Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps

Noorah Mshary; Hamdy M. Ahmed; Ahmed S. Ghanem; Noorah Mshary; Hamdy M. Ahmed; Ahmed S. Ghanem

doi:10.3934/math.2024477

AIMS Mathematics

2024, Volume 9, Issue 4: 9746-9769. doi: 10.3934/math.2024477

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Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt

Received: 12 December 2023 Revised: 01 February 2024 Accepted: 28 February 2024 Published: 11 March 2024
MSC : 34A08, 34K50, 93B05, 93E03

This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.
- impulsive effect,
- fractional calculus,
- stochastic differential system,
- control theory,
- semigroup theory
Citation: Noorah Mshary, Hamdy M. Ahmed, Ahmed S. Ghanem. Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps[J]. AIMS Mathematics, 2024, 9(4): 9746-9769. doi: 10.3934/math.2024477

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Abstract

This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.

References

[1]	Y. C. Guo, M. Q. Chen, X. B. Shu, F. Xu, The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm, Stoch. Anal. Appl., 39 (2021), 643–666. https://doi.org/10.1080/07362994.2020.1824677 doi: 10.1080/07362994.2020.1824677
[2]	B. P. Moghaddam, A. Mendes Lopes, J. A. Tenreiro Machado, Z. S. Mostaghim, Computational scheme for solving nonlinear fractional stochastic differential equations with delay, Stoch. Anal. Appl., 37 (2019), 893–908. https://doi.org/10.1080/07362994.2019.1621182 doi: 10.1080/07362994.2019.1621182
[3]	G. Shevchenko, Mixed fractional stochastic differential equations with jumps, Stochastics, 86 (2014), 203–217. https://doi.org/10.1080/17442508.2013.774404 doi: 10.1080/17442508.2013.774404
[4]	F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dyn. Nat. Soc., 2017 (2017), 5394528. https://doi.org/10.1155/2017/5394528 doi: 10.1155/2017/5394528
[5]	H. M. Ahmed, Sobolev-type nonlocal conformable stochastic differential equations, Bull. Iran. Math. Soc., 48 (2022), 1747–1761. https://doi.org/10.1007/s41980-021-00615-6 doi: 10.1007/s41980-021-00615-6
[6]	P. Balasubramaniam, Solvability of Atangana-Baleanu-Riemann (ABR) fractional stochastic differential equations driven by Rosenblatt process via measure of noncompactness, Chaos Soliton Fract., 157 (2022), 111960. https://doi.org/10.1016/j.chaos.2022.111960 doi: 10.1016/j.chaos.2022.111960
[7]	M. Abouagwa, J. Li, Stochastic fractional differential equations driven by Lévy noise under Carathéodory conditions, J. Math. Phys., 60 (2019), 022701. https://doi.org/10.1063/1.5063514 doi: 10.1063/1.5063514
[8]	H. M. Ahmed, H. M. El-Owaidy, M. A. Al-Nahhas, Neutral fractional stochastic partial differential equations with Clarke subdifferential, Appl. Anal., 100 (2021), 3220–3232. https://doi.org/10.1080/00036811.2020.1714035 doi: 10.1080/00036811.2020.1714035
[9]	K. Ramkumar, K. Ravikumar, A. Anguraj, H. M. Ahmed, Well posedness results for higher-order neutral stochastic differential equations driven by Poisson jumps and Rosenblatt process, Filomat, 35 (2021), 353–365. https://doi.org/10.2298/FIL2102353R doi: 10.2298/FIL2102353R
[10]	W. Hu, Q. X. Zhu, Stability analysis of impulsive stochastic delayed differential systems with unbounded delays, Syst. Control Lett., 136 (2020), 104606. https://doi.org/10.1016/j.sysconle.2019.104606 doi: 10.1016/j.sysconle.2019.104606
[11]	M. Feckan, J. R. Wang, Periodic impulsive fractional differential equations, Adv. Nonlinear Anal., 8 (2017), 482–496. https://doi.org/10.1515/anona-2017-0015 doi: 10.1515/anona-2017-0015
[12]	T. Sitthiwirattham, R. Gul, K. Shah, I. Mahariq, J. Soontharanon, K. J. Ansari, Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative, AIMS Mathematics, 7 (2022), 4017–4037. https://doi.org/10.3934/math.2022222 doi: 10.3934/math.2022222
[13]	S. A. Karthick, R. Sakthivel, F. Alzahrani, A. Leelamani, Synchronization of semi-Markov coupled neural networks with impulse effects and leakage delay, Neurocomputing, 386 (2020), 221–231. https://doi.org/10.1016/j.neucom.2019.12.097 doi: 10.1016/j.neucom.2019.12.097
[14]	D. Yang, J. R. Wang, Non-instantaneous impulsive fractional-order implicit differential equations with random effects, Stoch. Anal. Appl., 35 (2017), 719–741. https://doi.org/10.1080/07362994.2017.1319771 doi: 10.1080/07362994.2017.1319771
[15]	H. M. Ahmed, M. M. El-Borai, A. S. O. El Bab, M. E. Ramadan, Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion, Bound. Value Probl., 2020 (2020), 120. https://doi.org/10.1186/s13661-020-01418-0 doi: 10.1186/s13661-020-01418-0
[16]	H. M. Ahmed, Noninstantaneous impulsive conformable fractional stochastic delay integro-differential system with Rosenblatt process and control function, Qual. Theory Dyn. Syst., 21 (2022), 15. https://doi.org/10.1007/s12346-021-00544-z doi: 10.1007/s12346-021-00544-z
[17]	R. Dhayal, M. Malik, Approximate controllability of fractional stochastic differential equations driven by Rosenblatt process with non-instantaneous impulses, Chaos Soliton Fract., 151 (2021), 111292. https://doi.org/10.1016/j.chaos.2021.111292 doi: 10.1016/j.chaos.2021.111292
[18]	H. M. Ahmed, Conformable fractional stochastic differential equations with control function, Syst. Control Lett., 158 (2021), 105062. https://doi.org/10.1016/j.sysconle.2021.105062 doi: 10.1016/j.sysconle.2021.105062
[19]	G. J. Shen, R. Sakthivel, Y. Ren, M. Y. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collect. Math., 71 (2020), 63–82. https://doi.org/10.1007/s13348-019-00248-3 doi: 10.1007/s13348-019-00248-3
[20]	P. Muthukumar, K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order $1 < q < 2$ with infinite delay and Poisson Jumps, J. Dyn. Control Syst., 23 (2017), 213–235. https://doi.org/10.1007/s10883-015-9309-0 doi: 10.1007/s10883-015-9309-0
[21]	M. S. H. Ansari, M. Malik, D. Baleanu, Controllability of prabhakar fractional dynamical systems, Qual. Theory Dyn. Syst., 23 (2024), 63. https://doi.org/10.1007/s12346-023-00919-4 doi: 10.1007/s12346-023-00919-4
[22]	H. M. Ahmed, M. M. El-Borai, M. E. Ramadan, Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps, Adv. Differ. Equ., 2019 (2019), 82. https://doi.org/10.1186/s13662-019-2028-1 doi: 10.1186/s13662-019-2028-1
[23]	G. Gokul, R. Udhayakumar, Approximate controllability for Hilfer fractional stochastic non-instantaneous impulsive differential system with rosenblatt process and Poisson jumps, Qual. Theory Dyn. Syst., 23 (2024), 56. https://doi.org/10.1007/s12346-023-00912-x doi: 10.1007/s12346-023-00912-x
[24]	C. S. Varun Bose, V. Muthukumaran, S. Al-Omari, H. Ahmad, R. Udhayakumar, Study on the controllability of Hilfer fractional differential system with and without impulsive conditions via infinite delay, Nonlinear Anal. Model., 29 (2023), 166–188. https://doi.org/10.15388/namc.2024.29.33840 doi: 10.15388/namc.2024.29.33840
[25]	W. Kavitha Williams, V. Vijayakumar, R. Udhayakumar, S. K. Panda, K. S. Nisar, Existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order $1 < r < 2$, Numer. Methods Partial Differential Eq., 40 (2024), e22697. https://doi.org/10.1002/num.22697 doi: 10.1002/num.22697
[26]	N. Hakkar, R. Dhayal, A. Debbouche, D. F. M. Torres, Approximate controllability of delayed fractional stochastic differential systems with mixed noise and impulsive effects, Fractal Fract., 7 (2023), 104. https://doi.org/10.3390/fractalfract7020104 doi: 10.3390/fractalfract7020104
[27]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[28]	C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230–257. https://doi.org/10.1051/ps:2007037 doi: 10.1051/ps:2007037
[29]	E. H. Lakhel. 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. https://doi.org/10.1007/s12555-016-0363-5 doi: 10.1007/s12555-016-0363-5
[30]	H. B. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354. https://doi.org/10.1016/j.amc.2014.10.083 doi: 10.1016/j.amc.2014.10.083

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