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Recent advances in learning techniques have enabled the modelling of unknowndynamical systems directly from data. However, in many contexts, theselearning-based methods are short of safety guarantee and strict stabilityverification. To address this issue, this paper first approximates thepartially unknown nonlinear systems by using a learned state space withGaussian Processes and Chebyshev interpolants. A Sum-of-Squares Programmingbased approach is then proposed to synthesize a controller by searching anoptimal control Lyapunov Barrier function. In this way, we maximize theestimated region of attraction of partially unknown nonlinear systems, whileguaranteeing both safety and stability. It is shown that the proposed methodimproves the extrapolation performance, and at the same time, generates asignificantly larger estimated region of attraction.

Sum-of-Squares Program and Safe Learning On Maximizing the Region of Attraction of Partially Unknown Systems