• Previous Article
    A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case
  • EECT Home
  • This Issue
  • Next Article
    Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $
doi: 10.3934/eect.2021035
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

[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.   Google Scholar

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

[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.   Google Scholar

[52] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016.   Google Scholar
[1]

Yassine El Gantouh, Said Hadd. Well-posedness and approximate controllability of neutral network systems. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021018

[2]

Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024

[3]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure &amp; Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[4]

Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020100

[5]

Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020091

[6]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[7]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167

[8]

Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023

[9]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

[10]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure &amp; Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

[11]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[12]

Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310

[13]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure &amp; Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[14]

Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar, Chokkalingam Ravichandran. Results on controllability of non-densely characterized neutral fractional delay differential system. Evolution Equations & Control Theory, 2021, 10 (3) : 619-631. doi: 10.3934/eect.2020083

[15]

Venkatesan Govindaraj, Raju K. George. Controllability of fractional dynamical systems: A functional analytic approach. Mathematical Control & Related Fields, 2017, 7 (4) : 537-562. doi: 10.3934/mcrf.2017020

[16]

Therese Mur, Hernan R. Henriquez. Relative controllability of linear systems of fractional order with delay. Mathematical Control & Related Fields, 2015, 5 (4) : 845-858. doi: 10.3934/mcrf.2015.5.845

[17]

Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173

[18]

M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 809-820. doi: 10.3934/dcds.2001.7.809

[19]

T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure &amp; Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101

[20]

Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033

2020 Impact Factor: 1.081

Article outline

[Back to Top]