Periodic optical cruise of an atmospheric vehicle

Since the steady-state cruise path of an idealized point mass model of an atmospheric vehicle operating in the hypersonic flight regime is dynamically not fuel minimizing, closed periodic paths are numerically determined. By application of second-order conditions for local optimality, a periodic extremal path for a flat Earth is shown to be locally minimizing and produces an improvement in fuel usage of 4.2% over the steady-state cruise path. Application of these second variational conditions to extremal paths for the spherical Earth failed. Nevertheless, these paths produce improved fuel performance over the associated steady-state cruise path. UEL efficient cruise trajectories for aircraft have been a subject of continuous theoretical interest and are becoming one of practical interest as well. The analysis of aircraft trajectories for fuel minimization was first performed in a reduced state space.1"3 By neglecting the altitude and flight path angle dynamics, the equations of motion of a point mass representation of the aircraft motion reduces to the energy-state approximation where energy and fuel mass are the state variables, thrust and velocity (or altitude) are considered the control variables, and range is the independent variable. The first-order dynamics or rates are represented by the rate of change in the energy and fuel with respect to a change in range. The hodograph3 is formed by determining the boundary of reachable rates for admisible values of the control variables at a given energy value (the rates are independent of the fuel). The steady-state cruise fuel performance is given by the value of the fuel mass rate where the hodograph crosses the zero energy rate axis. If the hodograph is not convex so that a straight line tangent to two points on the hodograph (called the convex hull3) crosses the zero energy rate axis at a smaller value of fuel mass rate than does the hodograph, then the control variables at the points of tangency are used to form a chattering control sequence which theoretically will improve fuel performance over the steadystate cruise path. This chattering sequence, first discussed in Ref. 1, is an unrealizable infinite frequency control sequence between two thrust levels and two altitude and velocity points on the energy manifold where the aircraft is aerodynamic or propulsion efficient. This chattering cruise is also referred to as the relaxed steady state cruise in Ref. 2. Since velocity and altitude chattering is unrealizable, altitude has been added as a state variable in Ref. 4 and thrust and flight path angle are considered the control variables. In Ref. 4 the small angle approximation applied to the flight path angle results in flight path angle and thrust appearing linearly in the aircraft dynamic model and cost function. These control variables which lie interior to their admissible control sets along the extremizing steady-state cruise path form what is called a doubly singular arc in the calculus of variations.5'6 By applications of the matrix generalized Legendre-Clebsch condition due to Robbins, 7 it is demonstrated in Ref. 8 that the steady-state cruise path is not minimizing..