This paper investigates continuous-time and discrete-time firing-rate and Hopfield recurrent neural networks (RNNs), with applications in nonlinear control design and implicit deep learning. First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space, and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.

翻译：本文研究连续时间和离散时间发放率及Hopfield递归神经网络（RNNs），及其在非线性控制设计与隐式深度学习中的应用。首先，我们提出非线性分离原理，该原理保证收缩状态反馈控制器与收缩观测器互联系统的全局指数稳定性，同时包括鲁棒性与平衡点跟踪的参数扩展。其次，我们推导出保证发放率与Hopfield神经网络架构收缩性的严格线性矩阵不等式（LMI）条件。建立了这些认证条件间的结构关系——证明具有单调非递减激活函数的连续时间模型可最大化允许权重空间，并将这些稳定性保证扩展至互联系统与图RNNs。第三，结合分离原理与LMI框架解决RNN建模对象的输出参考跟踪问题。我们为反馈控制器与观测器提供了LMI综合方法，并严格设计了低增益积分控制器以消除稳态误差。最后，推导出收缩LMIs的精确无约束代数参数化方法，用于设计高表达能力隐式神经网络，在标准图像分类基准上实现了具有竞争力的精度与参数效率。