Fractional neutral stochastic integrodifferential equations with Caputo fractional derivative
Rosenblatt process, Poisson jumps and Optimal control
DOI:
https://doi.org/10.22199/issn.0717-6279-4329Keywords:
fractional neutral stochastic integrodifferential system, Rosenblatt process, Poisson jumps, optimal control, successive approximationAbstract
The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic integrodifferential equations driven by Rosenblatt process and Poisson jumps in Hilbert spaces. First we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach.
The results are formulated and proved by using the fractional calculus, solution operator and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations driven by Rosenblatt process and poisson jumps is also been presented. An example is provided to illustrate the theory.
References
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier Science, vol. 204, 2006.
I. Podlubny, Fractional Differential Equations. London: Academic Press, 1999.
F. Biagini, Y. Hu and B. Oksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. London: Springer, 2008.
X. Mao, Stochastic Differential Equations and Applications. Chichester: Horwood, 1997.
M. A. Ouahra, B. Boufoussi and E. Lakhel, “Existence and stability for stochastic impulsive neutral partial differential equations driven by Rosenblatt process with delay and Poisson jumps”, Communications on Stochastic Analysis, vol. 11, no. 1, pp. 99-117, 2017. https://doi.org/10.31390/cosa.11.1.07
G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. London: Cambridge University Press, 2014.
H. M. Ahmed, “Semilinear neutral fractional stochastic integrodifferential equations with nonlocal conditions”, Journal of Theoretical Probability, vol. 28, no. 2, pp. 667-680, 2015. https://doi.org/10.1007/s10959-013-0520-1
A. Anguraj and K. Ravikumar, “Existence and stability results for impulsive stochastic functional integrodifferential equations with Poisson jumps”, Journal of Applied Nonlinear Dynamics, vol. 8, no. 3, pp. 407-417, 2019. https://doi.org/10.5890/jand.2019.09.005
J. Luo and T. Taniguchi, “The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps”, Stochastics and Dynamics, vol. 9, no. 1, pp. 135-152, 2009. https://doi.org/10.1142/s0219493709002592
P. Tamilalagan and P. Balasubramaniam, Existence results for semilinear fractional stochastic evolution inclusions driven by Poisson jumps, In: P. N. Agrawal, R, N. Mohapatra, U. Singh, H. M. Srivastava, eds., Mathematical Analysis and Its Applications, Springer Proceedings in Mathematics and Statistics, vol. 143. Roorkee: Springer India, 2014.
F. A. Rihan, C. Rajivganthi and P. Muthukumar, “Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control”, Discrete Dynamics in Nature and Society, pp. 1-11, 2017. https://doi.org/10.1155/2017/5394528
P. Balasubramaniam, S. Saravanakumar and K. Ratnavelu, “Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps”, Stochastic Analysis and Applications, 2018. https://doi.org/10.1080/07362994.2018.1524303
H. M. Ahmed and M. M. El-Borai, “Hilfer fractional stochastic integrodifferential equations”, Applied Mathematics and Computation, vol. 331, pp. 182-189, 2018. https://doi.org/10.1016/j.amc.2018.03.009
F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Springer Science & Business Media, 2008.
C. A. Tudor, “Analysis of the Rosenblatt process”. ESAIM: Probability and Statistics, vol. 12, pp. 230-257, 2008. https://doi.org/10.1051/ps:2007037
M. Maejima and C. A. Tudor, “On the distribution of the Rosenblatt process”, Statistics & Probability Letters, vol. 83, no. 6, pp. 1490-1495, 2013. https://doi.org/10.1016/j.spl.2013.02.019
L. Urszula and H. Schattler, “Antiangiogenic therapy in cancer treatment as an optimal control proble”, SIAM Journal on Control and Optimization, vol. 46, no. 3, pp. 1052-1079, 2007. https://doi.org/10.1137/060665294
I. Area, F. NdaÏrou, J. J. Nieto, C. J. Silva and D. F. Torres, “Ebola model and optiimal control with vaccination constraints”, Journal of Industrial and Management Optimization, vol. 14, no. 2, pp. 427-446, 2018. https://doi.org/10.3934/jimo.2017054
P. Tamilalagan and P. Balasubramniam, “The solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps via resolvent operators The solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps via resolvent operators”, Applied mathematics and Optimization, vol. 77, no. 3, pp. 443-462, 2018.
S. Das, Functional Fractional Calculus. Springer-Verlag, Berlin, Heidelberg, 2011.
A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson and W. A. Wakeham, “A fractional differential equation for a MEMS viscometer used in the oil industry”, Journal of Computational and Applied Mathematics, vol. 229, pp. 373-381, 2009. https://doi.org/10.1016/j.cam.2008.04.018
W. G. Glöckle and T. F. Nonnenmacher, “A fractional calculus approach of self-similar protein dynamics”, Biophysical Journal, vol. 68, no. 1, pp. 46-53, 1995.https://doi.org/10.1016/S0006-3495(95)80157-8
H. Aicha, J. J. Nieto and D. Amar, “Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential”, Journal of Computational and Applied Mathematics, vol. 344, pp. 725-737, 2018. https://doi.org/10.1016/j.cam.2018.05.031
J. Wang, Y. Zhou, W. Wei and H. Xu, “Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls”, Computers and Mathematics with Applications, vol. 62, pp. 1427-1441, 2011. https://doi.org/10.1016/j.camwa.2011.02.040
E. J. Balder, “Necessary and sufficient conditions for L1-strong weak lower semicontinuity of integral functionals”, Nonlinear Analysis: Theory Methods and Applications, vol. 11, no. 12, pp. 1399-1404, 1987. https://doi.org/10.1016/0362-546X(87)90092-7
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”, Journal of Mathematical Physics, vol. 53, 073517, 2012. https://doi.org/10.1063/1.4739406
J. Dabas and A. Chauhan, “Existence and uniqueness of mild solution for an impulsive neutral fractional integro differential equations with infinite delay”, Mathematical and Computer Modelling, vol. 57, pp. 754-763, 2013. https://doi.org/10.1016/j.mcm.2012.09.001
K. Ramkumar, K. Ravikumar and S. Varshini, “Fractional neutral stochastic differential equations with Caputo fractional derivative: Fractional Brownian motion, Poisson jumps, and Optimal control”, Stochastic Analysis and Applications, vol. 39, no. 1, pp. 157-176, 2020. https://doi.org/10.1080/07362994.2020.1789476
Published
How to Cite
Issue
Section
Copyright (c) 2023 K. Ravikumar, K. Ramkumar, Hamdy Ahmed
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
