An investigation on boundary controllability for Sobolev-type neutral evolution equations of fractional order in Banach space

Yong-Ki Ma; Kamalendra Kumar; Rohit Patel; Anurag Shukla; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar; Yong-Ki Ma; Kamalendra Kumar; Rohit Patel; Anurag Shukla; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar

doi:10.3934/math.2022651

AIMS Mathematics

2022, Volume 7, Issue 7: 11687-11707. doi: 10.3934/math.2022651

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An investigation on boundary controllability for Sobolev-type neutral evolution equations of fractional order in Banach space

1.
Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, Republic of Korea
2.
Department of Mathematics, SRMS College of Engineering and Technology, Bareilly 243001, India
3.
Department of Mathematics, Government PG College Bisalpur, Pilibhit 244001, India
4.
Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj 209732, India
5.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
6.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India

Received: 13 November 2021 Revised: 06 April 2022 Accepted: 12 April 2022 Published: 15 April 2022
MSC : 34A08, 34K35, 49J15

The main focus of this paper is on the boundary controllability of fractional order Sobolev-type neutral evolution equations in Banach space. We show our key results using facts from fractional calculus, semigroup theory, and the fixed point method. Finally, we give an example to illustrate the theory we have established.
- fractional integrodifferential system,
- boundary controllability,
- fixed point theorem,
- nonlocal conditions
Citation: Yong-Ki Ma, Kamalendra Kumar, Rohit Patel, Anurag Shukla, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar. An investigation on boundary controllability for Sobolev-type neutral evolution equations of fractional order in Banach space[J]. AIMS Mathematics, 2022, 7(7): 11687-11707. doi: 10.3934/math.2022651

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Abstract

The main focus of this paper is on the boundary controllability of fractional order Sobolev-type neutral evolution equations in Banach space. We show our key results using facts from fractional calculus, semigroup theory, and the fixed point method. Finally, we give an example to illustrate the theory we have established.

References

[1]	H. M. Ahmed, Boundary controllability of impulsive nonlinear fractional delay integro-differential system, Cogent Eng., 3 (2016), 1215766. https://doi.org/10.1080/23311916.2016.1215766 doi: 10.1080/23311916.2016.1215766
[2]	H. M. Ahmed, M. M. El-Borai, M. E. Ramadan, Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps, Adv. Differ. Equ., 2019 (2019), 82. https://doi.org/10.1186/s13662-019-2028-1 doi: 10.1186/s13662-019-2028-1
[3]	H. M. Ahmed, Boundary controllability of nonlinear fractional integrodifferential systems, Adv. Differ. Equ., 2010 (2010), 279493. https://doi.org/10.1155/2010/279493 doi: 10.1155/2010/279493
[4]	K. Balachandran, E. R. Anandhi, J. P. Dauer, Boundary controllability of Sobolev-type abstract nonlinear integrodifferential systems, J. Math. Anal. Appl., 277 (2003), 446–464. https://doi.org/10.1016/S0022-247X(02)00522-X doi: 10.1016/S0022-247X(02)00522-X
[5]	K. Balachandran, A. Leelamani, A note on boundary controllability of neutral integrodifferential systems in Banach spaces, Nihonkai Math. J., 17 (2006), 89–101.
[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, K. S. Nisar, R. Udhayakumar, V. Vijayakumar, A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian J. Control, 2021. https://doi.org/10.1002/asjc.2650
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	H. O. Fattorini, Boundary control systems, SIAM J. Control Optim., 6 (1968), 349–384. https://doi.org/10.1137/0306025 doi: 10.1137/0306025
[14]	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
[15]	H. K. Han, J. Y. Park, Boundary controllability of differential equations with nonlocal condition, J. Math. Anal. Appl., 230 (1999), 242–250. https://doi.org/10.1006/jmaa.1998.6199 doi: 10.1006/jmaa.1998.6199
[16]	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
[17]	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
[18]	K. Kumar, R. Kumar, Boundary controllability of delay differential systems of fractional order with nonlocal condition, J. Appl. Nonlinear Dyn., 6 (2017), 465–472.
[19]	K. Kumar, R. Kumar, Boundary controllability of fractional order nonlocal semi-linear neutral evolution systems with impulsive condition, Discontinuity Nonlinearity Complexity, 8 (2019), 419–428.
[20]	K. Kumar, R. Patel, V. Vijayakumar, A. Shukla, C. Ravichandran, A discussion on boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations, Math. Methods Appl. Sci., 2022. https://doi.org/10.1002/mma.8117
[21]	I. Lasiecka, R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109–154. https://doi.org/10.1007/BF01442394 doi: 10.1007/BF01442394
[22]	F. Li, Nonlocal Cauchy problem for delay fractional integrodifferential equations of neutral type, Adv. Differ. Equ., 2012 (2012), 47. https://doi.org/10.1186/1687-1847-2012-47 doi: 10.1186/1687-1847-2012-47
[23]	Y. Li, B. Liu, Boundary controllability of non-linear stochastic differential inclusions, Appl. Anal., 87 (2008), 709–722. https://doi.org/10.1080/00036810802213231 doi: 10.1080/00036810802213231
[24]	J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer, 1972.
[25]	J. Liu, W. Xu, Q. Guo, Global attractiveness and exponential stability for impulsive fractional neutral stochastic evolution equations driven by fBm, Adv. Differ. Equ., 2020 (2020), 63. https://doi.org/10.1186/s13662-020-2520-7 doi: 10.1186/s13662-020-2520-7
[26]	J. Liu, W. Xu, An averaging result for impulsive fractional neutral stochastic differential equations, Appl. Math. Lett., 114 (2021), 106892. https://doi.org/10.1016/j.aml.2020.106892 doi: 10.1016/j.aml.2020.106892
[27]	R. M. Lizzy, K. Balachandran, Boundary controllability of nonlinear stochastic fractional systems in Hilbert spaces, Int. J. Appl. Math. Comput. Sci., 28 (2018), 123–133.
[28]	Y. K. Ma, K. Kumar, R. Patel, A. Shukla, V. Vijayakumar, Discussion on boundary controllability of nonlocal fractional neutral integrodifferential evolution systems, AIMS Math., 7 (2022), 7642–7656. https://doi.org/10.3934/math.2022429 doi: 10.3934/math.2022429
[29]	M. Mohan Raja, V. Vijayakumar, L. N. Huynh, R. Udhayakumar, K. S. Nisar, Results on the approximate controllability of fractional hemivariational inequalities 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
[30]	P. Muthukumar, K. Thiagu, Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order $1 < q < 2$ with infinite delay and Poisson jumps, J. Dyn. Control Syst., 23 (2017), 213–235. https://doi.org/10.1007/s10883-015-9309-0 doi: 10.1007/s10883-015-9309-0
[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]	M. Palanisamy, R. Chinnathambi, Approximate boundary controllability of Sobolev-type stochastic differential systems, J. Egypt. Math. Soc., 22 (2014), 201–208. https://doi.org/10.1016/j.joems.2013.07.005 doi: 10.1016/j.joems.2013.07.005
[33]	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
[34]	A. Pazy, Semigroup of linear operators and application to partial differential equations, New York: Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[35]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[36]	B. N. Sadovskii, A fixed-point principle, Funct. Anal. Appl., 1 (1967), 151–153. https://doi.org/10.1007/BF01076087 doi: 10.1007/BF01076087
[37]	S. Selvarasu, M. M. Arjunan, Approximate controllability of stochastic fractional neutral impulsive integro-differential systems with state dependent delay and Poisson jumps, J. Appl. Nonlinear Dyn., 8 (2019), 383–406.
[38]	A. Shukla, N. Sukavanam, D. N. Pandey, Approximate controllability of semilinear fractional control systems of order $\alpha\in(1, 2]$ with infinite delay, Mediterr. J. Math., 13 (2016), 2539–2550. https://doi.org/10.1007/s00009-015-0638-8 doi: 10.1007/s00009-015-0638-8
[39]	A. Shukla, N. Sukavanam, D. N. Pandey, 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, 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
[41]	A. Singh, A. Shukla, V. Vijayakumar, R. Udhayakumar, Asymptotic stability of fractional order $(1, 2]$ stochastic delay differential equations in Banach spaces, Chaos Solitons Fract., 150 (2021), 111095. https://doi.org/10.1016/j.chaos.2021.111095 doi: 10.1016/j.chaos.2021.111095
[42]	A. Shukla, V. Vijayakumar, K. S. Nisar, A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order $r\in(1, 2)$, Chaos Solitons Fract., 154 (2022), 111615. https://doi.org/10.1016/j.chaos.2021.111615 doi: 10.1016/j.chaos.2021.111615
[43]	A. Shukla, R. Patel, Existence and optimal control results for second-order semilinear system in Hilbert spaces, Circuits Syst. Signal Process., 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]	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
[46]	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
[47]	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
[48]	V. Vijayakumar, R. Murugesu, Controllability for a class of second-order evolution differential inclusions without compactness, Appl. Anal., 98 (2019), 1367–1385. https://doi.org/10.1080/00036811.2017.1422727 doi: 10.1080/00036811.2017.1422727
[49]	V. Vijayakumar, Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke's subdifferential type, Results Math., 73 (2018), 42. https://doi.org/10.1007/s00025-018-0807-8 doi: 10.1007/s00025-018-0807-8
[50]	V. Vijayakumar, R. Udhayakumar, S. K. Panda, K. S. Nisar, Results on approximate controllability of Sobolev type fractional stochastic evolution hemivariational inequalities, Numer. Methods Partial Differ. Equ., 2020. https://doi.org/10.1002/num.22690
[51]	V. Vijayakumar, R. Murugesu, M. Tamil Selvan, Controllability for a class of second order functional evolution differential equations without uniqueness, IMA J. Math. Control Inf., 36 (2019), 225–246. https://doi.org/10.1093/imamci/dnx048 doi: 10.1093/imamci/dnx048
[52]	J. Wang, L. Tian, Boundary controllability for the time-fractional nonlinear Korteweg-de Vries (KdV) equation, J. Appl. Anal. Comput., 10 (2020), 411–426. https://doi.org/10.11948/20180018 doi: 10.11948/20180018
[53]	L. W. Wang, Approximate controllability of boundary control systems with nonlinear boundary conditions, Appl. Mech. Mater., 538 (2014), 408–412. https://doi.org/10.4028/www.scientific.net/AMM.538.408 doi: 10.4028/www.scientific.net/AMM.538.408
[54]	W. K. Williams, V. Vijayakumar, R. Udhayakumar, S. K. Panda, K. S. Nisar, Existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order $1 < r < 2$, Numer. Methods Partial Differ. Equ., 2020. https://doi.org/10.1002/num.22697
[55]	X. Zhang, Y. Chen, Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order $\alpha$. The $0 < \alpha < 1$ case, ISA Trans., 82 (2018), 42–50. https://doi.org/10.1016/j.isatra.2017.03.008 doi: 10.1016/j.isatra.2017.03.008
[56]	J. X. Zhang, G. H. Yang, Low-complexity tracking control of strict-feedback systems with unknown control directions, IEEE T. Automat. Contr., 64 (2019), 5175–5182. https://doi.org/10.1109/TAC.2019.2910738https://doi.org/10.1109/TAC.2019.2910738 doi: 10.1109/TAC.2019.2910738
[57]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014.
[58]	Y. Zhou, Fractional evolution equations and inclusions: Analysis and control, New York: Elsevier, 2016.
[59]	Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077. https://doi.org/10.1016/j.camwa.2009.06.026 doi: 10.1016/j.camwa.2009.06.026

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