January  2019, 18(1): 455-478. doi: 10.3934/cpaa.2019023

Controllability for a class of semilinear fractional evolution systems via resolvent operators

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong 250014, China

* Corresponding author

Received  February 2018 Revised  March 2018 Published  August 2018

This paper deals with the exact controllability for a class of fractional evolution systems in a Banach space. First, we introduce a new concept of exact controllability and give notion of the mild solutions of the considered evolutional systems via resolvent operators. Second, by utilizing the semigroup theory, the fixed point strategy and Kuratowski's measure of noncompactness, the exact controllability of the evolutional systems is investigated without Lipschitz continuity and growth conditions imposed on nonlinear functions. The results are established under the hypothesis that the resolvent operator is differentiable and analytic, respectively, instead of supposing that the semigroup is compact. An example is provided to illustrate the proposed results.

Citation: Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023
References:
[1]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004.

[2]

B. AhmadJ. J. NietoA. Alsaedi and M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal., 13 (2012), 599-606.  doi: 10.1016/j.nonrwa.2011.07.052.

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C. Bucur, Local density of Caputo-stationary functions in the space of smooth functions, ESAIM Control Optim. Calc. Var., 23 (2017), 1361-1380.  doi: 10.1051/cocv/2016056.

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C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

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D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isreal J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.

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J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., Marcel Pekker, New York, 1980.

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K. Balachandran and J. Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal., 3 (2009), 363-367.  doi: 10.1016/j.nahs.2009.01.014.

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G. Da Prato and M. Iannelli, Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl., 112 (1985), 36-55.  doi: 10.1016/0022-247X(85)90275-6.

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A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450.  doi: 10.1016/j.camwa.2011.03.075.

[10]

M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fract., 14 (2002), 433-440.  doi: 10.1016/S0960-0779(01)00208-9.

[11]

E. Hern$\acute{a}$ndezD. O'Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73 (2010), 3462-3471.  doi: 10.1016/j.na.2010.07.035.

[12]

E. Hern$\acute{a}$ndezD. O'Regan and K. Balachandran, Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes mathematicae, 24 (2013), 68-82.  doi: 10.1016/j.indag.2012.06.007.

[13]

O. K. JaradatA. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69 (2008), 3153-3159.  doi: 10.1016/j.na.2007.09.008.

[14]

S. JiG. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217 (2011), 6981-6989. 

[15]

X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Autom. Contr., 62 (2017), 3618-3625.  doi: 10.1109/TAC.2017.2669580.

[16]

X. Li and S. Song, Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Trans. Neural Net. Learn. Sys., 24 (2013), 868-877. 

[17]

X. Li and S. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Contr., 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.

[18]

H. Li and Y. Wang, Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM J. Control Optim., 55 (2017), 3437-3457.  doi: 10.1137/16M1092581.

[19]

H. Li and Y. Wang, Further results on feedback stabilization control design of Boolean control networks, Automatica, 83 (2017), 303-308.  doi: 10.1016/j.automatica.2017.06.043.

[20]

X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.

[21]

H. LiL. Xie and Y. Wang, On robust control invariance of Boolean control networks, Automatica, 68 (2016), 392-396.  doi: 10.1016/j.automatica.2016.01.075.

[22]

H. LiL. Xie and Y. Wang, Output regulation of Boolean control networks, IEEE Trans. Autom. Contr., 62 (2017), 2993-2998.  doi: 10.1109/TAC.2016.2606600.

[23]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

[24]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[25]

R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results in Mathematics, 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.

[26]

I. StamovaT. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with Supremums, Inter. J. Adapt. Contr. Signal Proces., 28 (2014), 1227-1239.  doi: 10.1002/acs.2440.

[27]

J. Sprekels and E. Valdinoci, A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.

[28]

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.

[29]

J. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.

[30]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., 11 (2010), 4465-4475.  doi: 10.1016/j.nonrwa.2010.05.029.

show all references

References:
[1]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004.

[2]

B. AhmadJ. J. NietoA. Alsaedi and M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal., 13 (2012), 599-606.  doi: 10.1016/j.nonrwa.2011.07.052.

[3]

C. Bucur, Local density of Caputo-stationary functions in the space of smooth functions, ESAIM Control Optim. Calc. Var., 23 (2017), 1361-1380.  doi: 10.1051/cocv/2016056.

[4]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[5]

D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isreal J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.

[6]

J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., Marcel Pekker, New York, 1980.

[7]

K. Balachandran and J. Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal., 3 (2009), 363-367.  doi: 10.1016/j.nahs.2009.01.014.

[8]

G. Da Prato and M. Iannelli, Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl., 112 (1985), 36-55.  doi: 10.1016/0022-247X(85)90275-6.

[9]

A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450.  doi: 10.1016/j.camwa.2011.03.075.

[10]

M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fract., 14 (2002), 433-440.  doi: 10.1016/S0960-0779(01)00208-9.

[11]

E. Hern$\acute{a}$ndezD. O'Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73 (2010), 3462-3471.  doi: 10.1016/j.na.2010.07.035.

[12]

E. Hern$\acute{a}$ndezD. O'Regan and K. Balachandran, Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes mathematicae, 24 (2013), 68-82.  doi: 10.1016/j.indag.2012.06.007.

[13]

O. K. JaradatA. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69 (2008), 3153-3159.  doi: 10.1016/j.na.2007.09.008.

[14]

S. JiG. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217 (2011), 6981-6989. 

[15]

X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Autom. Contr., 62 (2017), 3618-3625.  doi: 10.1109/TAC.2017.2669580.

[16]

X. Li and S. Song, Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Trans. Neural Net. Learn. Sys., 24 (2013), 868-877. 

[17]

X. Li and S. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Contr., 62 (2017), 406-411.  doi: 10.1109/TAC.2016.2530041.

[18]

H. Li and Y. Wang, Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM J. Control Optim., 55 (2017), 3437-3457.  doi: 10.1137/16M1092581.

[19]

H. Li and Y. Wang, Further results on feedback stabilization control design of Boolean control networks, Automatica, 83 (2017), 303-308.  doi: 10.1016/j.automatica.2017.06.043.

[20]

X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63-69.  doi: 10.1016/j.automatica.2015.10.002.

[21]

H. LiL. Xie and Y. Wang, On robust control invariance of Boolean control networks, Automatica, 68 (2016), 392-396.  doi: 10.1016/j.automatica.2016.01.075.

[22]

H. LiL. Xie and Y. Wang, Output regulation of Boolean control networks, IEEE Trans. Autom. Contr., 62 (2017), 2993-2998.  doi: 10.1109/TAC.2016.2606600.

[23]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

[24]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[25]

R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results in Mathematics, 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.

[26]

I. StamovaT. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with Supremums, Inter. J. Adapt. Contr. Signal Proces., 28 (2014), 1227-1239.  doi: 10.1002/acs.2440.

[27]

J. Sprekels and E. Valdinoci, A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.

[28]

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.

[29]

J. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.

[30]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., 11 (2010), 4465-4475.  doi: 10.1016/j.nonrwa.2010.05.029.

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