Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Choque-Rivero A. E., Ornelas-Tellez F. Bounded finite-time stabilization of the prey – predator model via Korobov’s controllability function. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 1, pp. 76-87. DOI: 10.18500/1816-9791-2021-21-1-76-87, EDN: OPKVIW

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2021
Full text:
download
(downloads: 1482)
Полный текст в формате PDF(En):
download
(downloads: 60)
Language: 
English
Heading: 
Mechanics
Article type: 
Article
UDC: 
539.374
DOI: 
10.18500/1816-9791-2021-21-1-76-87
EDN: 
OPKVIW

Bounded finite-time stabilization of the prey – predator model via Korobov’s controllability function

Autors: 
Choque-Rivero Abdon E., Universidad Michoacana de San Nicolas de Hidalgo
Ornelas-Tellez Fernando, Universidad Michoacana de San Nicolas de Hidalgo
Abstract: 

The problem of finite-time stabilization for a Leslie-Gower prey – predator system through a bounded control input is solved. We use Korobov’s controllability function. The trajectory of the resulting motion is ensured for fulfilling a physical restriction that prey and predator cannot achieve negative values. For this purpose, a certain ellipse depending on given data and the equilibrium point of the considered system is constructed. Simulation results show the effectiveness of the proposed control methodology.

Key words: 
finite-time stabilization
Korobov’s controllability function
bounded control input
prey – predator model
References: 
  1. Collings J. B. The effects of the functional response on the bifurcation behavior of a mite predator – prey interaction model. Journal of Mathematical Biology, 1997, vol. 36, iss. 2, pp. 149–168. https://doi.org/10.1007/s002850050095
  2. Li Y., Xiao D. Bifurcations of a predator – prey system of Holling and Leslie types. Chaos, Solitons and Fractals, 2007, vol. 34, iss. 2, pp. 606–620. https://doi.org/10.1016/j.chaos.2006.03.068
  3. Jiang J., Song Y. Stability and bifurcation analysis of a delayed Leslie – Gower predator – prey system with nonmonotonic functional response. Abstract and Applied Analysis, 2013, vol. 2013, Article ID 152459. https://doi.org/10.1155/2013/152459
  4. Gakkhar S., Singh A. Complex dynamics in a prey predator system with multiple delays. Communications in Nonlinear Science and Numerical Simulation, 2012, vol. 17, iss. 2, pp. 914–929. https://doi.org/10.1016/j.cnsns.2011.05.047
  5. Leslie P., Gower J. The properties of a stochastic model for the predator – prey type of interaction between two species. Biometrika, 1960, vol. 47, iss. 3–4, pp. 219–234. https://doi.org/10.1093/biomet/47.3-4.219
  6. Pielou E. C. An Introduction to Mathematical Ecology. New York, Wiley-Interscience, 1969. 294 p.
  7. Korobov V. I. A general approach to the solution of the problem of synthesizing bounded controls in a control problem. Matematicheskii Sbornik (N. S.), 1979 , vol. 109, no. 4 (8), pp. 582–606 (in Russian). English transl.: Mathematics of the USSR-Sbornik, 1980, vol. 37, no. 4, pp. 535–557. https://doi.org/10.1070/SM1980v037n04ABEH002094
  8. Korobov V. I. Controllability Function Method. Moscow, Izhevsk, Institut komp’iuternykh issledovaniy, 2007. 576 p. (in Russian).
  9. Korobov V. I., Skoryk V. O. Construction of restricted controls for a non-equilibrium point in global sense. Vietnam Journal of Mathematics, 2015, vol. 43, iss. 2, pp. 459–469. https://doi.org/10.1007/s10013-015-0132-4
  10. Polyakov A., Efimov D., Perruquetti W. Finite-time stabilization using implicit Lyapunov function technique. IFAC Proceedings Volumes, 2013, vol. 46, iss. 23, pp. 140–145. https://doi.org/10.3182/20130904-3-FR-2041.00043
  11. Korobov V. I., Sklyar G. M. Methods for constructing positional controls, and a feasible maximum principle. Differential Equations, 1990, vol. 26, no. 11, pp. 1422–1431.
  12. Korobov V. I., Korotyaeva Y. V. Feedback control design for systems with x-discontinuous rigt-hand side. Journal of Optimization Theory and Applications, 2011, vol. 149, pp. 494– 512. https://doi.org/10.1007/s10957-011-9800-z
  13. Singh A. Stabilization of prey predator model via feedback control. In: J. Cushing, M. Saleem, H. Srivastava, M. Khan, M. Merajuddin, eds. Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics & Statistics, vol. 186. Springer, New Delhi, 2016, pp. 177–186. https://doi.org/10.1007/978-81-322-3640-5_10
  14. Kamenkov G. On stability of motion over a finite interval of time. Akad. Nauk SSSR. Prikladnaya Matematika i Mekhanika, 1953, vol. 17, pp. 529–540 (in Russian).
  15. Weiss L., Infante E. F. Finite time stability under perturbing forces and product spaces. IEEE Transactions on Automatic Control, 1967, vol. 12, iss. 1, pp. 54–59. https://doi.org/10.1109/TAC.1967.1098483
  16. LaSalle J., Letfschetz S. Stability by Liapunov’s Direct Method with Applications. New York, London, Academic Press, 1961. 134 p.
  17. Dorato P., Weiss L., Infante E. Comment on “Finite-time stability under perturbing forces and on product spaces”. IEEE Transactions on Automatic Control, 1967, vol. 12, iss. 3, pp. 340–340. https://doi.org/10.1109/TAC.1967.1098569
  18. Dorato P. An Overview of Finite-Time Stability. In: L. Menini, L. Zaccarian, C. T. Abdallah, eds. Current Trends in Nonlinear Systems and Control. Systems and Control: Foundations & Applications. Birkhauser Boston, 2006, pp. 185–194. https://doi.org/10.1007/0-8176-4470-9_10
  19. Bath S. P., Berstein D. S. Lyapunov analysis of finite-time differential equations. Proceedings of 1995 American Control Conference — ACC’95. Seattle, WA, USA, 1995, vol. 3, pp. 1831–1832. https://doi.org/10.1109/ACC.1995.531201
  20. Poznyak A. S., Polyakov A. Y., Strygin V. V. Analysis of finite-time convergence by the method of Lyapunov functions in systems with second-order sliding modes. Journal of Applied Mathematics and Mechanics, 2011, vol. 75, iss. 3, pp. 289–303. https://doi.org/10.1016/j.jappmathmech.2011.07.006
  21. Choque Rivero A. E., Korobov V. I., Skoryk V. O. The controllability function as the time of motion. I. Matematicheskaya Fizika, Analiz, Geometriya [Journal of Mathematical Physics, Analysis, Geometry], 2004, vol. 11, no. 2, pp. 208–225 (in Russian). English transl.: https://arxiv.org/abs/1509.05127
  22. Choque Rivero A. E., Korobov V. I., Skoryk V. O. The controllability function as the time of motion. II. Matematicheskaya Fizika, Analiz, Geometriya [Journal of Mathematical Physics, Analysis, Geometry], 2004, vol. 11, no. 3, pp. 341–354 (in Russian).
  23. Choque Rivero A. E. The controllability function method for the synthesis problem of a nonlinear control system. International Review of Automatic Control, 2008, vol. 1, no. 4, pp. 441–445.
  24. Yefimov N. A. Quadratic Forms and Matrices: An Introduction Approach. New York, London, Academic Press, 1964. 164 p.
Received: 
02.03.2020
Accepted: 
05.05.2020
Published: 
01.03.2021
Short text (in English):
download
(downloads: 81)
  • 1500 reads