We study the approximate controllability of a class of fractional partial neutral integro-differential inclusions with infinite delay in Hilbert spaces. By using the analytic α-resolvent operator and the fixed point theorem for discontinuous multivalued operators due to Dhage, a new set of necessary and sufficient conditions are formulated which guarantee the approximate controllability of the nonlinear fractional system. The results are obtained under the assumption that the associated linear system is approximately controllable. An example is provided to illustrate the main results.
References
1.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, “Theory and Applications of Fractional Differential Equations
,” North-Holland Mathematics Studies
, Vol. 204
, Elsevier
, Amsterdam, The Netherlands
.2.
Miller
, K. S.
, and Ross
, B.
, 1993
, An Introduction to the Fractional Calculus and Differential Equations
, Wiley
, New York
.3.
Podlubny
, I.
, 1999
, Fractional Differential Equations, Mathematics in Sciences and Engineering
, Vol. 198
, Academic
, San Diego, CA
.4.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, “Fractional Integrals and Derivatives
,” Theory and Applications
, Gordon and Breach
, Yverdon, Switzerland
.5.
Metzler
, F.
, Schick
, W.
, Kilian
, H. G.
, and Nonnemacher
, T. F.
, 1995
, “Relaxation in Filled Polymers: A Fractional Calculus Approach
,” J. Chem. Phys.
, 103
(7180
), pp. 7180
–7186
.10.1063/1.4703466.
Lakshmikantham
, V.
, 2008
, “Theory of Fractional Functional Differential Equations
,” Nonlinear Anal.
, 69
(10
), pp. 3337
–3343
.10.1016/j.na.2007.09.0257.
Lakshmikantham
, V.
, and Vatsala
, A. S.
, 2008
, “Basic Theory of Fractional Differential Equations
,” Nonlinear Anal.
, 69
(8
), pp. 2677
–2682
.10.1016/j.na.2007.08.0428.
Benchohra
, M.
, Henderson
, J.
, Ntouyas
, S. K.
, and Ouahab
, A.
, 2008
, “Existence Results for Fractional Order Functional Differential Equations With Infinite Delay
,” J. Math. Anal. Appl.
, 338
(2
), pp. 1340
–1350
.10.1016/j.jmaa.2007.06.0219.
Benchohra
, M.
, Henderson
, J.
, Ntouyas
, S. K.
, and Ouahab
, A.
, 2008
, “Existence Results for Fractional Functional Differential Inclusions With Infinite Delay and Application to Control Theory
,” Fract. Calc. Appl. Anal.
, 11
(1
), pp. 35
–56
.10.
El-Borai
, M. M.
, 2006
, “On Some Stochastic Fractional Integro-Differential Equations
,” Adv. Dyn. Syst. Appl.
, 1
(1
), pp. 49
–57
.11.
El-Borai
, M. M.
, 2002
, “Some Probability Densities and Fundamental Solutions of Fractional Evolution Equations
,” Chaos Solitons Fractals
, 14
(3
), pp. 433
–440
.10.1016/S0960-0779(01)00208-912.
El-Borai
, M. M.
, 2004
, “Semigroups and Some Nonlinear Fractional Differential Equations
,” Appl. Math. Comput.
, 149
(3
), pp. 823
–831
.10.1016/S0096-3003(03)00188-713.
Balachandran
, K.
, and Trujillo
, J. J.
, 2010
, “The Nonlocal Cauchy Problem for Nonlinear Fractional Integrodifferential Equations in Banach Spaces
,” Nonlinear Anal.
, 72
(12
), pp. 4587
–4593
.10.1016/j.na.2010.02.03514.
dos Santos
, J. P. C.
, Arjunan
, M. M.
, and Cuevas
, C.
, 2011
, “Existence Results for Fractional Neutral Integro-Differential Equations With State-Dependent Delay
,” Comput. Math. Appl.
, 62
(3
), pp. 1275
–1283
.10.1016/j.camwa.2011.03.04815.
Agarwal
, R. P.
, and Cuevas
, C.
, 2012
, “Analytic Resolvent Operator and Existence Results for Fractional Order Evolutionary Integral Equations
,” J. Abstr. Differ. Equations Appl.
, 2
(2
), pp. 26
–47
.16.
de Andrade
, B.
, and dos Santos
, J. P. C.
, 2012
, “Existence of Solutions for a Fractional Neutral Integro-Differential Equation With Unbounded Delay
,” Electron. J. Differ. Equations
, 2012
(90
), pp. 1
–13
.17.
Benchohra
, M.
, and Ntouyas
, S. K.
, 2002
, “Controllability for Functional Differential and Integrodifferential Inclusions
,” J. Optim. Theory Appl.
, 113
(3
), pp. 449
–472
.10.1023/A:101535250323318.
Mahmudov
, N. I.
, 2001
, “On Controllability of Linear Stochastic Systems in Hilbert Spaces
,” J. Math. Anal. Appl.
, 259
(1
), pp. 64
–82
.10.1006/jmaa.2000.738619.
Mahmudov
, N. I.
, and Denker
, A.
, 2000
, “On Controllability of Linear Stochastic Systems
,” Int. J. Control
, 73
(2
), pp. 144
–151
.10.1080/00207170021984920.
Dauer
, J. P.
, and Mahmudov
, N. I.
, 2002
, “Approximate Controllability of Semilinear Functional Equations in Hilbert Spaces
,” J. Math. Anal. Appl.
, 273
(2
), pp. 310
–327
.10.1016/S0022-247X(02)00225-121.
Balachandran
, K.
, and Park
, J. Y.
, 2009
, “Controllability of Fractional Integrodifferential Systems in Banach Spaces
,” Nonlinear Anal. Hybrid Syst.
, 3
(4
), pp. 363
–367
.10.1016/j.nahs.2009.01.01422.
Tai
, Z.
, and Wang
, X.
, 2009
, “Controllability of Fractional-Order Impulsive Neutral Functional Infinite Delay Integrodifferential Systems in Banach Spaces
,” Appl. Math. Lett.
, 22
(11
), pp. 1760
–1765
.10.1016/j.aml.2009.06.01723.
Debbouchea
, A.
, and Baleanu
, D.
, 2011
, “Controllability of Fractional Evolution Nonlocal Impulsive Quasilinear Delay Integro-Differential Systems
,” Comput. Math. Appl.
, 62
(3
), pp. 1442
–1450
.10.1016/j.camwa.2011.03.07524.
Yan
, Z.
, 2011
, “Controllability of Fractional-Order Partial Neutral Functional Integrodifferential Inclusions With Infinite Delay
,” J. Franklin Inst.
, 348
(8
), pp. 2156
–2173
.10.1016/j.jfranklin.2011.06.00925.
Triggiani
, R.
, 1977
, “A Note on the Lack of Exact Controllability for Mild Solutions in Banach Spaces
,” SIAM J. Control Optim.
, 15
(3
), pp. 407
–411
.10.1137/031502826.
Mahmudov
, N. I.
, 2003
, “Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces
,” SIAM J. Control Optim.
, 42
(5
), pp. 1604
–1622
.10.1137/S036301290139168827.
Klamka
, J.
, 2000
, “Constrained Approximate Controllability
,” IEEE Trans. Autom. Control
, 45
(9
), pp. 1745
–1749
.10.1109/9.88064028.
Sakthivel
, R.
, and Anandhi
, E. R.
, 2010
, “Approximate Controllability of Impulsive Differential Equations With State-Dependent Delay
,” Int. J. Control
, 83
(2
), pp. 387
–393
.10.1080/0020717090317134829.
Sakthivel
, R.
, Anandhi
, E. R.
, and Lee
, S. G.
, 2009
, “Approximate Controllability of Impulsive Differential Inclusions With Nonlocal Conditions
,” Dyn. Syst. Appl.
, 18
(3
), pp. 637
–654
.30.
Fu
, X.
, 2011
, “Approximate Controllability for Neutral Impulsive Differential Inclusions With Nonlocal Conditions
,” Dyn. Syst. Appl.
, 17
(3
), pp. 359
–386
.10.1007/s10883-011-9126-z31.
Rykaczewski
, K.
, 2012
, “Approximate Controllability of Differential Inclusions in Hilbert Spaces
,” Nonlinear Anal.
, 75
(5
), pp. 2701
–2712
.10.1016/j.na.2011.10.04932.
Sakthivel
, R.
, Ren
, Y.
, and Mahmudov
, N. I.
, 2011
, “On the Approximate Controllability of Semilinear Fractional Differential Systems
,” Comput. Math. Appl.
, 62
(3
), pp. 1451
–1459
.10.1016/j.camwa.2011.04.04033.
Sakthivel
, R.
, Suganya
, S.
, and Anthoni
, S. M.
, 2012
, “Approximate Controllability of Fractional Stochastic Evolution Equations
,” Comput. Math. Appl.
, 63
(3
), pp. 660
–668
.10.1016/j.camwa.2011.11.02434.
Kumar
, S.
, and Sukavanam
, N.
, 2012
, “Approximate Controllability of Fractional Order Semilinear Systems With Bounded Delay
,” J. Differ. Equations
, 252
(11
), pp. 6163
–6174
.10.1016/j.jde.2012.02.01435.
Sukavanam
, N.
, and Kumar
, S.
, 2011
, “Approximate Controllability of Fractional Order Semilinear Delay Systems
,” J. Optim. Theory Appl.
, 151
(2
), pp. 373
–384
.10.1007/s10957-011-9905-436.
Yan
, Z.
, 2012
, “Approximate Controllability of Partial Neutral Functional Differential Systems of Fractional Order With State-Dependent Delay
,” Int. J. Control
, 85
(8
), pp. 1051
–1062
.10.1080/00207179.2012.67551837.
Henderson
, J.
, and Ouahab
, A.
, 2009
, “Fractional Functional Differential Inclusions With Finite Delay
,” Nonlinear Anal.
, 70
(5
), pp. 2091
–2105
.10.1016/j.na.2008.02.11138.
Agarwal
, R. P.
, Belmekki
, M.
, and Benchohra
, M.
, 2009
, “A Survey on Semilinear Differential Equations and Inclusions Involving Riemann–Liouville Fractional Derivative
,” Adv. Differ. Equations
, 2009
(981728), pp. 1
–47
.39.
Yan
, Z.
, 2011
, “On a Nonlocal Problem for Fractional Integrodifferential Inclusions in Banach Spaces
,” Ann. Polonici Math.
, 101
(1
), pp. 87
–103
.10.4064/ap101-1-940.
Dhage
, B. C.
, 2006
, “Fixed-Point Theorems for Discontinuous Multi-Valued Operators on Ordered Spaces With Applications
,” Comput. Math. Appl.
, 51
(3–4
), pp. 589
–604
.10.1016/j.camwa.2005.07.01741.
42.
Deimling
, K.
, 1992
, Multi-Valued Differential Equations
, De Gruyter
, Berlin, Germany
.43.
Hu
, S.
, and Papageorgiou
, N.
, 1997
, Handbook of Multivalued Analysis
, Kluwer Academic Publishers
, Dordrecht/Boston, Springer
.10.1007/978-1-4615-6359-444.
Pazy
, A.
, 1983
, Semigroups of Linear Operators and Applications to Partial Differential Equations
, Springer
, New York
.10.1007/978-1-4612-5561-145.
Hale
, J. K.
, and Kato
, J.
, 1978
, “Phase Spaces for Retarded Equations With Infinite Delay
,” Funkcialaj Ekvacioj
, 21
(1
), pp. 11
–41
.46.
Lasota
, A.
, and Opial
, Z.
, 1965
, “An Application of the Kakutani-Ky Fan Theorem in the Theory of Ordinary Differential Equations
,” Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys.
, 13
, pp. 781
–786
.47.
Hino
, Y.
, Murakami
, S.
, and Naito
, T.
, 1991
, “Functional-Differential Equations With Infinite Delay
,” Lecture Notes in Mathematics
, Vol. 1473
, Springer
, Berlin, Germany
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