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

  • Kurt ChudejEmail author
  • 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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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  • Kurt Chudej
    • 1
    Email author
  • 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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