Previous article Next article A New Proof of the Lyapunov Convexity TheoremFabio TardellaFabio Tardellahttps://doi.org/10.1137/0328026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA new proof of the Lyapunov Theorem is given, based on the Shapley–Folkman Theorem, that does not require any tools of functional analysis.[1] Robert M. Anderson, An elementary core equivalence theorem, Econometrica, 46 (1978), 1483–1487 80d:90008 0394.90014 CrossrefISIGoogle Scholar[2] Gunnar Aronsson, Finite bang-bang controllability for certain non-linear systems, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 137–144 58:9481 0356.49007 CrossrefISIGoogle Scholar[3] David Blackwell, The range of certain vector integrals, Proc. Amer. Math. Soc., 2 (1951), 390–395 12,810d 0044.27702 CrossrefISIGoogle Scholar[4] C. Castaing and , M. Valadier, Convex analysis and measurable multifunctions, Springer-Verlag, Berlin, 1977vii+278, Lecture Notes in Mathematics 580 57:7169 0346.46038 CrossrefGoogle Scholar[5] Lamberto Cesari, Convexity of the range of certain integrals, SIAM J. Control, 13 (1975), 666–676 52:1494 0295.28016 LinkISIGoogle Scholar[6] N. Dinculeanu, Vector measures, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Oxford, 1967x+432 34:6011b CrossrefGoogle Scholar[7] Ivar Ekeland and , Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam, 1976ix+402 57:3931b 0322.90046 Google Scholar[8] Hubert Halkin, Liapounov's theorem on the range of a vector measure and Pontryagin's maximum principle, Arch. Rational Mech. Anal., 10 (1962), 296–304 29:2125 CrossrefISIGoogle Scholar[9] Hubert Halkin, On the necessary condition for optimal control of non-linear systems, J. Analyse Math., 12 (1964), 1–82 30:1883 0128.10103 CrossrefGoogle Scholar[10] Hubert Halkin, On a generalization of a theorem of Lyapunov, J. Math. Anal. Appl., 10 (1965), 325–329 10.1016/0022-247X(65)90127-7 30:4166 0133.07801 CrossrefISIGoogle Scholar[11] Hubert Halkin, Some further generalizations of a theorem of Lyapounov, Arch. Rational Mech. Anal., 17 (1964), 272–277 10.1007/BF00282290 30:4625 0126.28301 CrossrefISIGoogle Scholar[12] Paul R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc., 54 (1948), 416–421 9,574h 0033.05201 CrossrefISIGoogle Scholar[13] Joram Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech., 15 (1966), 971–972 34:7754 0152.24403 ISIGoogle Scholar[14] A. Lyapunov, Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 4 (1940), 465–478 2,315e 0024.38504 Google Scholar[15] Hans Richter, Verallgemeinerung eines in der Statistik benötigten Satzes der Masstheorie, Math. Ann., 150 (1963), 85–90 26:3851 0109.27801 CrossrefISIGoogle Scholar[16] Benyamin Shitovitz, Oligopoly in markets with a continuum of traders, Econometrica, 41 (1973), 467–501 55:14148 0281.90015 CrossrefISIGoogle Scholar[17] R. M. Starr, Quasi-equilibria in markets with non-convex preferences, Econometrica, 17 (1969), 25–38 0172.44602 CrossrefISIGoogle Scholar[18] H. J. Sussman, The “bang-bang” problem for certain control systems in ${\rm GL}(n,\,R)$, SIAM J. Control, 10 (1972), 470–476 46:9834 0242.49041 LinkISIGoogle Scholar[19] Gilead Tadmor, Functional-differential equations of retarded and neutral type: analytic solutions and piecewise continuous controls, J. Differential Equations, 51 (1984), 151–181 85f:34143 0547.34058 CrossrefISIGoogle Scholar[20] James A. Yorke, Another proof of the Liapunov convexity theorem, SIAM J. Control, 9 (1971), 351–353 46:1416 0216.55601 LinkISIGoogle ScholarKeywordsvector-valued measuresLyapunov Convexity Theoremconvex analysisintegrals of multi-functions Previous article Next article FiguresRelatedReferencesCited ByDetails Quantifiable & comparable evaluations of cyber defensive capabilities: A survey & novel, unified approachComputers & Security, Vol. 96 | 1 Sep 2020 Cross Ref The subdifferential of measurable composite max integrands and smoothing approximationMathematical Programming, Vol. 181, No. 2 | 24 October 2019 Cross Ref Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctionsApplied Mathematics and Nonlinear Sciences, Vol. 4, No. 2 | 23 August 2019 Cross Ref The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatmentPositivity, Vol. 17, No. 3 | 20 March 2012 Cross Ref A Unifed Approach to the Purification of Nash Equilibria in Large GamesSSRN Electronic Journal | 1 Jan 2006 Cross Ref An Elementary Proof of Lyapunov's TheoremThe American Mathematical Monthly, Vol. 112, No. 7 | 1 February 2018 Cross Ref Volume 28, Issue 2| 1990SIAM Journal on Control and Optimization253-501 History Submitted:01 September 1988Accepted:25 April 1989Published online:14 July 2006 InformationCopyright © 1990 Society for Industrial and Applied MathematicsKeywordsvector-valued measuresLyapunov Convexity Theoremconvex analysisintegrals of multi-functionsMSC codes28B0546010PDF Download Article & Publication DataArticle DOI:10.1137/0328026Article page range:pp. 478-481ISSN (print):0363-0129ISSN (online):1095-7138Publisher:Society for Industrial and Applied Mathematics