Existence Theorems for Optimal Solutions in Pontryagin and Lagrange Problems

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Previous article Next article Existence Theorems for Optimal Solutions in Pontryagin and Lagrange ProblemsLamberto CesariLamberto Cesarihttps://doi.org/10.1137/0303032PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[LA] Lamberto Cesari, Semicontinuità e convessità nel calcolo delle variazioni, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 389–423 MR0179649 0127.05501 Google Scholar[1B] Lamberto Cesari, Un teorema di esistenza in problemi di controlli ottimi, Ann. Scuola Norm. Sup. Pisa (3), 19 (1965), 35–78 MR0192936 0135.15803 Google Scholar[1C] Lamberto Cesari, An existence theorem in problems of optimal control, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 7–22 MR0187970 0135.15901 LinkGoogle Scholar[1D1] Lamberto Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I, Trans. Amer. Math. Soc., 124 (1966), 369–412 MR0203542 0145.12501 CrossrefISIGoogle Scholar[1D2] Lamberto Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. II. Existence theorems for weak solutions, Trans. Amer. Math. Soc., 124 (1966), 413–430 MR0203543 0145.12501 CrossrefISIGoogle Scholar[2A] A. F. Filippov, On certain questions in the theory of optimal control, Vestnik Moskov. Univ. Ser. Mat. Meh. Astronom., 2 (1959), 25–32, English transl., this Journal, 1 (1962), pp. 76–84. Google Scholar[2B] A. F. Filippov, Differential equations with multi-valued discontinuous right-hand side, Dokl. Akad. Nauk SSSR, 151 (1963), 65–68, English transl., Soviet Math. Dokl., 4 (1963), pp. 941–945. MR0151673 0141.27704 Google Scholar[3] R. V. Gamkrelidze, On sliding optimal states, Dokl. Akad. Nauk SSSR, 143 (1962), 1243–1245, English transl., Soviet Math. Dokl., 3 (1962), pp. 559–561. MR0137904 0131.32402 Google Scholar[4A] Edward James McShane, Curve-space topologies associated with variational problems, Ann. Scuola Norm. Super. Pisa (2), 9 (1940), 45–60 MR0001469 0022.23401 Google Scholar[4B] E. J. McShane, Generalized curves, Duke Math. J., 6 (1940), 513–536 10.1215/S0012-7094-40-00642-1 MR0002469 0023.39801 CrossrefGoogle Scholar[4C] E. J. McShane, Necessary conditions in generalized-curve problems of the calculus of variations, Duke Math. J., 7 (1940), 1–27 10.1215/S0012-7094-40-00701-3 MR0003478 0024.32503 CrossrefGoogle Scholar[4D] E. J. McShane, Existence theorems for Bolza problems in the calculus of variations, Duke Math. J., 7 (1940), 28–61 10.1215/S0012-7094-40-00702-5 MR0003479 0024.32504 CrossrefGoogle Scholar[4E] E. J. McShane, A metric in the space of generalized curves, Ann. of Math. (2), 52 (1950), 328–349 MR0037479 0039.11302 CrossrefISIGoogle Scholar[5] E. B. Lee and , L. Markus, Optimal control for nonlinear processes, Arch. Rational Mech. Anal., 8 (1961), 36–58 MR0128571 0099.08703 CrossrefISIGoogle Scholar[6] M. Nagumo, Über die gleichmässige Summierbarkeit und ihre Anwendung auf ein Variationsproblem, Japan J. Math., 6 (1929), 173–182 55.0156.03 CrossrefGoogle Scholar[7] Lucien W. Neustadt, The existence of optimal controls in the absence of convexity conditions, J. Math. Anal. Appl., 7 (1963), 110–117 10.1016/0022-247X(63)90081-7 MR0154768 0115.13304 CrossrefGoogle Scholar[8] L. S. Pontryagin, Optimal regulation processes, Uspehi Mat. Nauk, 14 (1959), 3–20, English transl., Automation Express, 1 (1959), pp. 15–17; 2 (1959), pp. 26–30. MR0120435 Google Scholar[9] L. S. Pontryagin, , V. G. Boltyanskii, , R. V. Gamkrelidze and , E. F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962viii+360, or Pergamon, London, 1964. MR0166037 0102.32001 Google Scholar[10] Emilio Roxin, The existence of optimal controls, Michigan Math. J., 9 (1962), 109–119 10.1307/mmj/1028998668 MR0136844 0105.07801 CrossrefGoogle Scholar[11A] Leonida Tonelli, Opere scelte. Vol. III: Calcolo delle variazioni, A cura dell'Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Ricerche, Edizioni Cremonese, Rome, 1962, 192–254, Sugli integrali del calcolo delle variazioni in forma ordinaria, Ann. Scuola Norm. Sup. Pisa (2), 3 (1934), pp. 401–450 MR0150020 0131.24103 Google Scholar[11B] L. Tonelli, Fondamenti di Calcolo delle Variazioni, Zanichelli, Bologna, 1921-1923 Google Scholar[11C] Leonida Tonelli, Opere scelte. Vol. III: Calcolo delle variazioni, A cura dell'Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Ricerche, Edizioni Cremonese, Rome, 1962iii+508 MR0150020 0131.24103 Google Scholar[12] J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111–128 10.1016/0022-247X(62)90033-1 MR0142020 0102.31801 CrossrefGoogle Scholar[13A] T. Ważewski, Sur une généralisation de la notion des solutions d'une équation au contingent, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 10 (1962), 11–15 MR0153944 0104.30404 Google Scholar[13B] T. Ważewski, Sur les systèmes de commande non linéaires dont le contredomaine de commande n'est pas forcément convexe, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 10 (1962), 17–21 MR0153945 0104.30501 Google Scholar[14] L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Compt. Rend. Soc. Sci. Lettr. Varsovie C1 III, 30 (1937), 212–234 0019.21901 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails A Survey of the Maximum Principles for Optimal Control Problems with State ConstraintsRichard F. Hartl, Suresh P. Sethi, and Raymond G. 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McShane18 July 2006 | SIAM Journal on Control, Vol. 5, No. 3AbstractPDF (4087 KB) Volume 3, Issue 3| 1965Journal of the Society for Industrial and Applied Mathematics Series A Control History Submitted:17 May 1965Published online:18 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0303032Article page range:pp. 475-498ISSN (print):0887-4603ISSN (online):2168-359XPublisher:Society for Industrial and Applied Mathematics