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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 4715861, 15 pages
https://doi.org/10.1155/2017/4715861
Review Article

Controllability Problem of Fractional Neutral Systems: A Survey

Institute of Automatic Control, Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland

Correspondence should be addressed to Michał Niezabitowski; lp.lslop@ikswotibazein.lahcim

Received 4 August 2016; Revised 25 October 2016; Accepted 30 October 2016; Published 18 January 2017

Academic Editor: Leonid Shaikhet

Copyright © 2017 Artur Babiarz and Michał Niezabitowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Zawiski, “On controllability and measures of noncompactness,” in Proceedings of the 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA '14), vol. 1637, pp. 1241–1246, Narvik, Norway, July 2014. View at Publisher · View at Google Scholar
  2. W. Paszke, P. Dabkowski, E. Rogers, and K. Gałkowski, “New results on strong practical stability and stabilization of discrete linear repetitive processes,” Systems & Control Letters, vol. 77, pp. 22–29, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. Wang, Z. Fan, and Y. Zhou, “Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 292–302, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. P. Balasubramaniam, V. Vembarasan, and T. Senthilkumar, “Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 35, no. 2, pp. 177–197, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. Liang and H. Yang, “Controllability of fractional integro-differential evolution equations with nonlocal conditions,” Applied Mathematics and Computation, vol. 254, pp. 20–29, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. P. Balasubramaniam and P. Tamilalagan, “Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function,” Applied Mathematics and Computation, vol. 256, pp. 232–246, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. Karthikeyan, K. Balachandran, and M. Sathya, “Controllability of nonlinear stochastic systems with multiple time-varying delays in control,” International Journal of Applied Mathematics and Computer Science, vol. 25, no. 2, pp. 207–215, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. X. Ding and J. J. Nieto, “Controllability and optimality of linear time-invariant neutral control systems with different fractional orders,” Acta Mathematica Scientia, vol. 35, no. 5, pp. 1003–1013, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Z. Liu and X. Li, “On the exact controllability of impulsive fractional semilinear functional differential inclusions,” Asian Journal of Control, vol. 17, no. 5, pp. 1857–1865, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. T. Mur and H. R. Henriquez, “Relative controllability of linear systems of fractional order with delay,” Mathematical Control and Related Fields, vol. 5, no. 4, pp. 845–858, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. B. Sikora, “Controllability of time-delay fractional systems with and without constraints,” IET Control Theory & Applications, vol. 10, no. 3, pp. 320–327, 2016. View at Publisher · View at Google Scholar · View at Scopus
  12. B. Sikora, “Controllability criteria for timedelay fractional systems with a retarded state,” International Journal of Applied Mathematics and Computer Science, vol. 26, no. 3, pp. 521–531, 2016. View at Google Scholar
  13. J. Klamka and B. Sikora, “New controllability criteria for fractional systems with varying delays,” in Theory and Applications of Non-integer Order Systems: 8th Conference on Non-Integer Order Calculus and Its Applications, Zakopane, Poland, vol. 407 of Lecture Notes in Electrical Engineering, pp. 333–344, Springer, Berlin, Germany, 2017. View at Publisher · View at Google Scholar
  14. L. Górniewicz, S. K. Ntouyas, and D. O'Regan, “Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces,” Reports on Mathematical Physics, vol. 56, no. 3, pp. 437–470, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. X. Fu, “Controllability of non-densely defined functional differential systems in abstract space,” Applied Mathematics Letters, vol. 19, no. 4, pp. 369–377, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. X. Fu and X. Liu, “Controllability of non-densely defined neutral functional differential systems in abstract space,” Chinese Annals of Mathematics Series B, vol. 28, no. 2, pp. 243–252, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. R. Sakthivel, Y. Ren, and N. I. Mahmudov, “Approximate controllability of second-order stochastic differential equations with impulsive effects,” Modern Physics Letters B, vol. 24, no. 14, pp. 1559–1572, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. R. Sakthivel, J. J. Nieto, and N. I. Mahmudov, “Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay,” Taiwanese Journal of Mathematics, vol. 14, no. 5, pp. 1777–1797, 2010. View at Google Scholar · View at MathSciNet · View at Scopus
  19. S. Ji, G. Li, and M. Wang, “Controllability of impulsive differential systems with nonlocal conditions,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6981–6989, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. J. Klamka, A. Babiarz, and M. Niezabitowski, “Banach fixed-point theorem in semilinear controllability problems—a survey,” Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 64, no. 1, pp. 21–35, 2016. View at Publisher · View at Google Scholar
  21. A. Babiarz, J. Klamka, and M. Niezabitowski, “Schauders fixed-point theorem in approximate controllability problems,” International Journal of Applied Mathematics and Computer Science, vol. 26, no. 2, pp. 263–275, 2016. View at Google Scholar
  22. X. Zhang, C. Zhu, and C. Yuan, “Approximate controllability of fractional impulsive evolution systems involving nonlocal initial conditions,” Advances in Difference Equations, vol. 2015, article 244, 2015. View at Publisher · View at Google Scholar
  23. T. Guendouzi and S. Farahi, “Approximate controllability of Sobolev-type fractional functional stochastic integro-differential systems,” Boletín de la Sociedad Matemática Mexicana, vol. 21, no. 2, pp. 289–308, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  24. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Elsevier Science, 1998.
  25. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006. View at MathSciNet
  26. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, CSP, 2009.
  27. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  28. C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  29. T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. D. Baleanu, Fractional Calculus: Models and Numerical Methods, World Scientific Publishing, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  31. T. Kaczorek and L. Sajewski, The Realization Problem for Positive and Fractional Systems, vol. 1 of Studies in Systems, Decision and Control, Springer, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  32. R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, 2014.
  33. O. Bachelier, P. Dabkowski, K. Gałkowski, and A. Kummert, “Fractional and nD systems: a continuous case,” Multidimensional Systems and Signal Processing, vol. 23, no. 3, pp. 329–347, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. Z. Yan, “Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay,” Journal of the Franklin Institute, vol. 348, no. 8, pp. 2156–2173, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. Z. Yan, “Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay,” International Journal of Control, vol. 85, no. 8, pp. 1051–1062, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. A. Debbouche and D. F. Torres, “Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions,” Applied Mathematics and Computation, vol. 243, pp. 161–175, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. Y. Zhou, V. Vijayakumar, and R. Murugesu, “Controllability for fractional evolution inclusions without compactness,” Evolution Equations and Control Theory, vol. 4, no. 4, pp. 507–524, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press, 2016.
  39. Z. Yan, “Existence results for fractional functional integrodifferential equations with nonlocal conditions in Banach spaces,” Annales Polonici Mathematici, vol. 97, no. 3, pp. 285–299, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. S. Liang and R. Mei, “Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions,” Advances in Difference Equations, vol. 2014, article 101, 2014. View at Publisher · View at Google Scholar
  42. A. Debbouche and J. J. Nieto, “Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls,” Applied Mathematics and Computation, vol. 245, pp. 74–85, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. A. Debbouche and D. F. Torres, “Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions,” Fractional Calculus and Applied Analysis, vol. 18, no. 1, pp. 95–121, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  44. Z. Liu and X. Li, “On the controllability of impulsive fractional evolution inclusions in Banach spaces,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 167–182, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. X. Zhang, X. Huang, and Z. Liu, “The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 775–781, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. J. Wang and Y. Zhou, “Complete controllability of fractional evolution systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4346–4355, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. X.-F. Zhou, J. Wei, 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  48. R. P. Agarwal, B. de Andrade, and G. Siracusa, “On fractional integro-differential equations with state-dependent delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1143–1149, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. B. de Andrade, S. Carvalho dos, and P. Jose, “Existence of solution for a fractional neutral integro-differential equation with unbounded delay,” Electronic Journal of Differential Equations, vol. 2012, article 90, 2012. View at Google Scholar
  50. M. Fečkan, J. Wang, and Y. Zhou, “Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 79–95, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  51. K. Yosida, Functional Analysis, Springer, Berlin, Germany, 2013.
  52. P. M. Fitzpatrick and W. V. Petryshyn, “Fixed point theorems for multivalued noncompact acyclic mappings,” Pacific Journal of Mathematics, vol. 54, no. 2, pp. 17–23, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  53. Y. Li, “Controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space,” Advances in Difference Equations, vol. 2014, article 234, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  54. R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21, Springer Science & Business Media, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  55. Z. Yan and X. Jia, “Approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces,” Advances in Difference Equations, vol. 2015, 2015. View at Publisher · View at Google Scholar
  56. A. Lasota and Z. Opial, “An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations,” Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 781–786, 1965. View at Google Scholar
  57. R. P. Agarwal and D. O'Regan, “Leray–Schauder and Krasnoselskii results for multivalued maps defined on pseudo-open subsets of a Fréchet space,” Applied Mathematics Letters, vol. 19, no. 12, pp. 1327–1334, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  58. B. C. Dhage, “Multivalued mapping and fixed point I,” Nonlinear Functional Analysis and Applications, vol. 10, pp. 359–378, 2005. View at Google Scholar
  59. K. Jeet and D. Bahuguna, “Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay,” Journal of Dynamical and Control Systems, vol. 22, no. 3, pp. 485–504, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  60. S. Kailasavalli, D. Baleanu, S. Suganya, and M. M. Arjunan, “Exact controllability of fractional neutral integro-differential systems with state-dependent delay in Banach spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, vol. 24, no. 1, pp. 29–55, 2016. View at Google Scholar · View at MathSciNet
  61. S. Suganya, M. M. Arjunan, and J. J. Trujillo, “Existence results for an impulsive fractional integro-differential equation with state-dependent delay,” Applied Mathematics and Computation, vol. 266, pp. 54–69, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  62. Z. Yan, “Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces,” IMA Journal of Mathematical Control and Information, vol. 30, no. 4, pp. 443–462, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  63. V. Vijayakumar, C. Ravichandran, and R. Murugesu, “Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay,” Nonlinear Studies, vol. 20, no. 4, pp. 513–532, 2013. View at Google Scholar · View at MathSciNet
  64. B. D. Andrade and J. P. C. Dos Santos, “Existence of solutions for a fractional neutral integrodifferential equation with unbounded delay,” Electronic Journal of Differential Equations, vol. 2012, no. 90, 13 pages, 2012. View at Google Scholar
  65. V. Vijayakumar, A. Selvakumar, and R. Murugesu, “Controllability for a class of fractional neutral integro-differential equations with unbounded delay,” Applied Mathematics and Computation, vol. 232, pp. 303–312, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  66. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  67. R. Ganesh, R. Sakthivel, Y. Ren, S. M. Anthoni, and N. I. Mahmudov, “Controllability of neutral fractional functional equations with impulses and infinite delay,” Abstract and Applied Analysis, vol. 2013, Article ID 901625, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  68. R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10334–10340, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  69. S. Liu, A. Debbouche, and J. Wang, “On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths,” Journal of Computational and Applied Mathematics, vol. 312, no. 1, pp. 47–57, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  70. J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  71. J. Wang, Y. Zhou, W. Wei, and H. Xu, “Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1427–1441, 2011. View at Publisher · View at Google Scholar · View at Scopus
  72. J. Wang, Y. Zhou, and M. Medveď, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31–50, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus