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激活函数的递归神经网络(RNN)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(QCs)相结合而推导得出。我们利用标量ReLU的已知性质，为重复ReLU建立了一类通用QCs。本文的稳定性与性能条件采用这些QCs以及ReLU RNN的"提升"表示形式。研究表明，标量ReLU满足的正齐次性质并未扩展重复ReLU的QCs类别。我们通过示例验证稳定性/性能条件，并分析提升时域的影响。