APPROXIMATE CONTROLLABILITY OF HILFER FRACTIONAL NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS

TitleAPPROXIMATE CONTROLLABILITY OF HILFER FRACTIONAL NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS
Publication TypeJournal Article
Year of Publication2018
AuthorsLV JINGYUN, YANG XIAOYUAN
JournalDynamic Systems and Applications
Volume27
Issue4
Start Page691
Pagination24
Date Published09/2018
ISSN1056-2176
AMS Subject Classification26A33, 34A08, 34K37, 34K50, 93B05
Abstract

In this paper, we investigate the approximate controllability of Hilfer fractionalneutral stochastic differential equations. Firstly, the existence and uniqueness of mild solutions for these equations are obtained by means of the Banach contraction mapping principle. Then, combining the techniques of stochastic analysis theory, fractional calculations and operator semigroup theory, a new set of sufficient conditions for approximate controllability of these equations is formulated. At last, an example is presented to illustrate the obtained results.
 

PDFhttps://acadsol.eu/dsa/articles/27/4/1.pdf
DOI10.12732/dsa.v27i4.1
Refereed DesignationRefereed
Full Text

REFERENCES
[1] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, Elsevier, New York (2006).
[2] Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Academic press (1998).
[3] Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 1063–1077 (2010).
[4] Zhou, Y., Zhang, L., Shen, X.H.: Existence of mild solutions for fractional evolution equations, J. Integral Equations Appl. 25 557–586 (2013).
[5] Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[6] Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput. 257 344–354 (2015).
[7] Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal. 263 476–510 (2012).
[8] Wang, J.R., Fec̆kan, M., Zhou, Y.: On the new concept solutions and existence results for impulsive fractional evolutions, Dynam. Part. Differ. Eq. 8 (4) 345–361 (2011).
[9] Zhou, Y.: Basic Theory of Fractional Differential Equations, World Scientific (2014).
[10] Zhou, Y., Jiao, F.: Nonlocal cauchy problem for fractional evolution equations, Nonlinear Anal: RWA 11 (5) 4465–4475 (2010).
[11] Hernández, E., O’Regan, D., Balachandran, K.: Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indag. Math. 24 (1) 68–82 (2013).
[12] Gou, H., Li, B.: Study on Sobolev type Hilfer fractional integro-differential equations with delay, J. Fixed Point Theory Appl. 20 (1) 44 (2018).
[13] Gou, H., Li, B.: Study a class of nonlinear fractional non-autonomous evolution equations with delay, J. Pseudo-Differ. Oper. 2 1–22 (2017).
[14] Wang, J.R.: Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput. 256 315–323 (2015).
[15] Li, K.: Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci. 38 (8) 1582–1591 (2015).
[16] Ahmed, H.M., El-Borai, M.M.: Hilfer fractional stochastic integro-differential equations, Appl. Math. Comput. 331 182–189 (2018).
[17] El-Borai, Mahmoud M., El-Nadi, Khairia El-Said, et al.: Semigroups and some fractional stochastic integral equations, Int. J. Pure Appl. Math. Sci., 3 (1) 47–52 (2006).
[18] Ahmed, H.M.: On some fractional stochastic integrodifferential equations in Hilbert spaces, Int. J. Math. Math. Sci., 2009 (2009) 8. 568078.
[19] Ahmed, H.M.: Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions, J. Theor. Probab., 26 (4) (2013).
[20] Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal. Theory Methods Appl., 81 70–86 (2013).
[21] Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim. 15 (3) 407–411 (1977).
[22] Farahi, S., Guendouzi, T.: Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Result. Math. 65 501–521 (2014).
[23] Mahmudov, N.I.: Existence and approximate controllability of sobolev type fractional stochastic evolution equations, Bull. Pol. Acad. Sci., Tech. Sci. 62 205–215 (2014).
[24] Rajivganthi, C., Muthukumar, P., Priya, B.G.: Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1 < α < 2, IMA J. Math. Control Inform. 33 685–699 (2016).
[25] Rajivganthi, C., Thiagu, K., Muthukumar, P. et al.: Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps, Appl. Math. 60 395–419 (2015).
[26] Sakthivel, R., Ganesh, R., Suganya, S.: Approximate controllability of fractional neutral stochastic system with infinite delay, Rep. Math. Phys. 70 291–311 (2012).
[27] Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl. 63 660–668 (2012).
[28] Slama, A., Boudaoui, A.: Approximate controllability of fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay, Ann. Appl. Math. 31 127–139 (2015).
[29] Zang, Y., Li, J.: Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Bound. Value Probl. 2013 1–13 (2013).
[30] Zhang, X., Zhu, C., Yuan, C.: Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay, Adv. Difference Equ. 2015 (2015).
[31] Ahmed, Hamdy M., El-Borai, Mahmoud M.: Hilfer fractional stochastic integrodifferential equations, Appl. Math. Comput. 331 182–189 (2018).
[32] Kruse, R.: Strong and weak approximation of semilinear stochastic evolution equations, Lecture Notes in Mathematics, 2093. Springer, (2014).
[33] Kruse, R., Larsson, S., et al.: Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise, Electron. J. Probab. 17 1–19 (2012).
[34] Dauer, J., Mahmudov, N.I., Matar, M.: Approximate controllability of backward stochastic evolution equations in Hilbert spaces, J. Math. Anal. Appl. 323, 42–56 (2006).
[35] Dauer, J., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces, J. Math. Anal. Appl. 290 373–394 (2004).