  This paper considers a stochastic control framework, in which the residual
model uncertainty of the dynamical system is learned using a Gaussian Process
(GP). In the proposed formulation, the residual model uncertainty consists of a
nonlinear function and state-dependent noise. The proposed formulation uses a
posterior-GP to approximate the residual model uncertainty and a prior-GP to
account for state-dependent noise. The two GPs are interdependent and are thus
learned jointly using an iterative algorithm. Theoretical properties of the
iterative algorithm are established. Advantages of the proposed state-dependent
formulation include (i) faster convergence of the GP estimate to the unknown
function as the GP learns which data samples are more trustworthy and (ii) an
accurate estimate of state-dependent noise, which can, e.g., be useful for a
controller or decision-maker to determine the uncertainty of an action.
Simulation studies highlight these two advantages.
