Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Approximate Controllability of Second Order Neutral Stochastic Integro Differential Equations with Impulses Driven By Fractional Brownian Motion

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 333--345 | DOI:10.5890/DNC.2021.06.012

S. Madhuri, Deekshitulu G.V.S.R.

Department of Mathematics, UCEK, JNTUK, Kakinada, A.P., India

Download Full Text PDF

 

Abstract

In this paper we introduce a class of second order neutral stochastic integro differential equations with impulses that are governed by fractional Brownian motion in Hilbert space. First, we establish the existence of mild solution using Banach fixed point theorem. Further approximate controllability for this system is formulated by assuming that the corresponding linear system is approximately controllable. The results are illustrated with example.

References

  1. [1] Arthi, G., Park, J.H., and Jung, H.Y. (2014), Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay, Applied Mathematics and Computation, 248, 328-341.
  2. [2] Caraballo, T., Garrido-Atienza, M.J., and Taniguchi, T. (2011), The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods $\&$ Applications, 74(11), 3671-3684.
  3. [3] Kolmogorov, A. (1940), Wienersche Spiralen und einige andere interessante Kurven im hilbertschen Raum. C. R. Dokl. Acad. USSS (NS) 26, 115-118.
  4. [4]  Mandelbrot, B.B. and Van, N. (1968), fractional Brownian motions, fractional noises, and applications, SIAM Review, 10, 422-437.
  5. [5] Arthi, G., Park, J.H., and Jung, H.Y. (2016), Existence and exponential stability for neutral stochastic integro differential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulation, 32, 145-157.
  6. [6] Boudaoui, A., Caraballo, T., and Ouahab, A. (2016), Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Applicable Analysis, 95(9), 2039-2062.
  7. [7]  Madhuri, S. and Deekshitulu, G.V.S.R. (2017), Linear quadratic optimal control of nonhomogeneous vector differential equations with FBM, Int.J.Dynam.Control, https.//doi.org/10.1007/s40435-017-0366-y.
  8. [8] Hale, J.K. and Lunel, S.M.V. (1991), Introduction to Functional Differential Equations. Springer, Berlin.
  9. [9]  Boudaoui, A. and Caraballo Garrido, T. (2017), Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion, Discrete $\&$ Continuous Dynamical Systems - B, 22(7), 2521-2541.
  10. [10] Boufoussi, B. and Hajji, S. (2017), Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion, Statistics $\&$ Probability Letters, 129, 222-229.
  11. [11] Chang, Y.K., Anguraj, A., Mallika Arjunan, M. (2008), Existence results for impulsive neutral fractional differential equations with infinite delay, Nonlinear Anal Hybrid Systems, 2(1), 209-218.
  12. [12] Lakhel, E. and McKibben, M.A. (2015), Controllability of INSDE Driven by fBm. Chapter 8 In book: Brownian motion: elements, dynamics, and applications. McKibben, M.A., Webster, M. (Eds). Nova Science Publishers, New York, pp. 131-148.
  13. [13] Ren, Y., Zhou, Q., and Chen L. (2011), Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl., 149, 315-331.
  14. [14] Revathi, P., et al. (2013), Existence and stability results for second-order stochastic equations driven by fractional Brownian motion, Transport Theory and Statistical Physics, 42.(6-7), 299-317.
  15. [15] Ren, Y. and Sakthivel, R. (2012), Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys., 53(7), 073517, 14 pp.
  16. [16]  Murty, K.N., Andreou, S., and Viswanadh, K.V.K. (2009), Qualitative properties of general first order matrix difference systems, 16(4).
  17. [17]  Mahmudov, N.I. (2003), Approximate controllability of semi linear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42, 1604-1622.
  18. [18] Triggiani, R. (1977), A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 407-411.
  19. [19] Jing, C.U.I. and Litan, Y.A.N. (2017), Controllability of neutral stochastic evolution equations driven by fractional brownian motion, Acta Mathematica Scientia, 37(1), 108-118.
  20. [20] Hassan Lakhel, E. (2016), Controllability of fractional stochastic neutral functional differential equations driven by fractional Brownian motion with infinite delay, arXiv preprint, arXiv:1604.04079.
  21. [21] Shi, J.X. and Wu, H.Q. (2017), p-th moment exponential convergence analysis for stochastic networked systems driven by fractional Brownian motion, Complex $\&$ Intelligent Systems, 1-11.