Kybernetika 55 no. 4, 675-689, 2019

On application of Rothe's fixed point theorem to study the controllability of fractional semilinear systems with delays

Beata SikoraDOI: 10.14736/kyb-2019-4-0675

Abstract:

The paper presents finite-dimensional dynamical control systems described by semilinear fractional-order state equations with multiple delays in the control and nonlinear function $f$. The relative controllability of the presented semilinear system is discussed. Rothe's fixed point theorem is applied to study the controllability of the fractional-order semilinear system. A control that steers the semilinear system from an initial complete state to a final state at time $t>0$ is presented. A numerical example is provided to illustrate the obtained theoretical results and a practical example is given to show a possible application of the study.

Keywords:

the Caputo derivative, delays in control, fractional systems, semilinear control systems, Rothe's fixed point theorem, pseudo-transition matrix

Classification:

93B05, 93C05, 93C10, 34G20

References:

  1. A. Babiarz, A. Czornik and M. Niezabitowski: Output controllability of the discrete-time linear switched systems. Nonlinear Analysis: Hybrid Systems 21 (2016), 1-10.   DOI:10.1016/j.nahs.2015.12.004
  2. A. Babiarz and M. Niezabitowski: Controllability Problem of Fractional Neutral Systems: A Survey. Math. Problems Engrg. 4715861 (2017), 1-15.   DOI:10.1155/2017/4715861
  3. K. Balachandran and J. Kokila: On the controllability of fractional dynamical systems. Int. J. Appl. Math. Comput. Sci. 22 (2012), 523-531.   DOI:10.2478/v10006-012-0039-0
  4. K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control. Commun. Nonlinear Sci. Numer. Simul. {\mi 17   CrossRef
  5. K. Balachandran, J. Y. Park and J. J. Trujillo: Controllability of nonlinear fractional dynamical systems. Nonlinear Analysis 75 (2012), 1919-1926.   DOI:10.1016/j.na.2011.09.042
  6. K. Balachandran and J. Kokila: Constrained controllability of fractional dynamical systems. Numer. Functional Anal. Optim. 34 (2013), 1187-1205.   DOI:10.1080/01630563.2013.778868
  7. K. Balachandran and J. Kokila: Controllability of fractional dynamical systems with prescribed controls. IET Control Theory Appl.7 (2013), 1242-1248.   DOI:10.1049/iet-cta.2012.0049
  8. K. Balachandran: Controllability of Nonlinear Fractional Delay Dynamical Systems with Multiple Delays in Control Lect. Notes Electr. Engrg. Theory and Applications of Non-integer Order Systems 407 (2016), 321-332.   DOI:10.1007/978-3-319-45474-0\_29
  9. Y. Q. Chen, H. S. Ahn and D. Xue: Robust controllability of interval fractional order linear time invariant systems. Signal Processes 86 (2006), 2794-2802.   DOI:10.1016/j.sigpro.2006.02.021
  10. W. Deng, C. Li and J. Lu: Stability analysis of linear fractional differential systems with multiple time delays. Nonlinear Dynamics 48 (2007), 409-416.   DOI:10.1007/s11071-006-9094-0
  11. G. Isac: On Rothe's fixed point theorem in a general topological vector space. An. St. Univ. Ovidius Constanta 12 (2004), 127-134.   CrossRef
  12. E. Iturriaga and H. Leiva: A characterization of semilinear surjective operators and applications to control problems. Appl. Math. 1 (2010), 265-273.   DOI:10.4236/am.2010.14033
  13. T. Kaczorek: Selected Problems of Fractional Systems Theory. Lect. Notes Control Inform. Sci. 2011.   DOI:10.1007/978-3-642-20502-6
  14. T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits. Studies in Systems, Decision and Control, 2015.   DOI:10.1007/978-3-319-11361-6
  15. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 2006.   DOI:10.1016/s0304-0208(06)x8001-5
  16. J. Klamka: Controllability of Dynamical Systems. Kluwer Academic Publishers, 1991.   CrossRef
  17. J. Klamka: Controllability and minimum energy control problem of fractional discrete-time systems. New Trends Nanotechology and Fractional Calculus Applications, Springer, 2010.   DOI:10.1007/978-90-481-3293-5\_45
  18. J. Klamka: Local controllability of fractional discrete-time semilinear systems. Acta Mechanica at Automatica 5 (2011), 55-58.   CrossRef
  19. J. Klamka: Controllability of dynamical systems. A survey. Bull. Pol. Ac.: Tech. Sci. 61 (2013), 335-342.   DOI:10.2478/bpasts-2013-0031
  20. J. Klamka, A. Czornik, M. Niezabitowski and A. Babiarz: Controllability and minimum energy control of linear fractional discrete-time infinite-dimensional systems. In: Proc. 11th IEEE International Conference on Control and Automation, Taiwan 2014, pp. 1210-1214.   DOI:10.1109/icca.2014.6871094
  21. J. Klamka and A. Czornik: Controllability problem of positive discrete fractional systems with constant delay. In: Proc. 17th International Carpathian Control Conference 2016, pp. 324-328.   DOI:10.1109/carpathiancc.2016.7501117
  22. J. Klamka and B. Sikora: New controllability Criteria for Fractional Systems with Varying Delays. Lect. Notes Electr. Engrg. Theory and Applications of Non-integer Order Systems 407 (2017), 333-344.   DOI:10.1007/978-3-319-45474-0\_30
  23. H. Leiva: Rothe's fixed point theorem and controllability of semilinear nonautonomous systems. Systems Control Lett. 67 (2014), 14-18.   DOI:10.1016/j.sysconle.2014.01.008
  24. W. L. Luyben: Process Modelling, Simulation and Control for Chemical Engineers. McGraw-Hill, Chemical Engineering Series, International Editions, 1990.   DOI:10.1002/pol.1973.130110416
  25. K. S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Calculus. Villey 1993.   CrossRef
  26. A. Monje, Y. Chen, B. M. Viagre, D. Xue and V. Feliu: Fractional-order Systems and Controls. Fundamentals and Applications. Springer-Verlag 2010.   DOI:10.1007/978-1-84996-335-0
  27. R. J. Nirmala, K. Balachandran, L. Rodriguez-Germa and J. J. Trujillo: Controllability of nonlinear fractional delay dynamical systems. Reports Math. Physics 77 (2016), 87-104.   DOI:10.1016/s0034-4877(16)30007-6
  28. Z. M. Odibat: Analytic study on linear systems of fractional differential equations. Computers Math. Appl. 59 (2010), 1171-1183.   DOI:10.1016/j.camwa.2009.06.035
  29. K. B. Oldham and J. Spanier: The Fractional Calculus. Academic Press 1974.   CrossRef
  30. I. Podlubny: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. In: Mathematics in Science and Engineering, Academic Press 1999.   CrossRef
  31. J. Sabatier, O. P. Agrawal and J. A Tenreiro Machado: Advances in Fractional Calculus. In: Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag 2007.   DOI:10.1007/978-1-4020-6042-7
  32. R. Sakthivel, Y. Ren and N.I. Mahmudov: On the approximate controllability of semilinear fractional differential systems. Computers Math. Appl. 62 (2011), 1451-1459.   DOI:10.1016/j.camwa.2011.04.040
  33. R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 3498-3508.   DOI:10.1016/j.cnsns.2013.05.015
  34. S. G. Samko, A. A. Kilbas and O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordan and Breach Science Publishers 1993.   CrossRef
  35. B. Sikora: Controllability of time-delay fractional systems with and without constraints. IET Control Theory Appl. 10 (2016), 320-327.   DOI:10.1049/iet-cta.2015.0935
  36. B. Sikora: Controllability criteria for time-delay fractional systems with a retarded state. Int. J. Appl. Math. Computer Sci. 26 (2016), 521-531.   DOI:10.1515/amcs-2016-0036
  37. B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control. Systems Control Lett. 106 (2017), 9-15.   DOI:10.1016/j.sysconle.2017.04.013
  38. B. Sikora and J. Klamka: Cone-type constrained relative controllability of semilinear fractional systems with delays. Kybernetika 53 (2017), 370-381.   DOI:10.14736/kyb-2017-2-0370
  39. J. D. R. Smart: Fixed Points Theorems. Cambridge University Press, 1974.   CrossRef
  40. W. Trzasko: Reachability and controllability of positive fractional discrete-time systems with delay. J. Automat. Mobile Robotics Intell. Systems 2 (2008), 43-47.   CrossRef
  41. J. Wang and Y. Zhou: Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 4346-4355.   DOI:10.1016/j.cnsns.2012.02.029
  42. J. Wei: The controllability of fractional control systems with control delay. Computers Math. Appl. 64 (2012), 3153-3159.   DOI:10.1016/j.camwa.2012.02.065