Stability of Gaussian process state space models

Gaussian Process State Space Models (GP-SSMs) are a non-parametric model class suitable to represent nonlinear dynamics. They become increasingly popular in datadriven modeling approaches, i.e. when no first-order physics-based models are available. Although a GP-SSM produces well-behaved approximations and gains increasing popularity, the fundamental system dynamics are just sparsely researched. In this paper, we present stability results for the GP-SSM depending on selected covariance function employing a deterministic point of view as widely done in the literature. The focus is set on the squared exponential function which is one of the most used covariance functions for nonlinear regression. We start with calculations according to the equilibrium points of GP-SSM and continue with conditions for stability.