In this paper, the control problem of nonlinear neutral fractional Volterra integrodifferential systems with implicit fractional derivative is established. Such kind of problems involve a number of problems on complex media. Sufficient conditions for controllability are obtained through the notions of a condensing map and measure of noncompactness of a set. A example to check the main results included in this paper is included.
AhmedEElgazzarAS (2007) On fractional order differential equations model for nonlocal epidemics. Physica A379(2): 607–614.
2.
AnichiniGContiGZeccaP (1986) A note on controllability of certain nonlinear systems. Note di Matematica6(1): 99–111.
3.
BagleyRLTorvikPJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology27(3): 201–210.
4.
BalachandranKBalasubramaniamP (1994) Controllability of nonlinear neutral Volterra integrodifferential systems. Journal of the Australian Mathematical Society36(1): 107–116.
5.
BalachandranKDauerJP (1987) Controllability of nonlinear systems via fixed point theorems. Journal of Optimization Theory and Applications53(3): 345–352.
6.
BalachandranKKokilaJ (2012a) On the controllability of fractional dynamical systems. International Journal of Applied Mathematics and Computer Science22(3): 523–531.
7.
BalachandranKKokilaJ (2013a) Constrained controllability of fractional dynamical systems. Numerical Functional Analysis and Optimization34(11): 1187–1205.
8.
BalachandranKKokilaJ (2013b) Controllability of nonlinear implicit fractional dynamical systems. IMA Journal of Applied Mathematics79(3): 562–570.
9.
BalachandranKKokilaJTrujilloJJ (2012b) Relative controllability of fractional dynamical systems with multiple delays in control. Computers and Mathematics with Applications64(10): 3037–3045.
10.
BalachandranKParkJYTrujilloJJ (2012c) Controllability of nonlinear fractional dynamical systems. Nonlinear Analysis: Theory, Methods and Applications75(4): 1919–1926.
11.
BalachandranKZhouYKokilaJ (2012d) Relative controllability of fractional dynamical systems with delays in control. Communications in Nonlinear Science and Numerical Simulation17(9): 3508–3520.
12.
BanksHTKentGA (1972) Control of functional differential equations of retarded and neutral type to target sets in function space. SIAM Journal of Control10(4): 562–593.
13.
BonillaBRiveroMRodríguez-GermaáLTrujilloJJ (2007) Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation187(1): 79–88.
14.
ChukwuEN (1987) On the Euclidean controllability of a neutral system with nonlinear base. Nonlinear Analysis: Theory, Methods and Applications11(1): 115–123.
15.
DackaC (1980) On the controllability of a class of nonlinear systems. IEEE Transaction on Automatic Control25(2): 263–266.
16.
El-SayedAMA (1996) Fractional-order diffusion-wave equation. International Journal of Theoretical Physics35(2): 311–322.
17.
GahlRD (1978) Controllability of nonlinear systems of neutral type. Journal of Mathematical Analysis and Applications63(1): 33–42.
18.
GyoriIWuJ (1991) A neutral equation arising form compartmental systems with pipes. Journal of Dynamics and Differential Equations3(2): 289–311.
19.
HaleJK (1972) Theory of Functional Differential Equations, New York: Springer-Verlag.
20.
HernándezEO’ReganDBalachandranK (2010) On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Analysis: Theory, Methods and Applications73(10): 3462–3471.
21.
JacobsMQLangenhopCE (1976) Criteria for function space controllability of linear neutral systems. SIAM Journal on Control and Optimization14(6): 1009–1048.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier.
24.
KlamkaJ (1993) Controllability of Dynamical Systems, Dordrecht: Kluwer Academic.
25.
KlamkaJ (2008) Constrained controllability of semi linear systems with delayed controls. Bulletin of the Polish Academy of Sciences: Technical Sciences56(4): 333–337.
26.
LakshmikanthamV (2008) Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods and Applications69(10): 3337–3343.
27.
MetzlerRKlafterJ (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports339(1): 1–77.
28.
MillerKSRossB (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley.
29.
OldhamKBSpanierJ (1974) The Fractional Calculus, London: Academic Press.
30.
OnwuatuJU (1984) On the null controllability in function space of nonlinear systems of neutral functional differential equations with limited controls. Journal of Optimization Theory and Applications42(3): 327–420.
31.
PodlubnyI (1999) Fractional Differential Equations, New York: Academic Press.
32.
SadovskiiJB (1972) Linear compact and condensing operator. Russian Mathematical Surveys27: 85–165.
