Envelopes, Conjugate Points, and Optimal Bang-Bang Extremals

  • H. J. Sussmann
Part of the Mathematics and Its Applications book series (MAIA, volume 29)


The theory of envelopes and conjugate points constitutes an important chapter of the classical Calculus of Variations. In Optimal Control theory, it is often desirable to get better information about optimal trajectories than that provided by the Pontryagin Maximum Principle. To do this, it is useful to have necessary conditions for optimality that exclude, for instance, bang-bang trajectories that have too many switchings, even if those trajectories satisfy the Maximum Principle. It turns out that, in a number of cases, this can be done by means of a suitable generalization of envelope theory. The purpose of this note is to outline one such generalization and to illustrate its use by proving theorems on the structure of optimal bang-bang trajectories for certain problems in two and three dimensions.


Optimal Trajectory Integral Curve Conjugate Point Admissible Pair Pontryagin Maximum Principle 
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Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • H. J. Sussmann
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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