Kybernetika 53 no. 1, 161-178, 2017

Relative controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control

Rajagopal Joice Nirmala and Krishnan BalachandranDOI: 10.14736/kyb-2017-1-0161

Abstract:

This paper describes the controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control. Necessary and sufficient conditions for the controllability criteria for linear fractional delay system are established. Further sufficient conditions for the controllability of nonlinear fractional delay integrodifferential system are obtained by using fixed point arguments. Examples are provided to illustrate the results.

Keywords:

controllability, fractional delay integrodifferential equation, Laplace transform, Mittag-Leffler function, Caputo fractional derivative

Classification:

34A08, 93B05

References:

  1. R. L. Bagley and P. J. Torvik: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 (1983), 201-210.   DOI:10.1122/1.549724
  2. R. L. Bagley and P. J. Torvik: Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23 (1985), 918-925.   DOI:10.2514/3.9007
  3. K. Balachandran: Global relative controllability of non-linear systems with time-varying multiple delays in control. Int. J. Control. 46 (1987), 193-200.   DOI:10.1080/00207178708933892
  4. K. Balachandran and J. P. Dauer: Controllability of perturbed nonlinear delay systems. IEEE Trans. Autom. Control. 32(1987), 172-174.   DOI:10.1109/tac.1987.1104536
  5. K. Balachandran, J. Kokila and J.J. Trujillo: Relative controllability of fractional dynamical systems with multiple delays in control. Comput. Math. Appl. 64 (2012), 3037-3045.   DOI:10.1016/j.camwa.2012.01.071
  6. K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control. Commun. Nonlinear. Sci. Numer. Simul. 17 (2012), 3508-3520.   DOI:10.1016/j.cnsns.2011.12.018
  7. K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with distributive delays in control. Comput. Math. Appl. 64(2012), 3201-3209.   DOI:10.1016/j.camwa.2011.11.061
  8. R. Bellman and K. L. Cooke: Differential Difference Equations. Academic Press, New York 1963.   DOI:10.1002/zamm.19650450612
  9. T. S. Chow: Fractional dynamics of interfaces between soft-nanoparticles and rough substrates. Physics Letter A 342 (2005), 148-155.   DOI:10.1016/j.physleta.2005.05.045
  10. J. P. Dauer and R. D. Gahl: Controllability of nonlinear delay systems. J. Optimiz. Theory. App. 21 (1977), 59-68.   DOI:10.1007/bf00932544
  11. J. P. Dauer: Nonlinear perturbations of quasi-linear control systems. J. Math. Anal. Appl. 54 (1976), 717-725.   DOI:10.1016/0022-247x(76)90191-8
  12. A. Halanay: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York 1966.   DOI:10.1016/s0076-5392(08)x6057-6
  13. J. Hale: Theory of Functional Differential Equations. Springer, New York 1977.   DOI:10.1007/978-1-4612-9892-2
  14. J. H. He: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods. Appl. Mech. Eng. 167 (1998), 57-68.   DOI:10.1016/s0045-7825(98)00108-x
  15. R. Joice Nirmala and K. Balachandran: Controllability of nonlinear fractional delay integrodifferential systems. J. Applied Nonlinear Dynamics 5 (2016), 59-73.   DOI:10.5890/dnc.2016.03.007
  16. R. Joice Nirmala, K. Balachandran, L. R. Germa and J. J. Trujillo: Controllability of nonlinear fractional delay dynamical systems. Rep. Math. Phys. 77 (2016), 87-104.   DOI:10.1016/s0034-4877(16)30007-6
  17. T. Kaczorek: Selected Problems of Fractional Systems Theory: Lecture Notes in Control and Information Science. Springer-Verlag, Berlin 2011.   DOI:10.1007/978-3-642-20502-6
  18. J. Klamka: Controllability of linear systems with time variable delay in control. Int. J. Control 24(1976), 869-878.   DOI:10.1080/00207177608932867
  19. J. Klamka: Relative controllability of nonlinear systems with delay in control. Automatica 12(1976), 633-634.   DOI:10.1016/0005-1098(76)90046-7
  20. A. Kilbas, H. M. Srivastava and J. J. Trujillo: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam 2006.   CrossRef
  21. J. T. Machado: Analysis and design of fractional order digital control systems. Systems Analysis, Modelling and Simulation 27 (1997), 107-122.   CrossRef
  22. J. T. Machado, A. C. Costa and M. D. Quelhas: Fractional dynamics in DNA. Commun. Nonlinear. Sci. Numer. Simul. 16 (2011), 2963-2969.   DOI:10.1016/j.cnsns.2010.11.007
  23. R. L Magin: Fractional calculus in bioengineering. Critical Rev. Biomed. Eng. 32 (2004), 1-377.   DOI:10.1615/critrevbiomedeng.v32.i1.10}, \href{http://dx.doi.org/10.1615/critrevbiomedeng.v32.i2.10}{DOI:10.1615/critrevbiomedeng.v32.i2.10}, \href{http://dx.doi.org/10.1615/critrevbiomedeng.v32.i34.10}{DOI:10.1615/critrevbiomedeng.v32.i34.10
  24. F. Mainardi: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, eds.), Springer-Verlag 1997, pp. 291-348.   DOI:10.1007/978-3-7091-2664-6_7
  25. R. Manzanilla, L. G. Marmol and C. J. Vanegas: On the controllability of differential equation with delayed and advanced arguments. Abstr. Appl. Anal. 2010 (2010), 1-16.   DOI:10.1155/2010/307409
  26. K. S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley and Sons, New York 1993.   CrossRef
  27. T. Mur and H. R. Henriquez: Relative controllability of linear systems of fractional order with delay. Math. Control. Relat. F 5(2015), 845-858.   DOI:10.3934/mcrf.2015.5.845
  28. M. N. Oguztoreli: Time-Lag Control Systems. Academic Press, New York 1966.   DOI:10.1016/s0076-5392(08)x6192-2
  29. K. B. Oldham and J. Spanier: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York 1974.   DOI:10.1016/s0076-5392(09)x6012-1
  30. M. D. Ortigueira: On the initial conditions in continuous time fractional linear systems. Signal Process 83 (2003), 2301-2309.   DOI:10.1016/s0165-1684(03)00183-x
  31. I. Podlubny: Fractional Differential Equations. Academic Press, New York 1999.   CrossRef
  32. I. Podlubny: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of their Solution and Some of their Applications. Academic Press, 1999.   DOI:10.1016/s0076-5392(99)x8001-5
  33. J. Sabatier, O. P. Agrawal and J. A. Tenreiro-Machado (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer-Verlag, New York 2007.   DOI:10.1007/978-1-4020-6042-7
  34. J. L. Schiff: The Laplace Transform: Theory and Applications. Springer, New York 1999.   DOI:10.1007/978-0-387-22757-3
  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. H. Smith: An Introduction to Delay Differential Equations with Application to the Life Sciences. Springer, New York 2011.   DOI:10.1007/978-1-4419-7646-8
  37. J. Wei: The controllability of fractional control systems with control delay. Comput. Math. Appl. 64 (2012), 3153-3159.   DOI:10.1016/j.camwa.2012.02.065