Major difficulties and challenges of visual servoing point stabilization focus on how to design controllers to efficiently complete the task while satisfying various constraints. This paper designs a hybrid control scheme that combines reinforcement learning and model predictive control to handle the point stabilization problem for image-based visual servoing systems of wheeled mobile robots with the visibility and input constraint. The image plane is divided into two parts: the deep-Q-network (DQN) part and the model predictive control (MPC) part. In the DQN part, a DQN controller incorporating two deep networks is designed and activated to pull the image coordinates of the feature into the MPC part. In the MPC part, a simplified-dual-neural-network-based MPC scheme (SDN-MPC) is proposed to precisely guide the robot to the desired position. The asymptotic stability of the SDN-MPC is proven for the point stabilization task. Finally, simulations and experiments are provided to verify the effectiveness, efficiency and robustness of the proposed hybrid controller by comparing with the MPC-only controller and DQN-only controller.