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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 521052, 8 pages
On Partial Complete Controllability of Semilinear Systems
Eastern Mediterranean University, Gazimagusa, North Cyprus, P.O. Box 95, Mersin 10, Turkey
Received 27 March 2013; Revised 28 May 2013; Accepted 11 June 2013
Academic Editor: Sakthivel Rathinasamy
Copyright © 2013 Agamirza E. Bashirov and Maher Jneid. This 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 first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential 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 sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.
A concept of controllability, defined by Kalman  in 1960 for finite dimensional control systems, is a property of attaining every point in the state space from every initial state point for a finite time. Further studies on this concept in infinite dimensional spaces demonstrated that it is suitable to consider its two versions: a stronger version of complete controllability and a weaker version of approximate controllability. The reason for these versions was the fact that many infinite dimensional control systems are not completely controllable while they are approximately controllable (see Fattorini  and Russell ). The necessary and sufficient conditions for complete and approximate controllability concepts are almost completely studied and presented in, for example, Curtain and Zwart , Bensoussan , Bensoussan et al. , Zabczyk , Bashirov , Klamka , and so forth for linear systems; Balachandran and Dauer [10, 11], Klamka , Mahmudov , Li and Yong , and so forth for nonlinear systems; Sakthivel et al. [15–17], Yan , and so forth for fractional differential systems; and Ren et al.  for differential inclusions.
Recently, in Bashirov et al. [20, 21] the partial controllability concepts were initiated. The idea of these concepts is that some control systems, including higher order differential equations, wave equations, and delay equations, can be written as a first-order differential equation only by enlarging the dimension of the state space. Therefore, the theorems on controllability, which are formulated for control systems in the form of first-order differential 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. The basic controllability conditions for linear systems, including resolvent conditions from Bashirov and Mahmudov  and Bashirov and Kerimov  (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 controllability of semilinear systems. The controllability concepts for semilinear systems are intensively discussed in the literature (see Balachandran and Dauer [10, 11], Klamka , Mahmudov , Sakthivel et al. [15, 17], and references therein). A basic tool of study in these works is fixed point theorems. In this paper, we also use one of the fixed point theorems, a contraction mapping theorem, and find a sufficient condition for the partial complete controllability of a semilinear control system.
The rest of this paper is organised in the following way. In Section 2 we set the problem, give basic definitions, and motivate the partial controllability concepts by considering a higher order differential 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.
2. Setting the Problem and Motivation
Consider the basic semilinear control system on the interval with , where and are state and control processes. We assume that the following conditions hold. (A)and are separable Hilbert spaces, is a closed subspace of , and is a projection operator from to ; (B) is a densely defined closed linear operator on , generating a strongly continuous semigroup , ; (C) is a bounded linear operator from to ; (D) is a nonlinear function from to , satisfying that(i) is continuous on ; (ii) is Lipschitz continuous with respect to and that is, for all , and , for some ; (E) is the space of all continuous functions from to . Define the controllability and -partial controllability operators and by where is the adjoint of . The -partial controllability operator becomes the controllability operator if (the identity operator). We will also assume the following condition;(F) is coercive; that is, there is such that for all .
Note that this condition implies the existence of as a bounded linear operator and . Respectively, the linear system associated with (1) (the case when ) is -partially complete controllable on the interval (see, Bashirov et al. [20, 21]).
The above conditions imply the existence of a unique continuous solution of (1) in the mild sense for every and (see Li and Yong  and Byszewski ); that is, there is a unique continuous function from to such that
Let Following Bashirov et al. , the semilinear control system (1) is said to be -partially complete controllable on if for all . Similarly, the semilinear system in (1) is said to be -partially approximate controllable on if for all , where is the closure of . If , these are just well-known complete and approximate controllability concepts, respectively. In this paper, we study the concept of -partial complete controllability.
The reason for studying -partial controllability concepts is that many systems can be written in the form of (1) if the original state space is enlarged. Therefore, suitable controllability concepts for such systems are the -partial controllability concepts with the operator projecting the enlarged state space to the original one. Here are some examples of such systems, which are discussed in Bashirov , Section 3.1.1, in more details.
Example 1. Consider the system assuming that its state space is the one-dimensional space . The ordinary controllability concepts for this system are the equality to or denseness in of the respective attainable set. We can write this system as the first-order differential equation if The state space of this system is the -dimensional Euclidean space and, respectively, its attainable set is a subset of . Therefore, the controllability concepts of the system for are stronger than those of the system for . But if we define the projection operator by then the -partial controllability concepts of the system for become the same as the ordinary controllability concepts of the system for .
Example 2. Consider the nonlinear wave equation where is a real-valued function of two variables and . The state space of this system is (the space of square integrable functions on ). This system can be written as the first-order abstract differential equation if where . The state space of the system for is the enlargement of the state space of the system for . This is a cost that is paid to bring the wave equation to the form of first-order differential equation. The ordinary controllability concepts for the system (11) are too strong for the system (10). If then -partially controllability concepts of the system for become ordinary controllability concepts of the system for .
Example 3. Consider the system which contains a simple distributed delay in the nonlinear term, assuming that is a real-valued function. Then the state space is . To bring this system to a system without delay, enlarge to and define -valued function Then for the above system can be written as the abstract system Similar to the previous examples, one can easily observe that the ordinary controllability concepts for the system (17) are too strong for the system (14), but the -partial controllability concepts of the system for with are exactly the ordinary controllability concepts of the system for .
These examples motivate a study of the partial controllability concepts. In this paper it is proved that under the conditions (A)–(F), the system in (1) is -partially complete controllable.
3. Main Result
Denote . Then is a Banach space with the norm
Lemma 4. Under the conditions (A), (B), and (C),
Proof. It is easy to see that and for all . Hence, Then This implies . The conclusion of the lemma regarding follows from .
The proof of the following lemma appears in different forms in several papers, for example, Mahmudov . Our proof is a minor modification of them.
Lemma 5. Assume that the conditions (A)–(F) hold and take arbitrary . Then for the operator , defined by where the following inequality holds: where
Lemma 6. Under the conditions (A)–(F), if then the operator , mapping into , has a unique fixed point .
Proof. By Lemma 5, is a contraction mapping. Also, the space is complete. Hence, has a fixed point.
Proof. Take any and . Show that there is such that . To this end, consider , defined as follows: Substituting (31) in (4) and applying Fubini’s theorem (see Bashirov , p. 45), we obtain According to Lemma 6, there is a unique pair , satisfying (31) and (32). So, . Furthermore, we have Thus, there is which steers to with . This means that the semilinear system (1) is -partially complete controllable on as desired.
Remark 8. Decomposing in the form where and are other components of besides and is an orthogonal complement of in , one can calculate where and . Therefore, the coercivity of implies the same of. But, the converse is not true. Theorem 7 is powerful in the cases when is coercive, but is not.
Example 9. Theorem 7 establishes just sufficient condition of -partial complete controllability. In this example we will demonstrate that this is not a necessary condition. We will consider a simple case of when -partial complete controllability reduces to complete controllability. Consider the one-dimensional control system
This is a linear system, and the controllability operator of this system is equal to
According to the theory of controllability for linear systems, this system is controllable (completely) for every .
Have another look at this system by writing it as where Here, satisfies the Lipschitz condition with . Also, , implying and . Furthermore, So, . Then the inequality (30) becomes The limit of the left-hand side in this inequality when is equal to . This means that there is a sufficiently large such that the conditions of Theorem 7 do not hold for this , although the system under consideration is completely controllable. Thus, Theorem 7 states a sufficient condition which is not a necessary condition.
We demonstrate the features of -partial complete controllability in the following examples of control systems.
Example 1. Consider the system of differential equations
on , where . Besides the complete controllability property, that is,
we can investigate the partial complete controllability property, that is,
We can write this system in as the following semilinear system:
It can be calculated that
The controllability operator is
Hence, is not coercive, and the conditions for complete controllability, based on coercivity of , fail for this example. Although system (42) can still be complete controllable for properly selected functions , we can investigate the partial complete controllability for this system being interested in just the first component of .
Let . Then This means that the linear system associated with the semilinear system (45) is -partially complete controllable. Furthermore, the inequality (30) becomes or, simplifying, This establishes a relation between Lipschitz coefficient and terminal time moment . Depending on , must be taken sufficiently large to satisfy (53). So, the system (42) is -partially complete controllable for the time if the Lipschitz coefficient , related to , satisfies (53).
Example 2. Delay equations are typical for application of partial controllability concepts. Consider a nonlinear delay equation
on , where , , and .
Similar to Example 3, introduce the function by This function satisfies Denote by , , the semigroup generated by the differential operator and let be the integral operator from to , defined by noticing that . Then for we can write system (54) as where assuming that The semigroup, generated by , has the form Therefore, the controllability operator for system (54) can be calculated as This is definitely not a coercive operator.
Taking into account that the original system is given by (54) and (59) is just representation of (54) in the standard form, which enlarges the original state space to , we observe that the complete controllability for system (54) is in fact -partial complete controllability for system (59) if Calculating partial controllability operator, we obtain which is coercive.
Furthermore, using we write the inequality (30) in the form If the Lipschitz coefficient of the function and terminal time moment satisfy this inequality, then system (54) is completely controllable, which in turn means that system (59) is -partially complete controllable.
In this paper a sufficient condition for partial complete controllability of a semilinear control system is proved. This is a continuation of the pioneering research that has been done by Bashirov et al. [20, 21] about partial controllability concepts. A research in this way, concerning partial complete and approximate controllability for semilinear deterministic and stochastic systems, has already been done and awaiting for publication. There are other kinds of systems which besides semilinearity include other features, for example, impulsiveness, fractional derivative issue, and so forth. The result of this paper can be extended to these systems as well.
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