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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Trajectory controllability of fractional-order α ∈(1,2] systems with delay

Journal of Applied Nonlinear Dynamics 7(2) (2018) 111--122 | DOI:10.5890/JAND.2018.06.001

V. Srinivasa; N. Sukavanam

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand-247667, India

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Abstract

This paper is concerned with trajectory controllability of a class of fractional-order systems of order α ∈ (1,2] with delay in state variable and with a nonlinear control term. Firstly, the existence and uniqueness of the system is proved under suitable conditions on the nonlinear term involving state variable. Then the trajectory controllability of this class of systems is studied using Mittag-Leffler functions and Gronwall-Bellman inequality. Finally, examples are given to illustrate the proposed theory.

References

  1. [1]  Bagley, R.L. and Torvik, P.J. (1986), On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30, 133-155.
  2. [2]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, no. 204, Elsevier: Amsterdam.
  3. [3]  Rivero, M., Trujillo, J.J., Vázquez, L., and Velasco, M.P. (2011), Fractional dynamics of populations, Applied Matheamtics and Computation, 218, 1089-1095.
  4. [4]  Pitcher, E. and Sewell, W.E. (1938), Existence theorems for solutions of differential equations of non-integral order, Bulletin of American Mathematical Society, 44(2), 100-107.
  5. [5]  Diethelm, K. (2010), The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004, Springer: Heidelberg.
  6. [6]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley: New York.
  7. [7]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press: San Diego.
  8. [8]  Benchohra, M., Henderson, J., Ntouyas, S.K., and Ouahab, A. (2008), Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338, 1340-1350.
  9. [9]  Lakshmikantham, V. (2008), Theory of fractional functional differential equations, Nonlinear Analysis, 69, 3337-3343.
  10. [10]  Maraaba, T.A., Jarad, F., and Baleanu, D. (2008), On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China Series A: Mathematics, 122 V. Srinivasan, N. Sukavanam / Journal of Applied Nonlinear Dynamics 7(2) (2018) 111–122 51(10), 1775-1786.
  11. [11]  Bhalekar, S. and Gejji, V.D. (2011), A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, Journal of Fractional Calculus and Applications, 1(5), 1-9.
  12. [12]  Adams, J.L. and Hartley, T.T. (2008), Finite time controllability of fractional order systems, Journal of Computational Nonlinear Dynamics, 3(2), 214021-214025.
  13. [13]  Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D. and Feliu, V. (2010), Fractional-order Systems and Controls Fundamentals and Applications, Advances in Industrial Control, no. 14-18, Springer: London.
  14. [14]  Surendra, K. and Sukavanam, N. (2012), Approximate controllability of fractional order semilinear systems with bounded delay, Journal of Differential Equations, 252, 6163-6174.
  15. [15]  Wei, J. (2012), The controllability of fractional control systems with control delay, Computers and Mathematics with Applications, 64, 3153-3159.
  16. [16]  Chalishajar, D.N., George,R.K., Nandakumaran, A.K., and Acharya, F.S. (2010), Trajectory controllability of nonlinear integro-differential systems, Journal of Franklin Institute, 347, 1065-1075.
  17. [17]  Bin, M. and Liu, Y. (2013), Trajectory controllability of semilinear differential evolution equations with impulses and delay, Open Journal of Applied Sciences, 3, 37-43.
  18. [18]  Klamka, J., Ferenstein, E., Babiarz, A., Czornik, A., and Niezabitowsk, M. (2015), Trajectory controllability of semilinear systems with delay in control and state, Applied Mechanics and Material, 789-790, 1045-1051.
  19. [19]  Govindaraj, V., Malik, M., and George, R.K. (2016), Trajectory controllability of fractional dynamical systems, Journal of Control and Decision, in press, DOI: 10.1080/23307706.2016.1249422
  20. [20]  Joshi, M.C., and Bose, R.K. (1985), Some topics in nonlinear functional analysis, Wiley Eastern Ltd.: New Delhi.
  21. [21]  McKibben, M.A. (2011), Discovering evolution equations with applications: Volume 1-Deterministic equations, Chapman & Hall/CRC: Boca Raton.
  22. [22]  Gómez-Aguilar, J.F. and Yépez-Martínez, H. and Calderón-Ramón, C. and Cruz-Orduña, I. and Escobar-Jiménez, R.F., and Olivares-Peregrino, V.H. (2015), Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel, 17, 6289-6303.