M. J. BIN, Y. L. LIU

42

The proof is completed.

5. Application

Example. We consider the following semilinear impul-

sive equation:

2

1

()() [cos()cos()]

3

[0]

(0)(0)1 2

() ()

()() (())

k

kkkk

wtxw utxxtwt

ttJ Ttt

wx xinkm

wtxtx inJ

wtxwtxI wtx

(5.1)

where is abounded domain in with smooth

boundary

(1

n

Rn

( )L

)

2

(0 )()xtx

2()XL

We take and define the operator

()

DAX X by

21

0

()()() ()DA HHAwAxDw

It is easily turned out that the operator A generates

equicontinuous -semigroup on X. re the control term

is linear,

0

C

)(()) (Btut ut

2

1

() [cos()cos()

3

t]

txxt wt

is Lipchitz conditions. It satisfies the conditions of

Theorem 4.1, then the problem (5.1) is T-controllable.

6. Acknowledgements

This work was financially supported by NNSF of China

Grant No.11271087, No.61263006, Guangxi Scientific

Experimental (China- ASEAN Research) Centre No. 20

120116, open fund of Guangxi Key laboratory of hybrid

computation and IC design analysis No.2012HC IC07,

and the Innovation Project of Guangxi Graduate Educa-

tion No. YCSZ2012062.

REFERENCES

[1] A. Anguraj and M. Mallika Arjunan, “Existence and

Uniqueness of Mild and Classical Solutions of Impulsive

Evolution Equations,” Electronic Journal of Differential

Equations, Vol. 111, 2005, pp. 1-8.

[2] A. Anguraj and M. Mallika Arjunan, “Existence Results

for an Impulsive Neutral Integro-differential Equations in

Banach Spaces,” Nonlinear Study, Vol. 16, No. 1, 2009,

pp. 33-48.

[3] D. D. Bainov and P. S. Simeonov, “Impulsive Differen-

tial Equations: Periodic Solutions and Applications,”

Longman Scientific and Technical Group, England, 1993.

[4] M. Benchohra J. Henderson and S. K. Ntouyas, “Exis-

tence Results for Impulsive Multivalued Semilinear Neu-

tral Functional Inclusions in Banach Spaces,” Journal of

Mathematical Analysis and Applications, Vol. 263, No. 2,

2001, pp. 763-780.

doi:10.1006/jmaa.2001.7663

[5] T. A. Burton and Colleen Kirk, “A Fixed Point Theorem

of Krasnoselskiii-Schaefer Type,” Mathematische

Nachrichten, Vol. 189, No. 1, 1998, pp. 23-31.

doi:10.1002/mana.19981890103

[6] S. Carl and S. Heikkila, “Fixed Point Theory in Ordered

Sets and Applications,” Springer, New York, Dordrecht,

Heidelberg, London, 2010.

[7] D. N. Chalishajar, R. K. George, A. K. Nandakumaran

and F. S. Acharya, “Trajectory Controllability of Nonlin-

ear Integro-differential System,” Journal of Frankin In-

situte., Vol. 347, No. 7, 2010, pp. 1065-1075.

doi:10.1016/j.jfranklin.2010.03.014

[8] L. Chen and G. Li, “Approximate Controllability of Im-

pulsive Differential Equations with Nonlocal Condi-

tions,” International Journal of Nonlinear Science, Vol.

10, 2010, pp. 438-446.

[9] Z. Fan, “Impulsive Problems for Semilinear Differential

Equations with Nonlocal Conditions,” Nonlinear Analysis,

Vol. 72, No. 2, 2010, pp. 1104-1109.

doi:10.1016/j.na.2009.07.049

[10] Z. Fan and G. Li, “Existence Results for Semilinear Dif-

ferential Equations with Nonlocal and Impulsive Condi-

tions,” Journal of Functional Analysis, Vol. 258, No. 5,

2010, pp. 1709-1727. doi:10.1016/j.jfa.2009.10.023

[11] E. Hernandez, M. Pierri and G. Goncalves, “Existence

Results for an Impulsive Abstract Partial Differential

Equation with State-dependent Delay,” Computer Math-

ematic Application, Vol. 52, No. 3-4, 2006, pp. 411-420.

doi:10.1016/j.camwa.2006.03.022

[12] E. Hernandez, M. M. Rabello and H. Henriaquez, “Exis-

tence of Solutions for Impulsive Partial Neutral Func-

tional Differential Equations,” Journal of Mathematic

Analysis and Application, Vol. 331, No. 2, 2007, pp.

1135-1158.

doi:10.1016/j.jmaa.2006.09.043

[13] S. Ji and S. Wen, “Nonlocal Cauchy Problem for Impul-

sive Differential Equations in Banach Spaces,” Interna-

tional Journal of Nonlinear Science, Vol. 10, 2010, pp.

88-95.

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,

“Theory of Impulsive Differential Equations,” World

Scientific, Singapore, 1989.

[15] M. Li, M. Wang and F. Zhang, “Controllability of Impul-

sive Functional Differential Systems in Banach Spaces,”

Chaos, Solitons and Fractals, Vol. 29, No. 1, 2006, pp.

175-181. doi:10.1016/j.chaos.2005.08.041

[16] Z. H. Liu and X. W. Li, “On the Controllability of Impul-

sive Fractional Evolution Inclusions in Banach Spaces,”

Journal of Optimizition Theory and Application, Vol. 156,

No. 1, 2013, pp. 167–182.

doi:10.1007/s10957-012-0236-x

[17] A. M. Samoilenko and N. A. Perestyuk, “Impulsive Dif-

ferential Equations,” World Scientific, Singapore, 1995.

[18] V. Obukhovski and P. Zecca, “Controllability for Sys-

tems Governed by Semilinear Differential Inclusions in a

Copyright © 2013 SciRes. OJAppS