Flexible robotic endoscopy have been employed in minimally invasive surgeries, however the non-linear deformable characteristics of the flexible endoscope complicate its control. This paper proposes an optimal teleoperation control method for flexible robotic endoscopy using neurodynamic optimization. A master-slave motion mapping strategy is designed in image space and the kinematic model of the flexible endoscope is established based on the constant curvature assumption. It obtains the mapping relationship between image feature velocity and actuation velocity. Considering robot physical constraints, an optimal control problem based on quadratic programming (QP) is constructed. A neurodynamic optimization method is designed to solve the complex QP problem. Experiments were conducted on a ureteroscope robot system to validate the effectiveness of the method. Experimental results indicate that the proposed method can significantly reduce human control errors and vibrations, maintaining target point tracking precision within 2.5%. The feasibility of the proposed method in ureteroscopic lithotripsy was also verified.