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References
N. I. Akhiezer,Calculus of Variations [in Russian], Vysshaya Shkola, Khar'kov (1981).
A. V. Arutyunov, “Convex properties of the Legendre transform,”Mat. Zametki,28, No. 2, 255–263.
V. G. Boltyanskii,Optimal Control of Discrete Systems [in Russian], Nauka, Moscow (1973).
Yu. D. Burago and V. A. Zalgaller, “Sufficient conditions for convexity,”Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,45, 3–52 (1974).
F. H. Clarke,Optimization and Nonsmooth Analysis, Wiley, New York (1973).
R. V. Gamkrelidze,Foundations of Optimal Control Theory [in Russian], Tbilisi Univ., Tbilisi (1975).
R. V. Gamkrelidze, A. A. Agrachev and S. A. Vakhrameev, “Ordinary differential equations on vector bundles and the chronological calculus,” in:Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Latest Achievements [in Russian], Vol. 35, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 3–107.
A. D. Ioffe and V. M. Tikhomirov,Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).
K. Leichtweiss,Convex Sets [Russian translation], Nauka, Moscow (1985).
Yung-Chen Lu,singularity Theory and an Introduction to Catastrophe Theory, Springer, New York (1976).
T. Motzkin, “Sur quelque proprietes characteristique des ensembles bornes non convexes,”Rend. Acad. Lincei,21, 773–779 (1935).
L. S. Pontryagin,Foundations of Combinatorial Topology [in Russian], Nauka, Moscow (1986).
L. S. Pontryagin,Selected Mathematical Works, Vol. 2 [in Russian], Nauka, Moscow (1988).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1976).
S. A. Vakhrameev, “Smooth control systems of constant rank and linearizable systems”, in:Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Latest Achievements [in Russian], Vol. 35, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 135–177.
S. A. Vakhrameev, “Morse theory and the Lyusternik Shnirel'man theory in geometrical control theory”, in:Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Latest Achievements [in Russian], Vol. 39, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1991), pp. 41–117.
S. A. Vakrameev, “On transversal convexity of reachable sets of a certain class of smooth control systems of a constant rank”,Dokl. Ross. Akad. Nauk,338, No. 1, 7–9 (1994).
S. A. Vakhrameev, “A theorem on the finiteness of the number of switchings for smooth control systems”,Usp. Mat. Nauk,49, No. 4, 197–198 (1994).
S. A. Vakhrameev, “Geometrical and topological methods in optimal control theory”,J. Math. Sci.,76, No. 5, 2555–2719 (1995).
S. A. Vakhrameev, “An existence theorem for the nonlinear time-optimal control problem in the class of bang-bang controls with a finite number of switchings”,Usp. Mat. Nauk,51, No. 2, 151–152 (1996).
S. A. Vakhrameev, “A bang-bang theorem with a finite number of switchings for smooth control systems”,J. Math. Sci.,85, No. 3, 2002–2016 (1997)
S. A. Vakhrameev, “Bang-bang theorems and related topics”,Tr. Mat. Inst. Ross. Akad. Nauk,220, 49–112 (1998).
S. A. Vakhrameev, “An existence theorem for the nonlinear time-optimal control problems”, in:International Conference Dedicated to the 90th Anniversary of the Birth of L.S. Pontryagin, Abstract of Reports, Optimal Control and Supplement, MGU, Moscow (1998), pp. 198–220.
S. A. Vakhrameev, “An existence theorem for the nonlinear time-optimal control problem”,Differents. Uravn. (in press).
S. A. Vakhrameev, “Morse lemmas for smooth functions on manifolds with corners”,J. Math. Sci. (in press).
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This work was supported by the Russian Foundation for Basic Research, project No. 96-01-00860.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 60, Pontryagin Conference-1, Optimal Control, 1998.
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Vakhrameev, S.A. A note on convexity in smooth nonlinear systems. J Math Sci 100, 2470–2490 (2000). https://doi.org/10.1007/BF02673837
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DOI: https://doi.org/10.1007/BF02673837