References
A. A. Agrachev, D. Palaschke, and S. Scholtes, “On Morse theory for piecewise smooth functions”, J. Dyn. Contr. Syst., 3, No. 4, 449–469 (1997).
A. A. Agrachev and S. A. Vakhrameev, “Morse theory in optimal control theory and mathematical programming”, in: Int. Soviet-Poland Symp. “Math. Meth. Optimal Control and Appl.”, Minsk, May 16–19, 1989, Abstracts of Reports [in Russian], Minsk (1989), pp. 7–8.
A. A. Agrachev and S. A. Vakhrameev, “Morse theory and optimal control problems”, in: Nonlinear Synthesis, Progress in System and Control Theory, Vol. 9, Birkhauser Verlag, Boston (1991), pp. 1–11.
N. A. Bobylev and Yu. M. Burman, “Morse lemmas for integral functionals”, Dokl. Akad. Nauk SSSR, 317, No. 2, 267–270 (1991).
J. M. Bonnisseau and B. Cornet, “Fixed point theorems and Morse’s lemma for Lipschitzian functions”, J. Math. Anal. Appl., 146, 318–332 (1990).
M. L. Bougeard, “Morse theory for some lower C 2-functions in finite dimension”, Math. Progr., 41, 141–159 (1988).
M. Sh. Farber, “On smooth sections of the intersection of multivalued mappings”, Izv. Akad. Nauk AzSSR, Ser. Fiz.-Tekh. Mat. Nauk, No. 6, 23–28 (1979).
M. Goretsky and R. McPherson, Stratified Morse Theory, Springer, Berlin (1988).
J. Milnor, Morse Theory, Princeton Univ. Press (1963).
R. S. Palais, “Morse theory on Hilbert manifolds”, Topology, 2, No. 4, 299–341 (1963).
R. S. Palais, “The Morse lemma for Banach spaces”, Bull. Amer. Math. Soc., 15, 185–212 (1970).
R. S. Palais and S. Smale, “A generalized Morse theory”, Bull. Amer. Math. Soc., 70, 165–171 (1964).
I. V. Skrypnik, Nonlinear Elliptic Equations of High Order [in Russian], Naukova Dumka, Kiev (1973).
A. Tromba, “Morse lemma in Banach spaces”, Proc. Amer. Math. Soc., 34, 396–402 (1972).
A. Tromba, “The Morse lemma on arbitrary Banach spaces”, Bull. Amer. Math. Soc., 79, 85–86 (1973).
S. A. Vakhrameev, “Hilbert manifolds with corners of finite codimension and optimal control theory”, in: Progress in Science and Technology, Series on Algebra, Topology, Geometry, Vol. 28 [in Russian], VINITI, Akad. Nauk SSSR (1990), pp. 96–171.
S. A. Vakhrameev, “Palais-Smale theory for manifolds with corners, I. Finite-codimensional case”, Usp. Mat. Nauk, 45, No. 4, 141–142 (1990).
S. A. Vakhrameev, “Morse index theorem for extremals of certain optimal control problems”, Dokl. Akad. Nauk SSSR, 317, No. 1, 11–15 (1991).
S. A. Vakhrameev, “Morse theory and Lyusternik-Shnirel’man theory in geometrical control theory”, in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Latest Achievements, Vol. 39, [in Russian], VINITI, Akad. Nauk SSSR, Moscow (1991), pp. 41–117.
S. A. Vakhrameev, “Lyusternik-Shnirel’man theory for transversely convex subsets of Hilbert manifolds and its application to optimal control theory”, Dokl. Akad. Nauk SSSR, 319, No. 1, 18–21 (1991).
S. A. Vakhrameev, “Morse theory for a certain class of optimal control problems”, Dokl. Ross. Akad. Nauk, 326, No. 3, 404–408 (1992).
S. A. Vakhrameev, “Critical point theory for smooth functions on Hilbert manifolds with signularities and its application to some optimal control problems”, J. Sov. Math., 67, No. 1, 2713–2811 (1993).
S. A. Vakhrameev, “On the transversal convexity of reachable sets of a certain class of smooth nonlinear systems”, Dokl. Ross. Akad. Nauk, 338, No. 1, 1–3 (1994).
S. A. Vakhrameev, “Geometrical and topological methods in optimal control theory”, J. Math. Sci., 76 No. 5, 2755–2719 (1995).
F. W. Warner, Foundation of Differential Manifolds and Lie Groups, Springer, New York (1983).
Additional information
This work was supported by the Russian Foundation for Basic Research, project No. 96-01-00860 and INTAS, project, No. 93-893.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 59, Dinamicheskie Sistemy-8, 1998.
Rights and permissions
About this article
Cite this article
Vakhrameev, S.A. Morse lemmas for smooth functions on manifolds with corners. J Math Sci 100, 2428–2445 (2000). https://doi.org/10.1007/s10958-000-0003-7
Issue Date:
DOI: https://doi.org/10.1007/s10958-000-0003-7