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Tr. Mat. Inst. Steklova, 2019, Volume 304, Pages 257–272 (Mi tm3983)  

Pontryagin's Direct Method for Optimization Problems with Differential Inclusion

E. S. Polovinkin

Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia

Abstract: We develop Pontryagin's direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler–Lagrange differential inclusion. The results are proved under maximally weak conditions, and very strong statements compared with the known ones are obtained; moreover, admissible velocity sets may be unbounded and nonconvex under a general hypothesis that the right-hand side of the differential inclusion is pseudo-Lipschitz. In the statements, we refine conditions on the Euler–Lagrange differential inclusion, in which neither the Clarke normal cone nor the limiting normal cone is used, as is common in the works of other authors. We also give an example demonstrating the efficiency of the results obtained.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00209a
This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00209a.


DOI: https://doi.org/10.4213/tm3983

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 304, 241–256

Bibliographic databases:

UDC: 517.977
Received: November 18, 2018
Revised: December 19, 2018
Accepted: January 17, 2019

Citation: E. S. Polovinkin, “Pontryagin's Direct Method for Optimization Problems with Differential Inclusion”, Optimal control and differential equations, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin, Tr. Mat. Inst. Steklova, 304, Steklov Math. Inst. RAS, Moscow, 2019, 257–272; Proc. Steklov Inst. Math., 304 (2019), 241–256

Citation in format AMSBIB
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\paper Pontryagin's Direct Method for Optimization Problems with Differential Inclusion
\inbook Optimal control and differential equations
\bookinfo Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 304
\pages 257--272
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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