Approximate controllability for mixed type non-autonomous fractional differential equations. (English) Zbl 1504.34228
Summary: In this paper, we discuss the approximate controllability for mixed type non-autonomous fractional differential equations. By using the Schauder fixed point theorem and two parameters \(\beta\)-resolvent family, we prove the approximate controllability of the control system. Last of all, the application of our main results is given.
MSC:
| 34K35 | Control problems for functional-differential equations |
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K30 | Functional-differential equations in abstract spaces |
| 93B05 | Controllability |
| 37C60 | Nonautonomous smooth dynamical systems |
Keywords:
mixed type non-autonomous fractional differential equations; approximate controllability; \(\beta\)-resolvent; mild solutionReferences:
| [1] | Mohan Raja, M.; Vijayakumar, V.; Udhayakumar, R.; Zhou, Y., A new approach on the approximate controllability of fractional differential evolution equations of order \(1 < r < 2\) in Hilbert spaces, Chaos, Soliton. Fract., 141, 110310 (2020) · Zbl 1496.34021 · doi:10.1016/j.chaos.2020.110310 |
| [2] | Mohan Raja, M.; Vijayakumar, V.; Udhayakumar, R., A new approach on approximate controllability of fractional evolution inclusions of order \(1 <r < 2\) with infinite delay, Chaos, Soliton. Fract., 141, 110343 (2020) · Zbl 1496.34120 · doi:10.1016/j.chaos.2020.110343 |
| [3] | Dineshkumar, C.; Udhayakumar, R.; Vijayakumar, V.; Shukla, A.; Nisar, KS, A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order \(r \in (1, 2)\) with delay, Chaos, Soliton. Fract., 153, 111565 (2021) · Zbl 1498.34210 · doi:10.1016/j.chaos.2021.111565 |
| [4] | Dineshkumar, C.; Nisar, KS; Udhayakumar, R.; Vijayakumar, V., New discussion about the approximate controllability of fractional stochastic differential inclusions with order \(1 < r < 2 \), Asian J. Control., 2021, 1-15 (2021) · Zbl 1540.93009 |
| [5] | Bezerra, FDM; Santos, LA, Fractional powers approach of operators for abstract evolution equations of third order in time Author links open overlay panel, J. Differ. Equations, 269, 7, 5661-5679 (2020) · Zbl 1468.34091 · doi:10.1016/j.jde.2020.04.020 |
| [6] | He, JW; Lizama, C.; Zhou, Y., The Cauchy problem for discrete time fractional evolution equations, J. Comput. Appl. Math., 370, 15, 112683 (2020) · Zbl 1444.35148 · doi:10.1016/j.cam.2019.112683 |
| [7] | Kavitha, K.; Vijayakumar, V.; Udhayakumar, R.; Sakthivel, N.; Nisar, KS, A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Math. Meth. Appl. Sci., 44, 4428-4447 (2021) · Zbl 1471.34148 · doi:10.1002/mma.7040 |
| [8] | Dineshkumar, C.; Nisar, KS; Udhayakumar, R.; Vijayakumar, V., A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian J. Control., 2021, 1-17 (2021) · Zbl 1496.34111 |
| [9] | Zhu, B.; Liu, LS; Wu, YH, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61, 73-79 (2016) · Zbl 1355.35196 · doi:10.1016/j.aml.2016.05.010 |
| [10] | Zhu, B.; Liu, LS; Wu, YH, Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Comput. Math. Appl., 78, 1811-1818 (2019) · Zbl 1442.35536 · doi:10.1016/j.camwa.2016.01.028 |
| [11] | Zhu, B., Han, B.Y.: Existence and Uniqueness of Mild Solutions for Fractional Partial Integro-Differential Equations. Mediterr. J. Math. 17(4), (2020). doi:10.1007/s00009-020-01550-2 · Zbl 1452.35248 |
| [12] | Chen, PY; Zhang, XP; Li, YX, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calc. Appl. Anal., 23, 1, 268-291 (2020) · Zbl 1441.34006 · doi:10.1515/fca-2020-0011 |
| [13] | Chen, P.Y., Zhang, X.P., Li, Y.X.: Non-autonomous evolution equations of parabolic type with non-instantaneous impulses. Mediterr. J. Math. 16(5), (2019). doi:10.1007/s00009-019-1384-0 · Zbl 1483.35333 |
| [14] | Chen, PY; Zhang, XP; Li, YX, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73, 794-803 (2017) · Zbl 1375.34115 · doi:10.1016/j.camwa.2017.01.009 |
| [15] | Brandibur, O.; Kaslik, E., Stability analysis of multi-term fractional-differential equations with three fractional derivatives, J. Math. Anal. Appl., 495, 2, 124751 (2021) · Zbl 1464.34017 · doi:10.1016/j.jmaa.2020.124751 |
| [16] | Yang, J.; Fečkan, M.; Wang, JR, Consensus problems of linear multi-agent systems involving conformable derivative, Appl. Math. Comput., 394, 125809 (2021) · Zbl 1508.93023 |
| [17] | Sathiyaraj, T.; Wang, JR; Balasubramaniam, P., Controllability and Optimal Control for a Class of Time-Delayed Fractional Stochastic Integro-Differential Systems, Appl. Math. Opt., 84, 2527-2554 (2021) · Zbl 1472.93016 · doi:10.1007/s00245-020-09716-w |
| [18] | He, WJ; Zhou, Y., Hölder regularity for non-autonomous fractional evolution equations, Fract. Calc. Appl. Anal., 25, 378-407 (2022) · Zbl 1503.35263 · doi:10.1007/s13540-022-00019-1 |
| [19] | Zhou, Y.; He, WJ, A Cauchy problem for fractional evolution equations with Hilfers fractional derivative on semi-infinite interval, Fract. Calc. Appl. Anal., 25, 924-961 (2022) · Zbl 1503.34038 · doi:10.1007/s13540-022-00057-9 |
| [20] | Samko, SG; Kilbas, AA; Marichev, OI, Fractional integrals and derivatives: Theory and Applications (1993), Amsterdam: Gordon and Breach, Amsterdam · Zbl 0818.26003 |
| [21] | Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69, 3692-3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004 |
| [22] | Debbouche, A.; Baleanu, D., Controllability of Fractional Evolution Nonlocal Impulsive Quasilinear Delay Integro-Differential Systems, Comput. Math. Appl., 62, 1442-1450 (2011) · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075 |
| [23] | Pazy, A., Semigroup of linear operators and applications to partial differential equations (1983), New York: Springer-Verlag, New York · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [24] | Grimer, R., Resolvent operators for integral equations in Banach spaces, Trans. Amer. Math. Soc., 48, 333-349 (1982) · Zbl 0493.45015 · doi:10.1090/S0002-9947-1982-0664046-4 |
| [25] | Pruss, J., On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations., 5, 211-236 (1983) · Zbl 0518.45008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
