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In this article, a recurrent neural network (RNN) is accelerated and applied to visual servo control of a physically constrained robotic flexible endoscope. The robotic endoscope consists of a patient side manipulator of the da Vinci Research Kit platform and a flexible endoscope working as an end-effector. To automate the robotic endoscope, kinematic modeling for visual servoing is conducted, leading to a quadratic programming (QP) control framework incorporating kinematic and physical constraints of the robotic endoscope. To solve the QP problem and realize the vision-based control, an RNN accelerated to finite-time convergence by a sign-bipower activation function (SBPAF) is proposed. The finite-time convergence of the RNN is theoretically proved in the sense of Lyapunov, showing that the SBPAF activated RNN exhibits a faster convergence speed as compared with its predecessor. To validate the efficacy of the RNN model and the control framework, simulations are performed using a simulated flexible endoscope in the robot operating system. Physical experiment is then further performed to verify the feasibility of the RNN model and the control framework. Both simulation and experimental results demonstrate that the proposed RNN solution is effective to achieve visual servoing and handle physical limits of the robotic endoscope simultaneously. [ABSTRACT FROM AUTHOR]