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

This article is cited in 14 scientific papers (total in 14 papers)

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.


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

Bibliographic databases:

UDC: 517.998
MSC: 49J24, 93C25, 34A60
Received: 28.12.2001

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
\by A.~A.~Tolstonogov
\paper Bogolyubov's theorem under constraints generated by a~controlled second-order evolution system
\jour Izv. RAN. Ser. Mat.
\yr 2003
\vol 67
\issue 5
\pages 177--206
\jour Izv. Math.
\yr 2003
\vol 67
\issue 5
\pages 1031--1060

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    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  mathnet  crossref  mathscinet  zmath  isi  elib
    2. A. A. Tolstonogov, “Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion”, Sb. Math., 196:2 (2005), 263–285  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. A. Tolstonogov, “Variational stability of optimal control problems involving subdifferential operators”, Sb. Math., 202:4 (2011), 583–619  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    7. S.A.. Timoshin, “Variational Stability of Some Optimal Control Problems Describing Hysteresis Effects”, SIAM J. Control Optim, 52:4 (2014), 2348  crossref  mathscinet  zmath  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    12. Aiki T., Timoshin S.A., “Relaxation For a Control Problem in Concrete Carbonation Modeling”, SIAM J. Control Optim., 55:6 (2017), 3489–3502  crossref  mathscinet  zmath  isi  scopus
    13. Jin N., Sun Sh., “On a Coupled System of Fractional Compartmental Models For a Biological System”, Adv. Differ. Equ., 2017, 146  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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