Optimisation problem for some class of hybrid differential-difference systems with delay

DOI: https://doi.org/10.33581/2520-6508-2021-1-6-17

Abstract

In the paper, the linear differential-difference dynamic systems with delayed arguments are considered. Such systems have a lot of application areas, in particular, processes with repetitive and learning structure. We apply the method of the separation hyperplane theorem for convex sets to establish optimality conditions for the control function to drive the trajectory to zero equilibrium state in the fastest possible way. For the special case of the integral control constraints, the proposed method is detailed to establish an analytical form of the optimal control function. The illustrative example is given to demonstrate the obtained results with the step-by-step calculation of the basic elements of the optimal control.

Author Biography

Michael P. Dymkov, Belarus State Economic University, 26 Partyzanski Avenue, Minsk 220070, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of higher mathematics, faculty of digital economy

References

  1. Wang J, He M, Xi J, Yang X. Suboptimal output consensus for time-delayed singular multi-agent systems. Asian Journal of Control. 2018;20(11):721–734. DOI: 10.1002/asjc.1592.
  2. Hale JK, Lunel SMV. Introduction to functional differential equations. New York: Springer-Verlag; 1993. 450 p. (Bloch A, Epstein CL, Goriely A, Greengard L, editors. Applied mathematical sciences; volume 99). DOI: 10.1007/978-1-4612-4342-7.
  3. Grigorieva EV, Kaschenko SA. Asymptotic representation of relaxation oscillations in lasers. Switzerland: Springer International Publishing; 2017. 230 p. DOI: 10.1007/978-3-319-42860-4.
  4. Zhang C, Wang X, Wang C, Zhou W. Synchronization of uncertain complex networks with time-varying node delay and multiple time-varying coupling delays. Asian Journal of Control. 2018;20(1):186–195. DOI: 10.1002/asjc.1539.
  5. Rogers E, Galkowski K, Owens DH. Control systems theory and applications for linear repetitive processes. Berlin: Springer Verlag; 2007. 456 p. (Allgöwer F, Morari M, editors. Lecture notes in control and information sciences; volume 349). DOI: 10.1007/978-3-540-71537-5.
  6. Dymkou S, Rogers E, Dymkov M, Galkowski K, Owens DH. Delay systems approach to linear differential repetitive processes. IFAC Proceedings Volumes. 2003;36(19):333–338. DOI: 10.1016/S1474-6670(17)33348-7.
  7. Dymkou S, Rogers E, Dymkov M, Galkowski K, Owens DH. An approach to controllability and optimization problems for repetitive processes. In: Stability and control processes (SCP-2005). Proceedings of international conference; Saint Petersburg, Russia. Volume II. [S. l.]: Saint Petersburg University; 2005. p. 1504–1516.
  8. Dymkou S. Graph and 2-D optimization theory and their application for discrete simulation of gas transportation networks and industrial processes with repetitive operations [dissertation]. Aachen: RWTH; 2006. 147 p.
  9. Marchenko VM. Hybrid discrete-continuous systems. II. Controllability and reachability. Differential Equations. 2013;49(1):112–125.
  10. Gabasov R, Kirillova FM. The qualitative theory of optimal processes. New York: M. Dekker; 1976. 640 p.
  11. Dymkou S, Dymkov M, Rogers E, Galkowski K. Optimal control of non-stationary differential linear repetitive processes. Integral Equations and Operator Theory. 2008;60:201–216.
  12. Dymkov M, Rogers E, Dymkou S, Galkowski K. Constrained optimal control theory for differential linear repetitive processes. SIAM Journal on Control and Optimization. 2008;47(1):396–420. DOI: 10.1137/060668298.
Published
2021-04-13
Keywords: differential-difference system, delayed argument, time optimal control problem
How to Cite
Dymkov, M. P. (2021). Optimisation problem for some class of hybrid differential-difference systems with delay. Journal of the Belarusian State University. Mathematics and Informatics, 1, 6-17. https://doi.org/10.33581/2520-6508-2021-1-6-17
Issue
No 1 (2021): Journal of the Belarusian State University. Mathematics and Informatics
Section
Differential Equations and Optimal Control

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