  The goal of this paper is to provide sufficient conditions for guaranteeing
the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability
({\delta}ISS) of Gated Recurrent Units (GRUs) neural networks. These
conditions, devised for both single-layer and multi-layer architectures,
consist of nonlinear inequalities on network's weights. They can be employed to
check the stability of trained networks, or can be enforced as constraints
during the training procedure of a GRU. The resulting training procedure is
tested on a Quadruple Tank nonlinear benchmark system, showing satisfactory
modeling performances.
