  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.
