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doi: 10.3934/eect.2021035
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Approximate controllability of fractional neutral evolution systems of hyperbolic type

1. 

School of Mathematical Sciences, Anhui University, Hefei 230039, China

2. 

School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

*Corresponding author: Xian-Feng Zhou

Received  December 2020 Revised  May 2021 Early access July 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (11471015 and 11601003); Natural Science Foundation of Anhui Province (1508085MA01, 1608085MA12, 1708085MA15 and 2008085QA19); Program of Natural Science Research for Universities of Anhui Province (KJ2016A023) and Program of Anhui Jianzhu University (K1930096)

In this paper, we deal with fractional neutral evolution systems of hyperbolic type in Banach spaces. We establish the existence and uniqueness of the mild solution and prove the approximate controllability of the systems under different conditions. These results are mainly based on fixed point theorems as well as constructing a Cauchy sequence and a control function. In the end, we give an example to illustrate the validity of the main results.

Citation: Xuan-Xuan Xi, Mimi Hou, Xian-Feng Zhou, Yanhua Wen. Approximate controllability of fractional neutral evolution systems of hyperbolic type. Evolution Equations and Control Theory, doi: 10.3934/eect.2021035
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Birkhauser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[2]

P. Y. ChenX. P. Zhang and Y. X. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.

[3]

J. Chang and H. Liu, Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the $\alpha$-norm, Nonlinear Anal., 71 (2009), 3759-3768.  doi: 10.1016/j.na.2009.02.035.

[4]

R. DhayalM. MalikS. AbbasA. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evol. Equ. Control Theory, 10 (2021), 411-429.  doi: 10.3934/eect.2020073.

[5]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321. 

[6]

X. Fu and R. Huang, Existence of solutions for neutral integro-differential equations with state-dependent delay, Appl. Math. Comput., 224 (2013), 743-759.  doi: 10.1016/j.amc.2013.09.010.

[7]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.

[8]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[9]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

K. Jeet and D. Bahuguna, Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay, J. Dyn. Control Syst., 22 (2016), 485-504.  doi: 10.1007/s10883-015-9297-0.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[12]

S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations, 252 (2012), 6163-6174.  doi: 10.1016/j.jde.2012.02.014.

[13]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.

[14]

S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA J. Math. Control Inform., 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007.

[15]

V. Kumar, M. Malik and A. Debbouche, Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects, J. Comput. Appl. Math., 383 (2021), 113158, 18 pp. doi: 10.1016/j.cam.2020.113158.

[16]

X. H. LiuJ. R. Wang and Y. Zhou, Approximate controllability for nonlocal fractional propagation systems of Sobolev type, J. Dyn. Control Syst., 25 (2019), 245-262.  doi: 10.1007/s10883-018-9409-8.

[17]

X. W. LiZ. H. LiuJ. Li and C. Tisdell, Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 229-242.  doi: 10.1007/s10473-019-0118-5.

[18]

Y. Luchko, Wave-diffusion dualism of the neutral-fractional processes, J. Comput. Phys., 293 (2015), 40-52.  doi: 10.1016/j.jcp.2014.06.005.

[19]

K. LiJ. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.  doi: 10.1016/j.jfa.2012.04.011.

[20]

Y. Li, Regularity of mild solutions for fractional abstract Cauchy problem with order $\alpha\in(1, 2)$, Z. Angew. Math. Phys., 66 (2015), 3283-3298.  doi: 10.1007/s00033-015-0577-z.

[21]

Y. LiH. Sun and Z. Feng, Fractional abstract Cauchy problem with order $\alpha\in(1, 2)$, Dyn. Partial Differ. Equ., 13 (2016), 155-177.  doi: 10.4310/DPDE.2016.v13.n2.a4.

[22]

Z. H. Liu and X. W. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.

[23]

K. LiJ. Peng and J. Gao, Controllability of nonlocal fractional defferential systems of order $\alpha\in(1, 2]$ in Banach spaces, Rep. Math. Phys., 71 (2013), 33-43.  doi: 10.1016/S0034-4877(13)60020-8.

[24]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.

[25] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. 
[26]

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

[27]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in Waves and Stability in Continuous Media, Bologna, 1993), Ser. Adv. Math. Appl. Sci., vol. 23, World Sci.Publ., River Edge, NJ, 1994,246–251.

[28]

M. F. Pinaud and H. R. Henr$\acute{i}$quez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269 (2020), 4609-4642.  doi: 10.1016/j.jde.2020.03.029.

[29]

T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.  doi: 10.1007/s11071-004-3751-y.

[30]

C. RavichandranN. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin. Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.

[31]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system, Acta. Mech., 120 (1997), 109-125.  doi: 10.1007/BF01174319.

[32]

R. SakthivelR. GaneshY. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3498-3508.  doi: 10.1016/j.cnsns.2013.05.015.

[33]

R. SakthivelN. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093.

[34]

G. J. ShenR. SakthivelY. Ren and M. Y. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collect. Math., 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3.

[35]

X. B. Shu and Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1<\alpha<2$, Comput. Math. Appl., 64 (2012), 2100-2110.  doi: 10.1016/j.camwa.2012.04.006.

[36]

A. ShuklaN. Sukavanam and 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.  doi: 10.1007/s00009-015-0638-8.

[37] D. R. Smart, Fixed Point Theorem, Cambridge University Press, London-New York, 1974. 
[38]

M. S. TavazoeiM. HaeriS. JafariS. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE T. Ind. Electron, 11 (2008), 4094-4101. 

[39]

N. H. TuanD. O'Regan and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory, 9 (2020), 773-793.  doi: 10.3934/eect.2020033.

[40]

C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.

[41]

V. VijayakumarR. Udhayakumar and K. Kavitha, On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evol. Equ. Control Theory, 10 (2021), 271-296.  doi: 10.3934/eect.2020066.

[42]

N. ValliammalC. Ravichandran and J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.

[43]

V. Vijayakumar, Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.  doi: 10.1080/00207179.2018.1433331.

[44]

V. VijayakumarA. Selvakumar and R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Appl. Math. Comput., 232 (2014), 303-312.  doi: 10.1016/j.amc.2014.01.029.

[45]

V. V. Vasil'evS. G. Krein and S. I. Piskarev, Semigroups of operators, cosine operator functions and linear differential equations, J. Soviet Math., 54 (1991), 1042-1129. 

[46]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[47]

M. Yang and Q. R. Wang, Approximate controllability of Caputo fractional neutral stochastic differential inclusions with state-dependent delay, IMA J. Math. Control Inform., 35 (2018), 1061-1085.  doi: 10.1093/imamci/dnx014.

[48]

H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21 (1983), 551-565.  doi: 10.1137/0321033.

[49]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.

[50]

Y. ZhouS. SuganyaM. M. Arjunan and B. Ahmad, Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces, IMA J. Math. Control. Inform., 36 (2019), 603-622.  doi: 10.1093/imamci/dnx060.

[51]

Y. Zhou and J. W. He, New results on controllability of fractional evolution systems with order $\alpha\in(1, 2)$, Evol. Equ. Control Theory, 9 (2020), 1-19. 

[52] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016. 

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Birkhauser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[2]

P. Y. ChenX. P. Zhang and Y. X. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.

[3]

J. Chang and H. Liu, Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the $\alpha$-norm, Nonlinear Anal., 71 (2009), 3759-3768.  doi: 10.1016/j.na.2009.02.035.

[4]

R. DhayalM. MalikS. AbbasA. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evol. Equ. Control Theory, 10 (2021), 411-429.  doi: 10.3934/eect.2020073.

[5]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321. 

[6]

X. Fu and R. Huang, Existence of solutions for neutral integro-differential equations with state-dependent delay, Appl. Math. Comput., 224 (2013), 743-759.  doi: 10.1016/j.amc.2013.09.010.

[7]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.

[8]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[9]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

K. Jeet and D. Bahuguna, Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay, J. Dyn. Control Syst., 22 (2016), 485-504.  doi: 10.1007/s10883-015-9297-0.

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[12]

S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations, 252 (2012), 6163-6174.  doi: 10.1016/j.jde.2012.02.014.

[13]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.

[14]

S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA J. Math. Control Inform., 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007.

[15]

V. Kumar, M. Malik and A. Debbouche, Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects, J. Comput. Appl. Math., 383 (2021), 113158, 18 pp. doi: 10.1016/j.cam.2020.113158.

[16]

X. H. LiuJ. R. Wang and Y. Zhou, Approximate controllability for nonlocal fractional propagation systems of Sobolev type, J. Dyn. Control Syst., 25 (2019), 245-262.  doi: 10.1007/s10883-018-9409-8.

[17]

X. W. LiZ. H. LiuJ. Li and C. Tisdell, Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 229-242.  doi: 10.1007/s10473-019-0118-5.

[18]

Y. Luchko, Wave-diffusion dualism of the neutral-fractional processes, J. Comput. Phys., 293 (2015), 40-52.  doi: 10.1016/j.jcp.2014.06.005.

[19]

K. LiJ. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.  doi: 10.1016/j.jfa.2012.04.011.

[20]

Y. Li, Regularity of mild solutions for fractional abstract Cauchy problem with order $\alpha\in(1, 2)$, Z. Angew. Math. Phys., 66 (2015), 3283-3298.  doi: 10.1007/s00033-015-0577-z.

[21]

Y. LiH. Sun and Z. Feng, Fractional abstract Cauchy problem with order $\alpha\in(1, 2)$, Dyn. Partial Differ. Equ., 13 (2016), 155-177.  doi: 10.4310/DPDE.2016.v13.n2.a4.

[22]

Z. H. Liu and X. W. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53 (2015), 1920-1933.  doi: 10.1137/120903853.

[23]

K. LiJ. Peng and J. Gao, Controllability of nonlocal fractional defferential systems of order $\alpha\in(1, 2]$ in Banach spaces, Rep. Math. Phys., 71 (2013), 33-43.  doi: 10.1016/S0034-4877(13)60020-8.

[24]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.

[25] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. 
[26]

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

[27]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in Waves and Stability in Continuous Media, Bologna, 1993), Ser. Adv. Math. Appl. Sci., vol. 23, World Sci.Publ., River Edge, NJ, 1994,246–251.

[28]

M. F. Pinaud and H. R. Henr$\acute{i}$quez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269 (2020), 4609-4642.  doi: 10.1016/j.jde.2020.03.029.

[29]

T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.  doi: 10.1007/s11071-004-3751-y.

[30]

C. RavichandranN. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin. Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.

[31]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system, Acta. Mech., 120 (1997), 109-125.  doi: 10.1007/BF01174319.

[32]

R. SakthivelR. GaneshY. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3498-3508.  doi: 10.1016/j.cnsns.2013.05.015.

[33]

R. SakthivelN. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093.

[34]

G. J. ShenR. SakthivelY. Ren and M. Y. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collect. Math., 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3.

[35]

X. B. Shu and Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1<\alpha<2$, Comput. Math. Appl., 64 (2012), 2100-2110.  doi: 10.1016/j.camwa.2012.04.006.

[36]

A. ShuklaN. Sukavanam and 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.  doi: 10.1007/s00009-015-0638-8.

[37] D. R. Smart, Fixed Point Theorem, Cambridge University Press, London-New York, 1974. 
[38]

M. S. TavazoeiM. HaeriS. JafariS. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE T. Ind. Electron, 11 (2008), 4094-4101. 

[39]

N. H. TuanD. O'Regan and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory, 9 (2020), 773-793.  doi: 10.3934/eect.2020033.

[40]

C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.

[41]

V. VijayakumarR. Udhayakumar and K. Kavitha, On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evol. Equ. Control Theory, 10 (2021), 271-296.  doi: 10.3934/eect.2020066.

[42]

N. ValliammalC. Ravichandran and J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.

[43]

V. Vijayakumar, Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.  doi: 10.1080/00207179.2018.1433331.

[44]

V. VijayakumarA. Selvakumar and R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Appl. Math. Comput., 232 (2014), 303-312.  doi: 10.1016/j.amc.2014.01.029.

[45]

V. V. Vasil'evS. G. Krein and S. I. Piskarev, Semigroups of operators, cosine operator functions and linear differential equations, J. Soviet Math., 54 (1991), 1042-1129. 

[46]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[47]

M. Yang and Q. R. Wang, Approximate controllability of Caputo fractional neutral stochastic differential inclusions with state-dependent delay, IMA J. Math. Control Inform., 35 (2018), 1061-1085.  doi: 10.1093/imamci/dnx014.

[48]

H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21 (1983), 551-565.  doi: 10.1137/0321033.

[49]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.

[50]

Y. ZhouS. SuganyaM. M. Arjunan and B. Ahmad, Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces, IMA J. Math. Control. Inform., 36 (2019), 603-622.  doi: 10.1093/imamci/dnx060.

[51]

Y. Zhou and J. W. He, New results on controllability of fractional evolution systems with order $\alpha\in(1, 2)$, Evol. Equ. Control Theory, 9 (2020), 1-19. 

[52] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016. 
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