Approximate controllability of fractional neutral evolution systems of hyperbolic type. (English) Zbl 1509.34076
In the paper, a certain abstract non-linear neutral functional fractional differential system of hyperbolic type is studied. By using standard fixed-point theorems, the authors prove the existence and uniqueness of a mild solution to the initial value problem for the consider system. Quite restrictive hypotheses in the existence theorems are probably caused by a very general form of the system studied. In the second part of the paper, the authors discuss the approximate controlability of the system studied.
Reviewer: Jiří Šremr (Brno)
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K35 | Control problems for functional-differential equations |
| 34K40 | Neutral functional-differential equations |
| 34K05 | General theory of functional-differential equations |
| 93B05 | Controllability |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
approximate controllability; mild solution; fractional neutral systems; existence and uniquenessReferences:
| [1] | W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, \(2^{nd}\) edition, Birkhauser Verlag, Basel, 2001. · Zbl 0978.34001 |
| [2] | P. Y. Chen; X. P. Zhang; Y. X. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26, 1-16 (2020) · Zbl 1439.34065 · doi:10.1007/s10883-018-9423-x |
| [3] | J. Chang; H. Liu, Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the \(\alpha \)-norm, Nonlinear Anal., 71, 3759-3768 (2009) · Zbl 1185.34112 · doi:10.1016/j.na.2009.02.035 |
| [4] | R. Dhayal; M. Malik; S. Abbas; A. Kumar; R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evol. Equ. Control Theory, 10, 411-429 (2021) · Zbl 1485.93064 · doi:10.3934/eect.2020073 |
| [5] | Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27, 309-321 (1990) · Zbl 0790.45009 |
| [6] | X. Fu; R. Huang, Existence of solutions for neutral integro-differential equations with state-dependent delay, Appl. Math. Comput., 224, 743-759 (2013) · Zbl 1334.34143 · doi:10.1016/j.amc.2013.09.010 |
| [7] | J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. · Zbl 0592.47034 |
| [8] | R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. · Zbl 0998.26002 |
| [9] | J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. · Zbl 0787.34002 |
| [10] | K. Jeet; D. Bahuguna, Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay, J. Dyn. Control Syst., 22, 485-504 (2016) · Zbl 1348.34133 · doi:10.1007/s10883-015-9297-0 |
| [11] | A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1092.45003 |
| [12] | S. Kumar; N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations, 252, 6163-6174 (2012) · Zbl 1243.93018 · doi:10.1016/j.jde.2012.02.014 |
| [13] | Y. Kian; M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20, 117-138 (2017) · Zbl 1362.35323 · doi:10.1515/fca-2017-0006 |
| [14] | S. Kumar; R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA J. Math. Control Inform., 37, 441-454 (2020) · Zbl 1447.93028 · doi:10.1093/imamci/dnz007 |
| [15] | V. Kumar, M. Malik and A. Debbouche, Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects, J. Comput. Appl. Math., 383 (2021), 113158, 18 pp. · Zbl 1448.93029 |
| [16] | X. H. Liu; J. R. Wang; Y. Zhou, Approximate controllability for nonlocal fractional propagation systems of Sobolev type, J. Dyn. Control Syst., 25, 245-262 (2019) · Zbl 07073340 · doi:10.1007/s10883-018-9409-8 |
| [17] | X. W. Li; Z. H. Liu; J. Li; C. Tisdell, Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci. Ser. B (Engl. Ed.), 39, 229-242 (2019) · Zbl 1499.34392 · doi:10.1007/s10473-019-0118-5 |
| [18] | Y. Luchko, Wave-diffusion dualism of the neutral-fractional processes, J. Comput. Phys., 293, 40-52 (2015) · Zbl 1349.35402 · doi:10.1016/j.jcp.2014.06.005 |
| [19] | K. Li; J. Peng; J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263, 476-510 (2012) · Zbl 1266.47066 · doi:10.1016/j.jfa.2012.04.011 |
| [20] | Y. Li, Regularity of mild solutions for fractional abstract Cauchy problem with order \(\alpha\in(1, 2)\), Z. Angew. Math. Phys., 66, 3283-3298 (2015) · Zbl 1348.34020 · doi:10.1007/s00033-015-0577-z |
| [21] | Y. Li; H. Sun; Z. Feng, Fractional abstract Cauchy problem with order \(\alpha\in(1, 2)\), Dyn. Partial Differ. Equ., 13, 155-177 (2016) · Zbl 1355.35193 · doi:10.4310/DPDE.2016.v13.n2.a4 |
| [22] | Z. H. Liu; X. W. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 53, 1920-1933 (2015) · Zbl 1326.34019 · doi:10.1137/120903853 |
| [23] | K. Li; J. Peng; J. Gao, Controllability of nonlocal fractional defferential systems of order \(\alpha\in(1, 2]\) in Banach spaces, Rep. Math. Phys., 71, 33-43 (2013) · Zbl 1285.93023 · doi:10.1016/S0034-4877(13)60020-8 |
| [24] | N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42, 1604-1622 (2003) · Zbl 1084.93006 · doi:10.1137/S0363012901391688 |
| [25] | F. Z. Mokkedem; X. L. Fu, Approximate controllability of semi-linear neutral integro-differential systems with finite delay, Appl. Math. Comput., 242, 202-215 (2014) · Zbl 1334.93033 · doi:10.1016/j.amc.2014.05.055 |
| [26] | F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in Waves and Stability in Continuous Media, Bologna, 1993), Ser. Adv. Math. Appl. Sci., vol. 23, World Sci.Publ., River Edge, NJ, 1994,246-251. |
| [27] | M. F. Pinaud; H. R. Henr \(\acute{i}\) quez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269, 4609-4642 (2020) · Zbl 1443.93035 · doi:10.1016/j.jde.2020.03.029 |
| [28] | T. Poinot; J. C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38, 133-154 (2004) · Zbl 1134.93324 · doi:10.1007/s11071-004-3751-y |
| [29] | C. Ravichandran; N. Valliammal; J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J. Franklin. Inst., 356, 1535-1565 (2019) · Zbl 1451.93032 · doi:10.1016/j.jfranklin.2018.12.001 |
| [30] | Y. A. Rossikhin; M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system, Acta. Mech., 120, 109-125 (1997) · Zbl 0901.73030 · doi:10.1007/BF01174319 |
| [31] | R. Sakthivel; R. Ganesh; Y. Ren; S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18, 3498-3508 (2013) · Zbl 1344.93019 · doi:10.1016/j.cnsns.2013.05.015 |
| [32] | R. Sakthivel; N. I. Mahmudov; J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218, 10334-10340 (2012) · Zbl 1245.93022 · doi:10.1016/j.amc.2012.03.093 |
| [33] | G. J. Shen; R. Sakthivel; Y. Ren; M. Y. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collect. Math., 71, 63-82 (2020) · Zbl 1450.34058 · doi:10.1007/s13348-019-00248-3 |
| [34] | X. B. Shu; Q. Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order \(1<\alpha<2\), Comput. Math. Appl., 64, 2100-2110 (2012) · Zbl 1268.34155 · doi:10.1016/j.camwa.2012.04.006 |
| [35] | A. Shukla; N. Sukavanam; D. N. Pandey, Approximate controllability of semilinear fractional control systems of order \(\alpha\in(1, 2]\) with infinite delay, Mediterr. J. Math., 13, 2539-2550 (2016) · Zbl 1355.34117 · doi:10.1007/s00009-015-0638-8 |
| [36] | M. S. Tavazoei; M. Haeri; S. Jafari; S. Bolouki; M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE T. Ind. Electron, 11, 4094-4101 (2008) |
| [37] | N. H. Tuan; D. O’Regan; T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory, 9, 773-793 (2020) · Zbl 1455.35295 · doi:10.3934/eect.2020033 |
| [38] | C. C. Travis; G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 32, 75-96 (1978) · Zbl 0388.34039 · doi:10.1007/BF01902205 |
| [39] | V. Vijayakumar; R. Udhayakumar; K. Kavitha, On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evol. Equ. Control Theory, 10, 271-296 (2021) · Zbl 1476.93084 · doi:10.3934/eect.2020066 |
| [40] | N. Valliammal; C. Ravichandran; J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40, 5044-5055 (2017) · Zbl 1385.34054 · doi:10.1002/mma.4369 |
| [41] | V. Vijayakumar, Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators, Internat. J. Control, 92, 2210-2222 (2019) · Zbl 1421.93023 · doi:10.1080/00207179.2018.1433331 |
| [42] | 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 |
| [43] | V. V. Vasil’ev; S. G. Krein; S. I. Piskarev, Semigroups of operators, cosine operator functions and linear differential equations, J. Soviet Math., 54, 1042-1129 (1991) · Zbl 0748.47038 |
| [44] | R. N. Wang; D. H. Chen; T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252, 202-235 (2012) · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048 |
| [45] | M. Yang; Q. R. Wang, Approximate controllability of Caputo fractional neutral stochastic differential inclusions with state-dependent delay, IMA J. Math. Control Inform., 35, 1061-1085 (2018) · Zbl 1417.93080 · doi:10.1093/imamci/dnx014 |
| [46] | H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21, 551-565 (1983) · Zbl 0516.93009 · doi:10.1137/0321033 |
| [47] | Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. · Zbl 1336.34001 |
| [48] | Y. Zhou; S. Suganya; M. M. Arjunan; B. Ahmad, Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces, IMA J. Math. Control. Inform., 36, 603-622 (2019) · Zbl 1475.93019 · doi:10.1093/imamci/dnx060 |
| [49] | Y. Zhou; J. W. He, New results on controllability of fractional evolution systems with order \(\alpha\in(1, 2)\), Evol. Equ. Control Theory, 9, 1-19 (2020) |
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.
