  This paper analyzes the stability-related properties of Long Short-Term
Memory (LSTM) networks and investigates their use as the model of the plant in
the design of Model Predictive Controllers (MPC). First, sufficient conditions
guaranteeing the Input-to-State stability (ISS) and Incremental Input-to-State
stability (dISS) of LSTM are derived. These properties are then exploited to
design an observer with guaranteed convergence of the state estimate to the
true one. Such observer is then embedded in a MPC scheme solving the tracking
problem. The resulting closed-loop scheme is proved to be asymptotically
stable. The training algorithm and control scheme are tested numerically on the
simulator of a pH reactor, and the reported results confirm the effectiveness
of the proposed approach.
