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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 531894, 11 pages
Approximate Controllability of Fractional Neutral Evolution Equations in Banach Spaces
Eastern Mediterranean University, Gazimagusa, TRNC, 10 Mersin, Turkey
Received 24 December 2012; Revised 13 March 2013; Accepted 13 March 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 N. I. Mahmudov. 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.
We discuss the approximate controllability of semilinear fractional neutral differential systems with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using Krasnoselkii's fixed-point theorem, fractional calculus, and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional neutral differential equations with infinite delay are formulated and proved. The results of the paper are generalization and continuation of the recent results on this issue.
1. Introduction and Preliminaries
Many social, physical, biological and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see, e.g., Samko et al.  and references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.
Nowadays, controllability theory for linear systems has already been well established, for finite and infinite dimensional systems; see, for instance, . Several authors have extended these concepts to infinite-dimensional systems represented by nonlinear evolution equations in infinite-dimensional spaces, see [3–30]. On the other hand, approximate controllability problems for fractional evolution equations in Hilbert spaces are not yet sufficiently investigated, and there are only few works on it [13–21, 28–30]. So far, the overwhelming majority of the approximate controllability results have only been available for semilinear evolution differential systems in Hilbert spaces, with the exception of the case of . Motivated by the fact that many partial fractional differential equations can be converted into fractional PDE in some Banach spaces, we consider that there is a realistic need to discuss the approximate controllability problem of fractional-order differential systems in Banach spaces. Note that our results are new even for the approximate controllability of fractional neutral differential equations with infinite delay in Hilbert spaces.
Consider the following fractional neutral evolution differential system with infinite delay: where the state takes values in a Banach space and the control function takes values in a Hilbert space . The functions will be specified in the sequel. Let denote , . Assume that is a continuous function satisfying . The Banach space induced by function is defined as follows: endowed with the norm . It should be mentioned that (approximate) controllability results for first- and second-order partial neutral functional differential equations with infinite delay were considered by Sakthivel et al. , Chalishajar , and Chalishajar and Acharya .
Throughout this paper, unless otherwise specified, the following notations will be used. Let be a separable reflexive Banach space, and let stand for its dual space with respect to the continuous pairing . We may assume, without loss of generality, that and are smooth and strictly convex, by virtue of renorming theorem (see, e.g., ). In particular, this implies that the duality mapping of into given by the following relations: is bijective, homogeneous, and demicontinuous, that is, continuous from with a strong topology into with weak topology and strictly monotonic. Moreover, is also duality mapping.
In this paper, we also assume that is the infinitesimal generator of a compact analytic semigroup , , of uniformly bounded linear operator in , that is, there exists such that for all . Without loss of generality, let , where is the resolvent set of . Then, for any , we can define by It follows that each is an injective continuous endomorphism of . Hence, we can define , which is a closed bijective linear operator in . It can be shown that each has dense domain and that for . Moreover, for every , and with , where and is the identity in .
We denote by the Banach space of equipped with norm for , which is equivalent to the graph norm of . Then, we have , for (with ), and the embedding is continuous. Moreover, has the following basic properties.
Lemma 1 (see ). has the following properties:(i) for each and .(ii) for each and .(iii)For every , is bounded in , and there exists such that (iv) is a bounded linear operator for in .
From Lemma 1(iv), since is a bounded linear operator for , there exists a constant such that for .
Let us recall the following known definitions in fractional calculus. For more details, see .
Definition 2. The fractional integral of order with the lower limit 0 for a function is defined as provided that the right-hand side is pointwise defined on , where is the gamma function.
Definition 3. Riemann-Liouville derivative of order with the lower limit 0 for a function can be written as
Definition 4. The Caputo derivative of order for a function can be written as
For , we define two families and of operators by where is the function of Wright type defined on which satisfies
Lemma 5. The operators and have the following properties:(i)for any fixed and any , one has the operators and which are linear and bounded operators; that is, for any (ii)the operators and are strongly continuous for all ;(iii) and are norm continuous in for ;(iv) and are compact operators in for ;(v)for every , the restriction of to and the restriction of to are norm continuous;(vi)for every , the restriction of to and the restriction of to are compact operators in ;(vii)for all and ,
In this paper, we adopt the following definition of mild solution of (1).
Definition 6. A solution is said to be a mild solution of (1) if for any the integral equation is satisfied.
Let be the state value of (14) at terminal time corresponding to the control . Introduce the set , which is called the reachable set of the system (14) at terminal time , and its closure in is denoted by .
Definition 7. The system (1) is said to be approximately controllable on if ; that is, given an arbitrary , it is possible to steer from the point to within a distance from all points in the state space at time .
To investigate the approximate controllability of the system (14), we assume the following conditions.(H1) is the infinitesimal generator of an analytic semigroup of bounded linear operators in , , is compact for , and there exists a positive constant such that .(H2)The function is continuous, and there exists some constant , such that is -valued and (H3)The function satisfies the following:(a) is continuous for each and for each is strongly measurable;(b) there is a positive integrable function and a continuous nondecreasing function such that for every , we have (H4)The following relationship holds: Here, .(Hac)For every converges to zero as in strong topology, where and is a solution of the equation
Let set be a seminorm defined by
Lemma 8 (see ). Assume that , then for all and
2. Existence Theorem
In order to formulate the controllability problem in the form suitable for application of fixed-point theorem, it is assumed that the corresponding linear system is approximately controllable. Then, it will be shown that the system (1) is approximately controllable if for all there exists a continuous function such that where Having noticed this fact, our goal in this section is to find conditions for solvability of (23). Note that it will be shown that the control in (23) drives the system (1) from to provided that the system (23) has a solution.
Theorem 9. Assume that assumptions (H1)–(H4) hold and . Then, there exists a solution to (23).
For all , consider the operator defined as follows: It will be shown that for all , the operator has a fixed point.
Suppose that , where is taken as for , while for , it is defined as . Set For any , we have Thus, is a Banach space. For each positive number , set It is clear that is bounded closed convex set in . For any , we see that Consider the maps defined by Obviously, the operator has a fixed point if and only if operator has a fixed point. In order to prove that has a fixed point we will employ the Krasnoselskii’s fixed-point theorem.
Lemma 10. Under assumptions (H1)–(H4), for any , there exists a positive number such that for all .
Proof. Let be fixed. If it is not true, then for each , there are functions , but . So for some , one can show that Let us estimate . By the assumption (H2), we have Using Lemma 5 and the Hölder inequality, one can deduce that Using the assumption (H3), one has Combining the estimates (32)–(36) yields On the other hand, Thus, Dividing both sides of (39) by and taking , we obtain that which is a contradiction by assumption (H4). Thus, for some .
Lemma 11. Let assumptions (H1)–(H4) hold. Then, is contractive.
Proof. Let . Then, Hence, where we have used the fact that . Thus, so is a contraction by assumption (H4).
Lemma 12. Let assumptions (H1)–(H4) hold. Then, maps bounded sets to bounded sets in .
Proof. By the similar argument as Lemma 10, we obtain which implies that .
Lemma 13. Let assumptions (H1)–(H4) hold. Then, the set is an equicontinuous family of functions on .
Proof. Let and such that for every with . For , , we have Applying (38) and the Hölder inequality, we obtain Therefore, for sufficiently small, the right-hand side of (47) tends to zero as . On the other hand, the compactness of , implies the continuity in the uniform operator topology. Thus, the set is equicontinuous.
Lemma 14. Let assumptions (H1)–(H4) hold. Then, maps onto a precompact set in .
Proof. Let be fixed and be a real number satisfying . For , define an operator on by Since , is a compact operator, the set is precompact in for every , . Moreover, for each , we have A similar argument as before where we have used the equality Form (49) to (50), one can see that for each , Therefore, there are relatively compact sets arbitrary close to the set . Hence, the set is also precompact in .
3. Main Results
Consider the following linear fractional differential system: The approximate controllability for the linear fractional system (53) is a natural generalization of approximate controllability of linear first-order control system. It is convenient at this point to introduce the controllability operator associated with (53) as respectively, where denotes the adjoint of and is the adjoint of . It is straightforward that the operator is a linear-bounded operator for .
Theorem 15 (see ). The following three conditions are equivalent: (i) is positive, that is, for all nonzero .(ii)For all converges to the zero as in the weak topology, where is a solution of the equation (iii)For all converges to the zero as in the strong topology.
Theorem 17 (see ). Let be a nonlinear operator. Assume that is a solution of the following equation: Then, there exists a subsequence of the sequence strongly converging to zero as .
We are now in a position to state and prove the main result of the paper.
Theorem 18. Let . Suppose that conditions (H1)–(H4) and (Hac) are satisfied. Besides, assume additionally that and is continuous from the weak topology of to the strong topology of . There exists such that
Then, the system (1) is approximately controllable on .
Proof. Let be a fixed point of in . Then, is a mild solution of (1) on under the control and satisfies the following equality: In other words, is a solution of the equation It follows that On the other hand, by (Hub), From (63) and (64), it follows that weakly as and by the assumption (Hgc) strongly as . Moreover, because of assumption (Hub), Consequently, the sequences are bounded. Then, there is a subsequence still denoted by which weakly converges to, say, in . Then, where as because of compactness of an operator . Then, by Theorem 17 as . This gives the approximate controllability. The theorem is proved.
Remark 19. Theorem 18 assumes that the operator generates a compact semigroup. If the compactness condition holds on the bounded operator that maps the control function or the generated -semigroup, then the controllability operator is also compact, and its inverse does not exist if the state space is infinite dimensional, and, consequently, the associated linear control system (53) is not exactly controllable. Therefore, the concept of complete controllability is too strong in infinite dimensional spaces, and the approximate controllability notion is more appropriate. Thus, Theorem 18 has no analogue for the concept of complete controllability.
In this section, we illustrate the obtained result. Let , and let be defined as follows: with domain Recall that is the infinitesimal generator of a strongly continuous semigroup , on which is analytic compact and self-adjoint, and the eigenvalues are , , with corresponding normalized eigenvectors and Moreover, the following hold.(a) is an orthonormal basis of .(b)If , then .(c)For , .(d)The operator is given as on the space .
For , consider the neutral system where is continuous functions. is a linear continuous mapping from to as follows:
To write the initial-boundary value problem (72) in the abstract form, we assume the following.(A1)The function is measurable and (A2)The function is measurable, , and let
From (A1), it is clear that is bounded linear operator on . Furthermore, , and . In fact, from the definition of and (A2), it follows that where . From (A2), we know that is a bounded linear operator with . Hence, , which implies the assertion. Moreover, assume that and satisfy conditions of Theorem 18. Thus, the problem (72) can be written in the abstract form Now, consider the associated linear system Show that it is approximately controllable on for . It is easy to see that if , then It follows that , and consequently, , which means that (79) is approximately controllable on . Therefore, from Theorem 18, the system (72) is approximately controllable .
In this paper, abstract results concerning the approximate controllability of fractional semilinear evolution systems with infinite delay in a separable reflexive Banach space are obtained. Approximate controllability result for semilinear systems is obtained by means of the Krasnoselskii’s fixed-point theorem under the compactness assumption. It is also proven that the controllability of the semilinear system is implied by the approximate controllability of the associated linear system under some natural conditions. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic evolution equations.
N. I. Mahmudov is thankful to the reviewers for making valuable suggestions leading to the better presentation of this paper.
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