Geodesic
Kelley Condition and Structure of Lagrange Manifold in a Neighborhood of a First-Order Singular Extremal
Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 19 (2006), pp. 5-44 
V. F. Borisov. Kelley Condition and Structure of Lagrange Manifold in a Neighborhood of a First-Order Singular Extremal. Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 19 (2006), pp. 5-44. http://geodesic.mathdoc.fr/item/CMFD_2006_19_a0/
@article{CMFD_2006_19_a0,
     author = {V. F. Borisov},
     title = {Kelley {Condition} and {Structure} of {Lagrange} {Manifold} in {a~Neighborhood} of {a~First-Order} {Singular} {Extremal}},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {5--44},
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     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_19_a0/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper considers optimal control problems linearly depending on the scalar control parameter in which there exist first-order singular extremals. The author proves a theorem on the structure of a generic Lagrange manifold (field of extremals) in a neighborhood of first-order singular extremals. As a consequence of this theorem, the author proves the optimality of singular extremals and nonsingular extremals in problems with fixed endpoints on small intervals of time. As an illustration, the paper presents constructions of Lagrange manifolds for the general linear-quadratic control problem with completely integrable linear system of differential constraints and for a certain problem of mathematical economics, a two-factor economic growth model with production function of the Cobb–Douglas type.

[1] Agrachev A. A., Gamkrelidze R. V., “Simplekticheskaya geometriya i neobkhodimye usloviya optimalnosti”, Mat. sb., 182:1 (1991), 36–54 | MR

[2] Agrachev A. A., Gamkrelidze R. V., “Printsip optimalnosti vtorogo poryadka dlya zadachi bystrodeistviya”, Mat. sb., 100:4 (1976), 610–643 | MR | Zbl

[3] Arnold V. I., Obyknovennye differentsialnye uravneniya, Nauka, M., 1980

[4] Bliss Dzh., Lektsii po variatsionnomu ischisleniyu, IL, M., 1950

[5] Zelikina L. F., “Mnogomernyi sintez i teoremy o magistralyakh”, Veroyatnostnye problemy upravleniya v ekonomike, Nauka, M., 1977, 33–114 | MR

[6] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[7] Nirenberg L., Lektsii po nelineinomu funktsionalnomu analizu, Mir, M., 1977 | MR | Zbl

[8] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1982 | MR | Zbl

[9] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1976 | Zbl

[10] Sternberg S., Lektsii po differentsialnoi geometrii, Mir, M., 1970 | MR | Zbl

[11] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[12] Dmitruk A. V., “Quadratic order conditions of a local minimum for singular extremals in a general optimal control problem”, Proc. Symp. Pure Math., 64, Amer. Math. Soc., Providence, Rhode Island, 1999 ; Differential Geometry and Control., Summer Research Inst. on Diff. Geometry and Control. (June 29–July 19, 1997), Univ. of Colorado, Boulder, 163–198 | MR | Zbl

[13] Goh B. S., “Necessary conditions for singular extremals involving multiple control variables”, SIAM J. Control., 4:4 (1966), 716–731 | DOI | MR | Zbl

[14] Kelley H. J., Kopp R. E., Moyer H. G., “Singular extremals”, Topics in Optimization, ed. Leitmann G., Academic Press, N.Y., 1967, 63–103 | MR

[15] Krener A. J., “The higher-order maximal principle and its application to singular extremals”, SIAM J. Control., 15:2 (1977), 256–293 | DOI | MR | Zbl

[16] Levitin E. S., Milyutin A. A., Osmolovskii N. P., “Conditions of higher order for a local minimum in extremal problems with constraints”, Russ. Math. Surv., 33:6 (1978), 138–168 | DOI | MR

[17] Robbins H. M., “A generalized Legendre–Clebsh condition for singular cases of optimal control”, IBM J. Res. Develop., 11 (1967), 361–372 | DOI | Zbl