A new supersonic unmanned aerial vehicle (UAV) with a powerless combustion chamber was proposed to solve the problem that the design was complicated and the fuel consumption was too high in order to improve the economic performance of small supersonic UAV. A trajectory optimization of the new UAV based on hp adaptive pseudo-spectral method was proposed to optimize the trajectory with the target of the minimum fuel consumption and minimum time in a flight section which shall make the UAV reach supersonic cruise flight by introducing the Mach subduction maneuver. The control variable and state variable were discretized, and the multi-constraint optimal control problem was transformed into a nonlinear programming problem by combining with the physical process of the UAV flight. The proposed optimal trajectory was compared with the conventional flight schemes, and the effects of critical design parameters on the optimal trajectory were analyzed. The simulation results show that the proposed method can effectively plot a feasible path which shall make the UAV climb from high subsonic to supersonic flight. The obtained minimum fuel consumption and minimum climb time are better than the traditional flight scheme. The minimum fuel consumption was reduced by about 11%, and the minimum climb time was reduced by about 46%.