Approximate Controllability for Time-Dependent Impulsive Neutral Stochastic Partial Differential Equations with Fractional Brownian Motion and Memory

Year 2020, Volume: 3 Issue: 3, 115 - 120, 29.09.2020

Ramkumar K K. Ravikumar Elsayed Elsayed A. Anguraj

https://doi.org/10.32323/ujma.588104

Abstract

In this manuscript, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and memory in Hilbert space. By using semigroup theory, stochastic analysis techniques and fixed point approach, we derive a new set of sufficient conditions for the approximate controllability of nonlinear stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate our results.                                                    

Keywords

Approximate controllability, Impulsive systems, Fractional Brownian motion

References

• [1] H. M. Ahmed, Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space, Adv. Difference Equ., 113 (2014), 1-11.
• [2] H. M. Ahmed, Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion, IMA J. Math. Control Inform, 32(4) (2015), 781-794
• [3] A. Boudaoui, 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.
• [4] G. Da Prato, J. Zabezyk, Stochastic Equations in Infinite Dimensions, Cambridge: University Press, Cambridge, UK, 44 (1992).
• [5] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci, Springer-Verlag, New York, 44 (1983).
• [6] V. Lakshmikantham, D. D. Bainor and P. S. Simeonnov, Theory of impulsive differential equations, World Scientific, (1989).
• [7] N. I. Mahmudov, S. Zorlu, Controllability of nonlinear stochastic systems, Int. J. Control, 76(2) (2003), 95-104.
• [8] N. I. Mahmudov, On controllability of linear stochastic system in Hilbert space, J. Math. Anal. Appl., 259 (2001), 64-82.
• [9] A. N. Kolmogorov, Wienerschc Spiralen and einige andere interessante Kurven in Hilbertsehen Raum, C.R.(Doklady)Acad.URSS(N.S), 26 (1940), 115-118.
• [10] B. B. Mandelbrot, J. W. Van Ness, Fractional Brownian motions, fractional noise and applications, SIAM Rev, 10 (1968), 422-437.
• [11] E. Lakhel, Controllability of neutral stochastic functional differential equations driven by fractional Brownian motion with infinite delay, Nonlinear Dyn. Syst. Theory, 17(3) (2017), 291-302.
• [12] E. Lakhel, Controllability of neutral stochastic functional integrodifferential equations driven by fractional Brownian motion, Stoch. Anal. Appl., 34(3) (2016), 427-440.
• [13] A. Anguraj, K. Ramkumar, Approximate controllability of semilinear stochastic integrodifferential system with nonlocal conditions, Fractal Fract, 2(4) (2018), 29.
• [14] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3498-3508.
• [15] M. Chen, Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motion, Stoch. Dyn., 15 (2015), 1-16.