On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps

Krishnan Thiagu

doi:doi:10.11648/j.ajam.20210902.13

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On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps

Krishnan Thiagu

Published in American Journal of Applied Mathematics (Volume 9, Issue 2)

Received: 4 February 2021 Accepted: 27 March 2021 Published: 26 April 2021

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Abstract

The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1C_a(t)}_t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.

Published in	American Journal of Applied Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.ajam.20210902.13
Page(s)	52-63
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Approximate Controllability, Fixed-Point Theorem, Fractional Stochastic Differential System, Hilbert Space, Poisson Jumps

References

[1]	K. J. Astrom, Introduction to stochastic control theory, AcademicPress, NewYork, 1970.
[2]	H. M. Ahmeda, M. M. El-Borai, H. M. El-Owaidy and A. S. Ghanema, Null controllability of fractional stochastic delay integrodifferential equations, J. Math. Computer Sci., 19 (2019), 143–150.
[3]	K. Aissani and M. Benchohra, Fractional Integro-Differential Equations with State-Dependent Delay Advances in Dynamical Systems and Applications 9, 1 (2014), 17–30.
[4]	B. D. Andrade and J. P. C. D. Santos, Existence of solutions for a fractional neutral integro-differential equations with unbounded delay, Electron. J. Differ. Equ., 90 (2012), 1-13.
[5]	R. P. Agarwal, J. P. C. dos Santos and C. Cuevas, Analytic resolvent operator and existence results for fractional integro-differetial equations, J. Abstr. Differ. Equ. Appl., 2, 2 (2012), 26-47.
[6]	E. G. Bazhlekova, Fractional evolution equtions in Banach spaces, Diss. University Press Facilities, Eindhoven University of Technology, 2001.
[7]	M. Benchohra and F. Berhoun, Impulsive fractional differential equations with state-dependent delay, Communications in Appl. Anal., 14, 2 (2010), 213–224.
[8]	O. Bolojan-Nica, G. Infante and R. Precup, Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal., 94 (2014), 231-242.
[9]	A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for linear deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808–-1821.
[10]	A. Boudaoui and A. Slama, Approximate Controllability of Nonlinear Fractional Impulsive Stochastic Differential Equations with Nonlocal Conditions and Infinite Delay, Nonlinear Dyn. Syst. Theory, 1 (2016) 35-48.
[11]	P. Balasubramaniam, V. Vembarasan and T. Senthilkumar, Approximate controallability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space, Numer. Funct. Anal. Optimi., 35 (2014), 177-197.
[12]	A. Chadha, S. N. Bora and R. Sakthivel, Approximate controllability of impulsive stochastic fractional differential equations with nonlocal conditions, Dyn. Syst. Appl., 1 (2018), 1-29.
[13]	R. Chaudhary and D. N. Pandey, Existence results for a class of impulsive neutral fractional stochastic integro-differential systems with state dependent delay, Stochastic Anal. Appl., 2019, Taylor and Francis Group.
[14]	J. Chu and P. J. Torres, Applications of schauder’s fixed point theorem to singular differential equations, Bull. London Math. Soc., 39 (2007), 653–660.
[15]	S. Deng, X. Shu and J. Mao Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point J. Math. Anal. Appl., !67 (2018), 398-420.
[16]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
[17]	H. O. Fattorimi, Second order linear differential equations in Banach spaces, North Holland, Amsterdam, New York, and London (1985).
[18]	B. Goldys and B. Malsowski, Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations, J. Funct. Anal., 226 (2005), 230–255.
[19]	T. Guendouzi and O. Benzatout, Existence of Mild Solutions for Impulsive Fractional Stochastic Differential Inclusions with State-Dependent Delay, Hindawi Publishing Corporation, Chinese Journal of Mathematics, Volume 2014, Article ID 981714, 13 pages.
[20]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Fumkcial Ekvac., 21 (1978), 11–41.
[21]	E. Hernandez and M. McKibben, On state-dependent delay partial neutral functional differential equations, Appl. Math. Comput., 186 (2007), 294–-301.
[22]	E. Hernandez, M. McKibben and H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Model., 49 (2009), 1260–1267.
[23]	E. Hernandez, M. Pierri and G. Goncalves, Existence Results for an Impulsive Abstract Partial Differential Equation with State-Dependent Delay, Com. Math. Appl., 52 (2006), 411–420.
[24]	E. Hernandez and H. R. Henriquez, Existence results for partial neutral functional differential equations with unbounded delay J. Math. Anal. Appl., 221 (1998), 452-475.
[25]	R. S. Jain and M. B. Dhakne, On global existence of solutions for abstract nonlinearfunctional integro-differential equations with nonlocal condition, Contempoary Math. and stat., 1 (2013), 44-53.
[26]	P. Kalamani, D. Baleanu, S. Selvarasu and M. Mallika Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions,. Adv. Differ. Equ.,(2016) 2016: 163.
[27]	L. Kexue, P. Jigen and G. Jinghuai, Controllability of nonlocal fractional differential systems of order α∈(1,2] in Banach spaces, Rep. Math. Physics, 71, 1 (2013), 33-43.
[28]	J. A. Kilbas, H. Srivastava and J. J. Trujillo, Theory and Applications of Fractional DifferentialEquations, Elsevier, Amesterdam, 2006.
[29]	W. Mao, L. Hu and X. Mao, Neutral stochastic functional differential equations with Levy jumps under the local Lipschitz condition, Advances in Difference Equations (2017) 2017: 57.
[30]	P. Muthukumar and C. Rajivganthi, Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces, J. Control Theory Appl., 11, 3 (2013), 351-358.
[31]	K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons Inc., NewYork, 1993.
[32]	P. Muthukumar and K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal stochastic differential equations of order 1
[33]	P. Muthukumar and K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order 1
[34]	B. Oksendal, Stochastic differential equations: An introduction with applications, Springer-verlag, 1995.
[35]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Academic Press, New York, 1999.
[36]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
[37]	Y. Ren and R. Sakthivel, 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 (2012), 073517.
[38]	C. Rajivganthi, K. Thiagu, P. Muthukumar and P. Balasubramaniam, Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps, Appl. Math., 4 (2015), 395-419.
[39]	R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. Con., 83 (2010), 387–393.
[40]	A. Slama and A. Boudaoui, Existence of solutions for nonlinear fractional impulsive stochastic differential equations with nonlocal conditions and infinite delay, Int. J. differ. eqn. appl., 4 (2014), 185–201.
[41]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publisher, Yverdon, 1993.
[42]	S. Selvarasu, P. Kalamani and M. Mallika Arjunan, Approximate controllability of impulsive fractional stochastic integro-differential systems with state-dependent delay and poisson jumps, J. Fractional Calculus and Applications, 2 (2018), 229-254.
[43]	R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results Math., 63 (2013), 949-963.
[44]	K. Thiagu and P. Muthukumar, Non-Lipschitz Sobolev Type Fractional Neutral Impulsive Stochastic Differential Equations with Fractional Stochastic Nonlocal Condition, Infinite Delay and Poisson Jumps, Compu. Appl. Math. Journal, 6, 1 (2020), 1-11.
[45]	N. Valliammal, C. Ravichandran, Z. Hammouch and H. M. Baskonu A New Investigation on Fractional-Ordered Neutral Differential Systems with State-Dependent Delay, Int. J. of Nonlinear Sci. Numer. simul., 20, 7-8 (2019).
[46]	J. Wang and H. M. Ahmed, Null controllability of nonlocal Hilfer fractional stochastic differential equations, Miskolc Math. Notes, 18 (2017), 1073-1083.
[47]	T. Yang, Impulsive systems and control: Theory and applications, Springer-Verlag, Berlin, 2001.
[48]	Z. Yan and H. Zhang, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with state-dependent delay, Elect. J. Diff. Equ., 206 (2013) 1-20.
[49]	Y. Zang and J. Li, Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Bound. Value Probl., 2013, 193.
[50]	X. Zhang, C. Zhu and C. Yuan, Approximate controllability of fractional impulsive evolution systems involving nonlocal initial conditions, Adv. Diff. Equ., 2015, pp 14.
[51]	X. Zhang, C. Zhu and C. Yuan, Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay, Adv. Diff. Equ., 2015, pp-12.

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APA Style

Krishnan Thiagu. (2021). On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. American Journal of Applied Mathematics, 9(2), 52-63. https://doi.org/10.11648/j.ajam.20210902.13

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ACS Style

Krishnan Thiagu. On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. Am. J. Appl. Math. 2021, 9(2), 52-63. doi: 10.11648/j.ajam.20210902.13

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AMA Style

Krishnan Thiagu. On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. Am J Appl Math. 2021;9(2):52-63. doi: 10.11648/j.ajam.20210902.13

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@article{10.11648/j.ajam.20210902.13,
  author = {Krishnan Thiagu},
  title = {On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {52-63},
  doi = {10.11648/j.ajam.20210902.13},
  url = {https://doi.org/10.11648/j.ajam.20210902.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210902.13},
  abstract = {The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.},
 year = {2021}
}

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TY  - JOUR
T1  - On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps
AU  - Krishnan Thiagu
Y1  - 2021/04/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210902.13
DO  - 10.11648/j.ajam.20210902.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 52
EP  - 63
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210902.13
AB  - The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.
VL  - 9
IS  - 2
ER  -

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Author Information

Krishnan Thiagu

Department of Mathematics, Kandaswami Kandar’s College, Velur (Namakkal), Tamilnadu, India

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APA Style

Krishnan Thiagu. (2021). On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. American Journal of Applied Mathematics, 9(2), 52-63. https://doi.org/10.11648/j.ajam.20210902.13

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ACS Style

Krishnan Thiagu. On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. Am. J. Appl. Math. 2021, 9(2), 52-63. doi: 10.11648/j.ajam.20210902.13

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AMA Style

Krishnan Thiagu. On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps. Am J Appl Math. 2021;9(2):52-63. doi: 10.11648/j.ajam.20210902.13

Copy | Download

@article{10.11648/j.ajam.20210902.13,
  author = {Krishnan Thiagu},
  title = {On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {52-63},
  doi = {10.11648/j.ajam.20210902.13},
  url = {https://doi.org/10.11648/j.ajam.20210902.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210902.13},
  abstract = {The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps
AU  - Krishnan Thiagu
Y1  - 2021/04/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210902.13
DO  - 10.11648/j.ajam.20210902.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 52
EP  - 63
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210902.13
AB  - The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.
VL  - 9
IS  - 2
ER  -

Copy | Download