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February  2022, 11(1): 239-258. doi: 10.3934/eect.2021001

Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives

Biao Zeng ,

Guangxi University for Nationalities, Faculty of Mathematics and Physics, Nanning 530006, Guangxi Province, P. R. China

* Corresponding author: Biao Zeng

Received  May 2020 Revised  October 2020 Published  February 2022 Early access  January 2021

Fund Project: The first author is supported by the Natural Science Foundation of Guangxi Province grant No. 2019GXNSFBA185005, the Start-up Project of Scientific Research on Introducing talents at school level in Guangxi University for Nationalities grant No. 2019KJQD04 and the Xiangsihu Young Scholars Innovative Research Team of Guangxi University for Nationalities grant No. 2019RSCXSHQN02

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.

Keywords: Impulsive delay evolution equation, Caputo fractional derivatives, feedback control, feasible pair, existence.
Mathematics Subject Classification: Primary: 34A08, 93B52; Secondary: 49J20, 49J53, 35R12.
Citation: Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations and Control Theory, 2022, 11 (1) : 239-258. doi: 10.3934/eect.2021001
References:
[1]

Y.-K. Chang, J. J. Nieto and Z.-H. Zhao, Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay, Nonlinear Anal.: Hybrid Systems, 4 (2010), 593-599.  doi: 10.1016/j.nahs.2010.03.006.

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[3]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4.

[4]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Weslwey, 1986.

[5]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[6]

M. I. Kamenskii, P. Nistri, V. V. Obukhovskii and P. Zecca, Optimal feedback control for a semilinear evolution equation, J. Optim. Theory Appl., 82 (1994), 503-517.  doi: 10.1007/BF02192215.

[7]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7, 2001. doi: 10.1515/9783110870893.

[8]

N. Kosmatov, Initial value problems of fractional order with fractional impulsive conditions, Results. Math., 63 (2013), 1289-1310.  doi: 10.1007/s00025-012-0269-3.

[9]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, vol. 204, Elservier Science B.V., Amsterdam, (2006).

[10]

X. J. Li and J. M. Yong, Optimal Control Theory for infinite Dimensional Systems, Birkhäuser, Boster, 1995. doi: 10.1007/978-1-4612-4260-4.

[11]

Z. Liu and X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1362-1373.  doi: 10.1016/j.cnsns.2012.10.010.

[12]

Z. Liu, X. Li and J. Sun, Controllability of nonlinear fractional impulsive evolution systems, J. Int. Equ. Appl., 25 (2013), 395-405.  doi: 10.1216/JIE-2013-25-3-395.

[13]

Z. Liu, S. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012.

[14]

A. L. Mees, Dynamics of Feedback Systems, John Wiley & Sons, Ltd., New York, 1981.

[15]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[19]

R. Sakthivel, Y. Ren and N. I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl., 62 (2011), 1451-1459.  doi: 10.1016/j.camwa.2011.04.040.

[20]

X. J. Wang and C. Z. Bai, Periodic boundary value problems for nonlinear impulsive fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, (2011), 1–15. doi: 10.14232/ejqtde.2011.1.3.

[21]

J. R. Wang, M. Fečkan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345-362.  doi: 10.4310/DPDE.2011.v8.n4.a3.

[22]

J. R. Wang, M. Fečkan and Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.

[23]

W. Wei and X. Xiang, Optimal feedback control for a class of nonlinear impulsive evolution equations, Chinese J. Engrg. Math., 23 (2006), 333-342. 

[24]

J. R. Wang, Y. Zhou and W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Contr. Lett., 61 (2012), 472-476.  doi: 10.1016/j.sysconle.2011.12.009.

[25]

C. Xiao, B. Zeng and Z. H. Liu, Feedback control for fractional impulsive evolution systems, Appl. Math. Comput., 268 (2015), 924-936.  doi: 10.1016/j.amc.2015.06.092.

[26]

H. P. Ye, J. M. Gao and Y. S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[27]

B. Zeng, Feedback control for non-stationary 3D Navier-Stokes-Voigt equations, Mathematics and Mechanics of Solids, 25 (2020), 2210-2221.  doi: 10.1177/1081286520926557.

[28]

B. Zeng, Feedback control systems governed by evolution equations, Optimization, 68 (2019), 1223-1243.  doi: 10.1080/02331934.2019.1578358.

[29]

B. Zeng and Z. H. Liu, Existence results for impulsive feedback control systems, Nonlinear Analysis: Hybrid Systems, 33 (2019), 1-16.  doi: 10.1016/j.nahs.2019.01.008.

[30]

W. Zhang and M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Model., 39 (2004), 479-493.  doi: 10.1016/S0895-7177(04)90519-5.

[31]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

[32]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theor., 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.

show all references

References:
[1]

Y.-K. Chang, J. J. Nieto and Z.-H. Zhao, Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay, Nonlinear Anal.: Hybrid Systems, 4 (2010), 593-599.  doi: 10.1016/j.nahs.2010.03.006.

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[3]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4.

[4]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Weslwey, 1986.

[5]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[6]

M. I. Kamenskii, P. Nistri, V. V. Obukhovskii and P. Zecca, Optimal feedback control for a semilinear evolution equation, J. Optim. Theory Appl., 82 (1994), 503-517.  doi: 10.1007/BF02192215.

[7]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7, 2001. doi: 10.1515/9783110870893.

[8]

N. Kosmatov, Initial value problems of fractional order with fractional impulsive conditions, Results. Math., 63 (2013), 1289-1310.  doi: 10.1007/s00025-012-0269-3.

[9]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, vol. 204, Elservier Science B.V., Amsterdam, (2006).

[10]

X. J. Li and J. M. Yong, Optimal Control Theory for infinite Dimensional Systems, Birkhäuser, Boster, 1995. doi: 10.1007/978-1-4612-4260-4.

[11]

Z. Liu and X. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1362-1373.  doi: 10.1016/j.cnsns.2012.10.010.

[12]

Z. Liu, X. Li and J. Sun, Controllability of nonlinear fractional impulsive evolution systems, J. Int. Equ. Appl., 25 (2013), 395-405.  doi: 10.1216/JIE-2013-25-3-395.

[13]

Z. Liu, S. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012.

[14]

A. L. Mees, Dynamics of Feedback Systems, John Wiley & Sons, Ltd., New York, 1981.

[15]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[19]

R. Sakthivel, Y. Ren and N. I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl., 62 (2011), 1451-1459.  doi: 10.1016/j.camwa.2011.04.040.

[20]

X. J. Wang and C. Z. Bai, Periodic boundary value problems for nonlinear impulsive fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, (2011), 1–15. doi: 10.14232/ejqtde.2011.1.3.

[21]

J. R. Wang, M. Fečkan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345-362.  doi: 10.4310/DPDE.2011.v8.n4.a3.

[22]

J. R. Wang, M. Fečkan and Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.

[23]

W. Wei and X. Xiang, Optimal feedback control for a class of nonlinear impulsive evolution equations, Chinese J. Engrg. Math., 23 (2006), 333-342. 

[24]

J. R. Wang, Y. Zhou and W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Contr. Lett., 61 (2012), 472-476.  doi: 10.1016/j.sysconle.2011.12.009.

[25]

C. Xiao, B. Zeng and Z. H. Liu, Feedback control for fractional impulsive evolution systems, Appl. Math. Comput., 268 (2015), 924-936.  doi: 10.1016/j.amc.2015.06.092.

[26]

H. P. Ye, J. M. Gao and Y. S. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[27]

B. Zeng, Feedback control for non-stationary 3D Navier-Stokes-Voigt equations, Mathematics and Mechanics of Solids, 25 (2020), 2210-2221.  doi: 10.1177/1081286520926557.

[28]

B. Zeng, Feedback control systems governed by evolution equations, Optimization, 68 (2019), 1223-1243.  doi: 10.1080/02331934.2019.1578358.

[29]

B. Zeng and Z. H. Liu, Existence results for impulsive feedback control systems, Nonlinear Analysis: Hybrid Systems, 33 (2019), 1-16.  doi: 10.1016/j.nahs.2019.01.008.

[30]

W. Zhang and M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Model., 39 (2004), 479-493.  doi: 10.1016/S0895-7177(04)90519-5.

[31]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

[32]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theor., 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.

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