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Approximate solution of the optimal control problem for non-linear differential inclusion on the semi-axes

  • Nina Kasimova EMAIL logo , Tetiana Zhuk and Iryna Tsyganivska
Published/Copyright: July 25, 2023
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Abstract

We consider the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters on semi-axes. Using the averaging method, we find an approximate solution for the optimal control of non-linear differential inclusions with fast-oscillating coefficients on a semi-axes. Thus, we prove the convergence of the optimal controls of the initial control problem to the optimal process of the averaged problem and the convergence of the corresponding cost functionals.

MSC 2020: 49J21; 34A45

Funding statement: The work is supported by the Ukrainian Government Scientific Research Grant No. 210BF38-01.

References

[1] J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Grundlehren Math. Wiss. 264, Springer, Berlin, 1984. 10.1007/978-3-642-69512-4Search in Google Scholar

[2] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. 10.1016/0022-247X(65)90049-1Search in Google Scholar

[3] V. I. Blagodat’skikh and A. F. Filippov, Differential inclusions and optimal control, Trudy Mat. Inst. Steklov. 169 (1985), 194–252. Search in Google Scholar

[4] S. Dashkovskiy and S. Pavlichkov, Constructive design of adaptive controllers for nonlinear MIMO systems with arbitrary switchings, IEEE Trans. Automat. Control 61 (2016), no. 7, 2001–2007. 10.1109/TAC.2015.2491679Search in Google Scholar

[5] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side (in Russian), “Nauka”, Moscow, 1985. Search in Google Scholar

[6] O. A. Kapustian, O. V. Kapustyan, A. Ryzhov and V. Sobchuk, Approximate optimal control for a parabolic system with perturbations in the coefficients on the half-axis, Axioms 11 (2022), 10.3390/axioms11040175. 10.3390/axioms11040175Search in Google Scholar

[7] E. A. Kapustyan and A. G. Nakonechnyĭ, Minimax problems of pointwise observation for a parabolic boundary value problem, J. Automat. Inform. Sci. 34 (2002), no. 5, 52–63. 10.1615/JAutomatInfScien.v34.i5.60Search in Google Scholar

[8] O. D. Kichmarenko, Application of the averaging method to optimal control problem of system with fast parameters, Int. J. Pure Appl. Math. 115 (2017), no. 1, 93–114. 10.12732/ijpam.v115i1.8Search in Google Scholar

[9] O. D. Kichmarenko, Application of the averaging method to optimal control problems for ordinary differential equations on the semiaxis (in Ukrainian), Ukraïn. Mat. Zh. 70 (2018), no. 5, 642–654; translation in Ukrainian Math. J. 70 (2018), no. 5, 739–753. Search in Google Scholar

[10] O. D. Kichmarenko, O. A. Kapustian, N. V. Kasimova and T. Y. Zhuk, Optimal control problem for a differential inclusion with rapidly oscillating coefficients on the semiaxis, J. Math. Sci. (N. Y.) 272 (2023), no. 2, 267–277. 10.1007/s10958-023-06415-zSearch in Google Scholar

[11] O. D. Kichmarenko, N. V. Kasimova and T. Y. Zhuk, Approximate solution of the optimal control problem for differential inclusion with fast oscillating coefficients, Res. Math. Mech. 6 (2021), 38–54. Search in Google Scholar

[12] O. D. Kichmarenko and O. Stanzhytskyi, Sufficient conditions for the existence of optimal controls for some classes of functional-differential equations, Nonlinear Dyn. Syst. Theory 18 (2018), no. 2, 196–211. 10.18514/MMN.2019.2739Search in Google Scholar

[13] O. G. Nakonechniĭ, O. A. Kapustyan and A. O. Chikrīĭ, Approximate guaranteed mean-square estimates for functionals from solutions of parabolic problems with rapidly oscillating coefficients under nonlinear observations (in Ukrainian), Kibernet. Sistem. Anal. 55 (2019), no. 5, 95–105; translation in Cybernet. Systems Anal. 55 (2019), no. 5, 785–795. Search in Google Scholar

[14] T. V. Nosenko and O. M. Stanzhits’kiĭ, The averaging method in some optimal control problems (in Ukrainian), Nelīnīĭnī Koliv. 11 (2008), no. 4, 512–519; translation in Nonlinear Oscil. (N. Y.) 11 (2008), no. 4, 539–547. Search in Google Scholar

[15] V. A. Plotnikov, A. V. Plotnikov and A. N. Vityuk, Differential Equations with Multi-Valued Right-Hand Side (in Russian), “AstroPrint”, Odessa, 1999. Search in Google Scholar

[16] N. V. Plotnikova, The Krasnosel’skiĭ-Kreĭn theorem for differential inclusions (in Russian), Differ. Uravn. 41 (2005), no. 7, 997–1000; translation in Differ. Equ. 41 (2005), no. 7, 1049–1053. Search in Google Scholar

[17] R. Suttner and S. Dashkovskiy, Robustness and averaging properties of a large-amplitude, high-frequency extremum seeking control scheme, Automatica J. IFAC 136 (2022), Paper No. 110020. 10.1016/j.automatica.2021.110020Search in Google Scholar

[18] N. V. Zadoianchuk and P. O. Kasyanov, Dynamics of solutions of a class of second-order autonomous evolution inclusions (in Russian), Kibernet. Sistem. Anal. 2012 (2012), no. 3, 111–126; translatiom in Cybernet. Systems Anal. 48 (2012), no. 3, 414–428. Search in Google Scholar

[19] T. Zhuk, N. Kasimova and A. Ryzhov, Application of the averaging method to the optimal control problem of non-linear differential inclusions on the finite interval, Axioms 11 (2022), 10.3390/axioms11110653. 10.3390/axioms11110653Search in Google Scholar

Received: 2023-02-24
Accepted: 2023-04-03
Published Online: 2023-07-25
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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