Abstract
The article deals with the approximate controllability of Atangana–Baleanu semilinear control systems. The outcomes are derived by applying Gronwall’s inequality and Cauchy sequence, and avoid the use of the fixed point theorem. We have also included an example for the validation of theoretical results.
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: None declared.
-
Conflict of interest statement: This work does not have any conflicts of interest.
References
[1] O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynam., vol. 29, pp. 145–155, 2012. https://doi.org/10.1023/A:1016539022492.10.1023/A:1016539022492Search in Google Scholar
[2] P. Bedi, A. Kumar, T. Abdeljawad, and A. Khan, “Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations,” Adv. Differ. Equ., vol. 155, pp. 1–16, 2020. https://doi.org/10.1186/s13662-020-02615-y.Search in Google Scholar
[3] P. Bedi, A. Khan, A. Kumar, and T. Abdeljawad, “Computational study of fractional order vector borne diseases model,” Fractals, vol. 30, no. 5, pp. 1–13, 2022. https://doi.org/10.1142/S0218348X22401491.Search in Google Scholar
[4] P. Bedi, A. Kumar, T. Abdeljawad, and A. Khan, “Mild solutions of coupled hybrid fractional order system with Caputo-Hadamard derivatives,” Fractals, vol. 20, no. 6, pp. 1–10, 2021. https://doi.org/10.1142/S0218348X21501589.Search in Google Scholar
[5] A. Devi and A. Kumar, “Hyers-Ulam stability and existence of solution for hybrid fractional differential equation with p-Laplacian operator,” Chaos, Solit. Fractals, vol. 156, pp. 1–8, 2021. https://doi.org/10.1016/j.chaos.2022.111859.Search in Google Scholar
[6] A. Devi, A. Kumar, T. Abdeljawad, and A. Khan, “Stability analysis of solutions and existence theory of fractional Lagevin equation,” Alex. Eng. J., vol. 60, no. 4, pp. 3641–3647, 2021. https://doi.org/10.1016/j.aej.2021.02.011.Search in Google Scholar
[7] M. Francesco, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Italy, World Scientific, 2010.Search in Google Scholar
[8] F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solit. Fractals, vol. 7, no. 9, pp. 1461–1477, 1996. https://doi.org/10.1016/0960-0779(95)00125-5.Search in Google Scholar
[9] H. Richard, Fractional Calculus: An Introduction for Physicists, Singapore, World Scientific, 2014.Search in Google Scholar
[10] W. K. Williams, V. Vijayakumar, R. Udhayakumar, and K. S. Nisar, “A new study on existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces,” Numer. Methods Part. Differ. Equ., vol. 37, no. 2, pp. 949–961, 2020. https://doi.org/10.1002/num.22560.Search in Google Scholar
[11] W. K. Williams, V. Vijayakumar, R. Udhayakumar, S. K. Panda, and K. S. Nisar, “Existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order 1 < r < 2,” Numer. Methods Part. Differ. Equ., pp. 1–21, 2020. https://doi.org/10.1002/num.22697.Search in Google Scholar
[12] A. Atangana and D. Balneau, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model,” Therm. Sci., vol. 20, no. 2, pp. 763–769, 2016. https://doi.org/10.48550/arXiv.1602.03408.Search in Google Scholar
[13] A. Atangana and R. T. Alqahtani, “New numerical method and application to keller-segel model with fractional order derivative,” Chaos, Solit. Fractals, vol. 116, pp. 14–21, 2018. https://doi.org/10.1016/j.chaos.2018.09.013.Search in Google Scholar
[14] S. Ucar, E. Ucar, N. Ozdemir, and Z. Hammouch, “Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative,” Chaos, Solit. Fractals, vol. 118, pp. 300–306, 2019. https://doi.org/10.1016/j.chaos.2018.12.003.Search in Google Scholar
[15] D. Aimene, D. Baleanu, and D. Seba, “Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay,” Chaos, Solit. Fractals, vol. 128, pp. 51–57, 2019. https://doi.org/10.1016/j.chaos.2019.07.027.Search in Google Scholar
[16] A. Atangana and I. Koca, “Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order,” Chaos, Solit. Fractals, vol. 89, pp. 447–454, 2016. https://doi.org/10.1016/j.chaos.2016.02.012.Search in Google Scholar
[17] A. Devi and A. Kumar, “Existence and uniqueness results for integro fractional differential equations with Atangana-Baleanu fractional derivative,” J. Math. Ext., vol. 15, no. 5, pp. 1–24, 2021.Search in Google Scholar
[18] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, and A. Shukla, “A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay,” Chaos, Solit. Fractals, vol. 157, pp. 1–21. https://doi.org/10.1016/j.chaos.2022.111916.Search in Google Scholar
[19] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, et al.., “A note on existence and approximate controllability outcomes of Atangana-Baleanu neutral fractional stochastic hemivariational inequality,” Results Phys., vol. 38, p. 105647, 2022. https://doi.org/10.1016/j.rinp.2022.105647.Search in Google Scholar
[20] C. Dineshkumar, K. S. Nisar, R. Udhayakumar, and V. Vijayakumar, “New discussion about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2,” Asian J. Control, vol. 24, no. 5, pp. 2519–2533, 2022. https://doi.org/10.1002/asjc.2663.Search in Google Scholar
[21] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, and K. S. Nisar, “A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions,” Asian J. Control, vol. 24, no. 5, pp. 2378–2394, 2022. https://doi.org/10.1002/asjc.2650.Search in Google Scholar
[22] F. Jarad, T. Abdeljawad, and Z. Hammouch, “On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative,” Chaos, Solit. Fractals, vol. 117, pp. 16–20, 2018. https://doi.org/10.1016/j.chaos.2018.10.006.Search in Google Scholar
[23] M. Johnson, V. Vijayakumar, K. S. Nisar, A. Shukla, T. Botmart, and V. Ganesh, “Results on the approximate controllability of Atangana-Baleanu fractional stochastic delay integrodifferential systems,” Alex. Eng. J., vol. 62, pp. 211–222, 2022. https://doi.org/10.1016/j.aej.2022.06.038.Search in Google Scholar
[24] A. Kumar and D. N. Pandey, “Existence of mild solution of Atangana-Baleanu fractional differential equations with non-instantaneous impulses and with non-local conditions,” Chaos, Solit. Fractals, vol. 132, pp. 1–4, 2020. https://doi.org/10.1016/j.chaos.2019.109551.Search in Google Scholar
[25] Y. K. Ma, C. Dineshkumar, V. Vijayakumar, R. Udhayakumar, A. Shukla, and K. S. Nisar, “Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions,” Ain Shams Eng. J., p. 210188, 2202. https://doi.org/10.1016/j.asej.2022.101882.Search in Google Scholar
[26] K. M. Owolabi and A. Atangana, “On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems,” Chaos, vol. 29, no. 2, pp. 1–19, 2019. https://doi.org/10.1063/1.5085490.Search in Google Scholar PubMed
[27] C. Ravichandran, K. Logeswari, and F. Jarad, “New results on existence in the frame-work of Atangana-Baleanu derivative for fractional integro-differential equations,” Chaos, Solit. Fractals, vol. 125, pp. 194–200, 2019. https://doi.org/10.1016/j.chaos.2019.05.014.Search in Google Scholar
[28] K. M. Saad, A. Atangana, and D. Baleanu, “New fractional derivatives with non-singular kernel applied to the burgers equation,” Chaos, vol. 28, no. 6, pp. 1–7, 2018, https://doi.org/10.1063/1.5026284.Search in Google Scholar PubMed
[29] K. M. Saad, D. Baleanu, and A. Atangana, “New fractional derivatives applied to the korteweg-de vries and korteweg-de vries-burgers equations,” Comput. Appl. Math., vol. 37, no. 4, pp. 5203–5216, 2018. https://doi.org/10.1007/s40314-018-0627-1.Search in Google Scholar
[30] W. K. Williams and V. Vijayakumar, “Discussion on the controllability results for fractional neutral impulsive Atangana-Baleanu delay integro-differential systems,” Math. Methods Appl. Sci., pp. 1–16, 2021, https://doi.org/10.1002/mma.7754.Search in Google Scholar
[31] R. E. Kalman, “Controllability of linear dynamical systems,” Contrib. Differ. Equ., vol. 1, pp. 190–213, 1963.Search in Google Scholar
[32] P. Balasubramaniam, J. Y. Park, and P. Muthukumar, “Approximate controllability of neutral stochastic functional differential systems with infinite delay,” Stoch. Anal. Appl., vol. 28, pp. 389–400, 2010. https://doi.org/10.1186/s13662-015-0368-z.Search in Google Scholar
[33] H. R. Henriquez and E. M. Hernandez, “Approximate controllability of second-order distributed implicit functional systems,” Nonlinear Anal., vol. 70, pp. 1023–1039, 2019. https://doi.org/10.1016/j.na.2008.01.029.Search in Google Scholar
[34] K. Kavitha, V. Vijayakumar, R. Udhayakumar, N. Sakthivel, and K. S. Nisar, “A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay,” Math. Methods Appl. Sci., vol. 44, no. 6, pp. 4428–4447, 2021. https://doi.org/10.1002/mma.7040.Search in Google Scholar
[35] K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar, and R. Udhayakumar, “Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type,” Chaos, Solit. Fractals, vol. 151, pp. 1–8, 2021. https://doi.org/10.1016/j.chaos.2021.111264.Search in Google Scholar
[36] M. Mohan Raja, V. Vijayakumar, L. N. Huynh, R. Udhayakumar, and K. S. Nisar, “Results on the approximate controllability of fractional hemivariational inequalities of order 1 < r < 2,” Adv. Differ. Equ., vol. 237, pp. 1–25, 2021. https://doi.org/10.1186/s13662-021-03373-1.Search in Google Scholar
[37] M. Mohan Raja, V. Vijayakumar, and R. Udhayakumar, “A new approach on approximate controllability of fractional evolution inclusions of order 1 < r < 2 with infinite delay,” Chaos, Solit. Fractals, vol. 141, pp. 1–13, 2020. https://doi.org/10.1016/j.chaos.2020.110343.Search in Google Scholar
[38] M. Mohan Raja and V. Vijayakumar, “Optimal control results for Sobolev-type fractional mixed Volterra-Fredholm type integrodifferential equations of order 1 < r < 2 with sectorial operators,” Optim. Control Appl. Methods, vol. 43, no. 5, pp. 1314–1327, 2022. https://doi.org/10.1002/oca.2892.Search in Google Scholar
[39] M. Mohan Raja and V. Vijayakumar, “New results concerning to approximate controllability of fractional integrodifferential evolution equations of order 1 < r < 2,” Numer. Methods Part. Differ. Equ., vol. 38, no. 3, pp. 509–524, 2022. https://doi.org/10.1002/num.22653.Search in Google Scholar
[40] R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate controllability of semilinear fractional differential systems,” Comput. Math. Appl., vol. 62, pp. 1451–1459, 2011. https://doi.org/10.1016/j.camwa.2011.04.040.Search in Google Scholar
[41] A. Shukla, N. Sukavanam, and D. N. Pandey, “Complete controllability of semilinear stochastic systems with delay in both state and control,” Math. Rep., vol. 18, pp. 247–259, 2016.10.1093/imamci/dnw059Search in Google Scholar
[42] A. Shukla, N. Sukavanam, and D. N. Pandey, “Approximate controllability of semilinear stochastic control system with nonlocal conditions,” Nonlinear Dynam. Syst. Theor., vol. 15, no. 3, pp. 321–333, 2018. https://doi.org/10.1142/S1793557118500882.Search in Google Scholar
[43] N. Sukavanam and S. Kumar, “Approximate controllability of fractional order semilinear delay systems,” J. Optim. Theor. Appl., vol. 252, no. 11, pp. 73–78, 2011. https://doi.org/10.1007/s10957-011-9905-4.Search in Google Scholar
[44] N. Sukavanam and S. Tafesse, “Approximate controllability of a delayed semilinear control system with growing nonlinear term,” Nonlinear Anal., vol. 74, pp. 6868–6875, 2011. https://doi.org/10.1016/j.na.2011.07.009.Search in Google Scholar
[45] V. Vijayakumar, “Approximate controllability results for analytic resolvent integro differential inclusions in Hilbert spaces,” Int. J. Control, vol. 91, no. 1, pp. 204–214, 2018. https://doi.org/10.1080/00207179.2016.1276633.Search in Google Scholar
[46] V. Vijayakumar, C. Ravichandran, and R. Murugesu, “Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay,” Nonlinear Stud., vol. 20, no. 4, pp. 513–532, 2013. https://doi.org/10.1093/imamci/dns033.Search in Google Scholar
[47] V. Vijayakumar, C. Ravichandran, and R. Murugesu, “Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in Banach spaces,” Dyn. Continuous Discrete Impuls. Syst., vol. 20, nos. 4–5b, pp. 485–502, 2013.Search in Google Scholar
[48] V. Vijayakumar and R. Murugesu, “Controllability for a class of second order evolution differential inclusions without compactness,” Hist. Anthropol., vol. 98, no. 7, pp. 1367–1385, 2019. https://doi.org/10.1080/00036811.2017.1422727.Search in Google Scholar
[49] V. Vijayakumar, “Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type,” Results Math., vol. 73, no. 42, pp. 1–23, 2018. https://doi.org/10.1007/s00025-018-0807-8.Search in Google Scholar
[50] I. Podlubny, “An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,” Math. Sci. Eng., vol. 198, p. 1999. https://doi.org/10.1016/s0076-5392(99)x8001-5.Search in Google Scholar
[51] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol 44, New York, NY, Springer, 1983.10.1007/978-1-4612-5561-1Search in Google Scholar
[52] G. Bahaa and A. Hamiaz, “Optimality conditions for fractional differential inclusions with nonsingular Mittag-Leffler kernel,” Adv. Differ. Equ., vol. 257, no. 1, pp. 1–26, 2018. https://doi.org/10.1186/s13662-018-1706-8.Search in Google Scholar
[53] X. B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Anal., vol. 74, no. 5, pp. 2003–2011, 2011. https://doi.org/10.1016/j.na.2010.11.007.Search in Google Scholar
[54] L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” J. Math. Anal. Appl., vol. 162, pp. 494–505, 1991. https://doi.org/10.1016/0022-247X(91)90164-U.Search in Google Scholar
[55] L. Byszewski and H. Akca, “On a mild solution of a semilinear functional-differential evolution nonlocal problem,” J. Appl. Math. Stoch. Anal., vol. 10, no. 3, pp. 265–271, 1997. https://doi.org/10.1155/S1048953397000336.Search in Google Scholar
[56] A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential Systems,” Comput. Math. Appl., vol. 62, pp. 1442–1450, 2011. https://doi.org/10.1016/j.camwa.2011.03.075.Search in Google Scholar
[57] G. M. Mophou and G. M. N’Guerekata, “Existence of mild solution for some fractional differential equations with nonlocal conditions,” Semigr. Forum, vol. 79, no. 2, pp. 322–335, 2009. https://doi.org/10.1007/s00233-008-9117-x.Search in Google Scholar
[58] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Anal. R. World Appl., vol. 11, pp. 4465–4475, 2010. https://doi.org/10.1016/j.nonrwa.2010.05.029.Search in Google Scholar
[59] K. Naito, “Controllability of semilinear control systems dominated by the linear part,” SIAM J. Control Optim., vol. 25, no. 3, pp. 715–722, 1987. https://doi.org/10.1137/0325040.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
