рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 52

On Controllability and Stabilization of Nonlinear Continuous-discrete Dynamic Systems

Author(s)
Svetlana V. Akmanova1

1Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation

Abstract
The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.
About the Authors
Svetlana V. Akmanova, Cand. Sci. (Pedagog.), Assoc. Prof., Nosov Magnitogorsk State Technical University, Magnitogorsk, 455000, Russian Federation, svet.akm 74@mail.ru
For citation
Akmanova S. V. On Controllability and Stabilization of Nonlinear Continuous-discrete Dynamic Systems. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 52, pp. 3–20. (in Russian)

https://doi.org/10.26516/1997-7670.2025.52.3

Keywords
continuous-discrete system, discrete system, hybrid system, controlled system, stabilized system
UDC
517.938
MSC
34A34, 37N35, 34D20
DOI
https://doi.org/10.26516/1997-7670.2025.52.3
References
  1. Akmanova S.V. Stabilization of Nonlinear Continuous-Discrete Dynamic Systems with a Constant Sampling Step. Automation and Remote Control, 2024, vol. 85, no. 9, pp. 864–878. https://doi.org/10.31857/S0005117924090022
  2. Akulenko L.D. Asymptotic methods of optimal control. Moscow, Nauka Publ., 1987, 368 p. (in Russian)
  3. Ananevsky I.M. Synthesis of control of dynamic systems based on the method of Lyapunov functions. Modern Mathematics. Fundamental directions, 2011, vol. 42, pp. 23–29. (in Russian)
  4. Anokhin N.V. Control of nonlinear mechanical systems with a deficit of control actions in the vicinity of the equilibrium position. Cand. sci. diss. abstr. (Phys.- Math.). Moscow, 2014, 72 p. (in Russian)
  5. Akhundov A.A. Controllability of linear hybrid systems. Controllable systems, 1975, no. 14, pp. 4–10. (in Russian)
  6. Berezin I.C., Zhidkov N.P. Methods of calculations. Moscow, Fizmatlit Publ., vol. 2, 1959, 620 p. (in Russian)
  7. Bortakovskii A.S. Sufficient Optimality Conditions for Hybrid Systems of Variable Dimension. Proceedings of the Steklov Institute of Mathematics, 2020, vol. 308, pp. 79–91. https://doi.org/10.1134/S0081543820010071
  8. Bortakovskii A.S., Panteleev A.V. Sufficient conditions for optimal control of continuously discrete systems. Avtom. Telemekh., 1987, no. 7, pp. 57–66. (in Russian)
  9. Kwakernaak H., Sivan R. Linear optimal control systems. Moscow, Mir Publ., 1977, 650 p.(in Russian)
  10. Logunova O.S., Agapitov E.B., Barankova I.I. et al. Mathematical models for studying the thermal state of bodies and controlling thermal processes. Electrical systems and complexes, 2019. no. 2 (43). pp. 25–34.(in Russian) https://doi.org/10.18503/2311-8318-2019-2(43)-25-34
  11. Maksimov V.P. Continuous-discrete dynamic models. Ufa Mathematical Journal, 2021, vol. 13, no. 3, pp. 95–103. https://doi.org/10.13108/2021-13-3-95
  12. Maksimov V.P., Chadov A.L. Hybrid models in problems of economic dynamics. Bulletin of Perm University, 2011, no. 2 (9), pp. 13–23. (in Russian)
  13. Marchenko V.M., Borkovskaya I.M. Stability and stabilization of linear hybrid discrete-continuous stationary systems. Proceedings of BSTU. Physical and mathematical sciences and computer science, 2012, no. 6, pp. 7–10.
  14. Propoi A.I. Elements of the theory of optimal discrete processes. Moscow, Nauka Publ., 1973, 256 p. (in Russian)
  15. Rasina I.V. Iterative Optimization Algorithms for Discrete-Continuous Processes. Automation and Remote Control, 2012, vol. 73, no. 10, pp. 1591-1603. http://doi.org/10.1134/S0005117912100013
  16. Сazanova L.A. On control of discrete systems. Cand. sci. diss. abstr. (Phy.-Math.). Yekaterinburg, 2002, 95 p. (in Russian)
  17. Chebykin L.S. Optimality conditions for discrete-continuous systems. Optimal control of systems with uncertain information, Sverdlovsk, USC of the USSR Academy of Sciences Publ., 1980, pp. 132–140. (in Russian)
  18. Yumagulov M.G., Akmanova S.V. On stability of equilibria of nonlinear continuous-discrete dynamical systems. Ufa Mathematical Journal, 2023, vol. 15, no. 2, pp. 85–99. https://doi.org/10.13108/2023-15-2-85
  19. Akmanova S.V., Kopylova N.A. The Stability of Continuous-Discrete Dynamical Systems under Fast Switching. Lobachevskii Journal of Mathematics, 2023, vol. 44, no. 5, pp. 1826–1832. https:// doi.org/10.1134/S1995080223050074
  20. La Salle J.P. The Stability and Control of Discrete Processes. New York, SpringerVerlag Inc. Publ., 1986, 150 p.
  21. Karafyllis L., Krstic M. Nonlinear Stabilization Under Sampled and Delayed Measurements, and With Inputs Subject to Delay and Zero-Order Hold. IEEE Transact. Autom. Control, 2012, vol. 57, no. 5, pp. 1141–1154.

Full text (russian)