  This paper presents a method to design a min-norm Control Lyapunov Function
(CLF)-based stabilizing controller for a control-affine system with uncertain
dynamics using Gaussian Process (GP) regression. In order to estimate both
state and input-dependent model uncertainty, we propose a novel compound kernel
that captures the control-affine nature of the problem. Furthermore, by the use
of GP Upper Confidence Bound analysis, we provide probabilistic bounds of the
regression error, leading to the formulation of a CLF-based stability chance
constraint which can be incorporated in a min-norm optimization problem. We
show that this resulting optimization problem is convex, and we call it
Gaussian Process-based Control Lyapunov Function Second-Order Cone Program
(GP-CLF-SOCP). The data-collection process and the training of the GP
regression model are carried out in an episodic learning fashion. We validate
the proposed algorithm and controller in numerical simulations of an inverted
pendulum and a kinematic bicycle model, resulting in stable trajectories which
are very similar to the ones obtained if we actually knew the true plant
dynamics.
