Fuel-optimal periodic solutions to hypersonic cruise with active engine cooling

In this brief paper, it is shown that the active cooling of convectively-heated engine and airframe components in hypersonic vehicles may be approximately modeled and the associated questions of fuel-optimality are tractable using variational methods. Assuming a fuel-consumption which is linear with respect to throttle setting, it is shown that a first-order model of the cooling cycle results in a variational Hamiltonian which is piecewise linear in throttle, which is nonsmooth at a single interior equilibrium throttle setting. This special Hamiltonian control structure is exploited in the solution of the acceleration-constrained, fuel-optimal periodic cruise problem. This work should be seen as a first step towards a more theoretical understanding of the coupling which exists between thermal-optimal and fuel-optimal hypersonic flight.