Results on controllability for Sobolev type fractional differential equations of order $ 1 &lt; r &lt; 2 $ with finite delay

Yong-Ki Ma; Marimuthu Mohan Raja; Kottakkaran Sooppy Nisar; Anurag Shukla; Velusamy Vijayakumar; Yong-Ki Ma; Marimuthu Mohan Raja; Kottakkaran Sooppy Nisar; Anurag Shukla; Velusamy Vijayakumar

doi:10.3934/math.2022568

AIMS Mathematics

2022, Volume 7, Issue 6: 10215-10233. doi: 10.3934/math.2022568

Previous Article Next Article

Research article Special Issues

Results on controllability for Sobolev type fractional differential equations of order $ 1 < r < 2 $ with finite delay

1.
Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, Republic of Korea
2.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India
3.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
4.
Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj 209732, India

Received: 13 November 2021 Revised: 13 March 2022 Accepted: 15 March 2022 Published: 22 March 2022
MSC : 34A08, 26A33, 93B05, 46E36, 47D09

In this article, exact controllability results for Sobolev fractional delay differential system of $ 1 < r < 2 $ are investigated. Fractional analysis, cosine and sine function operators, and Schauder's fixed point theorem are applied to verify the main results of this study. To begin, we use sufficient conditions to explore the controllability for fractional evolution differential system with finite delay. Lastly, an example is provided to illustrate the obtained theoretical results.
- fractional differential systems,
- controllability,
- Mainardi's Wright-type function,
- Sobolev type,
- mild solutions
Citation: Yong-Ki Ma, Marimuthu Mohan Raja, Kottakkaran Sooppy Nisar, Anurag Shukla, Velusamy Vijayakumar. Results on controllability for Sobolev type fractional differential equations of order $ 1 < r < 2 $ with finite delay[J]. AIMS Mathematics, 2022, 7(6): 10215-10233. doi: 10.3934/math.2022568

Related Papers:

Abstract

In this article, exact controllability results for Sobolev fractional delay differential system of $ 1 < r < 2 $ are investigated. Fractional analysis, cosine and sine function operators, and Schauder's fixed point theorem are applied to verify the main results of this study. To begin, we use sufficient conditions to explore the controllability for fractional evolution differential system with finite delay. Lastly, an example is provided to illustrate the obtained theoretical results.

References

[1]	R. P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equ., 2009 (2009), 1–47. https://doi.org/10.1155/2009/981728 doi: 10.1155/2009/981728
[2]	W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2 Eds., Birkhauser Verlag, 2011.
[3]	P. Balasubramaniam, P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function, Appl. Math. Comput., 256 (2015), 232–246. https://doi.org/10.1016/j.amc.2015.01.035 doi: 10.1016/j.amc.2015.01.035
[4]	M. Bohner, O. Tunç, C. Tunç, Qualitative analysis of caputo fractional integro-differential equations with constant delays, Comp. Appl. Math., 40 (2021), 40. https://doi.org/10.1007/s40314-021-01595-3 doi: 10.1007/s40314-021-01595-3
[5]	Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Soliton Fract., 33 (2007), 1601–1609. https://doi.org/10.1016/j.chaos.2006.03.006 doi: 10.1016/j.chaos.2006.03.006
[6]	C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, A. Shukla, A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order $1 < r < 2$, Math. Comput. Simulat., 190 (2021), 1003–1026. https://doi.org/10.1016/j.matcom.2021.06.026 doi: 10.1016/j.matcom.2021.06.026
[7]	C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, A. Shukla, K. S. Nisar, A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order $r \in (1, 2)$ with delay, Chaos Soliton Fract., 153 (2021), 111565. https://doi.org/10.1016/j.chaos.2021.111565 doi: 10.1016/j.chaos.2021.111565
[8]	H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, Elsevier Science, 1985.
[9]	M. Fečkan, J. Wang, Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators, J. Optim. Theory Appl., 156 (2013), 79–95. https://doi.org/10.1007/s10957-012-0174-7 doi: 10.1007/s10957-012-0174-7
[10]	J. W. Hanneken, D. M. Vaught, B. N. Narahari Achar, Enumeration of the real zeros of the Mittag-Leffler function $E_\alpha(z)$, $1 < \alpha < 2$, In: J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in Fractional Calculus, Dordrecht: Springer, 2007, 15–26. https://doi.org/10.1007/978-1-4020-6042-7_2
[11]	A. Haq, N. Sukavanam, Existence and approximate controllability of Riemann-Liouville fractional integrodifferential systems with damping, Chaos Soliton Fract., 139 (2020), 110043. https://doi.org/10.1016/j.chaos.2020.110043 doi: 10.1016/j.chaos.2020.110043
[12]	A. Haq, Partial-approximate controllability of semi-linear systems involving two Riemann-Liouville fractional derivatives, Chaos Soliton Fract., 157 (2022), 111923. https://doi.org/10.1016/j.chaos.2022.111923 doi: 10.1016/j.chaos.2022.111923
[13]	A. Haq, N. Sukavanam, Partial approximate controllability of fractional systems with Riemann-Liouville derivatives and nonlocal conditions, Rend. Circ. Mat. Palermo, Ser. 2, 70 (2021), 1099–1114. https://doi.org/10.1007/s12215-020-00548-9 doi: 10.1007/s12215-020-00548-9
[14]	A. Haq, N. Sukavanam, Controllability of second-order nonlocal retarded semilinear systems with delay in control, Appl. Anal., 99 (2020), 2741–2754. https://doi.org/10.1080/00036811.2019.1582031 doi: 10.1080/00036811.2019.1582031
[15]	A. Haq, N. Sukavanam, Mild solution and approximate controllability of retarded semilinear systems with control delays and nonlocal conditions, Numer. Funct. Anal. Optim., 42 (2021), 721–737. https://doi.org/10.1080/01630563.2021.1928697 doi: 10.1080/01630563.2021.1928697
[16]	J. W. He, Y. Liang, B. Ahmad, Y. Zhou, Nonlocal fractional evolution inclusions of order $\alpha \in (1, 2)$, Mathematics, 209 (2019), 209. https://doi.org/10.3390/math7020209 doi: 10.3390/math7020209
[17]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control, 2021. https://doi.org/10.1002/asjc.2549
[18]	K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar, R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Soliton Fract., 151 (2021), 111264. https://doi.org/10.1016/j.chaos.2021.111264 doi: 10.1016/j.chaos.2021.111264
[19]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[20]	V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
[21]	J. H. Lightbourne, S. Rankin, A partial functional differential equation of Sobolev type, J. Math. Anal. Appl., 93 (1983), 328–337. https://doi.org/10.1016/0022-247X(83)90178-6 doi: 10.1016/0022-247X(83)90178-6
[22]	Z. Liu, X. Li, On the exact controllability of impulsive fractional semilinear functional differential inclusions Asian J. Control, 17 (2015), 1857–1865. https://doi.org/10.1002/asjc.1071 doi: 10.1002/asjc.1071
[23]	M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, Results on the existence and controllability of fractional integro-differential system of order $1 < r < 2$ via measure of noncompactness, Chaos Soliton Fract., 139 (2020), 110299. https://doi.org/10.1016/j.chaos.2020.110299 doi: 10.1016/j.chaos.2020.110299
[24]	M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1 < r < 2$ in Hilbert spaces, Chaos Soliton Fract., 141 (2020), 110310. https://doi.org/10.1016/j.chaos.2020.110310 doi: 10.1016/j.chaos.2020.110310
[25]	M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, A new approach on approximate controllability of fractional evolution inclusions of order $1 < r < 2$ with infinite delay, Chaos Soliton Fract., 141 (2020), 110343. https://doi.org/10.1016/j.chaos.2020.110343 doi: 10.1016/j.chaos.2020.110343
[26]	M. Mohan Raja, V. Vijayakumar, New results concerning to approximate controllability of fractional integro-differential evolution equations of order $1 < r < 2$, Numer. Methods Partial Differ. Equ., 2020. https://doi.org/10.1002/num.22653 doi: 10.1002/num.22653
[27]	M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, K. S. Nisar, Results on existence and controllability results for fractional evolution inclusions of order $1 < r < 2$ with Clarke's subdifferential type, Numer. Methods Partial Differ. Equ., 2020. https://doi.org/10.1002/num.22691 doi: 10.1002/num.22691
[28]	M. Mohan Raja, V. Vijayakumar, L. N. Huynh, R. Udhayakumar, K. S. Nisar, New discussion on nonlocal controllability for fractional evolution system of order $1 < r < 2$, Adv. Differ. Equ., 2021 (2021), 237. https://doi.org/10.1186/s13662-021-03373-1 doi: 10.1186/s13662-021-03373-1
[29]	M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, S. Rezapour, New discussion on nonlocal controllability for fractional evolution system of order $1 < r < 2$, Adv. Differ. Equ., 2021 (2021), 481. https://doi.org/10.1186/s13662-021-03630-3 doi: 10.1186/s13662-021-03630-3
[30]	M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, N. Sakthivel, K. Kaliraj, Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order $r \in (1, 2)$, Optim. Contr. Appl. Methods, 2022. https://doi.org/10.1002/oca.2867 doi: 10.1002/oca.2867
[31]	K. S. Nisar, V. Vijayakumar, Results concerning to approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral delay differential system, Math. Methods. Appl. Sci., 44 (2021), 13615–13632. https://doi.org/ 10.1002/mma.7647 doi: 10.1002/mma.7647
[32]	R. Patel, A. Shukla, S. S. Jadon, Existence and optimal control problem for semilinear fractional order (1, 2] control system, Math. Methods. Appl. Sci., 2020. https://doi.org/10.1002/mma.6662 doi: 10.1002/mma.6662
[33]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to method of their solution and some of their applications, San Diego: Academic Press, 1999.
[34]	G. Rahman, K. S. Nisar, S. Khan, D. Baleanu, V. Vijayakumar, On the weighted fractional integral inequalities for Chebyshev functionals, Adv. Differ. Equ., 2021 (2021), 1–19. https://doi.org/10.1186/s13662-020-03183-x doi: 10.1186/s13662-020-03183-x
[35]	R. Sakthivel, R. Ganesh, Y. Ren, S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci., 18 (2013), 3498–3508. https://doi.org/10.1016/j.cnsns.2013.05.015 doi: 10.1016/j.cnsns.2013.05.015
[36]	R. Sakthivel, N. I. Mahmudov, J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334–10340. https://doi.org/10.1016/j.amc.2012.03.093 doi: 10.1016/j.amc.2012.03.093
[37]	T. Sathiyaraj, P. Balasubramaniam, The controllability of fractional damped stochastic integrodifferential systems, Asian J. Control, 19 (2017), 1455–1464. https://doi.org/10.1002/asjc.1453 doi: 10.1002/asjc.1453
[38]	A. Shukla, N. Sukavanam, D. N. Pandey, Controllability of semilinear stochastic control system with finite delay, IMA J. Math. Control Inf., 35 (2018), 427–449. https://doi.org/10.1093/imamci/dnw059 doi: 10.1093/imamci/dnw059
[39]	A. Shukla, N. Sukavanam, Complete controllability of semi-linear stochastic system with delay, Rend. Circ. Mat. Palermo, 64 (2015), 209–220. https://doi.org/10.1007/s12215-015-0191-0 doi: 10.1007/s12215-015-0191-0
[40]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of fractional semilinear stochastic system of order $\alpha\in (1, 2]$, J. Dyn. Control Syst., 23 (2017), 679–691. https://doi.org/10.1007/s10883-016-9350-7 doi: 10.1007/s10883-016-9350-7
[41]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of semilinear fractional stochastic control system, Asian-Eur. J. Math., 11 (2018), 1850088. https://doi.org/10.1142/S1793557118500882 doi: 10.1142/S1793557118500882
[42]	A. Shukla, N. Sukavanam, D. N. Pandey, Controllability of semilinear stochastic system with multiple delays in control, IFAC Proc. Vol., 47 (2014), 306–312. https://doi.org/10.3182/20140313-3-IN-3024.00107 doi: 10.3182/20140313-3-IN-3024.00107
[43]	A. Shukla, R. Patel, Existence and optimal control results for second-order semilinear system in Hilbert spaces, Circuits Syst. Signal Proccess., 40 (2021), 4246–4258. https://doi.org/10.1007/s00034-021-01680-2 doi: 10.1007/s00034-021-01680-2
[44]	A. Shukla, R. Patel, Controllability results for fractional semilinear delay control systems, J. Appl. Math. Comput., 65 (2021), 861–875. https://doi.org/10.1007/s12190-020-01418-4 doi: 10.1007/s12190-020-01418-4
[45]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of semilinear fractional control systems of order $\alpha\in (1, 2)$, Proc. Conf. Control Appl., Soc. Ind. Appl. Math., 2015,175–180. https://doi.org/10.1137/1.9781611974072.25 doi: 10.1137/1.9781611974072.25
[46]	A. Singh, A. Shukla, V. Vijayakumar, R. Udhayakumar, Asymptotic stability of fractional order $(1, 2]$ stochastic delay differential equations in Banach spaces, Chaos Soliton Fract., 150 (2021), 111095. https://doi.org/10.1016/j.chaos.2021.111095 doi: 10.1016/j.chaos.2021.111095
[47]	C. C. Travis, G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Sci., 32 (1978), 75–96. https://doi.org/10.1007/BF01902205 doi: 10.1007/BF01902205
[48]	C. Tunc, A. K. Golmankhaneh, On stability of a class of second alpha-order fractal differential equations, AIMS Math., 5 (2020), 2126–2142. https://doi.org/10.3934/math.2020141 doi: 10.3934/math.2020141
[49]	O. Tunc, Ö. Atan, C. Tunc, J. C. Yao, Qualitative analyses of integro-fractional differential equations with Caputo derivatives and retardations via the Lyapunov-Razumikhin method, Axioms, 10 (2021), 58. https://doi.org/10.3390/axioms10020058 doi: 10.3390/axioms10020058
[50]	V. Vijayakumar, C. Ravichandran, K. S. Nisar, K. D. Kucche, New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order $1 < r < 2$, Numer. Methods Partial Differ. Equ., 2021. https://doi.org/10.1002/num.22772 doi: 10.1002/num.22772
[51]	V. Vijayakumar, S. K. Panda, K. S. Nisar, H. M. Baskonus, Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay, Numer. Methods Partial Differ. Equ., 37 (2021), 1200–1221. https://doi.org/10.1002/num.22573 doi: 10.1002/num.22573
[52]	V. Vijayakumar, R. Udhayakumar, C. Dineshkumar, Approximate controllability of second order nonlocal neutral differential evolution inclusions, IMA J. Math. Control Inf., 38 (2021), 192–210. https://doi.org/10.1093/imamci/dnaa001 doi: 10.1093/imamci/dnaa001
[53]	J. R. Wang, Z. Fan, Y. Zhou, Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, J. Optim. Theory Appl., 154 (2012), 292–302. https://doi.org/10.1007/s10957-012-9999-3 doi: 10.1007/s10957-012-9999-3
[54]	W. K. Williams, V. Vijayakumar, R. Udhayakumar, 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 Partial Differ. Equ., 37 (2021), 949–961. https://doi.org/10.1002/num.22560 doi: 10.1002/num.22560
[55]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014. https://doi.org/10.1142/9069
[56]	Y. Zhou, Fractional evolution equations and inclusions: Analysis and control, Elsevier, 2016. https://doi.org/10.1016/C2015-0-00813-9
[57]	Y. Zhou, J. W. He, New results on controllability of fractional evolution systems with order $\alpha \in (1, 2)$, Evol. Equ. Control The., 10 (2021), 491–509. https://doi.org/10.3934/eect.2020077 doi: 10.3934/eect.2020077
[58]	Y. Zhou, L. Zhang, X. H. Shen, Existence of mild solutions for fractional evolution equations, J. Int. Equ. Appl., 25 (2013), 557–585. https://doi.org/10.1216/JIE-2013-25-4-557 doi: 10.1216/JIE-2013-25-4-557
[59]	Z. F. Zhang, B. Liu, Controllability results for fractional functional differential equations with nondense domain, Numer. Funct. Anal. Optim., 35 (2014), 443–460. https://doi.org/10.1080/01630563.2013.813536 doi: 10.1080/01630563.2013.813536

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)