Abstract
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]).
The author was partially supported by NSF Grants GP-6247 and GP-6325 and by Grant AF-AFOSR-359-66.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Albrecht, Topics in Control Theory, Lecture Notes in Mathematics, Springer-Verlag, to appear.
H. Halkin, On the Necessary Condition for Optimal Control of Nonlinear Systems, J. Anal. Math., 12, 1–82 (1963).
E.B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley and Sons, 1967.
E. Roxin, A Geometric Interpretation of Pontryagin’s Maximum Principle, in Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, 1963.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1969 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Albrecht, F. (1969). Control vector fields on manifolds and attainability. In: Kuhn, H.W., Szegö, G.P. (eds) Mathematical Systems Theory and Economics I / II. Lecture Notes in Operations Research and Mathematical Economics, vol 11/12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46196-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-46196-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-04635-6
Online ISBN: 978-3-642-46196-5
eBook Packages: Springer Book Archive