Approximate Controllability of Fractional Order Retarded Semilinear Control Systems

Affiliation(s)

Department of Mathematics, Haramaya University, Dire Dawa, Ethiopia.

Department of Mathematics, Indian Institute Technology Roorkee (IITR), Roorkee, India.

Department of Mathematics, Haramaya University, Dire Dawa, Ethiopia.

Department of Mathematics, Indian Institute Technology Roorkee (IITR), Roorkee, India.

ABSTRACT

In this paper, approximate controllability of fractional
order retarded semilinear systems is studied when the nonlinear term satisfies
the newly formulated bounded integral contractor-type conditions. We have shown
the existence and uniqueness of the mild solution for the fractional order
retarded semilinear systems using an iterative procedure approach. Finally, we
obtain the approximate controllability results of the system under simple
condition.

KEYWORDS

Approximate Controllability; Fractional Order; Existence and Uniqueness, Retarded; Semilinear System; Integral Contractor

Approximate Controllability; Fractional Order; Existence and Uniqueness, Retarded; Semilinear System; Integral Contractor

Cite this paper

S. Tafesse and N. Sukavanam, "Approximate Controllability of Fractional Order Retarded Semilinear Control Systems,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 3, 2013, pp. 153-160. doi: 10.4236/ijmnta.2013.23020.

S. Tafesse and N. Sukavanam, "Approximate Controllability of Fractional Order Retarded Semilinear Control Systems,"

References

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[2] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, 2003, pp. 3413-3442.

[3] J. R. Wang and Y. Zhou, “Complete Controllability of Fractional Evolution Systems,” Communication in Nonlinear Science and Numerical Simulation, Vol. 17, No. 11, 2012, pp. 4346-4355. doi:10.1016/j.cnsns.2012.02.029

[4] 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, 2009, pp. 1760-1765. doi:10.1016/j.aml.2009.06.017

[5] R. Sakthivel, Y. Ren and N. I. Mahmudov, “On the Approximate Controllability of Semilinear Fractional Differential Systems,” Computers and Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 1451-1459.

[6] S. Kumar and N. Sukavanam, “Approximate Controllability of Fractional Order Semilinear Systems with Bounded Delay,” Journal of Differential Equations, Vol. 252, No. 11, 2012, pp. 6163-6174. doi:10.1016/j.jde.2012.02.014

[7] Y. Zhou and F. Jiao, “Existence of Mild Solutions for Fractional Neutral Evolutions,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1063-1077. doi:10.1016/j.camwa.2009.06.026

[8] M. M. El-Borai, “Probability Densities and Fundamental Solutions of Fractional Evolution Equations,” Chaos Solitons and Fractals, Vol. 14, No. 3, 2002, pp. 433-440. doi:10.1016/S0960-0779(01)00208-9

[9] M. Altman, “Inverse Differentiability Contractors and Equations in Banach Spaces,” Studia Mathematica, Vol. 46, No. 1, 1973, pp. 1-15.

[10] R. K. George, D. N. Chalishajar and A. K. Nandakumaran, “Exact Controllability of the Nonlinear Third-Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007 pp. 1028-1044. doi:10.1016/j.jmaa.2006.10.084

[11] L. Wang, “Approximate Controllability of Delayed Semilinear Control Systems,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2005, No. 1, 2005, pp. 67-76.

[1] S. Das, “Functional Fractional Calculus,” 2nd Edition, Springer-Verlag, Berlin, Heldelberg, 2011, doi:10.1007/978-3-642-20545-3

[2] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, 2003, pp. 3413-3442.

[3] J. R. Wang and Y. Zhou, “Complete Controllability of Fractional Evolution Systems,” Communication in Nonlinear Science and Numerical Simulation, Vol. 17, No. 11, 2012, pp. 4346-4355. doi:10.1016/j.cnsns.2012.02.029

[4] 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, 2009, pp. 1760-1765. doi:10.1016/j.aml.2009.06.017

[5] R. Sakthivel, Y. Ren and N. I. Mahmudov, “On the Approximate Controllability of Semilinear Fractional Differential Systems,” Computers and Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 1451-1459.

[6] S. Kumar and N. Sukavanam, “Approximate Controllability of Fractional Order Semilinear Systems with Bounded Delay,” Journal of Differential Equations, Vol. 252, No. 11, 2012, pp. 6163-6174. doi:10.1016/j.jde.2012.02.014

[7] Y. Zhou and F. Jiao, “Existence of Mild Solutions for Fractional Neutral Evolutions,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1063-1077. doi:10.1016/j.camwa.2009.06.026

[8] M. M. El-Borai, “Probability Densities and Fundamental Solutions of Fractional Evolution Equations,” Chaos Solitons and Fractals, Vol. 14, No. 3, 2002, pp. 433-440. doi:10.1016/S0960-0779(01)00208-9

[9] M. Altman, “Inverse Differentiability Contractors and Equations in Banach Spaces,” Studia Mathematica, Vol. 46, No. 1, 1973, pp. 1-15.

[10] R. K. George, D. N. Chalishajar and A. K. Nandakumaran, “Exact Controllability of the Nonlinear Third-Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007 pp. 1028-1044. doi:10.1016/j.jmaa.2006.10.084

[11] L. Wang, “Approximate Controllability of Delayed Semilinear Control Systems,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2005, No. 1, 2005, pp. 67-76.