  Recent advances in learning techniques have enabled the modelling of unknown
dynamical systems directly from data. However, in many contexts, these
learning-based methods are short of safety guarantee and strict stability
verification. To address this issue, this paper first approximates the
partially unknown nonlinear systems by using a learned state space with
Gaussian Processes and Chebyshev interpolants. A Sum-of-Squares Programming
based approach is then proposed to synthesize a controller by searching an
optimal control Lyapunov Barrier function. In this way, we maximize the
estimated region of attraction of partially unknown nonlinear systems, while
guaranteeing both safety and stability. It is shown that the proposed method
improves the extrapolation performance, and at the same time, generates a
significantly larger estimated region of attraction.
