Periodic Optimal Hypersonic Scramjet Cruise

Fuel‐optimal periodic cruise paths are determined numerically for a hypersonic vehicle with more realistic aerodynamic and scramjet engine models than used in previous work. A modification of the shooting method, called the minimizing‐boundary‐condition method, is used to increase the region of convergence of the solution to the two‐point boundary‐value problem required to satisfy the first‐order necessary conditions. The resulting extremal trajectories are shown to be locally minimizing by a second‐order sufficiency condition. Little fuel improvement is obtained over the full‐throttle steady‐state cruise if the model with an engine‐off‐drag penalty is used. However, altitude changes of over 25 per cent from its lowest value on the periodic path are obtained. If the model without engine‐off‐drag penalty is used, the fuel improvement for the optimal periodic cruise is 5 per cent over the full‐throttle steady‐state cruise.