Taiwanese Journal of Mathematics

THE SOLVABILITY AND OPTIMAL CONTROLS OF IMPULSIVE FRACTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS

Xiuwen Li and Zhenhai Liu

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Abstract

In this paper, we deal with the impulsive control systems of fractional order and their optimal controls in Banach spaces. We firstly show the existence and uniqueness of mild solutions for a broad class of impulsive fractional infinite dimensional control systems under suitable assumptions. Then by using generally mild conditions of cost functionals, we extend the existence result of optimal controls to the impulsive fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

Article information

Source
Taiwanese J. Math., Volume 19, Number 2 (2015), 433-453.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133639

Digital Object Identifier
doi:10.11650/tjm.19.2015.3131

Mathematical Reviews number (MathSciNet)
MR3332306

Zentralblatt MATH identifier
1357.49017

Subjects
Primary: 34K37: Functional-differential equations with fractional derivatives 35R11: Fractional partial differential equations

Keywords
impulsive differential equations fractional derivatives mild solutions optimal controls

Citation

Li, Xiuwen; Liu, Zhenhai. THE SOLVABILITY AND OPTIMAL CONTROLS OF IMPULSIVE FRACTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS. Taiwanese J. Math. 19 (2015), no. 2, 433--453. doi:10.11650/tjm.19.2015.3131. https://projecteuclid.org/euclid.twjm/1499133639


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References

  • E. Balder, Necessary and sufficient conditions for $L^{1}$-strong-weak lower semicontinuity of integral functional, Nonlinear Anal.: RWA., 11 (1987), 1399-1404.
  • D. Baleanu and A. K. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal.: RWA., 11(1) (2010), 288-292.
  • J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Comput. Model., 57 (2013), 754-763.
  • A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450.
  • A. Debbouche and M. M. El-Borai, Weak almost periodic and optimal mild solutions of fractional evolution equations, Electronic J. Diff. Equ., 46 (2009), 1-8.
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis $($Theory$)$, Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
  • A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in: North-Holland Mathematics Studies, Vol. 204, Elservier Science B. V., Amsterdam, 2006.
  • V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682.
  • J. T. Liang, Y. L. Liu and Z. H. Liu, A class of BVPS for first order impulsive integro-differential equations, Appl. Math. Comput., 218(7) (2011), 3667-3672.
  • Z. H. Liu, J. F. Han and L. J. Fang, Integral boundary value problems for first order integro-differential equations with impulsive integral conditions, Comput. Math. Appl., 61(10) (2011), 3035-3043.
  • Z. H. Liu and X. W. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1362-1373.
  • Z. H. Liu and X. W. Li, On the controllability of impulsive fractional evolution inclusions in Banach spaces, J. Optim. Theory Appl., 156 (2013), 167-182.
  • Z. H. Liu and S. Migórski, Analysis and control of differential inclusions with anti-periodic conditions, Proceedings of the Royal Society of Edinburgh, 144A(3) (2014), 591-602.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • Z. Tai and X. Wang, Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces, Appl. Math. Lett., 22(11) (2009), 1760-1765.
  • J. R. Wang, M. Fe\~ckan and Y. Zhou, Relaxed controls for nonlinear fractional impulsive evolution equations, J. Optim. Theory Appl., 156 (2013), 13-32.
  • J. R. Wang, Y. Zhou and Mi. Medved, On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. Optim. Theory Appl., 152 (2012), 31-50.
  • J. R. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal: RWA., 12 (2011), 262-272.
  • W. Wei, X. Xiang and Y. Peng, Nonlinear impulsive integro-differential equation of mixed type and optimal controls, Optim., 55 (2006), 141-156.
  • Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.