C&SA | Contents

Volume 62 >>> № 3 MAY — JUNE 2026

-->

UDC 519.8

A.Yu. Mishchuk
Lutsk Pedagogical Institute, Lutsk, Ukraine,
anton.mi.ju@gmail.com

A.M. Shutovskyi
Lesya Ukrainka Volyn National University, Lutsk, Ukraine,
sh93ar@gmail.com

ON ONE EXTREMAL PROBLEM FOR AN ORTHOGONAL POLYNOMIAL SYSTEM

Abstract. A generalized Poisson integral in an orthogonal polynomial system is proposed as a mathematical model of admissible control strategies under the action of a three-dimensional control vector. An optimality criterion is formulated as an extremal problem for the deviation of a generalized Poisson-type operator from a critical optimal state, which is represented by functions belonging to a Holder class. The solution of the resulting optimization problem is given in the form of an asymptotic equality describing the behavior of the controlled dynamical system over the entire set of admissible controls.

Keywords: control parameter, Holder class functions, optimality criterion, asymptotic equality.

full text

REFERENCES

1. Kondratenko Yu.P., Kuntsevich V.M., Chikrii A.A., Gubarev V.F. Recent developments in automatic control systems. Recent Developments in Automatic Control Systems, 2024. P. 1–492.
2. Baranovska L. Method of upper and lower resolving functions for pursuit differential-difference games with pure delay. Recent Developments in Automatic Control Systems, 2024. P. 131–143.
3. Chikriy A.A., Baranovskaya L.V., Chikriy Al.A. An approach game problem under the failure of controlling devices. Journal of Automation and Information Sciences. 2000. Vol. 32, N 5. P. 1–8. https://doi.org/10.1615/JAutomatInfScien.v32.i5.10.
4. Vlasenko L.A., Rutkas A.G. Optimal control of a class of random distributed Sobolev type systems with aftereffect. Journal of Automation and Information Sciences. 2013. Vol. 45, N 9. P. 65–76. https://doi.org/10.1615/JAutomatInfScien.v45.i9.60.
5. Chikrij A.A., Baranovskaya L.V., Chikrij Al.A. The game problem of approach under the condition of failure of controlling devices. Problemy Upravleniya i Informatiki (Avtomatika). 1997. N 4. P. 5–13.
6. Vlasenko L.A., Rutkas A.G., Chikrii A.O. Functional-differential games with nonatomic difference operator. Ukrainian Mathematical Journal. 2022. Vol. 74, N 2. P. 186–202. https://doi.org/10.1007/s11253-022-02057-7.
7. Rutkas A., Vlasenko L. On a differential game in a nondamped distributed system. Mathematical Methods in the Applied Sciences. 2019. Vol. 42, N 18. P. 6155–6164. https://doi.org/10.1002/mma.5712.
8. Vlasenko L.A., Rutkas A.G., Semenets V.V., Chikrii A.A. On the optimal impulse control in descriptor systems. Journal of Automation and Information Sciences. 2019. Vol. 51, N 5. P. 1–15. https://doi.org/10.1615/JAutomatInfScien.v51.i5.10.
9. Bushev D.M., Kal’chuk I.V., Kharkevych Yu.I. Approximation of isometric function spaces by entire functions of exponential type. Carpathian Mathematical Publications. 2025. Vol. 17, N 2. P. 679–692. https://doi.org/10.15330/cmp.17.2.679-692.
10. Vlasenko L.A., Chikrii A.A. The method of resolving functionals for a dynamic game in a Sobolev system. Journal of Automation and Information Sciences. 2014. Vol. 46, N 7. P. 1–11. https://doi.org/10.1615/JAutomatInfScien.v46.i7.10.
11. Vlasenko L.A., Samoilenko A.M. Optimal control with impulsive component for systems described by implicit parabolic operator differential equations. Ukrainian Mathematical Journal. 2009. Vol. 61, N 8. P. 1250–1263. https://doi.org/10.1007/s11253-010-0274-1.
12. Abdullayev F.G., Bushev D.M., Imashkyzy M., Kharkevych Yu.I. Isometry of the subspaces of solutions of systems of differential equations to the spaces of real functions. Ukrainian Mathematical Journal. 2020. Vol. 71, N 8. P. 1153–1172. https://doi.org/10.1007/s11253-019-01705-9.
13. Kharkevych Yu.I., Stepaniuk T.A. Approximate properties of Abel–Poisson integrals on classes of differentiable functions defined by moduli of continuity. Carpathian Mathematical Publications. 2023. Vol. 15, N 1. P. 286–294. https://doi.org/10.15330/cmp.15.1.286-294.
14. Shutovskyi A.M. Integral representations of polyharmonic operators. Cybernetics and Systems Analysis. 2025. Vol. 61, N 3. P. 450–460. https://doi.org/10.1007/s10559-025-00782-y.
15. Bushev D.M., Kharkevych Yu.I. Approximation of classes of periodic multivariable functions by linear positive operators. Ukrainian Mathematical Journal. 2006. Vol. 58, N 1. P. 12–21. https://doi.org/10.1007/s11253-006-0048-y.
16. Shutovskyi A.M. Some representations of triharmonic functions. Cybernetics and Systems Analysis. 2024. Vol. 60, N 6. P. 991–1000. https://doi.org/10.1007/s10559-024-00735-x.
17. Zhyhallo K.M., Kharkevych Yu.I. On some approximate properties of biharmonic Poisson integrals in the integral metric. Carpathian Mathematical Publications. 2024. Vol. 16, N 1. P. 303–308. https://doi.org/10.15330/cmp.16.1.303-308.
18. Shutovskyi A.M., Kharkevych Yu.I. On saturation of one linear method for summing Fourier series given by a function set. Researches in Mathematics. 2025. Vol. 33, N 1. P. 122–136. https://doi.org/10.15421/242508.
19. Kharkevych Yu.I. On approximation of the quasi-smooth functions by their Poisson type integrals. Journal of Automation and Information Sciences. 2017. Vol. 49, N 10. P. 74–81. https://doi.org/10.1615/JAutomatInfScien.v49.i10.80.
20. Mishchuk A.Yu., Shutovskyi A.M. Optimization properties of generalized Chebyshev–Poisson integrals. Cybernetics and Systems Analysis. 2024. Vol. 60, N 4. P. 613–620. https://doi.org/10.1007/s10559-024-00700-8.
21. Zhyhallo T.V., Kharkevych Yu.I. Approximating properties of biharmonic Poisson operators in the classes . Ukrainian Mathematical Journal. 2017. Vol. 69, N 5. P. 757–765. https:// doi.org/10.1007/s11253-017-1393-8.
22. Shutovskyi A.M. Optimization properties of the Taylor formula for the triharmonic Poisson integral. Cybernetics and Systems Analysis. 2025. Vol. 61, N 5. P. 827–836. https://doi.org/10.1007/s10559-025-00814-7.
23. Vlasenko L.A., Rutkas A.G. On a differential game in a system described by an implicit differential-operator equation. Differential Equations. 2015. Vol. 51, N 6. P. 798–807. https:// doi.org/10.1134/S0012266115060117.
24. Vlasenko L.A., Rutkas A.G. On the optimal control of systems described by implicit evolution equations. Differential Equations. 2009. Vol. 45, N 3. P. 429–438. https://doi.org/10.1134/S0012266109030124.
25. Vlasenko L.A., Rutkas A.G. Optimal control of undamped Sobolev-type retarded systems. Mathematical Notes. 2017. Vol. 102, N 3–4. P. 297–309. https://doi.org/10.1134/S0001434617090012.
26. Bondarenko M.F., Vlasenko L.A., Rutkas A.G. Discrete optimal control of descriptor systems with variable parameters. Journal of Automation and Information Sciences. 2011. Vol. 43, N 5. P. 1–9. https://doi.org/10.1615/JAutomatInfScien.v43.i5.10.
27. Kharkevych Yu.I., Shutovskyi A.M. Approximation of functions from Hlder class by biharmonic Poisson integrals. Carpathian Mathematical Publications. 2024. Vol. 16, N 2. P. 631–637. https://doi.org/10.15330/cmp.16.2.631-637.
28. Zhyhallo K.M., Kharkevych Yu.I. Approximation of differentiable periodic functions by their biharmonic Poisson integrals. Ukrainian Mathematical Journal. 2002. Vol. 54, N 9. P. 1462–1470. https://doi.org/10.1023/A:1023463801914.
29. Zhyhallo T.V., Kharkevych Yu.I. Approximation of functions from the class by Poisson integrals in the uniform metric. Ukrainian Mathematical Journal. 2009. Vol. 61, N 12. P. 1893–1914. https://doi.org/10.1007/s11253-010-0321-y.
30. Zhyhallo T.V., Kharkevych Yu.I. Approximation of –differentiable functions by Poisson integrals in the uniform metric. Ukrainian Mathematical Journal. 2009. Vol. 61, N 11. P. 1757–1779. https://doi.org/10.1007/s11253-010-0311-0.
31. Shapovalov D.V., Shutovskyi A.M. Asymptotic properties of a certain system of orthogonal polynomials. Cybernetics and Systems Analysis. 2025. Vol. 61, N 6. P. 1014–1023. https://doi.org/10.1007/s10559-025-00834-3.
32. Shutovskyi A., Kharkevych Yu. Structural connection between solutions to biharmonic equation and Laplace equation. Studies in Systems, Decision and Control. 2026. Vol. 609. P. 185–203. https://doi.org/10.1007/978-3-031-97529-5_12.
33. Chikrii A. Control of moving object groups in a conflict situation. Recent Developments in Automatic Control Systems, 2024. P. 3–29.