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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Controllability Criteria for Nonlinear Higher Order Fractional Delay Dynamical Systems with Impulses

Discontinuity, Nonlinearity, and Complexity 8(2) (2019) 225--239 | DOI:10.5890/DNC.2019.06.009

M. Sivabalan, K. Sathiyanathan

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore - 641020, India

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Abstract

This paper is concerned with the controllability of nonlinear higher order fractional delay with impulses, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional delay dynamical systems with impulses has proved, and a sufficient condition for the corresponding nonlinear systems has obtained. Examples has given to verify the results.

References

  1. [1]  Balachandran, K. and Divya, S. (2017), Controllability of nonlinear neutral fractional integrodifferential systems with infinite delay, Journal of Applied Nonlinear Dynamics, 6, 333-344.
  2. [2]  Balachandran, K. and Divya, S. (2016), Relative controllability of nonlinear neutral fractional volterra integrodifferential systems with multiple delays in control, Journal of Applied Nonlinear Dynamics, 5, 147-160.
  3. [3]  Zhang, H., Cao, J.D., and Jiang, W. (2013), Controllability criteria for linear fractional differential systems with state delay and impulses, Journal of Applied Mathematics, Article ID 146010, 9 pages.
  4. [4]  Joice Nirmala, R. and Balachandran, K. (2016), Controllability of nonlinear fractional delay integrodifferential system, Discontinuity, Nonlinearity, and Complexity, 5, 59-73.
  5. [5]  Joice Nirmala, R. and Balachandran, K. (2016), Controallability of fractional nonlinear systems in Banach spaces, Journal of Applied Nonlinear Dynamics, 5, 485-494.
  6. [6]  Joice Nirmala, R. and Balachnadran, K. (2016), Controllability of nonlinear fractional delay integrodifferential systems, Discontinuity, Nonlinearity, and Complexity, 5, 59-73.
  7. [7]  Sivabalan, M. and Sathiyanathan, K (2017), Controllability results for nonlinear higher order fractional delay dynamical systems with distributed delays in control, Global Journal of Pure and Applied Mathematics, 13, 7969-7989.
  8. [8]  Sivabalan,M., Sivasamy, R., and Sathiyanathan, K. (2019), Controllability results for nonlinear higher order fractional delay dynamical systems with control delay, Journal of Applied Nonlinear Dynamics, 8, 211-232.
  9. [9]  Das, S. (2008), Functional fractional calculus for system identification and controls, Berlin, Springer-Verlag.
  10. [10]  Manabe, S. (1961), The non-integer integral and its application to control systems, English Translation Journal Japan, 6, 83-87.
  11. [11]  Magin, R.L. (2004), Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, 32, 1-377.
  12. [12]  Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., and Feliu, V. (2010), Fractional order systems and controls fundamentals and applications, London, Springer.
  13. [13]  Ortigueira, M.D. (2003), On the initial conditions in continuous time fractional linear systems, Signal Process, 83, 2301-2309.
  14. [14]  Bagley, R.L. and Torvik, A. (1983), A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27, 201-210.
  15. [15]  Bagley, R.L. and Torvik, A. (1985), Fractional calculus in the transient analysis of viscoelastically damped structures, American Institute of Aeronautics and Astronautics, 23, 918-925.
  16. [16]  Sabatier, J., Agarwal, O.P., and Tenreiro Machado, J.A. (2007), Advances in fractional calculus, Theoretical developments and applications in physics and engineering, Springer-Verlag.
  17. [17]  Chow, T.S. (2005), Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Physics Letters A, 342, 148-155.
  18. [18]  Hilfer, R. (2000), Applications of Fractional Calculus in Physics, World Scientific Publisher, Singapore.
  19. [19]  Mainardi, F. (1997), Fractional calculus: some basic problems in continuum and statistical mechanics, A Carpinteri, F. Mainardi (Eds.), Fractals and Fractional calculus in Continuum Mechaics, Springer-Verlag, New York, 291-348.
  20. [20]  He, J.H. (1998), Nonlinear oscillation with fractional derivative and its applications, International Conferences on Vibrating Engineering '98, Dalian, China, 288-291.
  21. [21]  Bainov, D. and Simeonov, P. (1993), Impulsive differential equations: Periodic solutions and applications, JohnWiley and Sons, New York, USA.
  22. [22]  Bechohra, M., Henderson, J., and Ntouyas, S. (2006), Impulsive differential equations and inclusions, Hindawai Publishing Corporation, New York, USA.
  23. [23]  Feckan,M., Zhou, Y., andWang, J. (2012), On the concept and existence of soltion for impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulations, 17, 3050-3060.
  24. [24]  Balachandran, K., Kiruthika, S., and Trujillo, J.J. (2011), Existence results for fractional impulsive integrodifferential equations in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 16, 1970-1977.
  25. [25]  Guo, T.L. (2012), Controllability and observability of impulsive fractional linear time invariant system, Computers and Mathematics with Applications, 64, 3171-3181.
  26. [26]  Li, Y. (2014), Controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, Advances in Difference Equations, 2014, 234.
  27. [27]  Yan, Z.M. and Lu, F.X. (2017), Approximate controllability of a multi valued fractional impulsive stochastic partial integrodifferential equation with infinite delay, Applied Mathematics and Computation, 292, 425-447.
  28. [28]  Gu, K.Q., Kharitonov, L.V., and Chen, J. (2003), Stability of time-delay systems, Birkhauser, Basel.
  29. [29]  John, C. and Loiseau, J.J. (2007), Applications of time delay systems, Springer-Verlag, Berlin Heidelberg.
  30. [30]  Wu, M., He, Y., and She, J.H. (2010), Stability analysis and robust control of time delay systems, Springer, Berlin.
  31. [31]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier, Amsterdam.
  32. [32]  Schiff, J.L. (1999), The laplace transform - theory and applications, Springer, New York.
  33. [33]  Smart, D.R. (1974), Fixed point theorems, Cambridge Tract in Mathematics, Cambridge University Press, London-New York.

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