Averaging method in the problem of optimal control for a perturbed parabolic equation

  • O. V. Kapustyan Taras Shevchenko National University of Kyiv
  • O. M. Stanzhytskyi Taras Shevchenko National University of Kyiv
  • I. D. Fartushny National technical University of Ukraine "KPI named after I. Sikorskyi", Kyiv
Keywords: optimal control, averaging, parabolic equation

Abstract

UDC 517.9

We consider the optimal control problem formed by a parabolic nonlinear equation with rapidly oscillating coefficients, an additive control function, and coercive cost functional. It is proved that the optimal value of the perturbed problem is close to the optimal value for the corresponding problem with averaged coefficients.

References

N. N. Bogolyubov, YU. A. Mitropol'skij, Asimptoticheskie metody v teorii nelinejnyh kolebanij, Nauka, Moskva (1963).

N. N. Bogolyubov, YU. A. Mitropol'skij, A. M. Samojlenko, Metod uskorennoj skhodimosti v nelinejnoj mekhanike, Nauk. dumka, Kiev (1969).

A. M. Samoilenko, A. N. Stanzhitskii, On averaging differential equations on an infinite interval, Differents. Uravneniya, 42, № 4, 476 – 482 (2006), https://doi.org/10.1134/S0012266106040070

J. A. Sanders, F. Verhulst, Averaging methods in nonlinear dynamical systems, Springer, New York (1985), https://doi.org/10.1007/978-1-4757-4575-7

T. V. Nosenko, O. M. Stanzhyts’kyi, Averaging method in some problems of optimal control, Nonlinear Oscillations, 11, № 4, 539 – 547 (2008), https://doi.org/10.1007/s11072-009-0049-5

O. Kichmarenko, O. Stanzhytskyi, Sufficient conditions for the existence of optimal controls for some classes of functional-differential gathers, Nonlinear Dyn. and Syst. Theory, 18, № 2, 196 – 211 (2018), https://doi.org/10.1007/s11072-009-0049-5

O. A. Kapustyan, A. V. Sukretna, Nablizhenij userednenij sintez v zadachi optimal'nogo keruvannya dlya parabolichnogo rivnyannya, Ukr. mat. zhurn., 56, № 10, 1653 – 1664 (2004).

O. V. Kapustyan, O. A. Kapustian, A. V. Sukretna, Approximate stabilization for a nonlinear parabolic boundary-value problem, Ukr. Math. J., 63, № 5, 759 – 767 (2011), https://doi.org/10.1007/s11253-011-0540-x

O. G. Nakonechnyi, O. A. Kapustian, A. O. Chikrii, Approximate guaranteed mean square estimates of functionals on solutions of parabolic problems with fast oscillating coefficients under nonlinear observations, Cybernet. and System Anal., 55, № 5, 785 – 795 (2019), https://doi.org/10.1007/s10559-019-00189-6

O. V. Kapustyan, P. O. Kasyanov, J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9, 2257 – 2264 (2015), https://doi.org/10.12785/amis

V. V. Chepyzhov, M. I. Vishik, Attractors of equations of mathematical physics, Amer. Math. Soc. (2002).

ZH.-L. Lions, Optimal'noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, Moskva (1972).

Published
09.08.2022
How to Cite
Kapustyan , O. V., O. M. Stanzhytskyi, and I. D. Fartushny. “Averaging Method in the Problem of Optimal Control for a Perturbed Parabolic Equation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 7, Aug. 2022, pp. 973 -79, doi:10.37863/umzh.v74i7.7016.
Section
Research articles