The main focus of this paper is on developing a nonlinear controller for a hypersonic aircraft using the Sum-Of-Squares (SOS) approach. In particular, the longitudinal dynamics of the aircraft are studied using time-scale decomposition, where the fast dynamics consist of the pitch rate dynamics and the slow dynamics include the rest of the states. For the slow dynamics, the SOS technique is applied for control design, which uses the pilot commanded altitude and velocity as inputs to derive engine throttle and a commanded pitch rate. The SOS controller is designed by following recent results on the dual problem of the Lyapunov theorem, which allows the joint search of a Lyapunov function and a nonlinear controller using semidefinite programs. Then, the commanded pitch rate derived from the slow dynamics is fed into the fast dynamics and a simple inversion of the pitch rate dynamics is used to derive the elevator deflection. Simulation results are presented to evaluate the stability and performance of the controller, as well as the robustness with respect to the parameter uncertainties in aerodynamic coefficients. The results are also compared with those generated by a controller designed using Nonlinear Dynamics Inversion (NDI) for both time scales.