• Previous Article
    A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case
  • EECT Home
  • This Issue
  • Next Article
    Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
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. BalasubramaniamN. KumaresanK. 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. DeschR. 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. RavichandranN. 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. RenH. 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. ValliammalC. 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. BalasubramaniamN. KumaresanK. 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. DeschR. 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. RavichandranN. 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. RenH. 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. ValliammalC. 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]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[2]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[3]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017

[4]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[5]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73

[6]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201

[7]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[8]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[9]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[10]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[11]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[12]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[13]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[14]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[15]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[16]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[18]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[19]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015

[20]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (80)
  • HTML views (168)
  • Cited by (0)

Other articles
by authors

[Back to Top]