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 Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 2, Pages 284–312 (Mi izv1480)

Quadratic conditions for a Pontryagin minimum in an optimum control problem linear in the control. I: A decoding theorem

A. V. Dmitruk

Abstract: The general optimum control problem considered here is linear in the control and without constraints on the control. Quadratic (i.e., second-order) necessary and sufficient conditions are given for the problem to have a minimum in the class of variations bounded in modulus by an arbitrary constant and having small integral. These conditions are stronger than the previously known conditions for a weak minimum, and, like the latter conditions, constitute an adjoining pair, i.e., the sufficient condition differs from the necessary condition only in the strengthening of an inequality.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1987, 28:2, 275–303

Bibliographic databases:

UDC: 517.97
MSC: Primary 49B10; Secondary 34H05

Citation: A. V. Dmitruk, “Quadratic conditions for a Pontryagin minimum in an optimum control problem linear in the control. I: A decoding theorem”, Izv. Akad. Nauk SSSR Ser. Mat., 50:2 (1986), 284–312; Math. USSR-Izv., 28:2 (1987), 275–303

Citation in format AMSBIB
\Bibitem{Dmi86} \by A.~V.~Dmitruk \paper Quadratic conditions for a~Pontryagin minimum in an optimum control problem linear in the control.~I: A~decoding theorem \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1986 \vol 50 \issue 2 \pages 284--312 \mathnet{http://mi.mathnet.ru/izv1480} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=842584} \zmath{https://zbmath.org/?q=an:0682.49020|0611.49014} \transl \jour Math. USSR-Izv. \yr 1987 \vol 28 \issue 2 \pages 275--303 \crossref{https://doi.org/10.1070/IM1987v028n02ABEH000882} 

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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. A. V. Dmitruk, “Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in the control. II. Theorems on weakening equality constraints”, Math. USSR-Izv., 31:1 (1988), 121–141
2. A.V. Dmitruk, “Quadratic order conditions of a local minimum for abnormal extremals”, Nonlinear Analysis: Theory, Methods & Applications, 30:4 (1997), 2439
3. Daniel Hoehener, “Variational Approach to Second-Order Optimality Conditions for Control Problems with Pure State Constraints”, SIAM J. Control Optim, 50:3 (2012), 1139
4. M. Aronna, J. Bonnans, Andrei Dmitruk, Pablo Lotito, “Quadratic order conditions for bang-singular extremals”, NACO, 2:3 (2012), 511
5. M. Soledad Aronna, J. Frédéric Bonnans, Pierre Martinon, “A Shooting Algorithm for Optimal Control Problems with Singular Arcs”, J Optim Theory Appl, 2013
6. J. Frédéric Bonnans, “Optimal control of a semilinear parabolic equation with singular arcs”, Optimization Methods and Software, 2013, 1
7. Hélène Frankowska, Daniela Tonon, “Pointwise Second-Order Necessary Optimality Conditions for the Mayer Problem with Control Constraints”, SIAM J. Control Optim, 51:5 (2013), 3814
8. L. V. Lokutsievskii, “The Hamiltonian property of the flow of singular trajectories”, Sb. Math., 205:3 (2014), 432–458
9. L. V. Lokutsievskii, “Singular regimes in controlled systems with multidimensional control in a polyhedron”, Izv. Math., 78:5 (2014), 1006–1027
10. M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136
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