  Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems
often used to model sequence-to-sequence maps. RNNs have excellent expressive
power but lack the stability or robustness guarantees that are necessary for
many applications. In this paper, we formulate convex sets of RNNs with
stability and robustness guarantees. The guarantees are derived using
incremental quadratic constraints and can ensure global exponential stability
of all solutions, and bounds on incremental $ \ell_2 $ gain (the Lipschitz
constant of the learned sequence-to-sequence mapping). Using an implicit model
structure, we construct a parametrization of RNNs that is jointly convex in the
model parameters and stability certificate. We prove that this model structure
includes all previously-proposed convex sets of stable RNNs as special cases,
and also includes all stable linear dynamical systems. We illustrate the
utility of the proposed model class in the context of non-linear system
identification.
