Three-Dimensional Air Combat: Numerical Solution of Complex Differential Games

  • R. Lachner
  • M. H. Breitner
  • H. J. Pesch
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 3)


Complex pursuit-evasion games with complete information under state variable inequality constraints are investigated. By exploitation of Isaacs’ minimax principle, necessary conditions of first and second order are derived for the optimal trajectories. These conditions give rise to multipoint boundary-value problems, which yield open-loop representations of the optimal strategies along the optimal trajectories. The multipoint boundary-value problems are accurately solved by the multiple shooting method. The computed open-loop representations can thereafter be used to synthesize the optimal strategies globally.

As an illustrative example, the evasion of an aircraft from a pursuing missile is investigated. The flight of the aircraft is restricted by various control variable inequality contraints and by a state variable inequality constraint for the dynamic pressure. The optimal trajectories exhibit boundary arcs with regular and singular constrained controls. The influence of various singular surfaces in the state space including a low-dimensional universal surface is discussed.


Differential games pursuit-evasion games singular surfaces multipoint boundary-value problems multiple shooting method 


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Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • R. Lachner
    • 1
  • M. H. Breitner
    • 1
  • H. J. Pesch
    • 1
  1. 1.Institute of MathematicsClausthal University of TechnologyClausthal-ZellerfeldGermany

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