Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 521052, 8 pages http://dx.doi.org/10.1155/2013/521052 Research Article On Partial Complete Controllability of Semilinear Systems Agamirza E. Bashirov and Maher Jneid Eastern Mediterranean University, Gazimagusa, North Cyprus, P.O. Box 95, Mersin 10, Turkey Correspondence should be addressed to Agamirza E. Bashirov; agamirza.bashirov@emu.edu.tr Received 27 March 2013; Revised 28 May 2013; Accepted 11 June 2013 Academic Editor: Sakthivel Rathinasamy Copyright © 2013 A. E. Bashirov and M. Jneid. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many control systems can be written as a frst-order diferential equation if the state space enlarged. Terefore, general conditions on controllability, stated for the frst-order diferential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufcient condition for the partial complete controllability of semilinear control system is proved. Te result is demonstrated through examples. 1. Introduction A concept of controllability, defned by Kalman [1] in 1960 for fnite dimensional control systems, is a property of attaining every point in the state space from every initial state point for a fnite time. Further studies on this concept in infnite dimensional spaces demonstrated that it is suitable to con- sider its two versions: a stronger version of complete control- lability and a weaker version of approximate controllability. Te reason for these versions was the fact that many infnite dimensional control systems are not completely controllable while they are approximately controllable (see Fattorini [2] and Russell [3]). Te necessary and sufcient conditions for complete and approximate controllability concepts are almost completely studied and presented in, for example, Curtain and Zwart [4], Bensoussan [5], Bensoussan et al. [6], Zabczyk [7], Bashirov [8], Klamka [9], and so forth for linear systems; Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Li and Yong [14], and so forth for nonlinear systems; Sakthivel et al. [15–17], Yan [18], and so forth for fractional diferential systems; and Ren et al. [19] for diferential inclusions. Recently, in Bashirov et al. [20, 21] the partial controlla- bility concepts were initiated. Te idea of these concepts is that some control systems, including higher order diferential equations, wave equations, and delay equations, can be written as a frst-order diferential equation only by enlarging the dimension of the state space. Terefore, the theorems on controllability, which are formulated for control systems in the form of frst-order diferential equation, are too strong for them because they involve the enlarged state space. In such cases the partial controllability concepts became preferable, which assume the original state space. Te basic controllability conditions for linear systems, including resol- vent conditions from Bashirov and Mahmudov [22] and Bashirov and Kerimov [23] (see also [24–26]), are extended to partial controllability concepts by just a replacement of the controllability operator by its partial version. In this paper our aim is to study the partial complete con- trollability of semilinear systems. Te controllability concepts for semilinear systems are intensively discussed in the litera- ture (see Balachandran and Dauer [10, 11], Klamka [12], Mah- mudov [13], Sakthivel et al. [15, 17], and references therein). A basic tool of study in these works is fxed point theorems. In this paper, we also use one of the fxed point theorems, a contraction mapping theorem, and fnd a sufcient condition for the partial complete controllability of a semilinear control system. Te rest of this paper is organised in the following way. In Section 2 we set the problem, give basic defnitions, and moti- vate the partial controllability concepts by considering a higher order diferential equation, a wave equation, and a delay equation. Section 3 contains the proof of the main result. In Section 4, we demonstrate the main result in the examples. Finally, Section 5 contains directions of further research regarding partial controllability concepts.