Controllability of fractional impulsive neutral stochastic functional differential equations via Kuratowski measure of noncompactness

Volume 10, Issue 7, pp 3903--3915 http://dx.doi.org/10.22436/jnsa.010.07.43

Publication Date: July 27, 2017 Submission Date: July 03, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2166 Downloads
• 3636 Views

Authors

Junhao Hu - School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China. Jiashun Yang - School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China. Chenggui Yuan - Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK.

Abstract

In this paper, the controllability problem for a class of fractional impulsive neutral stochastic functional differential equations is considered in infinite dimensional space. By using Kuratowski measure of noncompactness and Mönch fixed point theorem, the sufficient conditions of controllability of the equations are obtained under the assumption that the semigroup generated by the linear part of the equations is not compact. At the end, an example is provided to illustrate the proposed result.

Share and Cite

Keywords

• Controllability
• fractional differential equations
• impulsive stochastic differential equations
• Kuratowski measure of noncompactness
• Mönch fixed point theorem.

MSC

•  26A33
•  65C30
•  93B05

References

• [1] R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973–1033.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] P. Balasubramaniam, J. P. Dauer, Controllability of semilinear stochastic delay evolution equations in Hilbert spaces, Int. J. Math. Math. Sci., 31 (2002), 157–166.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] P. Balasubramaniam, J. Y. Park, P. Muthukumar, Approximate controllability of neutral stochastic functional differential systems with infinite delay, Stoch. Anal. Appl., 28 (2010), 389–400.
  • View Article
  • MathSciNet
  • Google Scholar


• [4] H.-B. Bao, J.-D. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732–1743.
  • View Article
  • Google Scholar


• [5] S. Das, D. Pandey, N. Sukavanam, Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1509–1523.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. Dehici, N. Redjel, Measure of noncompactness and application to stochastic differential equations , Adv. Difference Equ., 2016 (2016), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] S. Duan, J.-H. Hu, Y. Li, Exact controllability of nonlinear stochastic impulsive evolution differential inclusions with infinite delay in Hilbert spaces, Int. J. Nonlinear Sci. Numer. Simul., 12 (2011), 23–33.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] T. Guendouzi, Existence and controllability of fractional-order impulsive stochastic system with infinite delay, Discuss. Math. Differ. Incl. Control Optim., 33 (2013), 65–87.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] D. J. Guo, Impulsive integral equations in Banach spaces and applications, J. Appl. Math. Stochastic Anal., 5 (1992), 111–122.
  • View Article
  • Google Scholar


• [10] H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., (1983), 1351–1371.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] L.-R. Huang, F.-Q. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automat. Control, 53 (2008), 1718–1723 .
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] Y.-G. Kao, Q.-X. Zhu, W.-H. Qi, Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching, Appl. Math. Comput., 271 (2015), 795–804.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
  • MathSciNet


• [14] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ (1989)
  • Google Scholar


• [15] K.-X. Li, J.-G. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941–959.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] X.-R. Mao, Razumikhin-type theorems on exponential stability of stochastic functional-differential equations, Stochastic Process. Appl., 65 (1996), 233–250.
  • View Article
  • Google Scholar


• [17] X.-R. Mao, Stochastic differential equations and their applications, Second edition, Woodhead Publishing Limited, Cambridge (2007)
  • Google Scholar


• [18] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
  • MathSciNet
  • Google Scholar


• [19] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985–999.
  • View Article
  • Google Scholar


• [20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1983)
  • View Article
  • MathSciNet


• [21] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [22] M. D. Quinn, N. Carmichael, An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim., 7 (1984/85), 197–219.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] R. Sakthivel, R. Ganesh, S. Suganya, Approximate controllability of fractional neutral stochastic system with infinite delay, Rep. Math. Phys., 70 (2012), 291–311.
  • View Article
  • MathSciNet
  • Google Scholar


• [24] R. Sakthivel, N. I. Mahmudov, J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334–10340.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] R. Sakthivel, S. Suganya, S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660–668.
  • View Article
  • Google Scholar


• [26] R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optimization, 15 (1977), 407–411.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] R. Triggiani, ddendum: ”A note on the lack of exact controllability for mild solutions in Banach spaces” , [SIAM J. Control Optim., 15 (1977), 407–411], SIAM J. Control Optim., 18 (1980), 98–99.
  • Google Scholar


• [28] Q. Wang, X.-Z. Liu, Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method, Appl. Math. Lett., 20 (2007), 839–845.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] T. Yang, Impulsive systems and control: theory and applications, Nova Science Publishers, Inc., New York (2001)
  • Google Scholar


• [30] R.-P. Ye, Existence of solutions for impulsive partial neutral functional differential equation with infinite delay, Nonlinear Anal., 73 (2010), 155–162.
  • View Article
  • MathSciNet
  • Google Scholar


• [31] Y.-C. Zang, J.-P. Li, Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Bound. Value Probl., 2013 (2013), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077.
  • View Article
  • Google Scholar