This paper presents sufficient conditions for the stability and $\ell_2$-gain performance of recurrent neural networks (RNNs) with ReLU activation functions. These conditions are derived by combining Lyapunov/dissipativity theory with Quadratic Constraints (QCs) satisfied by repeated ReLUs. We write a general class of QCs for repeated RELUs using known properties for the scalar ReLU. Our stability and performance condition uses these QCs along with a "lifted" representation for the ReLU RNN. We show that the positive homogeneity property satisfied by a scalar ReLU does not expand the class of QCs for the repeated ReLU. We present examples to demonstrate the stability / performance condition and study the effect of the lifting horizon.

翻译：本文提出了具有ReLU激活函数的循环神经网络（RNN）的稳定性与$\ell_2$-增益性能的充分条件。这些条件通过将Lyapunov/耗散理论与重复ReLU满足的二次约束（QC）相结合推导得出。我们利用标量ReLU的已知性质，为重复ReLU构建了一类通用的二次约束。所提出的稳定性与性能条件结合了这些二次约束以及ReLU RNN的"提升"表示。研究表明，标量ReLU满足的正齐次性并不扩展重复ReLU的二次约束类别。我们通过示例验证了稳定性/性能条件，并分析了提升时间窗口的影响。