Controllability Analysis of Nonlinear Neutral-type Fractional-order Differential Systems with State Delay and Impulsive Effects

  • B. Sundara Vadivoo
  • Raja Ramachandran
  • Jinde Cao
  • Hai Zhang
  • Xiaodi Li
Regular Paper Control Theory and Applications
  • 36 Downloads

Abstract

This paper is concerned with the controllability problem of nonlinear neutral-type fractional differential systems with state delay and impulsive effects. By using the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function and Laplace transform, a new set of sufficient conditions are obtained for the considered system to be controllable. Finally, two numerical examples are given to demonstrate the validity of the obtained theoretical results.

Keywords

Caputo fractional derivative controllability fractional integro-differential equations impulses neutraltype state delay 

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References

  1. [1]
    J. Shen and J. Cao, “Necessary and sufficient conditions for consensus of delayed fractional-order systems,” Asian Journal of Control, vol. 14, no. 6, pp. 1690–1697, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    C. Huang, J. Cao, M. Xiao, A. Alsaedi, and F.E. Alsaadi, “Controlling bifurcation in a delayed fractional predatorprey system with incommensurate orders,” Applied Mathematics and Computation, vol. 293, pp. 293–310, 2017. [click]MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Huang, J. Cao, M. Xiao, A. Alsaedi, and T. Hayat, “Bifurcations in a delayed fractional complex-valued neural network,” Applied Mathematics and Computation, vol. 292, pp. 210–227, 2017. [click]MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Hu, J. Cao, and A. Hu, “Exponential stability of discrete-time recurrent neural networks with time-varying delays in the leakage terms and linear fractional uncertainties,” IMA Journal of Mathematical Control and Information, vol. 31, pp. 345–362, 2014. [click]MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.MATHGoogle Scholar
  6. [6]
    K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.MATHGoogle Scholar
  7. [7]
    I. Podlubny, Fractinal Differential Equations, vol.198 of Mathematics in Science and Engineering. Technical University of Kosice, Kosice, Slovak Rebublic, 1999.Google Scholar
  8. [8]
    T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 2011.CrossRefMATHGoogle Scholar
  9. [9]
    K. Oldham and J. Spanier, The fractional Calculus, Academic Press, New York, 1974.MATHGoogle Scholar
  10. [10]
    S. A. Ammour, S. Djennoune, and M. Bettayeb, “A slidingmode control for linear fractional systems with input and state delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2310–2318, 2009.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    K. Balachandran, J. Y. Park, and J. J. Trujillo, “Controllability of nonlinear fractional dynamical systems,” Nonlinear Analysis, Theory, Methods and Applications. An International Multidisciplinary Journal Series A: Theory and Methods, vol. 75, no. 4, pp. 1919–1926, 2012.MathSciNetMATHGoogle Scholar
  12. [12]
    K. Balachandran, J. Kokila, and J. J. Trujillo, “Relative controllability of fractional dynamical systems with multiple delays in control,” Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3037–3045, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    K. Balachandran, Y. Zhou, and J. Kokila, “Relative controllability of fractional dynamical systems with distributed delays in control,” Computers and Mathematics with Applications, vol. 64, pp. 3201–3209, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    K. Balachandran and S. Divya, “Controllability of nonlinear implicit fractional integro-differential systems,” International Journal of Applied Mathematics and Computer Science, vol. 24, no. 4, pp. 713–722, 2014. [click]MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Debbouche and D. Baleanu, “Exact null controllability for fractional nonlocal integro-differential equations via implicit evolution system,” Journal of Applied Mathematics, vol. 2012, Article ID931975, 17 pages, 2012. [click]CrossRefMATHGoogle Scholar
  16. [16]
    T. L. Guo, “Controllability and observability of impulsive fractional linear time-invariant system,” Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3171–3182, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    W. Jiang and W. Z. Song, “Controllability of singular systems with control delay,” Automatica, vol. 37, no. 11, pp. 1873–1877, 2001. [click]CrossRefMATHGoogle Scholar
  18. [18]
    N. I. Mahmudov, “Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID502839, 9 pages, 2013. [click]MathSciNetMATHGoogle Scholar
  19. [19]
    X. F. Zhou, W. Jiang, and L. G. Hu, “Controllability of a fractional linear time-invariant neutral dynamical system,” Applied Mathematics Letters, vol. 26, no. 4, pp. 418–424, 2013. [click]MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    I. Györi and J. Wu, “A neutral equation arising from compartmental systems with pipes,” Journal of Dynamics and Differential Equations, vol. 3, no. 2, pp. 289–311, 1991. [click]MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    X. Li and S. Song, “Stabilization of delay systems: delaydependent impulsive control,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 406–411, 2017. [click]MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    X. Li and J. Cao, “An impulsive delay inequality involving unbounded time-varying delay and applications,” IEEE Transactions on Automatic Control, DOI: 10.1109/TAC.2017.2669580. In press 2017.Google Scholar
  23. [23]
    J. Cao and R. Li, “Fixed-time synchronization of delayed memristor-based recurrent neural networks,” Science China Information Sciences, vol. 60, no. 3, Article ID032201, 2017.Google Scholar
  24. [24]
    Y. Wan, J. Cao, G. Wen, and W. Yu, “Robust fixed-time synchronization of delayed Cohen-Grossberg neural networks,” Neural Networks, vol. 73, pp. 86–94, 2016. [click]CrossRefGoogle Scholar
  25. [25]
    J. Cao, R. Rakkiyappan, K. Maheswari, and A. Chandrasekar, “Exponential H filtering analysis for discretetime switched neural networks with random delays using sojourn probabilities,” Science China Technological Sciences. vol. 59, no. 3, pp. 387–402, 2016. [click]CrossRefGoogle Scholar
  26. [26]
    F. Chen and Y. Zhou, “Attractivity of fractional functional differential equations,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1359–1369, 2011. [click]MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    E. Kaslik and S. Sivasundaram, “Analytical and numerical methods for the stability analysis of linear fractional delay differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 16, pp. 4027–4041, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    X. Zhang, “Some results of linear fractional order timedelay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407–411, 2008. [click]MathSciNetCrossRefGoogle Scholar
  29. [29]
    Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for p-type fractional neutral differential equations,” Nonlinear Analysis: Theory, Methods and Applications. An International Multidisciplinary Journal: Series A: Theory and Methods, vol. 71, no. 7–8, pp. 2724–2733, 2009.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    X. Li, M. Bohner, and C. Wang, “Impulsive differential equations: periodic solutions and applications,” Automatica, vol. 52, pp. 173–178, 2015.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    X. Li and J. Wu, “Stability of nonlinear differential systems with state-dependent delayed impulses,” Automatica, vol. 64, pp. 63–69, 2016. [click]MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    X. Li, X. Zhang, and S. Song, “Effect of delayed impulses on input-to-state stability of nonlinear systems,” Automatica, vol. 76, pp. 378–382, 2017. [click]MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    Q. Zhu and J. Cao, “Stability of Markovian jump neural networks with impulse control and time varying delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2259–2270, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, John Wiley and Sons, New York, NY, USA, 1993.MATHGoogle Scholar
  35. [35]
    M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, NY, USA, 2, 2006.CrossRefMATHGoogle Scholar
  36. [36]
    M. Feickan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. [click]MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    R. E. Kalman, Y. C. Ho, and K. S. Narendra, “Controllability of linear dynamical systems,” Contributions to Differential Equations, vol. 1, no. 2, pp. 189–213, 1963.MathSciNetMATHGoogle Scholar
  38. [38]
    H. Shi, G. M. Xie, and W. G. Luo, “Controllability of linear discrete-time systems with both delayed states and delayed inputs,” Abstract and Applied Analysis, vol. 2013, Article ID 975461, 5 pages, 2013. [click]MathSciNetMATHGoogle Scholar
  39. [39]
    H. Shi and G. Xie, “Controllability and observability criteria for linear piecewise constant impulsive systems,” Journal of Applied Mathematics, vol. 2012, Article ID 182040, 24 pages, 2012. [click]MathSciNetMATHGoogle Scholar
  40. [40]
    X. J. Wan, Y. P. Zhang, and J. T. Sun, “Controllability of impulsive neutral functional differential inclusions in Banach spaces,” Journal of Applied Mathematics, vol. 2013, Article ID 861568, 8 pages, 2013. [click]Google Scholar
  41. [41]
    L. Zhang, Y. Ding, T. Wang, L. Hu, and K. Hao, “Controllability of second-order semilinear impulsive stochastic neutral functional evolution equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 748091, 13 pages, 2012. [click]MathSciNetMATHGoogle Scholar
  42. [42]
    H. Zhang, J. Cao, and W. Jiang, “Controllability criteria for linear fractional differential systems with state delay and impulses,” Journal of Applied Mathematics, vol. 2013, Article ID 146010, 9 pages, 2013. [click]MathSciNetMATHGoogle Scholar
  43. [43]
    S. MarirMohammed, C. Djillali, and Bouagada, “A novel approach of admissibility for singular linear continuoustime fractional-order systems,” Int. Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 959–964, 2017. [click]CrossRefGoogle Scholar
  44. [44]
    R. L. Bagley and R. A. Calico, “New admissibility conditions for singular linear continuous-time fractional-order systems,” Journal of the Franklin Institute, vol. 354, no. 2, pp. 752–766, 2017. [click]MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • B. Sundara Vadivoo
    • 1
  • Raja Ramachandran
    • 2
  • Jinde Cao
    • 3
    • 4
  • Hai Zhang
    • 5
  • Xiaodi Li
    • 6
  1. 1.Department of MathematicsAlagappa UniversityKaraikudiIndia
  2. 2.Ramanujan Centre for Higher MathematicsAlagappa UniversityKaraikudiIndia
  3. 3.School of Mathematics, and Research Center for Complex Systems and Network SciencesSoutheast UniversityNanjingChina
  4. 4.School of Mathematics and StatisticsShandong Normal UniversityJinanChina
  5. 5.School of Mathematics and Computation ScienceAnqing Normal UniversityAnqingChina
  6. 6.School of Mathematics and StatisticsShandong Normal UniversityJi’nanChina

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