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access icon free Stability analysis of fractional differential time-delay equations

This study provides a novel analytical approach to studying the solutions and stability of fractional differential delay equations without using Lyapunov function method. By applying the properties of Caputo fractional derivatives, the Laplace transform and the Mittag–Leffler function, the authors first provide an explicit formula and solution bounds for the solutions of linear fractional differential delay equations. Then, they prove new sufficient conditions for exponential boundedness, asymptotic stability and finite-time stability of such equations. The results are illustrated by numerical examples.

References

    1. 1)
      • 3. Kilbas, A.A., Srivastava, H., Trujillo, J.: ‘Theory and applications of fractional differential equations’ (Elsevier Science, Amsterdam, 2006).
    2. 2)
      • 32. Rakkiyappan, R., Velmurugan, G., Cao, J.: ‘Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays’, Nonlinear Dyn., 2014, 78, pp. 28232836.
    3. 3)
      • 33. Hei, X., Wu, R.: ‘Finite-time stability of impulsive fractional-order systems with time-delay’, Appl. Math. Model., 2016, 40, pp. 42854290.
    4. 4)
      • 8. Lazarevi, M.P., Spasi, A.M.: ‘Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach’, Math. Comput. Model., 2009, 49, pp. 475481.
    5. 5)
      • 9. Deng, W., Li, C., Lu, J.: ‘Stability analysis of linear fractional differential system with multiple time delays’, Nonlinear Dyn., 2007, 48, pp. 409416.
    6. 6)
      • 28. Li, M., Wang, J.: ‘Finite time stability of fractional delay differential equations’, Appl. Math. Lett., 2017, 64, pp. 170176.
    7. 7)
      • 13. Li, C.P., Zhang, F.R.: ‘A survey on the stability of fractional differential equations’, Eur. Phys. J., Spec. Top., 2011, 193, pp. 2747.
    8. 8)
      • 26. Wang, J., Zhang, Y.: ‘On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives’, Appl. Math. Lett., 2015, 39, pp. 8590.
    9. 9)
      • 25. Li, K., Peng, J.: ‘Laplace transform and fractional differential equations’, Appl. Math. Lett., 2011, 24, pp. 20192023.
    10. 10)
      • 2. Podlubny, I.: ‘Fractional differential equations’ (Academic Press, San Diego, 1999).
    11. 11)
      • 18. Li, Y., Chen, Y.Q., Podlubny, I.: ‘Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Lefflers stability’, Comput. Math. Appl., 2010, 59, pp. 18101821.
    12. 12)
      • 16. Hale, J.K., Verduyn Lunel, S.M.: ‘Introduction to functional differential equations’ (Springer Science-Business Media, New York, 1993).
    13. 13)
      • 31. Lazarevic, M., Debeljkovic, D.: ‘Finite-time stability analysis of linear autonomous fractional order systems with delayed state’, Asian J. Control, 2005, 7, pp. 440447.
    14. 14)
      • 29. Krol, K.: ‘Asymptotic properties of fractional delay differential equations’, Appl. Math. Comput., 2011, 218, pp. 15151532.
    15. 15)
      • 27. Wang, J., Ibrahim, A.G., l Feckan, M.: ‘Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces’, Appl. Math. Comput., 2015, 257, pp. 103118.
    16. 16)
      • 34. Pachpatte, B.G.: ‘Inequalities for differential and integral equations’ (Academic Press, New York, 1998).
    17. 17)
      • 4. Wang, J., Zhou, Y., Feckan, M.: ‘Nonlinear impulsive problems for fractional differential equations and Ulam stability’, Comput. Math. Appl., 2012, 64, pp. 33893405.
    18. 18)
      • 15. Kaslik, E., Sivasundaram, S.: ‘Analytical and numerical methods for the stability analysis of linear fractional delay differential equations’, J. Comput. Appl. Math., 2012, 236, pp. 40274041.
    19. 19)
      • 22. Wen, Y., Zhou, X.F., Zhang, Z., et al: ‘Lyapunov method for nonlinear fractional differential systems with delay’, Nonlinear Dyn., 2015, 74, pp. 10151025.
    20. 20)
      • 23. Gao, Z., Liao, X.: ‘Robust stability criterion of fractional-order functions for interval fractional-order systems’, IET Control Theory Appl., 2013, 7, pp. 6067.
    21. 21)
      • 5. Yang, H., Jiang, B.: ‘Stability of fractional-order switched non-linear systems’, IET Control Theory Appl., 2016, 10, pp. 965970.
    22. 22)
      • 12. De la Sen, M.: ‘About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory’, Fixed Point Theory Appl., 2011, 2011, ID 867932, pp. 119.
    23. 23)
      • 17. Erdelyi, A.: ‘Higher transcendental functions’ (McGraw-Hill, New York, 1953).
    24. 24)
      • 20. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., et al: ‘Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2015, 22, pp. 650659.
    25. 25)
      • 14. Cermak, J., Hornicek, J., Kisela, T.: ‘Stability regions for fractional differential systems with a time delay’, Commun. Nonlinear Sci. Numer. Simul., 2016, 31, pp. 108123.
    26. 26)
      • 6. Ge, F., Kou, C.: ‘Stability analysis by Krasnoselskii fixed point theorem for nonlinear fractional differential equations’, Appl. Math. Comput., 2015, 257, pp. 308316.
    27. 27)
      • 21. Sabatier, J., Moze, M., Farges, C.: ‘LMI stability conditions for fractional order systems’, Comput. Math. Appl., 2010, 59, pp. 15941609.
    28. 28)
      • 24. Agarwal, R., O'Regan, D., Hristova, S., et al: ‘Practical stability with respect to initial time difference for Caputo fractional differential equations’, Commun. Nonlinear Sci. Numer. Simul., 2017, 42, pp. 106120.
    29. 29)
      • 10. Johnson, M.A.: ‘Stability of small periodic waves in fractional KdV-type equations’, SIAM J. Math. Anal., 2013, 45, pp. 31683193.
    30. 30)
      • 19. Baleanu, D., Sadati, S.J., Ghaderi, R., et al: ‘Razumikhin stability theorem for fractional systems with delay’, Abs. Appl. Anal., 2010, 2010, ID 124812, pp. 19.
    31. 31)
      • 1. Kiryakova, V.: ‘Generalized fractional calculus and applications’ (Longman Group UK Limited, Essex, UK, 1994).
    32. 32)
      • 30. Yang, Z.Z., Cao, J.: ‘Initial value problems for arbitrary order fractional differential equations with delay’, Commun. Nonlinear Sci. Numer. Simul., 2013, 18, pp. 29933005.
    33. 33)
      • 7. Ye, H., Gao, J., Ding, Y.: ‘A generalized Gronwall inequality and its application to a fractional differential equation’, J. Math. Anal. Appl., 2007, 328, pp. 10751081.
    34. 34)
      • 11. Chen, Y., Moore, K.L.: ‘Analytical stability bound for a class of delayed fractional-order dynamic systems’, Nonlinear Dyn., 2002, 29, pp. 191200.
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