Null controllability of fractional stochastic delay integro-differential equations

Volume 19, Issue 3, pp 143--150 http://dx.doi.org/10.22436/jmcs.019.03.01

Publication Date: May 15, 2019 Submission Date: September 12, 2017 Revision Date: December 31, 2018 Accteptance Date: April 30, 2019

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)

• 2237 Downloads
• 3894 Views

Authors

Hamdy M. Ahmed - Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt. Mahmoud M. El-Borai - Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt. H. M. El-Owaidy - Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt. A. S. Ghanem - Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt.

Abstract

Sufficient conditions for exact null controllability of semilinear fractional stochastic delay integro-differential equations in Hilbert space are established. The required results are obtained based on fractional calculus, semigroup theory, Schauder's fixed point theorem and stochastic analysis. In the end, an example is given to show the application of our results.

Share and Cite

Keywords

• Fractional calculus
• null controllability
• fractional stochastic delay integrodifferential equation
• Schauder's fixed point theorem

MSC

•  26A33
•  60H10
•  34K50
•  93B05

References

• [1] O. Abu Arqub, Solutions of time--fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space, Numer. Methods Partial Differential Equations, 34 (2018), 1759--1780
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] O. Abu Arqub, M. Al-Smadi, Atangana--Baleanu fractional approach to the solutions of Bagley--Torvik and Painlevé equations in Hilbert space, Chaos Solitons Fractals, 117 (2018), 161--167
  • View Article
  • MathSciNet
  • Google Scholar


• [3] O. Abu Arqub, M. Al-Smadi, Numerical algorithm for solving time--fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions, Numer. Methods Partial Differential Equations, 34 (2018), 1577--1597
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] O. Abu Arqub, B. Maayah, Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana--Baleanu fractional operator, Chaos Solitons Fractals, 117 (2018), 117--124
  • View Article
  • MathSciNet
  • Google Scholar


• [5] O. Abu Arqub, Z. Odibat, M. Al-Smadi, Numerical solutions of time--fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates, Nonlinear Dyn., 94 (2018), 1819--1834
  • View Article
  • Google Scholar


• [6] H. M. Ahmed, Controllability of fractional stochastic delay equations, Lobachevskii J. Math., 30 (2009), 195--202
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] H. M. Ahmed, On some fractional stochastic integrodifferential equations in Hilbert spaces, Int. J. Math. Math. Sci., 2009 (2009), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [8] H. M. Ahmed, Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space, Adv. Difference Equ., 2014 (2014), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] H. M. Ahmed, M. M. El-Borai, Hilfer fractional stochastic integro-differential equations, Appl. Math. Comput., 331 (2018), 182--189
  • View Article
  • MathSciNet
  • Google Scholar


• [10] A. Ali, B. Samet, K. Shah, R. A. Khan, Existence and stability of solution to a toppled systems of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] A. Ali, K. Shah, R. A. Khan, Existence of solution to a coupled system of hybrid fractional differential equations, Bull. Math. Anal. Appl., 9 (2017), 9--18
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. Al-Smadi, O. A. Arqub, Computational algorithm for solving fredholm time--fractional partial integrodifferential equations of dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280--294
  • View Article
  • MathSciNet
  • Google Scholar


• [13] K. Balachandran, P. Balasubramaniam, J. P. Dauer, Local null controllability of nonlinear functional differential systems in Banach space, J. Optim. Theory Appl., 88 (1996), 61--75
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] K. Balachandran, J.-H. Kim, Sample controllability of nonlinear stochastic integrodifferential systems, Nonlinear Anal. Hybrid Syst., 4 (2010), 543--549
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J. P. Dauer, P. Balasubramaniam, Null controllability of semilinear integrodifferential systems in Banach spaces, Appl. Math. Lett., 10 (1997), 117--123
  • View Article
  • MathSciNet
  • Google Scholar


• [16] J. P. Dauer, N. I. Mahmudov, Exact null controllability of semilinear integrodifferential systems in Hilbert spaces, J. Math. Anal. Appl., 299 (2004), 322--332
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] X. L. Fu, Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B (Engl. Ed.), 33 (2013), 747--757
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] P. Kumama, A. Ali, K. Shah, R. A. Khan, Existence results and Hyers--Ullam stability to a class of nonlinear arbitrary order differential equations, J. Nonlinear Sci. Appl., 2017 (2017), 2986--2997
  • View Article
  • Google Scholar


• [19] P. Kumlin, A note on fixed point theory, Chalmers and GU, Gothenburg (2004)
  • View Article
  • Google Scholar


• [20] C. M. Marle, Measures et Probabilités, Hermann, Paris (1974)
  • View Article
  • Google Scholar


• [21] K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, John Wiley \& Sons, New York (1993)
  • View Article
  • MathSciNet
  • Google Scholar


• [22] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] I. Podlubny, Fractional differential equations, Academic press, San Diego (1999)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [25] W. R. Schneider, W. Wyss, Fractional diffusion and wave equation, J. Math. Phys., 30 (1989), 134--144
  • View Article
  • MathSciNet
  • Google Scholar


• [26] K. Shah, A. Ali, R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multi--point boundary value problems, Bound. Value Probl., 2016 (2016), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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