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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 for Time-dependent Impulsive Neutral Stochastic Partial Differential Equations with Fractional Brownian Motion and Poisson Jumps

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 227--235 | DOI:10.5890/DNC.2021.06.005

K. Ramkumar , K. Ravikumar, A. Anguraj

Department of Mathematics, PSG College of Arts & Science, Coimbatore, 641 014, India

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Abstract

In this paper, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and Poisson jumps in Hilbert space. The results are obtained by using semigroup theory, stochastic analysis, 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.

References

  1. [1]  Ahmed, H.M. (2014), Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space, Advances in Difference Equations, 113, 1-11.
  2. [2]  Ahmed, H.M. (2014), Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion, IMA Journal of Mathematical control and Information, 1-14.
  3. [3]  Boudaoui, A. and Lakhel, E. (2018), Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay, Differ. Equ. Dyn. Syst, 28(1-2), 247-263.
  4. [4]  Boudaoui, A. and Slama, A. (2014), Approximate controllability of stochastic inpulsive integrodifferential systems with infinite delay, Advances in Differential Equations and Control Processes, 13, 1-19.
  5. [5]  Mandelbrot, B.B. and Van Ness, J.W. (1968), Fractional Brownian motions, fractional noise and applications, SIAM Rev, 10, 422-437.
  6. [6]  Lakhel, E. (2017), Controllability of neutral stochastic functional differential equations driven by fractional Brownian motion with infinite delay, Nonlinear Dynamics and Systems Theory, 17(3), 291-302.
  7. [7]  Anguraj, A and Ramkumar, K. (2018), Approximate controllability of semilinear stochastic integrodifferential system with nonlocal conditions, fractal Fract, 2(4), 29.
  8. [8]  Mahmudov, N.I. and Zorlu, S. (2003), Controllability of nonlinear stochastic systems, Int. J. Control, 76(2), 95-104.
  9. [9]  Mahmudov, N.I. (2001), On controllability of linear stochastic system in Hilbert space, J. Math. Anal. Appl, 259, 64-82.
  10. [10]  Lakshmikantham, V., Bainor, D.D., and Simeonnov, P.S. (1989), Theory of impulsive differential equations, World Scientific.
  11. [11]  Lakhel, E.H. (2016), Exponential stability for stochastic neutral functional differential equations driven by Rosenblatt process with delay and Poisson jumps, Random Operators and Stochastic Equations, 24(2), 113-127.
  12. [12]  Sakthivel, R. and Ren, R. (2012), Exponential stability of second order stochastic evolution equations with Poisson jumps, Communications in Nonlinear Science and Numerical Simulations, 17, 4517-4523.
  13. [13]  Sakthivel, R. and Ren, Y. (2011), Complete controllability of stochastic evolution equations with jumps, Reports on Mathematical Physics, 68, 163-174.
  14. [14]  Kolmogorov, A.N. (1940), Wienerschc Spiralen and einige andere interessante Kurven in Hilbertsehen Raum. C.R.(Doklady)Acad.URSS(N.S), 26, 115-118.
  15. [15]  Lakhel, E. (2016), Controllability of neutral stochastic functional integrodifferential equations driven by fractional Brownian motion, Stoch. Anal. Appl, 34(3), 427-440.
  16. [16]  Chen, M. (2015), Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motion, Stoch. Dyn, 15, 1-16.
  17. [17]  Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, 44.
  18. [18]  Da Prato, G. and Zabezyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge: University Press, Cambridge, UK, 44.
  19. [19]  Kurma, S. and Sukavanam, N. (2012), Approximate controllability of fractional order semilinear system with bounded delay, J. Differential Equations, 252, 6163-6174.
  20. [20]  Klamka, J. (2013), Controllability of dynamical systems, A survey. Bull. Pol. Acad. Sci. Tech. Sci, 61, 221-229.
  21. [21]  Park, J.Y., Balachandran, K., and Arthi, G. (2009), Controllability of impulsive neutral integrodifferential systems with infinite delay in Banach spaces, Nonlinear Analysis: Hybrid Systems, 3, 184-194.
  22. [22]  Sakthivel, R., Ganesh, R., Ren, Y., and Anthoni, S.M. (2013), Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul, 18, 3498-3508.
  23. [23]  Sakthivel, R. and Luo, J.W. (2009), Asymptotic stability of impulsive stochastic partial differential equations, Statist. Probab. Lett, 79, 1219-1223.
  24. [24]  Balachandran, K., Kim, J.H., and Karthikeyan, S. (2007), Controllability of semilinear stochastic integrodifferential equations, Kybernetika, 43, 31-44.