Morse Theory and Optimal Control Problems
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.
KeywordsOptimal Control Problem Homotopic Type Tangent Cone Morse Theory Morse Function
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- 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.Google Scholar
- 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)Google Scholar
- 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.Google Scholar
- Goresky, M. and MacPherson, R. Stratified Morse Theory. Springer (1988), New York, etc.Google Scholar
- Milnor, J. Morse Theory. Princeton Univ. Press (1963), Princeton, New Jersey, USA.Google Scholar
- Palais, R. and Smale, S. A generalized Morse theory. Bull. Amer. Math. Soc. Vol. 79. (1964) pp. 165–171.Google Scholar
- Palais, R. Morse Theory on Hilbert manifold. Topology. Vol. 2 (1963) pp. 165–171.Google Scholar
- 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.)Google Scholar
- Vakhrameev, S.A. Palais-Smale theory for manifolds with corners. I. The case of finite co-dimension. Uspeki mat. nauk. to appear.Google Scholar
- Vakhrameev, S.A. Hilbert manifolds with corners of finite co-dimension and optimal control theory. (in preparation).Google Scholar