-
Previous Article
Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity
- EECT Home
- This Issue
-
Next Article
Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids
Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives
Guangxi University for Nationalities, Faculty of Mathematics and Physics, Nanning 530006, Guangxi Province, P. R. China |
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.
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. Google Scholar |
[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. Google Scholar |
[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. |
[1] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[2] |
Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049 |
[3] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282 |
[4] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[5] |
Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021006 |
[6] |
Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 |
[7] |
Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 |
[8] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 |
[9] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[10] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[11] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[12] |
Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 |
[13] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[14] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[15] |
Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 |
[16] |
Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128 |
[17] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[18] |
Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495 |
[19] |
Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326 |
[20] |
Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]