# American Institute of Mathematical Sciences

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
 [1] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [2] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [3] Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104 [4] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [5] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [6] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [7] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [8] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [9] Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 [10] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [11] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [12] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [13] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [14] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [15] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [16] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [17] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [18] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [19] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [20] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

2019 Impact Factor: 0.953