Approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by Rosenblatt process via resolvent operators

Essozimna Kpizim; Bertin Dehigbe; Ramkumar Kasinathan; Ravikumar Kasinathan; Mamadou Abdoul Diop

doi:10.56754/0719-0646.2503.467

DOI:

https://doi.org/10.56754/0719-0646.2503.467

Abstract

In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

Keywords

Approximate controllability , fixed point theorem , Rosenblatt process , stochastic integrodifferential equations , resolvent operator , non-instantaneous impulses

Mathematics Subject Classification:

93B05 , 47H10 , 47G20 , 35A50 , 60G20

Essozimna Kpizim Numerical Analysis and Computer Science Laboratory, Department of Mathematics, Gaston Berger University of Saint-Louis, UFR Applied Sciences and Technologies B.P:234, Saint-Louis, Senegal. https://orcid.org/0009-0008-6525-434X
Bertin Dehigbe Institut de Mathématiques et de Sciences Physiques, URMPM B.P. 613, Porto-Novo, Bénin. https://orcid.org/0009-0006-0234-8117
Ramkumar Kasinathan Department of Mathematics, PSG college of Arts and Science, Coimbatore, 641 046, India. https://orcid.org/0000-0002-3624-5363
Ravikumar Kasinathan Department of Mathematics, PSG college of Arts and Science, Coimbatore, 641 046, India. https://orcid.org/0000-0002-5635-8517
Mamadou Abdoul Diop Numerical Analysis and Computer Science Laboratory, Department of Mathematics, Gaston Berger University of Saint-Louis, UFR Applied Sciences and Technologies B.P:234, Saint-Louis, Senegal – UMMISCO UMI 209 IRD/UPMC, Bondy, France. https://orcid.org/0000-0002-3420-2124

Pages: 467–495
Date Published: 2023-12-30
Vol. 25 No. 3 (2023)

S. H. Abid, S. Q. Hasan, and U. J. Quaez, “Approximate controllability of fractional Sobolev type stochastic differential equations driven by mixed fractional Brownian motion,” Journal of Mathematical Sciences and Applications, vol. 3, no. 1, pp. 3–11, 2015.

P. Abry and V. Pipiras, “Wavelet-based synthesis of the Rosenblatt process,” Signal Process., vol. 86, no. 9, pp. 2326–2339, 2006, doi: 10.1016/j.sigpro.2005.10.021.

A. Anguraj and A. Vinodkumar, “Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional differential equations with infinite delays,” Electron. J. Qual. Theory Differ. Equ., 2009, Art. ID 67.

A. E. Bashirov and N. I. Mahmudov, “On concepts of controllability for deterministic and stochastic systems,” SIAM J. Control Optim., vol. 37, no. 6, pp. 1808–1821, 1999, doi: 10.1137/S036301299732184X.

B. Bayour and D. F. M. Torres, “Existence of solution to a local fractional nonlinear differential equation,” J. Comput. Appl. Math., vol. 312, pp. 127–133, 2017, doi: 10.1016/j.cam.2016.01.014.

T. Caraballo and K. Liu, “Exponential stability of mild solutions of stochastic partial differential equations with delays,” Stochastic Anal. Appl., vol. 17, no. 5, pp. 743–763, 1999, doi: 10.1080/07362999908809633.

D. Chalishajar, R. Kasinathan, R. Kasinathan, and G. Cox, “Existence uniqueness and stability of nonlocal neutral stochastic differential equations with random impulses and Poisson jumps,” Results in Nonlinear Analysis, vol. 5, no. 3, pp. 250–262, 2022.

D. Chalishajar, R. Kasinathan, R. Kasinathan, and M. A. Diop, “Optimal control for neutral stochastic systems with infinite time delay and deviated argument driven by Rosenblatt process,” Results in Control and Optimization, vol. 9, 2022, Art. ID 100181.

J. Cui and L. Yan, “Existence result for fractional neutral stochastic integro-differential equations with infinite delay,” J. Phys. A, vol. 44, no. 33, 2011, Art. ID 335201.

R. F. Curtain and P. L. Falb, “Stochastic differential equations in Hilbert space,” J. Differential Equations, vol. 10, pp. 412–430, 1971, doi: 10.1016/0022-0396(71)90004-0.

G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., ser. Encyclopedia of Mathematics and its Applications. Cambridge, England: Cambridge University Press, 2014, vol. 152.

A. Debbouche and V. Antonov, “Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces,” Chaos Solitons Fractals, vol. 102, pp. 140–148, 2017, doi: 10.1016/j.chaos.2017.03.023.

W. Desch, R. Grimmer, and W. Schappacher, “Some considerations for linear integrodifferential equations,” Journal of Mathematical Analysis and Applications, vol. 104, no. 1, pp. 219–234, 1984, doi: 10.1016/0022-247X(84)90044-1.

R. Dhayal, M. Malik, and S. Abbas, “Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps,” Asian J. Control, vol. 23, no. 6, pp. 2669–2680, 2021, doi: 10.1002/asjc.2389.

M. Dieye, M. A. Diop, and K. Ezzinbi, “On exponential stability of mild solutions for some stochastic partial integrodifferential equations,” Statist. Probab. Lett., vol. 123, pp. 61–76, 2017, doi: 10.1016/j.spl.2016.10.031.

C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, and K. S. Nisar, “A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems,” Chaos Solitons Fractals, vol. 142, 2021, Art. ID 110472.

M. A. Diop, K. Ezzinbi, and M. M. Zene, “Existence and stability results for a partial impulsive stochastic integro-differential equation with infinite delay,” SeMA J., vol. 73, no. 1, pp. 17–30, 2016, doi: 10.1007/s40324-015-0053-x.

N. T. Dung, “Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space,” Stochastics, vol. 87, no. 1, pp. 142–159, 2015, doi: 10.1080/17442508.2014.924938.

S. Farahi and T. Guendouzi, “Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions,” Results Math, vol. 65, no. 3-4, pp. 501–521, 2014, doi: 10.1007/s00025-013-0362-2.

R. C. Grimmer, “Resolvent operators for integral equations in a Banach space,” Trans. Amer. Math. Soc., vol. 273, no. 1, pp. 333–349, 1982, doi: 10.2307/1999209.

M. H. M. Hamit, I. M. Barka, M. A. Diop, and K. Ezzinbi, “Controllability of impulsive stochastic partial integrodifferential equation with noncompact semigroups,” Discuss. Math. Differ. Incl. Control Optim., vol. 39, no. 2, pp. 159–180, 2019.

M. H. M. Hassan, M. A. Diop, R. Kasinathan, and R. Kasinathan, “Existence, global attracting sets and exponential decay of solution to stochastic functional integro-differential equations driven by Rosenblatt process,” Electron. J. Math. Anal. Appl., vol. 8, no. 2, pp. 38–59, 2020.

R. Kasinathan, R. Kasinathan, M. H. M. Hamit, and M. A. Diop, “Exponential behavior of neutral impulsive stochastic integro-differential equations driven by poisson jumps and rosenblatt process,” J. Differential Equations, vol. 7, no. 7, pp. 1–21, 2020, doi: 10.1515/msds- 2020-0001.

A. N. Kolmogoroff, “Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum,” C. R. (Doklady) Acad. Sci. URSS (N.S.), vol. 26, pp. 115–118, 1940.

J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal problems for integrodifferential equations,” Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., vol. 15, no. 6, pp. 815–824, 2008.

A. Lin, Y. Ren, and N. Xia, “On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators,” Math. Comput. Modelling, vol. 51, no. 5-6, pp. 413–424, 2010, doi: 10.1016/j.mcm.2009.12.006.

N. I. Mahmudov and A. Denker, “On controllability of linear stochastic systems,” Internat. J. Control, vol. 73, no. 2, pp. 144–151, 2000, doi: 10.1080/002071700219849.

B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev., no. 1968, pp. 422–437, 10, doi: 10.1137/1010093.

A. Mane, K. H. Bete, C. Ogouyandjou, and M. A. Diop, “Controllability results for a nonlocal impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps,” J. Nonlinear Anal. Optim., vol. 9, no. 1, pp. 67–83, 2018.

F. Z. Mokkedem and X. Fu, “Approximate controllability of semi-linear neutral integro-differential systems with finite delay,” Appl. Math. Comput., vol. 242, pp. 202–215, 2014, doi: 10.1016/j.amc.2014.05.055.

B. Radhakrishnan and K. Balachandran, “Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay,” Nonlinear Anal. Hybrid Syst., vol. 5, no. 4, pp. 655–670, 2011, doi: 10.1016/j.nahs.2011.05.001.

R. Sakthivel, R. Ganesh, Y. Ren, and S. M. Anthoni, “Approximate controllability of nonlinear fractional dynamical systems,” Commun. Nonlinear Sci. Numer. Simul., vol. 18, no. 12, pp. 3498–3508, 2013, doi: 10.1016/j.cnsns.2013.05.015.

R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate controllability of semilinear fractional differential systems,” Comput. Math. Appl., vol. 62, no. 3, pp. 1451–1459, 2011, doi: 10.1016/j.camwa.2011.04.040.

G. Shen and Y. Ren, “Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space,” J. Korean Statist. Soc., vol. 44, no. 1, pp. 123–133, 2015, doi: 10.1016/j.jkss.2014.06.002.

G. Shen, R. Sakthivel, Y. Ren, and M. Li, “Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process,” Collect. Math., vol. 71, no. 1, pp. 63–82, 2020, doi: 10.1007/s13348-019-00248-3.

M. Taqqu, “Weak convergence to fractional Brownian motion and to the Rosenblatt process,” Advances in Appl. Probability, vol. 7, no. 2, pp. 249–249, 1975, doi: 10.2307/1426060.

C. A. Tudor, “Analysis of the Rosenblatt process,” ESAIM Probab. Stat., vol. 12, pp. 230–257, 2008, doi: 10.1051/ps:2007037.

S. Varshini, K. Banupriya, K. Ramkumar, and K. Ravikumar, “Existence, uniqueness and stability results for neutral stochastic differential equations with random impulses,” Filomat, vol. 37, no. 3, pp. 979–987, 2023, doi: 10.2298/fil2303979v.

Z. Yan and F. Lu, “Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay,” Appl. Math. Comput., vol. 292, pp. 425–447, 2017, doi: 10.1016/j.amc.2016.06.035.

Z. Yan and X. Yan, “Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay,” Collect. Math., vol. 64, no. 2, pp. 235–250, 2013, doi: 10.1007/s13348-012-0063-2.