  This paper proposes a Recurrent Neural Network (RNN) controller for
lane-keeping systems, effectively handling model uncertainties and
disturbances. First, quadratic constraints cover the nonlinearities brought by
the RNN controller, and the linear fractional transformation method models the
dynamics of system uncertainties. Second, we prove the robust stability of the
lane-keeping system in the presence of uncertain vehicle speed using a linear
matrix inequality. Then, we define a reachable set for the lane-keeping system.
Finally, to confirm the safety of the lane-keeping system with tracking error
bound, we formulate semidefinite programming to approximate the outer set of
the reachable set. Numerical experiments demonstrate that this approach
confirms the stabilizing RNN controller and validates the safety with an
untrained dataset with untrained varying road curvatures.
