Abstract and Applied Analysis

Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

Abstract

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 374938, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511976

Digital Object Identifier
doi:10.1155/2013/374938

Mathematical Reviews number (MathSciNet)
MR3081596

Zentralblatt MATH identifier
1291.34129

Citation

Zhou, Xian-Feng; Liu, Song; Jiang, Wei. Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay. Abstr. Appl. Anal. 2013 (2013), Article ID 374938, 7 pages. doi:10.1155/2013/374938. https://projecteuclid.org/euclid.aaa/1393511976

References

• K. Balachandran, S. Kiruthika, and J. J. Trujillo, “Remark on the existence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2244–2247, 2012.
• M. Benchohra and F. Berhoun, “Impulsive fractional differential equations with variable times,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1245–1252, 2010.
• J. Cao and H. Chen, “Impulsive fractional differential equations with nonlinear boundary conditions,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 303–311, 2012.
• Y.-K. Chang, A. Anguraj, and K. Karthikeyan, “Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 71, no. 10, pp. 4377–4386, 2009.
• Y.-K. Chang, V. Kavitha, and M. M. Arjunan, “Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators,” Nonlinear Analysis. Hybrid Systems. An International Multidisciplinary Journal, vol. 4, no. 1, pp. 32–43, 2010.
• T. L. Guo and W. Jiang, “Impulsive fractional functional differential equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3414–3424, 2012.
• G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 72, no. 3-4, pp. 1604–1615, 2010.
• M. H. M. Rashid and A. Al-Omari, “Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3493–3503, 2011.
• X.-B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 74, no. 5, pp. 2003–2011, 2011.
• G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 74, no. 3, pp. 792–804, 2011.
• J. R. Wang, Y. Zhou, and M. Fečkan, “Nonlinear impulsive problems for fractional differential equations and Ulam stability,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3389–3405, 2012.
• J. R. Wang, X. Li, and W. Wei, “On the natural solution of an impulsive fractional differential equation of order $q\in (1,2)$,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4384–4394, 2012.
• 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. An International Multidisciplinary Journal, vol. 4, no. 4, pp. 775–781, 2010.
• Z. Yan, “Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2252–2262, 2011.
• J. R. Wang, Y. Zhou, and M. Fečkan, “On recent developments in the theory of boundary value problems for impulsive fractional differential equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3008–3020, 2012.
• J. R. Wang, M. Fečkan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345–361, 2011.
• T. L. Guo, “Controllability and observability of impulsive fractional linear time-invariant system,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3171–3182, 2012.
• Z. Tai and S. Lun, “On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 25, no. 2, pp. 104–110, 2012.
• Z. Tai, “Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2158–2161, 2011.
• 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.
• 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.
• T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, Germany, 2011.
• Y. Q. Chen, H. S. Ahn, and D. Xue, “Robust controllability of interval fractional order linear time invariant systems,” Signal Process, vol. 86, pp. 2794–2802, 2006.
• J. Wei, “The controllability of fractional control systems with control delay,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3153–3159, 2012.
• A. B. Shamardan and M. R. A. Moubarak, “Controllability and observability for fractional control systems,” Journal of Fractional Calculus, vol. 15, pp. 25–34, 1999.
• J. L. Adams and T. F. Hartley, “Finite time controllability of fractional order systems,” Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, Article ID 021402, 2008.
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
• 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.