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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 5, Pages 177–206 (Mi izv456)

Bogolyubov's theorem under constraints generated by a controlled second-order evolution system

A. A. Tolstonogov

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

Abstract: We prove an analogue of Bogolyubov's theorem with constraints in the form of a controlled second-order evolution system. The main assertion of this theorem deals with relations between the values of an integral functional that is non-convex with respect to control on the solutions of a controlled system with non-convex constraints on the control and the values of the functional convexified with respect to control on the solutions of a controlled system with convexified constraints. This theorem also establishes relations between the solutions of non-convex and convexified controlled systems. We apply the theorem to the problem of minimizing a non-convex integral functional on the solutions of a non-convex controlled system. We consider in detail an example of a non-linear hyperbolic system.

DOI: https://doi.org/10.4213/im456

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English version:
Izvestiya: Mathematics, 2003, 67:5, 1031–1060

Bibliographic databases:

UDC: 517.998
MSC: 49J24, 93C25, 34A60

Citation: A. A. Tolstonogov, “Bogolyubov's theorem under constraints generated by a controlled second-order evolution system”, Izv. RAN. Ser. Mat., 67:5 (2003), 177–206; Izv. Math., 67:5 (2003), 1031–1060

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv456
• https://doi.org/10.4213/im456
• http://mi.mathnet.ru/eng/izv/v67/i5/p177

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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. A. Tolstonogov, “Properties of attainable sets of evolution inclusions and control systems of subdifferential type”, Siberian Math. J., 45:4 (2004), 763–784
2. A. A. Tolstonogov, “Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion”, Sb. Math., 196:2 (2005), 263–285
3. A. A. Tolstonogov, “Variational stability of optimal control problems involving subdifferential operators”, Sb. Math., 202:4 (2011), 583–619
4. A.A. Tolstonogov, “Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system”, Nonlinear Analysis: Theory, Methods & Applications, 2012
5. S.A. Timoshin, A.A. Tolstonogov, “Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect”, Nonlinear Analysis: Theory, Methods & Applications, 75:15 (2012), 5884
6. Xiaoyou Liu, Zhenhai Liu, “Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative”, Abstract and Applied Analysis, 2012 (2012), 1
7. S.A.. Timoshin, “Variational Stability of Some Optimal Control Problems Describing Hysteresis Effects”, SIAM J. Control Optim, 52:4 (2014), 2348
8. Ahmad B., Alsaedi A., Nazemi S.Z., Rezapour Sh., “Some Existence Theorems For Fractional Integro-Differential Equations and Inclusions With Initial and Non-Separated Boundary Conditions”, Bound. Value Probl., 2014, 249
9. Liu X., Liu Zh., Fu X., “Relaxation in Nonconvex Optimal Control Problems Described By Fractional Differential Equations”, J. Math. Anal. Appl., 409:1 (2014), 446–458
10. Liu X., Xu Y., “Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System”, Fract. Calc. Appl. Anal., 19:1 (2016), 94–115
11. Tolstonogov A.A., “Relaxation in Nonconvex Optimal Control Problems Containing the Difference of Two Subdifferentials”, SIAM J. Control Optim., 54:1 (2016), 175–197
12. Aiki T., Timoshin S.A., “Relaxation For a Control Problem in Concrete Carbonation Modeling”, SIAM J. Control Optim., 55:6 (2017), 3489–3502
13. Jin N., Sun Sh., “On a Coupled System of Fractional Compartmental Models For a Biological System”, Adv. Differ. Equ., 2017, 146
14. Timoshin S.A., Aiki T., “Extreme Solutions in Control of Moisture Transport in Concrete Carbonation”, Nonlinear Anal.-Real World Appl., 47 (2019), 446–459
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