Journal of Applied Mathematics

Approximate Controllability of Fractional Integrodifferential Evolution Equations

R. Ganesh, R. Sakthivel, N. I. Mahmudov, and S. M. Anthoni

Full-text: Open access

Abstract

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory, p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 291816, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807989

Digital Object Identifier
doi:10.1155/2013/291816

Mathematical Reviews number (MathSciNet)
MR3056238

Zentralblatt MATH identifier
1266.93019

Citation

Ganesh, R.; Sakthivel, R.; Mahmudov, N. I.; Anthoni, S. M. Approximate Controllability of Fractional Integrodifferential Evolution Equations. J. Appl. Math. 2013 (2013), Article ID 291816, 7 pages. doi:10.1155/2013/291816. https://projecteuclid.org/euclid.jam/1394807989


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References

  • R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095–1100, 2010.
  • J. Tenreiro Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • J. R. 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.
  • 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.
  • L. Shen and J. Sun, “Relative controllability of stochastic nonlinear systems with delay in control,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2880–2887, 2012.
  • J. Klamka, “Stochastic controllability of systems with multiple delays in control,” International Journal of Applied Mathematics and Computer Science, vol. 19, no. 1, pp. 39–47, 2009.
  • J. Klamka, “Constrained exact controllability of semilinear systems,” Systems & Control Letters, vol. 47, no. 2, pp. 139–147, 2002.
  • X. J. Wan, Y. P. Zhang, and J. T. Sun, “Controllability of impulsive neutral functional differential inclusions in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 861568, 8 pages, 2013.
  • J. Klamka, “Constrained controllability of semilinear systems with delayed controls,” Bulletin of the Polish Academy of Sciences, vol. 56, no. 4, pp. 333–337, 2008.
  • J. Klamka, “Constrained controllability of semilinear systems with delays,” Nonlinear Dynamics, vol. 56, no. 1-2, pp. 169–177, 2009.
  • Z. Tai and X. Wang, “Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1760–1765, 2009.
  • S. Kumar and N. Sukavanam, “Approximate controllability of fractional order semilinear systems with bounded delay,” Journal of Differential Equations, vol. 252, no. 11, pp. 6163–6174, 2012.
  • A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442–1450, 2011.
  • J. Klamka, “Local controllability of fractional discrete-time semilinear systems,” Acta Mechanica et Automatica, vol. 5, no. 2, pp. 55–58, 2011.
  • J. Klamka, “Controllability and minimum energy control problem of fractional discrete-time systems,” in New Trends in Nanotechnology and Fractional Calculus Applications, D. Baleanu, Z. B. Guvenc, and J. A. Tenreiro Machado, Eds., pp. 503–509, Springer, New York, NY, USA, 2010.
  • 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.
  • N. I. Mahmudov and A. Denker, “On controllability of linear stochastic systems,” International Journal of Control, vol. 73, no. 2, pp. 144–151, 2000.
  • P. Muthukumar and P. Balasubramaniam, “Approximate controllability of mixed stochastic Volterra-Fredholm type integrodifferential systems in Hilbert space,” Journal of the Franklin Institute, vol. 348, no. 10, pp. 2911–2922, 2011.
  • R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate controllability of semilinear fractional differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1451–1459, 2011.
  • 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.