Optimal Trajectory for a Hypersonic Cruise Missile with a Nonconvex Control Set

This paper focuses on the range maximization problem for hypersonic cruise missiles (HCMs). The controls of the HCM under investigation are the angle of attack and the fuel flow of the scramjet engine. When fuel is burned, the angle of attack is strictly restricted under the engine's limitations. Therefore, a novel control relation is introduced, allowing for a broader range of admissible angle-of-attack values during non-fuel-burning periods. Our analysis primarily examines two methods. The first is based on a vanishing constraint formulation, which implies that the control set is nonconvex and the linear independence constraint qualification is invalid. The second method employs an extended variable time transformation (VTT) approach to reformulate the problem. The advantage of the VTT formulation in terms of convergence to an optimal solution is demonstrated using a simplified problem. Finally, we present a comparison of both formulations for the range maximization problem of the HCM, highlighting the merits of the VTT method.