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 Mat. Sb., 2005, Volume 196, Number 2, Pages 117–138 (Mi msb1269)

Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion

A. A. Tolstonogov

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: An analogue of the classical theorem of Bogolyubov with non-convex constraint is proved. The constraint is the solution set of a differential inclusion with non-convex lower semicontinuous right-hand side. As an application we study the interrelation between the solutions of the problem of minimizing an integral functional with non-convex integrand on the solutions of the original inclusion and the solutions of the relaxation problem.

DOI: https://doi.org/10.4213/sm1269

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English version:
Sbornik: Mathematics, 2005, 196:2, 263–285

Bibliographic databases:

UDC: 517.972
MSC: Primary 49J45; Secondary 34A60, 49J24

Citation: A. A. Tolstonogov, “Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion”, Mat. Sb., 196:2 (2005), 117–138; Sb. Math., 196:2 (2005), 263–285

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1269
• https://doi.org/10.4213/sm1269
• http://mi.mathnet.ru/eng/msb/v196/i2/p117

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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. Jiang Y.-r., Huang N.-j., Zhang Q.-f., Shang Ch.-ch., “Relaxation in Nonconvex Optimal Control Problems Governed By Evolution Inclusions With the Difference of Two Clarke'S Subdifferentials”, Int. J. Control
2. D. V. Khlopin, “Euler's broken lines in systems with Carathéodory conditions”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S141–S158
3. D. V. Khlopin, “Euler polygons in systems with time-measurable right-hand side”, Differ. Equ., 44:12 (2008), 1711–1720
4. Tolstonogov A.A., “Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values”, Topology Appl., 155:8 (2008), 898–905
5. S. A. Timoshin, A. A. Tolstonogov, “Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect”, Nonlinear Anal., 75:15 (2012), 5884
6. Xiaoyou Liu, Zhenhai Liu, “Relaxation control for a class of evolution hemivariational inequalities”, Isr. J. Math., 202:1 (2014), 35–58
7. Liu X. Xu Y., “Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System”, Fract. Calc. Appl. Anal., 19:1 (2016), 94–115
8. Tolstonogov A.A., “Relaxation in Nonconvex Optimal Control Problems Containing the Difference of Two Subdifferentials”, SIAM J. Control Optim., 54:1 (2016), 175–197
9. Li X., Liu Zh., “Relaxation in Nonconvex Optimal Control Problems For Nonautonomous Fractional Evolution Equations”, Pac. J. Optim., 13:3 (2017), 443–462
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