AccScience Publishing / IJOCTA / Volume 13 / Issue 2 / DOI: 10.11121/ijocta.2023.1348
RESEARCH ARTICLE

A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators

Kalimuthu Kaliraj1* Kothandapani Muthuvel1,2
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1 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, Tamil Nadu, India
2 Department of Mathematics, Guru Nanak College, Chennai 600 042, Tamil Nadu, India
IJOCTA 2023, 13(2), 193–204; https://doi.org/10.11121/ijocta.2023.1348
Submitted: 23 December 2022 | Accepted: 24 April 2023 | Published: 19 July 2023
© 2023 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

Keywords
Fractional derivatives and integrals
Approximate controllability
Sectorial operators
Fixed point techniques
Fractional stochastic system
Conflict of interest
The authors declare they have no competing interests.
References

[1] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J.(2006). Theory and applications of fractional dif- ferential equations, Elsevier, Amsterdam.

[2] Miller, K.S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differ- ential Equations, Wiley, New York.

[3] Podlubny, I. (1999). Fractional differential equa- tions: An introduction to fractional derivatives, fractional differential equations, to method of their solution and some of their applications, San Diego, CA: Academic Press.

[4] Zhou, Y. (2014). Basic Theory of Fractional Dif- ferential Equations, World Scientific, Singapore. https://doi.org/10.1142/10238.

[5] Zhou, Y. (2015). Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York.

[6] Dineshkumar, C., Nisar, K.S., Udhayakumar, R., & Vijayakumar, V. (2021). New discussion about the approximate controllability of frac- tional stochastic differential inclusions with or- der 1 < r < 2. Asian Journal of Control, 1-15. https://doi.org/10.1002/num.22698.

[7] Kaliraj, K., Lakshmi Priya, P.K., & Ravichan- dran, C. (2022). An explication of finite-time sta- bility for fractional delay model with neutral im- pulsive conditions. Qualitative Theory of Dynam- ical Systems, 21, 1-17. Doi:10.1007/s12346-022- 00694-8.

[8] Kavitha, K., Vijayakumar, V., Udhayakumar, R., & Ravichandran, C. (2021). Results on con- trollability of Hilfer fractional differential equa- tions with infinite delay via measures of non- compactness. Asian Journal of Control, 24, 1-10. https://doi.org/10.1002/asjc.2549.

[9] Lakshmi Priya, P.K., & Kaliraj, K. (2022). An application of fixed point technique of Rothe’s- type to interpret the controllability criteria of neu- tral nonlinear fractional order impulsive system. Chaos, Solitons & Fractals, 164, 112647.

[10] Mohan Raja, M., & Vijayakumar, V. (2022). Existence results for Caputo fractional mixed Volterra-Fredholm-type integrodifferential inclu- sions of order r ∈ (1, 2) with sectorial op- erators. Chaos, Solitons & Fractals, 159, 1-8. https://doi.org/10.1016/j.chaos.2022.112127.

[11] Mohan Raja, M., Vijayakumar, V., Shukla, A., Nisar, K.S., & Baskonus, H.M. (2022). On the approximate controllability results for fractional integrodifferential systems of order 1 < r <2 with sectorial operators. Journal of Compu- tational and Applied Mathematics, 415, 1-12. https://doi.org/10.1016/j.cam.2022.114492.

[12] Sakthivel, R., Ganesh, R., & Anthoni, S.M.(2013). Approximate controllability of frac- tional nonlinear differential inclusions. Applied Mathematics and Computation, 225, 708-717. https://doi.org/10.1016/j.amc.2013.09.068.

[13] Shu, L., Shu, X.B., & Mao, J. (2019). Ap- proximate controllability and existence of mild solutions for Riemann-Liouville fractional sto- chastic evolution equations with nonlocal con- ditions of order 1 < α < 2. Fractional Calculus & Applied Analysis, 22, 1086-1112. https://doi.org/10.1515/fca-2019-0057.

[14] Shu, X.B., & Wang, Q. (2012). The exis- tence and uniqueness of mild solutions for frac- tional differential equations with nonlocal con- ditions of order 1 < α < 2. Computers and Mathematics with Applications, 64, 2100-2110. https://doi.org/10.1016/j.camwa.2012.04.006.

[15] Mohan Raja, M., Vijayakumar, V., & Udhayaku- mar, R. (2020). Results on the existence and con- trollability of fractional integro-differential system of order 1 < r < 2 via measure of noncompact- ness. Chaos, Solitons & Fractals, 139, 110299. https://doi.org/10.1016/j.chaos.2020.110299.

[16] Wang, J., & Zhou, Y. (2011). Ex- istence and controllability results for fractional semilinear differential inclu- sions. Nonlinear Analysis, 12, 3642-3653. https://doi.org/10.1016/j.nonrwa.2011.06.021.

[17] El-Sayed, A.M.A., & Ibrahim, A.G. (1995). Mul- tivalued fractional differential equations of arbi- trary orders. Applied Mathematics and Compu- tation, 68, 15-25. https://doi.org/10.1016/0096- 3003(94)00080-N.

[18] Ito, K., Jin, B., & Takeuchi, T. (2015), On the sectorial property of the Caputo derivative oper- ator. Applied Mathematics Letters, 47, 43-46.

[19] Wang, J.R., Ibrahim, A.G., & Feckan, M.(2015). Nonlocal impulsive fractional dif- ferential inclusions with fractional sectorial operators on Banach spaces. Applied Math- ematics and Computation, 257, 103-118. https://doi.org/10.1016/j.amc.2014.04.093.

[20] Agarwal, R.P., Bashir, A., Alsaedi, A., & Shahzad, N. (2012). On the dimension of the so- lution set for semilinear fractional differential in- clusions. Abstract and Applied Analysis, 10, 1-10. https://doi.org/10.1155/2012/305924.

[21] Benchohra, M., & Ziane, M. (2013). Impulsive evolution inclusions with state-dependent delay and multivalued jumps. Electronic Journal of Qualitative Theory of Differential Equations, 42, 1-21. https://doi.org/10.14232/ejqtde.2013.1.42.

[22] He, J.W., Liang, Y., Ahmad, B., & Zhou, Y.(2019). Nonlocal fractional evolution inclusions of order α ∈ (1, 2). Mathematics, 209, 1-17. https://doi.org/10.3390/math7020209.

[23] Mohan Raja, M., Vijayakumar, V., & Udhayaku- mar, R. (2020). A new approach on approx- imate controllability of fractional evolution in- clusions of order 1 < r < 2 with infinite delay. Chaos Solitons & Fractals, 141, 110343. https://doi.org/10.1016/j.chaos.2020.110343

[24] Mohan Raja, M., Vijayakumar, V., Ud- hayakumar, R., & Zhou, Y. (2020). A new approach on the approximate controllabil- ity of fractional differential evolution equa- tions of order 1 < r < 2 in Hilbert spaces. Chaos Solitons & Fractals, 141, 110310. https://doi.org/10.1016/j.chaos.2020.110310.

[25] Dhayal, R., Malik, M., & Abbas, S. (2021). Solv- ability and optimal controls of non-instantaneous impulsive stochastic fractional differential equa- tion of order q ∈ (1, 2). Stochastics, 93(5), 780– 802. http://www.aimspress.com/article/10.3934/math.2019.3.663.

[26] Shu, X.B., Lai, Y., & Chen, Y. (2011). The existence of mild solutions for im- pulsive fractional partial differential equa- tions. Nonlinear Analysis, 74, 2003-2011. https://doi.org/10.1016/j.na.2010.11.007.

[27] Dineshkumar, C., Nisar, K.S., Udhayakumar, R., & Vijayakumar, V. (2021). A discus- sion on approximate controllability of Sobolev- type Hilfer neutral fractional stochastic differen- tial inclusions. Asian Journal of Control, 1-17. https://doi.org/10.1016/j.chaos.2020.110472.

[28] Singh, A., Shukla, A., Vijayakumar, V., & Udhayakumar, R. (2021). Asymptotic sta- bility of fractional order (1,2] stochastic de- lay differential equations in Banach spaces. Chaos Solitons & Fractals, 150, 111095. https://doi.org/10.1016/j.chaos.2021.111095.

[29] Kavitha, K., Vijayakumar, V., Anurag, S., Nisar, K.S., & Udhayakumar, R. (2021). Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type. Chaos Solitons & Fractals, 151, 111264. https://doi.org/10.1016/j.chaos.2021.111264.

[30] Ma, Y.K., Kavitha, K., Albalawi, W., Shukla, A., Nisar, K.S., & Vijayakumar, V. (2022). An anal- ysis on the approximate controllability of Hilfer fractional neutral differential systems in Hilbert spaces. Alexandria Engineering Journal, 61(9), 7291-7302.

[31] Shukla, A., Sukavanam, N., & Pandey, D.N.(2015). Complete controllability of semi-linear stochastic system with delay. Rendicondi del Circolo Matematico di Palermo, 64, 209–220. https://doi.org/10.1007/s12215-015-0191-0.

[32] Vijayakumar, V., Nisar, K.S., Chalishajar, D., Shukla, A., Malik, M., Alsaadi, A., & Aldosary, S.F. (2022). A note on ap- proximate controllability of fractional semilin- ear integrodifferential control systems via re- solvent operators. Fractal and Fractional, 6(2). https://doi.org/10.3390/fractalfract6020073.

[33] Shu, X.B., & Xu, F. (2014). Upper and lower solution method for factional evolu- tion equations with order 1 < α < 2. Korean Mathematical Society, 51, 1123-1139. https://doi.org/10.4134/JKMS.2014.51.6.1123.

[34] Deimling, K. (1992). Multivalued Dif- ferential Equations. De Gruyter, Berlin, https://doi.org/10.1515/9783110874228.

[35] Chang, Y., & Nieto, J.J. (2009). Existence of solutions for impulsive neutral integro- differential inclusions with nonlocal initial condi- tions via fractional operators. Numerical Func- tional Analysis and Optimization, 30, 227–244. https://doi.org/10.1080/01630560902841146.

[36] Bohnenblust, H.F., & Karlin, S. (1950). On a theorem of Ville, Contributions to the theory of games. Annals of Mathematics Studies, 24, Princeton University Press, Princeton, N. J., 155– 160. DOI: 10.4236/am.2013.47A008.

[37] Mahmudov, N.I. (2001). Controllability of linear stochastic systems in Hilbert spaces. Journal of Mathematical Analysis and Applications, 259(1), 64-82. https://doi.org/10.1006/jmaa.2000.7386.

[38] Byszewski, L., & Akca, H. (1997). On a mild so- lution of a semilinear functional-differential evolu- tion nonlocal problem. Journal of Applied Math- ematics and Stochastic Analysis, 10, 265–271. https://doi.org/10.1155/S1048953397000336.

[39] Mohan Raja, M., & Vijayakumar, V. (2022). Op- timal control results for Sobolev-type fractional mixed Volterra-Fredholm type integrodifferential equations of order 1 < r < 2 with sectorial opera- tors. Optimal Control Applications and Methods, 43, 1-17, https://doi.org/10.1002/oca.2892.

[40] Wang, X., & Shu, X.B. (2015). The existence of positive mild solutions for fractional differential evolution equations with nonlocal conditions of order 1 < α < 2. Advances in Difference Equa- tions, 159, 1-15. https://doi.org/10.1186/s13662- 015-0461-3.

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