in the phase-space domain. Together, these properties yield enhanced convergence behavior and improved robustness over traditional neural control approaches. To rigorously assess its merits, the performance of the Q-ICHN is evaluated through a large-scale benchmark involving 20 established control methods, including Sliding Mode Control (SMC), Model Predictive Control (MPC), and Backstepping. The experimental results obtained across 20 heterogeneous scenarios demonstrate that the proposed model achieves a 48% reduction in Mean Squared Error (MSE) relative to the classical ICHN. In addition, the Q-ICHN exhibits improved smoothness, reflected in a 30% reduction in jerk with respect to high-gain robust controllers, and enhanced reliability, validated by superior spectral purity and a 34% reduction in integrated variance under stochastic perturbations. Collectively, these results underscore the potential of quantum-inspired activation mechanisms to favorably balance control responsiveness and harmonic stability, providing a robust framework for handling both continuous dynamics and impulsive effects.