American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
Go
American Journal of Applied Mathematics and Statistics. 2015, 3(4), 168-176
DOI: 10.12691/ajams-3-4-7
Open AccessArticle

Approximate Controllability of Fractional Stochastic Perturbed Control Systems Driven by Mixed Fractional Brownian Motion

Salah H. Abid1, , Sameer Q. Hasan1 and Uday J. Quaez1

1Mathematics department, Education College, Al-Mustansiriya University, Baghdad, Iraq

Pub. Date: August 13, 2015

Cite this paper:
Salah H. Abid, Sameer Q. Hasan and Uday J. Quaez. Approximate Controllability of Fractional Stochastic Perturbed Control Systems Driven by Mixed Fractional Brownian Motion. American Journal of Applied Mathematics and Statistics. 2015; 3(4):168-176. doi: 10.12691/ajams-3-4-7

Abstract

In this paper, the approximate controllability of nonlinear Fractional order 0<α<1 Riemann-Liouville type stochastic perturbed control systems driven by mixed fractional Brownian motion in a real separable Hilbert spaces has been studied by using Krasnoselskii's fixed point theorem, stochastic analysis theory, fractional calculus and some sufficient conditions.

Keywords:
approximate controllability mixed fractional brownian motion fixed point theorem perturbed control systems mild solution control function

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Abid S. H., Hasan S. Q. and Quaez U. J. “Approximate controllability of Fractional Stochastic Integro-Differential Equations Driven by Mixed Fractional Brownian Motion”, American Journal of Mathematics and Statistics 2015, Vo. 2, PP:72-81, 2015.
 
[2]  Ahmed M. Hamdy, “Approximate Controllability Of Impulsive Neutral Stochastic Differential Equations With Fractional Brownian Motion in A Hilbert Space”, Advances in Difference Equations, Springer Open Journal, 2014:113, 2014.
 
[3]  Balachandran K., Kiruthika S. and Trujillo J. “On Fractional Impulsive Equations Of Sobolev Type with Nonlocal Conditions in Banach Spaces”, Computers and Mathematics with Applications, No. 62, PP: 1157-1165, 2011.
 
[4]  Engel K. J, and Nagel R. “One Parameter Semigroup For Linear Evolution Equations”, Springer-Verlag, New York, Berlin, 2000.
 
[5]  Gani J., Heyde C.C., Jagers P. and Kurtz T.G., “Probability and Its Applications”, Springer-Verlag London Limited, 2008.
 
[6]  Grippenberg, G. and Norros I., “On The Prediction Of Fractional Brownian Motion”, Journal of Applied Probability, Vol. 33, No. 2,PP: 400-410, 1996.
 
[7]  Guendouzi T. and Idrissi S., “Approximate Controllability of Fractional Stochastic Functional Evolution Equations Driven By A Fractional Brownian Motion”, Romai J., Vo.8, No.2,PP:103-117, 2012.
 
[8]  Kerboua M., Debbouche A. and Baleanu D., “Approximate Controllability Of Sobolev Type Fractional Stochastic Nonlocal Nonlinear Differential Equations in Hilbert Spaces”, Electronic Journal Of Qualitative Theory of Differential Equations, No. 58, PP: 1-16, 2014.
 
[9]  Li C., Qian D. and Chen Y., “On Riemann- Liouville and Caputo Derivatives”, Hindawi Publishing Corporation, Vol. 2011, Article ID 562494, 15 pages.2011.
 
[10]  Li K., “Stochastic Delay Fractional Evolution Equations Driven By Fractional Brownian Motion”, Mathematical Method in The Applied Sciences, 2014.
 
[11]  Madsen Henrik, “ ito integrals”, 2006.
 
[12]  Mahmudov N. and Zorlu S., “Approximate Controllability Of Fractional Integro-Differential Equations Involving Nonlocal Initial Conditions”, Boundary Value Problems, Springer Open Journal, 2013:118, 2013.
 
[13]  Mahmudov N., “Controllability of Linear Stochastic Systems in Hilbert Spaces”, Journal of Mathematical Analysis and Applications Vo. 259, PP: 64-82, 2001.
 
[14]  Mishura Y. S., “Stochastic Calculus for Fractional Brownian Motion and Related Processes”, Lect,Notes in Math., 1929, Springer, 2008.
 
[15]  Nourdin I.., “Select Aspects of fractional Brownian Motion”, Springer-Verlag Italia, 2012.
 
[16]  Nualart D., “Fractional Brownian motion: stochastic calculus and Applications”, Proceedings of the International Congress of Mathematicians, Madrid, Spain, European Mathematical Society, 2006.
 
[17]  Pazy, A., “Semigroup of Linear Operator and Applications to Partial Differential Equations”, Springer-Verlag, New York, 1983.
 
[18]  Podlubny I., “Fractional Differential Equations”, Academic Press, San Diego. California, USA, 1999.
 
[19]  Sakthivel R., “Approximate Controllability Of Impulsive Stochastic Evolution Equations”, Funkcialaj Ekvacioj, Vol. 52(2009), PP:381-393, 2009.
 
[20]  Zhou Y., Wang J. and Feckan M. “Controllability Of Sobolev Type Fractional Evolution systems ”, Dynamics of PDE, Vol. 11, No. 1, PP: 71-87, 2014.