This paper proposes a control design framework with guarantees for systems, which contains learning-based control elements. The framework is based on a supervisory control structure, which contains a supervisor, a robust Linear Parameter Varying (LPV) controller and the learning-based control elements. This paper presents the design of a lateral path following control for driver assistance systems, which is aided with a learning-based agent. In the paper the formulation of the supervisor, the design method of the robust LPV controller are provided, while the learning-based agent via an imitation learning process is considered to be given. The effectiveness of the method on driver-in-the-loop simulation scenario is demonstrated. It is shown that the proposed control system is able to provide guarantee on the limitation of the path following error, and to guarantee transition between the driver steering actuation and the automated control intervention.