Abstract
It is well known that Morse Theory is a very flexible tool for dealing with nonlinear problems of analysis and topological problems. The main purpose of the present paper is to describe a modification of this theory which can be used for the study of optimal control problems. The necessity of such a modification is related to the fact that for these problems the inequality constraints are typical (for example, control constraints, phase constraints, etc.) The inequalities destroy the smooth structure and hence the necessity to construct the theory for spaces with singularities. We encounter this situation in the case of optimal control problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
Agrachev, A.A. and Vakhrameev, S.A. Nonlinear control systems of constant rank, and the bang-bang conditions for extremal controls. Soviet Math. Dokl. Vol. 30 (1984) No. 3, pp. 620–624.
Agrachev, A.A. and Vakhrameev, S.A. The linear on the control systems of constant rank, and the bang-bang conditions for extremal control. Uspeki mat. nauk. Vol. 41 (1986) No. 6, pp. 163–164 (in Russian)
Agrachev, A.A. and Vakhrameev, S.A. Morse Theory in Optimal Control and Mathematical Programming. Proc. Int. Soviet-Poland Workshop, Minsk, May 16–19, 1989. Minsk (1989) pp. 7–8.
Goresky, M. and MacPherson, R. Stratified Morse Theory. Springer (1988), New York, etc.
Milnor, J. Morse Theory. Princeton Univ. Press (1963), Princeton, New Jersey, USA.
Palais, R. and Smale, S. A generalized Morse theory. Bull. Amer. Math. Soc. Vol. 79. (1964) pp. 165–171.
Palais, R. Morse Theory on Hilbert manifold. Topology. Vol. 2 (1963) pp. 165–171.
Vakhrameev, S.A. The smooth control systems of constant rank and the linearized systems. Itogi Nauki: Sov. Probi. Mat. Nov. dost., Vol. 35 (1989) VINITI, Moscow, pp. 135–178. (Engl. transl. to appear in J. Soviet Math.)
Vakhrameev, S.A. Palais-Smale theory for manifolds with corners. I. The case of finite co-dimension. Uspeki mat. nauk. to appear.
Vakhrameev, S.A. Hilbert manifolds with corners of finite co-dimension and optimal control theory. (in preparation).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Agrachev, A.A., Vakhrameev, S.A. (1991). Morse Theory and Optimal Control Problems. In: Byrnes, C.I., Kurzhansky, A.B. (eds) Nonlinear Synthesis. Progress in Systems and Control Theory, vol 9. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-2135-5_1
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2135-5_1
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3484-1
Online ISBN: 978-1-4757-2135-5
eBook Packages: Springer Book Archive