  This paper introduces new model parameterizations for learning dynamical
systems from data via the Koopman operator, and studies their properties.
Whereas most existing works on Koopman learning do not take into account the
stability or stabilizability of the model -- two fundamental pieces of prior
knowledge about a given system to be identified -- in this paper, we propose
new classes of Koopman models that have built-in guarantees of these
properties. These models are guaranteed to be stable or stabilizable via a
novel {\em direct parameterization approach} that leads to {\em unconstrained}
optimization problems with respect to their parameter sets. To explore the
representational flexibility of these model sets, we establish novel
theoretical connections between the stability of discrete-time Koopman
embedding and contraction-based forms of nonlinear stability and
stabilizability. The proposed approach is illustrated in applications to stable
nonlinear system identification and imitation learning via stabilizable models.
Simulation results empirically show that the learning approaches based on the
proposed models outperform prior methods lacking stability guarantees.
