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We propose an iterative method to safely learn the unmodeled dynamics of anonlinear system using Bayesian Gaussian process (GP) models with polynomialkernel functions. The method maintains safety by ensuring that the system statestays within the region of attraction (ROA) of a stabilizing control policywhile collecting data. A quadratic programming based exploration control policyis computed to keep the exploration trajectory inside an inner-approximation ofthe ROA and to maximize the information gained from the trajectory. A prior GPmodel, which incorporates prior information about the unknown dynamics, is usedto construct an initial stabilizing policy. As the GP model is updated withdata, it is used to synthesize a new policy and a larger ROA, which increasesthe range of safe exploration. The use of polynomial kernels allows us tocompute ROA inner-approximations and stabilizing control laws for the modelusing sum-of-squares programming. We also provide a probabilistic guarantee ofsafety which ensures that the policy computed using the learned modelstabilizes the true dynamics with high confidence.

Bayesian Safe Learning and Control with Sum-of-Squares Analysis and Polynomial Kernels