Flexible robotic manipulators are increasingly utilized in the aerospace industry, leading to continuous advancements in their control methodologies. While PID controllers have traditionally been predominant in this field, there is a shift toward exploring and implementing new control strategies. This paper employs the finite element method to develop the dynamics of a single-link flexible robotic manipulator, calculating the generalized inertia and stiffness matrices for a specified manipulator length. It also formulates both the linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controllers. The system is linearized to create a state-space representation, and the dynamic responses, including tip deflections and velocities, are analyzed under a bang-bang torque scenario. The study examines LQR and LQG controllers with both partial and full state feedback. The tip deflections of the manipulator are compared to those produced by a PID controller. The performance of all controllers is evaluated based on the time required for tip stabilization, with conclusions drawn from the collected data.