Flexible endoscopic robots, with their continuum structural characteristics, demonstrate unique advantages in minimally invasive surgery. However, the inherent nonlinear deformation features of continuum structures pose significant challenges to motion control precision. To address this technical bottleneck, this paper proposes an optimal teleoperation control method for flexible endoscopic robots based on neurodynamic optimization. First, a master-slave motion mapping mechanism in image space is established, coupled with a kinematic model of the flexible endoscope, to achieve accurate mapping between image feature velocities and driving velocities. Second, joint motion constraints are incorporated to formulate the robot control as a quadratic programming based optimal control problem, which is efficiently solved using a neurodynamic-based real-time solver. Experimental validation is conducted on a ureteroscopic robotic platform. Results demonstrate that the proposed method effectively suppresses manual operation errors and velocity oscillations, maintaining target tracking errors within 2.5% while significantly enhancing the accuracy and stability of instrument manipulation during lithotripsy procedures.