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

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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. 
@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},
     publisher = {mathdoc},
     volume = {19},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_19_a0/}
}
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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/