Instationary Heat-Constrained Trajectory Optimization of a Hypersonic Space Vehicle by ODE–PDE-Constrained Optimal Control

  • Kurt Chudej
  • Hans Josef Pesch
  • Markus Wächter
  • Gottfried Sachs
  • Florent Le Bras
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)


During ascent and reentry of a hypersonic space vehicle into the atmosphere of any heavenly body, the space vehicle is subjected, among others, to extreme aerothermic loads. Therefore, an efficient, sophisticated and lightweight thermal protection system is determinative for the success of the entire mission. For a deeper understanding of the conductive, convective and radiative heating effects through a thermal protection system, a mathematical model is investigated which is given by an optimal control problem subject to not only the usual dynamic equations of motion and suitable control and state variable inequality constraints but also an instationary quasi-linear heat equation with nonlinear boundary conditions. By this model, the temperature of the heat shield can be limited in certain critical regions. The resulting ODE–PDE-constrained optimal control problem is solved by a second-order semi-discretization in space of the quasi-linear parabolic partial differential equation yielding a large-scale nonlinear ODE-constrained optimal control problem with additional state constraints for the heat load. Numerical results obtained by a direct collocation method are presented, which also include those for active cooling of the engine by the liquid hydrogen fuel. The aerothermic load and the fuel loss due to engine cooling can be considerably reduced by optimization.


Optimal Control Problem Stagnation Point Nonlinear Boundary Condition Hypersonic Vehicle Active Cool 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    T. Barth, M. Ohlberger: Finite volume methods: foundation and analysis. In: E. Stein, R. de Borst, T. J. R. Hughes (Eds.): Encyclopedia of Computational Mechanics, Volume 1, Fundamentals. John Wiley and Sons, Weinheim, Germany, 439–474, 2004.Google Scholar
  2. 2.
    R. Bayer, G. Sachs: Optimal return-to-base cruise of hypersonic carrier vehicles, Z. Flugwiss. Weltraumforsch. 19 (1995) 47–54.Google Scholar
  3. 3.
    M. Bouchez, S. Beyer, G. Cahuzac: PTAH-SOCR fuel cooled composite material structure for dual mode ramjet and liquid rocket engines, Proc. of the 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Fort Lauderdale, USA, 2004. AIAA 2004-3653.Google Scholar
  4. 4.
    W. Buhl, K. Ebert, H. Herbst: Optimal ascent trajectories for advanced launch vehicles, AIAA Fourth International Aerospace Planes Conf., Orlando, Florida, 1992. AIAA-92-5008.Google Scholar
  5. 5.
    R. Bulirsch, K. Chudej: Combined optimization of trajectory and stage separation of a hypersonic two-stage space vehicle, Z. Flugwiss. Weltraumforsch. 19 (1995) 55–60.Google Scholar
  6. 6.
    K. Chudej, M. Wächter, F. Le Bras: Verringerung der thermischen Belastung eines Hyperschall-Flugsystems durch Trajektorienoptimierung, Proc. Appl. Math. Mech. 5 (2005) 803–804.CrossRefGoogle Scholar
  7. 7.
    J. Drexler: Untersuchung optimaler Aufstiegsbahnen raketengetriebener Raumtransporter-Oberstufen, PhD Thesis, Technische Universität München, Faculty of Mechanical Engineering, Munich, Germany, 1995.Google Scholar
  8. 8.
    M. Dinkelmann: Reduzierung der thermischen Belastung eines Hyperschallflugzeugs durch optimale Bahnsteuerung. PhD thesis, Technische Universität München, Faculty for Mechanical Engineering, Munich, Germany, 1997.Google Scholar
  9. 9.
    M. Dinkelmann, M. Wächter, G. Sachs: Modelling and simulation of unsteady heat transfer effects on trajectory optimization of aerospace vehicles, Math. Comput. in Simulat. 53 (2002) 389–394.CrossRefGoogle Scholar
  10. 10.
    M. Dinkelmann, M. Wächter, G. Sachs: Modelling of heat transfer and vehicle dynamics for thermal load reduction by hypersonic flight optimization, Math. and Comp. Model. Dyn. Systems 8 (2002) 237–255.MATHCrossRefGoogle Scholar
  11. 11.
    D. Glass, A. Dilley, H. Kelley: Numerical analysis of convection/transpiration cooling, Proc. 9th International Space Planes and Hypersonic Systems and Technologies Conference, Norfolk, USA, 1999. AIAA 99-4911.Google Scholar
  12. 12.
    W. W. Hager: Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik 87 (2000) 247–282.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Jänsch, A. Markl: Trajectory optimization and guidance for a Hermes-type reentry vehicle, in Proc. of the AIAA Guidance, Navigation, and Control Conference, New Orleans, USA, 1991. AIAA-91-2659.Google Scholar
  14. 14.
    C. Jänsch, K. Schnepper, K. H. Well: Trajectory optimization of a transatmospheric vehicle, Proc. American Control Conf., Boston, (1991), 2232–2237.Google Scholar
  15. 15.
    H. Kreim, B. Kugelmann, H. J. Pesch, M. Breitner: Minimizing the maximum heating of a reentering space shuttle: An optimal control problem with multiple control constraints, Optim. Contr. Appl. Met. 17 (1996) 45–69.MATHCrossRefGoogle Scholar
  16. 16.
    C. Krishnaprakas: Efficient solution of spacecraft thermal models using preconditioned conjugate gradient methods, J. Spacecraft Rockets 35(1998) 760–764.CrossRefGoogle Scholar
  17. 17.
    H. Kuczera, H. Hauck, P. Sacher: The German hypersonics technology programme-status 1993 and perspectives, Proc. of the 5th AIAA/DGLR International Aerospace Planes and Hypersonics Technologies Conference, Munich, Germany, 1993. AIAA-93-5159.Google Scholar
  18. 18.
    F. Le Bras: Trajectoires optimales en vol hypersonique tenant compte de léchauffement instationaire de l’avion. Rapport de stage d’option scientifique, Ecole Polytechnique Paris, France, and Universität Bayreuth, Germany, 2004.Google Scholar
  19. 19.
    W. S. Martinson, P. I. Barton: A differentiation index for partial differential-algebraic equations, SIAM J. Sci. Comput. 21 (1999) 2295–2315.CrossRefMathSciNetGoogle Scholar
  20. 20.
    E. Meese, H. Nørstrud: Simulation of convective heat flux and heat penetration for a spacecraft at re-entry, Aerosp. Sci. and Technol. 6 (2002) 185–194.MATHCrossRefGoogle Scholar
  21. 21.
    A. Miele: Flight Mechanics I, Theory of Flight Paths. Addison-Wesley, Reading, 1962.Google Scholar
  22. 22.
    A. Miele, W. Y. Lee, G. D. Wu: Ascent Performance Feasibility of the National Aerospace Plane, Atti della Accademia delle Scienze di Torino 131 (1997) 91–108.Google Scholar
  23. 23.
    A. Miele, S. Mancus: Optimal Ascent Trajectories and Feasibility of Next-Generation Orbital Spacecraft, J. Optimi. Theory App. 95 (1997) 467–499.MATHCrossRefGoogle Scholar
  24. 24.
    A. Miele, S Mancuso: Design Feasibility via Ascent Optimality for Next-Generation Spacecraft, Acta Astronaut. 45/11 (1999) 655–668.CrossRefGoogle Scholar
  25. 25.
    H. J. Pesch: Numerische Berechnung optimaler Steuerungen mit Hilfe der Mehrzielmethode dokumentiert am Problem der Rückführung eines Raumgleiters unter Berücksichtigung von Aufheizungsbegrenzungen, Diploma Thessis, Universität Köln, Department of Mathematics, Cologne, Germany, 1973.Google Scholar
  26. 26.
    H. J. Pesch, A. Rund, W. von Wahl, S. Wendl: On a Prototype Class of ODE-PDE State-Constrained Optimal Control Problems. Part 1: Analysis of the State-unconstrained Problems. Part 2: The State-constrained Problems, submitted. AQ4Google Scholar
  27. 27.
    V. Rausch, C.McClinton: NASA’s Hyper-X program, Proc. of the 51th International Astronautical Congress, IAF-00-V.4.01, Rio de Janeiro, Brasil, 2000.Google Scholar
  28. 28.
    J. Ring: Flight trajectory control for thermal abatement of hypersonic vehicles, Proc. of the 2nd AIAA International Aerospace Planes Conference, Orlando, USA, 1990.Google Scholar
  29. 29.
    G. Sachs, M. Dinkelmann: Reduction of coolant fuel losses in hypersonic flight by optimal trajectory control, J. Guid. Control Dynam. 19 (1996) 1278–1284.MATHCrossRefGoogle Scholar
  30. 30.
    G. Sachs, W. Schoder: Optimal separation of lifting vehicles in hypersonic flight, AIAA Guidance, Navigation, and Control Conference, New Orleans, LA, Aug. 12–14, 1991, Technical Papers, Vol. 1 (A91-49578 21-08). Washington, DC, American Institute of Aeronautics and Astronautics, 1991, p. 529–536.Google Scholar
  31. 31.
    O. v. Stryk: Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen. Fortschritt-Berichte VDI, Reihe 8: Meß-, Steuerungs- und Regeltechnik, No. 441, VDI Verlag, Düsseldorf, Germany, 1995.MATHGoogle Scholar
  32. 32.
    O. v. Stryk: User’s Guide for DIRCOL. A direct Collocation method for the numerical solution of optimal control problems. Version 1.2. Lehrstuhl für Höhere Mathematik und Numerische Mathematik, Technische Universität München, Munich, Germany, 1995.Google Scholar
  33. 33.
    F. Tröltzsch: Optimalsteuerung bei partiellen Differentialgleichungen. Vieweg, Wiesbaden, Germany, 2005.Google Scholar
  34. 34.
    M. Wächter: Optimalflugbahnen im Hyperschall unter Berücksichtigung der instationären Aufheizung. PhD Thesis, Technische Universität München, Faculty of Mechanical Engineering, Munich, Germany, 2004.Google Scholar
  35. 35.
    M. Wächter, G. Sachs: Unsteady heat load reduction for a hypersonic vehicle with a multipoint approach. In: Optimal Control.Workshop at the University of Greifswald, 1.–3.10.2002, Report of the Collaborative Research Center 255 of the Deutsche Forschungsgemeinschaft (German Research Foundation), Technische Universität München, Munich, Germany, 2002, pp. 15–26.Google Scholar
  36. 36.
    R. Windhorst, M. D. Ardema, J. V. Bowles: Minimum heating reentry trajectories for advanced hypersonic launch vehicles, Proc. of the AIAA Guidance, Navigation, and Control Conference, New Orleans, USA, 1997. AIAA-97-3535.Google Scholar
  37. 37.
    N. X. Vinh, A. Busemann, R. D. Culp: Hypersonic and Planetary Entry Flight Mechanics. The University of Michigan Press, Ann Arbor, 1980.Google Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  • Kurt Chudej
    • 1
  • Hans Josef Pesch
    • 1
  • Markus Wächter
    • 2
  • Gottfried Sachs
    • 3
  • Florent Le Bras
    • 4
  1. 1.Lehrstuhl für IngenieurmathematikUniversität BayreuthBayreuthGermany
  2. 2.German Institute of Science and TechnologySingaporeSingapore
  3. 3.Lehrstuhl für Flugmechanik und FlugregelungTechnische Universität MünchenMünchenGermany
  4. 4.Laboratoire de Recherches Balistiques et Aérodynamiques, Délégation Générale pour l’Armementformerly: École Polytechnique ParisVernonFrance

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