  Contraction theory formulates the analysis of nonlinear systems in terms of
Jacobian matrices. Although this provides the potential to develop a linear
matrix inequality (LMI) framework for nonlinear control design, conditions are
imposed not on controllers but on their partial derivatives, which makes
control design challenging. In this paper, we illustrate this so-called
integrability problem can be solved by a non-standard use of Gaussian process
regression (GPR) for parameterizing controllers and then establish an LMI
framework of contraction-based control design for nonlinear discrete-time
systems, as an easy-to-implement tool. Later on, we consider the case where the
drift vector fields are unknown and employ GPR for functional fitting as its
standard use. GPR describes learning errors in terms of probability, and thus
we further discuss how to incorporate stochastic learning errors into the
proposed LMI framework.
