Abstract
This is a short overview of the connections of the Lyapunov Convexity Theorem with the modern sections of analysis, geometry, and optimal control.
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References
A. A. Lyapunov, “On Completely Additive Set Functions. I,” Izv. Akad. Nauk SSSR Ser. Mat. 4, 465–478 (1940).
A. A. Lyapunov, “On Completely Additive Set Functions. II,” Izv. Akad. Nauk SSSR Ser. Mat. 10(3), 277–279 (1946).
K. I. Chuĭkina, “On Additive Vector-Functions,” Dokl. Akad. Nauk SSSR 76, 801–804 (1951).
E. V. Glivenko, “On the Ranges of Additive Vector-Functions,” Mat. Sb. 34(76), 407–416 (1954).
Yu. G. Reshentnyak and V. A. Zalgaller, “On Rectifiable Curves, Additive Vector-Functions, and Mixing of Straight Line Segments,” Vestnik Leningrad. Gos. Univ. 2, 45–65 (1954).
Z. Artstein, “Yet Another Proof of the Lyapunov Convexity Theorem,” Proc. Amer. Math. Soc. 108(1), 89–91 (1990).
E. Bolker, “A Class of Convex Bodies,” Trans. Amer. Math. Soc. 145, 323–345 (1969).
J. Elton and Th. Hill, “A Generalization of Lyapunov Convexity Theorem to Measures with Atoms,” Proc. Amer. Math. Soc. 99(2), 97–304 (1987).
P. Goodey and W. Weil, “Zonoids and Generalizations,” in Handbook of Convex Geometry, Vol. B (North-Holland, Amsterdam, 1993), pp. 1296–1326.
H. Halkin, “A Generalization of La Salle’s Bang-Bang Principle,” SIAM J. Control and Optimization 2, 199–202 (1965).
H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control (Academic Press, New York, 1969).
J. P. LaSalle, “The Time Optimal Control Problem,” in Contributions to the Theory of Non-Linear Oscillations, Vol. 5 (Princeton Univ. Press, Princeton, 1960), pp. 1–24.
N. Levinson, “Minimax, Liapunov, and ‘Bang-Bang,” J. Differential Equations 2, 218–241 (1966).
J. Lindenstrauss, “A Short Proof of Liapounoff’s Convexity Theorem,” J. Math. Mech. 15, 971–972 (1966).
L. W. Neustadt, “The Existence of Optimal Control in the Absence of Convexity,” J. Math. Anal. Appl. 7, 110–117 (1963).
R. J. Nunke and L. J. Savage, “On the Set of Values of a Nonatomic, Finitely Additive, Finite Measure,” Proc. Amer. Math. Soc. 3(2), 217–218 (1952).
C. Olech, “Extremal Solutions of a Control System,” J. Differential Equations 2, 74–101 (1966).
Handbook of Measure Theory, Vols. 1 and 2, Ed. by E. Pap (North Holland, Amsterdam, 2002).
A. Pietsch, History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007).
D. Ross, “An Elementary Proof of Lyapunov’s Theorem,” Amer. Math. Monthly 112(7), 651–653 (2005).
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On the centenary of the birth of A. A. Lyapunov
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Kutateladze, S.S. Lyapunov’s Convexity Theorem, zonoids, and bang-bang. J. Appl. Ind. Math. 5, 163–164 (2011). https://doi.org/10.1134/S1990478911020025
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DOI: https://doi.org/10.1134/S1990478911020025