Controllability problem for fractional integrodifferential evolution systems of mixed type with the measure of noncompactness
© Qin et al.; licensee Springer. 2014
Received: 24 February 2014
Accepted: 10 July 2014
Published: 18 August 2014
We consider the controllability problem for a class of fractional evolution systems of mixed type in an infinite dimensional Banach space. The existence of mild solutions and controllability results are discussed by a new estimation technique of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. However, the main results do not need any restrictive conditions on estimated parameters of the measure of noncompactness. Since we do not assume that the semigroup is compact and other conditions are more general, the outcomes we obtain here improve and generalize many known controllability results. An example is also given to demonstrate the applications of our main results.
MSC:26A33, 34B15, 93B05, 93C25.
The concept of controllability was firstly introduced by Kalman in 1960. There has been a significant development in controllability results of systems represented by differential equations, integrodifferential equations, impulsive equations, differential inclusions, neutral differential equations and delay differential equations in Banach spaces. Most of the previous results require the assumption that the operator semigroup is compact. Using a compact analytic semigroup and a nonlinear alterative of Leray-Schauder type for multivalued maps due to O’Regan, Yan  established sufficient conditions for the controllability of fractional order partial neutral functional integrodifferential inclusion with infinite delay. Balachandran and Park  studied the controllability of fractional integrodifferential systems in Banach spaces, and an example with a compact analytic semigroup was also given. Based upon Bohnenblust-Karlin’s fixed point theorem and a compact semigroup, Chang  investigated a controllability result of mixed Volterra-Fredholm type integrodifferential inclusions in Banach spaces. Chalishajar  considered sufficient conditions for semilinear mixed Volterra-Fredholm type integrodifferential systems in a Banach space via a compact semigroup. Hernández and O’Regan  pointed out that the controllability results will be restricted to the finite dimensional space when the compactness of a semigroup and some other assumptions are satisfied. So, many researchers have tried to get sufficient conditions guaranteeing the controllability results of various systems without involving the compactness of a semigroup.
By using the fractional power of operators and Sadovskii’s fixed point theorem, they obtained the complete controllability of fractional neutral differential systems in an abstract space without involving the compactness of characteristic solution operators, but the main results require that the set is relatively in a Banach space for arbitrary and (see () in ).
in a Banach space. With the help of two new characteristic solution operators and their properties, such as compactness and boundedness, the controllability results for fractional evolution equations were obtained by the Schauder fixed point theorem. Later, researchers have always tried to avoid the compactness of a semigroup via the measure of noncompactness. Ji et al. considered the controllability of impulsive functional differential equations with nonlocal conditions by the measure of noncompactness and the Mönch fixed point theorem. Machado et al.  obtained the controllability results for a class of abstract impulsive mixed-type functional integrodifferential equations with finite delay in Banach spaces, sufficient conditions for controllability were obtained by the Mönch fixed point theorem via the measure of noncompactness and semigroup theory. Those results in [8, 9] do not assume the compactness of the evolution system, but restrictive conditions on the estimated parameter of the measure of noncompactness are required.
Chen and Li  studied a nonlocal problem for fractional evolution equations of mixed type in a Banach space, the existence of mild solutions and positive mild solutions was obtained by utilizing the measure of noncompactness and a new fixed point theorem.
Here , , and , .
In the present paper, we introduce a suitable concept of mild solutions for abstract control system (1.1). Under some necessary conditions on the characteristic solution operators and , we obtain the sufficient conditions of controllability results for system (1.1) when the operator , , is not compact. The methods we use are a new estimation technique of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. The main results do not require any restrictive conditions on estimated parameters of the measure of noncompactness, i.e., parameters () do not appear in inequality (3.1) and any other inequalities, which is the main difference between our study and the previous results, and also the main contribution of this paper.
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used later to prove our main results. In Section 3, we discuss the controllability results for system (1.1). At last, an example is provided to illustrate the theory in Section 4. Section 5 is a conclusion.
2 Preliminaries and lemmas
For the convenience of the readers, we shall recall here some necessary definitions from fractional calculus theory and some properties of the measure of noncompactness, one can refer to the monographs by Podlubny , Miller and Ross  and Deimling .
In this paper, we denote by a Banach space with the norm . Assume that is another Banach space, denotes the space of bounded linear operators from to . We also use to denote the norm of f whenever , . Let denote the Banach space of functions which are Bochner integrable normed by . represents a Banach space endowed with supnorm, i.e., for .
where is the Euler gamma function.
where the function has absolutely continuous derivatives up to order .
Definition 2.3 (see )
Lemma 2.1 (see )
where M1 is a positive constant to be specified later.
- (2)The operators () and () are strongly continuous. Therefore, for all and , one has(2.4)
if and only if is relatively compact.
Lemma 2.4 (see )
Lemma 2.5 (see )
Lemma 2.6 (see )
Lemma 2.7 (Fixed point theorem with respect to a convex-power condensing operator, see )
then Q has at least one fixed point in D.
Definition 2.4 The fractional system (1.1) is said to be controllable on the interval I if, for every , there exists a control such that a mild solution x of system (1.1) satisfies .
3 Main results
(H2) The nonlinearity satisfies the Carathéodory type conditions, that is, is strong measure for all , and is continuous for a.e. .
(H3) For , there exist constants and functions such that for a.e. and all satisfying ,Moreover, there exists a constant such that
(H4) There exist constants () such that for any bounded and countable sets () and a.e. ,
(H5) The nonlocal term is compact and continuous, there exist a constant and a nondecreasing continuous function such that, for some and all ,
(H6) The linear operator is bounded, defined byhas an inverse operator which takes values in , and there exist two positive constants such that
then the fractional evolution system (1.1) is controllable on I.
the operator Q has a fixed point, which is a mild solution of fractional evolution system (1.1). Note that and Definition 2.4, which means that system (1.1) is controllable on I.
Step 1. Q maps bounded sets into bounded sets.
By (3.1) and (3.4), we know that . Therefore, Q maps bounded sets into bounded sets.
Step 2. Q is continuous in .
Therefore, as , that is, Q is continuous.
Step 3. is equicontinuous.
It can be easily seen that , and tend to 0. Therefore, for any , as , which means that is equicontinuous.
Step 4. is a convex-power condensing operator.
Therefore is a convex-power condensing operator, Q has at least one fixed point, which is a mild solution of system (1.1). By Definition 2.4, system (1.1) is controllable on I. □
In order to obtain more controllability results, we replace conditions (H3) and (H5) by the following hypotheses:
for all and a.e. .
then the fractional evolution system (1.1) is controllable on I.
Using a similar method as in the previous proof, we can get that the fractional evolution system (1.1) is controllable on I. □
4 An example
has a bounded invertible operator defined by . If we can verify that all the conditions of Theorem 3.1 and inequality (3.1) are satisfied, the fractional control system (1.1) is controllable on , then the fractional control system (4.1) is controllable.
In this paper, using the uniform boundedness, equicontinuity of an operator semigroup and a fixed theorem with respect to a convex-power condensing operator, we have obtained the controllability of abstract fractional evolution control systems in a Banach space. It is well known that the compactness conditions of the operator semigroup can be weakened to equicontinuity. However, we have not only implemented a concrete assumption on the compactness condition via a parameter estimator but also removed the estimated parameter constraints; the sufficient conditions of controllability for various semilinear evolution systems are hence weakened. The conclusion of this paper is one of the most important developments in the aspect of imposing the necessary condition of controllability.
The authors thank the anonymous referees and handing editor of the manuscript for their valuable suggestions and fruitful comments.
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