Control vector fields on manifolds and attainability

  • Felix Albrecht
Conference paper
Part of the Lecture Notes in Operations Research and Mathematical Economics book series (LNE, volume 11/12)


This paper contains a coordinate-free proof of the following statement: if (ξ, U) is a given control vector field on a finite-dimensional manifold of class C3, then any control u ε U steering a point of the manifold to the boundary of its set of attainability satisfies the Pontryagin Maximum Principle. Similar results for control processes in Euclidean spaces (under various restrictions) have been proved by different methods (see e.g. [2], [3], [1]).


Vector Field Integral Curve Pontryagin Maximum Principle Control Space Continuous Linear Mapping 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Albrecht, Topics in Control Theory, Lecture Notes in Mathematics, Springer-Verlag, to appear.Google Scholar
  2. 2.
    H. Halkin, On the Necessary Condition for Optimal Control of Nonlinear Systems, J. Anal. Math., 12, 1–82 (1963).MathSciNetCrossRefGoogle Scholar
  3. 3.
    E.B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley and Sons, 1967.Google Scholar
  4. 4.
    E. Roxin, A Geometric Interpretation of Pontryagin’s Maximum Principle, in Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, 1963.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1969

Authors and Affiliations

  • Felix Albrecht

There are no affiliations available

Personalised recommendations