Controllability for a class of semilinear fractional evolution systems via resolvent operators. (English) Zbl 06969373
Summary: 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.
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 93B05 | Controllability |
| 34K30 | Functional-differential equations in abstract spaces |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional evolution systems; exact controllability; resolvent operators; Kuratowski’s measure of noncompactness; fixed point theoremsReferences:
| [1] | D. Araya; C. Lizama, 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 |
| [2] | B. Ahmad; J. J. Nieto; A. Alsaedi; M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal., 13, 599-606 (2012) · Zbl 1238.34008 · 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, 1361-1380 (2017) · Zbl 1396.26013 · 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. · Zbl 1377.35002 |
| [5] | D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isreal J. Math., 108, 109-138 (1998) · Zbl 0922.47048 · 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. · Zbl 0441.47056 |
| [7] | K. Balachandran; J. Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal., 3, 363-367 (2009) · Zbl 1175.93028 · doi:10.1016/j.nahs.2009.01.014 |
| [8] | G. Da Prato; M. Iannelli, Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl., 112, 36-55 (1985) · Zbl 0583.45009 · doi:10.1016/0022-247X(85)90275-6 |
| [9] | A. Debbouche; D. Baleanu, 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 |
| [10] | M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fract., 14, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9 |
| [11] | E. Hern \(\acute{a}\) ndez; D. O’Regan; K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73, 3462-3471 (2010) · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035 |
| [12] | E. Hern \(\acute{a}\) ndez; D. O’Regan; K. Balachandran, Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes mathematicae, 24, 68-82 (2013) · Zbl 1254.34011 · doi:10.1016/j.indag.2012.06.007 |
| [13] | O. K. Jaradat; A. Al-Omari; S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008 |
| [14] | S. Ji; G. Li; M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217, 6981-6989 (2011) · Zbl 1219.93013 |
| [15] | X. Li; J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Autom. Contr., 62, 3618-3625 (2017) · Zbl 1370.34131 · doi:10.1109/TAC.2017.2669580 |
| [16] | X. Li; 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, 868-877 (2013) |
| [17] | X. Li; S. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Contr., 62, 406-411 (2017) · Zbl 1359.34089 · doi:10.1109/TAC.2016.2530041 |
| [18] | H. Li; Y. Wang, Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM J. Control Optim., 55, 3437-3457 (2017) · Zbl 1373.93249 · doi:10.1137/16M1092581 |
| [19] | H. Li; Y. Wang, Further results on feedback stabilization control design of Boolean control networks, Automatica, 83, 303-308 (2017) · Zbl 1373.93259 · doi:10.1016/j.automatica.2017.06.043 |
| [20] | X. Li; J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64, 63-69 (2016) · Zbl 1329.93108 · doi:10.1016/j.automatica.2015.10.002 |
| [21] | H. Li; L. Xie; Y. Wang, On robust control invariance of Boolean control networks, Automatica, 68, 392-396 (2016) · Zbl 1334.93057 · doi:10.1016/j.automatica.2016.01.075 |
| [22] | H. Li; L. Xie; Y. Wang, Output regulation of Boolean control networks, IEEE Trans. Autom. Contr., 62, 2993-2998 (2017) · Zbl 1369.93345 · doi:10.1109/TAC.2016.2606600 |
| [23] | I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. · Zbl 0924.34008 |
| [24] | J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993. · Zbl 0784.45006 |
| [25] | R. Sakthivel; Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results in Mathematics, 63, 949-963 (2013) · Zbl 1272.34105 · doi:10.1007/s00025-012-0245-y |
| [26] | I. Stamova; T. Stamov; X. Li, Global exponential stability of a class of impulsive cellular neural networks with Supremums, Inter. J. Adapt. Contr. Signal Proces., 28, 1227-1239 (2014) · Zbl 1338.93316 · doi:10.1002/acs.2440 |
| [27] | J. Sprekels; E. Valdinoci, A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55, 70-93 (2017) · Zbl 1394.49033 · doi:10.1137/16M105575X |
| [28] | V. Vijayakumar; A. Selvakumar; R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Appl. Math. Comput., 232, 303-312 (2014) · Zbl 1410.93025 · doi:10.1016/j.amc.2014.01.029 |
| [29] | J. Wang; Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simulat., 17, 4346-4355 (2012) · Zbl 1248.93029 · doi:10.1016/j.cnsns.2012.02.029 |
| [30] | Y. Zhou; F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., 11, 4465-4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029 |
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