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doi: 10.3934/eect.2020096

## Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$

 1 Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 2390 Marrakesh, Morocco

* Corresponding author: Soufiane Mouchtabih

Received  May 2019 Revised  September 2020 Published  October 2020

This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than $1/2$. We employ the theory of resolvent operators developed by [10] combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result.

Citation: Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$. Evolution Equations & Control Theory, doi: 10.3934/eect.2020096
##### References:
 [1] H. M. Ahmed and J. Wang, Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, Bull. Iran. Math. Soc., 44 (2018), 673-690.  doi: 10.1007/s41980-018-0043-8.  Google Scholar [2] M. A. Alqudah, C. Ravichandran, T. Abdeljawad and N. Valliammal, New results on Caputo fractional-order neutral differential inclusions without compactness, Adv Differ Equ, 2019 (2019), Paper No. 528, 14 pp. doi: 10.1186/s13662-019-2455-z.  Google Scholar [3] P. Balasubramaniam, N. Kumaresan, K. Ratnavelu and P. Tamilalagan, Local and global existence of mild solution for impulsive fractional stochastic differential equations, Bull. Malays. Math. Sci. Soc., 38 (2015), 867-884.  doi: 10.1007/s40840-014-0054-4.  Google Scholar [4] A. Boudaoui and E. Lakhel, Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay, Differ Equ Dyn Syst, 26 (2018), 247-263.  doi: 10.1007/s12591-017-0401-7.  Google Scholar [5] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional Brownian motion with Hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9.  Google Scholar [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3.  Google Scholar [7] A. Chadha and D. N. Pandey, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal., 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [8] J. Cui and L. Yan, Controllability of neutral stochastic evolution equations driven by fractional Brownian motion, Acta Mathematica Scientia B, 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9.  Google Scholar [9] W. Desch, R. Grimmer and W. Schappacher, Some consideration for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.  doi: 10.1016/0022-247X(84)90044-1.  Google Scholar [10] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [11] K. Jothimani, K. Kaliraj, Z. Hammouch and C. Ravichandran, New results on controllability in the framework of fractional integro-differential equations with nondense domain, Eur. Phys. J. Plus, 134 (2019). Google Scholar [12] E. H. Lakhel, Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion, Stoch. Ana. and App., 34 (2016), 427-440.  doi: 10.1080/07362994.2016.1149718.  Google Scholar [13] E. Lakhel, Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay, Le Matematiche, 73 (2018), 319-339.  doi: 10.4418/2018.73.2.6.  Google Scholar [14] E. Lakhel and M. A. McKibben, Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses, Int. J. Control Autom. Syst., 17 (2019), 286-297.  doi: 10.1007/s12555-016-0363-5.  Google Scholar [15] J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl, 2013 (2013), 16pp. doi: 10.1186/1687-1812-2013-66.  Google Scholar [16] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [17] J. Pruss, Evolutionary Integral Equations and Applications, Birkhauser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [18] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J.Franklin I., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.  Google Scholar [19] Y. Ren, H. Dai and R. Sakthivel, Approximate controllability of stochastic differential system driven by a Levy process, Inter nat. J. Control, 86 (2013), 1158-1164.  doi: 10.1080/00207179.2013.786188.  Google Scholar [20] T. Sathiyaraj and P. Balasubramaniam, Controllability of fractional neutral stochastic integro-differential inclusions of order $p\in(0, 1]$, $q\in(1, 2]$ with fractional Brownian motion, Eur. Phys. J. Plus 131, 357 (2016). Google Scholar [21] P. Tamilalagan and P. Balasubramanniam, Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators, Int. J. of Control., 90 (2017), 1713-1727.  doi: 10.1080/00207179.2016.1219070.  Google Scholar [22] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.  Google Scholar [23] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155.  Google Scholar

show all references

##### References:
 [1] H. M. Ahmed and J. Wang, Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, Bull. Iran. Math. Soc., 44 (2018), 673-690.  doi: 10.1007/s41980-018-0043-8.  Google Scholar [2] M. A. Alqudah, C. Ravichandran, T. Abdeljawad and N. Valliammal, New results on Caputo fractional-order neutral differential inclusions without compactness, Adv Differ Equ, 2019 (2019), Paper No. 528, 14 pp. doi: 10.1186/s13662-019-2455-z.  Google Scholar [3] P. Balasubramaniam, N. Kumaresan, K. Ratnavelu and P. Tamilalagan, Local and global existence of mild solution for impulsive fractional stochastic differential equations, Bull. Malays. Math. Sci. Soc., 38 (2015), 867-884.  doi: 10.1007/s40840-014-0054-4.  Google Scholar [4] A. Boudaoui and E. Lakhel, Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay, Differ Equ Dyn Syst, 26 (2018), 247-263.  doi: 10.1007/s12591-017-0401-7.  Google Scholar [5] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional Brownian motion with Hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9.  Google Scholar [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3.  Google Scholar [7] A. Chadha and D. N. Pandey, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal., 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [8] J. Cui and L. Yan, Controllability of neutral stochastic evolution equations driven by fractional Brownian motion, Acta Mathematica Scientia B, 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9.  Google Scholar [9] W. Desch, R. Grimmer and W. Schappacher, Some consideration for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.  doi: 10.1016/0022-247X(84)90044-1.  Google Scholar [10] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [11] K. Jothimani, K. Kaliraj, Z. Hammouch and C. Ravichandran, New results on controllability in the framework of fractional integro-differential equations with nondense domain, Eur. Phys. J. Plus, 134 (2019). Google Scholar [12] E. H. Lakhel, Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion, Stoch. Ana. and App., 34 (2016), 427-440.  doi: 10.1080/07362994.2016.1149718.  Google Scholar [13] E. Lakhel, Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay, Le Matematiche, 73 (2018), 319-339.  doi: 10.4418/2018.73.2.6.  Google Scholar [14] E. Lakhel and M. A. McKibben, Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses, Int. J. Control Autom. Syst., 17 (2019), 286-297.  doi: 10.1007/s12555-016-0363-5.  Google Scholar [15] J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl, 2013 (2013), 16pp. doi: 10.1186/1687-1812-2013-66.  Google Scholar [16] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [17] J. Pruss, Evolutionary Integral Equations and Applications, Birkhauser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [18] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J.Franklin I., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.  Google Scholar [19] Y. Ren, H. Dai and R. Sakthivel, Approximate controllability of stochastic differential system driven by a Levy process, Inter nat. J. Control, 86 (2013), 1158-1164.  doi: 10.1080/00207179.2013.786188.  Google Scholar [20] T. Sathiyaraj and P. Balasubramaniam, Controllability of fractional neutral stochastic integro-differential inclusions of order $p\in(0, 1]$, $q\in(1, 2]$ with fractional Brownian motion, Eur. Phys. J. Plus 131, 357 (2016). Google Scholar [21] P. Tamilalagan and P. Balasubramanniam, Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators, Int. J. of Control., 90 (2017), 1713-1727.  doi: 10.1080/00207179.2016.1219070.  Google Scholar [22] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.  Google Scholar [23] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155.  Google Scholar
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