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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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