A new approach is presented to solve periodic optimal control problems. As an application of the approach, fueloptimal periodic control problems for a hypersonic vehicle are solved. The model of the vehicle was constructed by using numerical data and gures from available space plane literature. In particular, heating-rate and load-factor constraints are considered to make this model more realistic than other previous models. These constraints increase the dif culty in obtaining a numerical solution and also increase the sensitivity to the initial guess for convergence. By assuming the shape of an altitude pro le as a sinusoidal function of range and by using a bang-bang thrust control, a suboptimal solution is obtained for the vehicle without any constraints. This suboptimal solution serves as a very good initial guess for the optimal solution generated by the minimizing-boundary-conditionmethod. The optimal solution shows a fuel saving of 8.12% over the steady-state cruise, with a maximum heating rate of 1202.4 W/cm 2 , and with a maximum load factor of 8.27. Constraints for a heating rate and a load factor are then added to the problem. With a maximum heating rate of 400 W/cm 2 , the fuel saving reduces to 2.45%. With a load factor of 7, the fuel saving does not change much from the nonconstrained solution. An optimal periodic-cruise solution with maximum heating rate of 1158.0 W/cm 2 and simultaneously with maximum load factor of 7 is also determined with a fuel saving of 8.09%.
NomenclatureA e = inlet area, m 2 A w = wing area, m 2 a = speed of sound at sea level or at the standard temperature, m/s b j = lapse rate between j th and ( j C 1)th junction points, where j is 0, 2, 4, and 6 C = load factor constraint function C D = drag coef cient C D0 = zero-lift drag coef cient C L = lift coef cient C L 0 = zero angle-of-attack lift coef cient C L ® = lift-curve slope C T max = maximum thrust coef cient D = drag, N f = system dynamics vector g = gravity acceleration at sea level, m/s 2 h = altitude, km h a = amplitude corresponding to frequency ! of a speci ed sinusoidal altitude curve h b = amplitude correspondingto frequency 2! of a speci ed sinusoidal altitude curve h c = offset of a speci ed sinusoidal altitude curve I sp = speci c impulse, s J = cost function K = induced drag parameter L = lift, N M = Mach number (de ned as a normalized velocity by a constant speed of sound at sea level) m = mass of the vehicle, kg n = load factor n max = speci ed maximum load factor P Q = heating rate, W/cm 2 P Q max = speci ed maximum heating rate, W/cm 2 q = dynamic pressure, N/m 2 R = speci c gas constant for air, J/kgK R 0 = radius of the Earth, km ¤ Part-time Associate Professor, School of Aerospace Engineering. Senior Member AIAA. † Graduate Student, School of Aerospace Engineering. r = range, km r d = coordinate of range where a speci ed throttle falls from 1 to 0 r u = coordinate of range where a speci ed throttle rises from 0 to 1 r 1 = coordinate of range of an entry point to a boundary S = heating-rate constraint...
