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 Mat. Sb., 1991, Volume 182, Number 1, Pages 36–54 (Mi msb1273)

Symplectic geometry and conditions necessary conditions for optimality

A. A. Agrachev, R. V. Gamkrelidze

Abstract: With the help of a symplectic technique the concept of a field of extremals in the classical calculus of variations is generalized to optimal control problems. This enables us to get new optimality conditions that are equally suitable for regular, bang-bang, and singular extremals. Special attention is given to systems of the form $\dot x=f_0(x)+uf_1(x)$ with a scalar control. New pointwise conditions for optimality and sufficient conditions for local controllability are obtained as a consequence of the general theory.

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English version:
Mathematics of the USSR-Sbornik, 1992, 72:1, 29–45

Bibliographic databases:

UDC: 517.97
MSC: 49K15, 58F05

Citation: A. A. Agrachev, R. V. Gamkrelidze, “Symplectic geometry and conditions necessary conditions for optimality”, Mat. Sb., 182:1 (1991), 36–54; Math. USSR-Sb., 72:1 (1992), 29–45

Citation in format AMSBIB
\Bibitem{AgrGam91} \by A.~A.~Agrachev, R.~V.~Gamkrelidze \paper Symplectic geometry and conditions necessary conditions for optimality \jour Mat. Sb. \yr 1991 \vol 182 \issue 1 \pages 36--54 \mathnet{http://mi.mathnet.ru/msb1273} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1098838} \zmath{https://zbmath.org/?q=an:0776.49014|0729.49018} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..72...29A} \transl \jour Math. USSR-Sb. \yr 1992 \vol 72 \issue 1 \pages 29--45 \crossref{https://doi.org/10.1070/SM1992v072n01ABEH002137} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992JF72300002} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Pukhlikov A., “The Poisson Theorem in Optimal-Control Problems”, Differ. Equ., 29:11 (1993), 1685–1690
2. Pllkhlikov A., “The Geometry of Discontinuous Systems Near an Elliptic Singularity”, Differ. Equ., 35:11 (1999), 1516–1529
3. Pukhlikov A., “A Gauss-Ostrogradskii Theorem for Integral Transforms Over the Euler Characteristic - to the Memory of Professor Anatolii Platonovich Prudnikov”, Integral Transform. Spec. Funct., 9:4 (2000), 299–312
4. Pukhlikov A., “The Geometry of Discontinuous Systems Near a Hyperbolic Singular Point of the Sliding Field”, Differ. Equ., 37:3 (2001), 377–400
5. Eduardo Martínez, “Reduction in optimal control theory”, Reports on Mathematical Physics, 53:1 (2004), 79
6. Pukhlikov A., “Geometry of Discontinuous Systems Near a Manifold of Singularities of Third Order”, Differ. Equ., 41:12 (2005), 1710–1716
7. V. F. Borisov, “Kelley Condition and Structure of Lagrange Manifold in a Neighborhood of a First-Order Singular Extremal”, Journal of Mathematical Sciences, 151:6 (2008), 3431–3472
8. A. A. Agrachev, R. V. Gamkrelidze, “The Pontryagin Maximum Principle 50 years later”, Proc. Steklov Inst. Math. (Suppl.), 253, suppl. 1 (2006), S4–S12
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