Shimmy vibration of aircraft nose landing gear is damped and controlled using a nonlinear control which is optimal and robust against parametric uncertainties and external disturbances. Shimmy vibration is the lateral and torsional vibrations in the wheel of the aircraft that is self-excited and causes instability in high speed performances which can damage the landing gear of the aircraft, its fuselage and even may result in hurting the passengers. Thus, control and damping of this vibration are extremely important. In this paper a robust optimal controller is designed by integrating sliding mode control (SMC) together with State-Dependent Riccati Equation (SDRE) to prevent the shimmy vibrations in aircraft nose landing gear. The SDRE compensator controls the nonlinear system in an optimal way while the sliding mode controller guarantees its stability against uncertainties and disturbances. The proposed controller can effectively suppress the shimmy vibration of the landing gear with variable taxiing velocity and wheel caster length. To verify the optimal performance and robustness of the proposed controller, vibration response of the system is simulated by MATLAB software and its performance and efficiency are verified using comparative analysis. Considerable improvement can be seen in the performance of the closed loop system since not only the vibrations are effectively damped but also the consumption of energy is minimized.