Approximate controllability for abstract measure differential systems
Introduction
The study of abstract measure differential systems was initiated by Sharma [1] in 1970s. From then on, although properties of abstract measure differential systems have been researched by various authors, only some limited results on abstract measure differential systems can be found, such as existence [1], [2], uniqueness [2], [3], and extremal solutions [2], [4], [5]. There was also a great deal of research on abstract measure integro-differential systems [6], [7]. The work on abstract measure differential systems is still rare.
Controllability is one of the primary problems in control theory. It has great industrial and theoretical meaning. Since the controllability notion has extensive industrial applications (see [8], [9] and references therein), there has been a great deal of research on it [10], [11], [12], [13], [14], [15], [16]. Actually, the research results on controllability were various. Respondek in [10] considered controllability with the cone type control constraints, which enabled to consider that some physical quantities cannot be negative e.g. forces. Authors in Refs. [11], [12] studied controllability with compact constraints, which can consider both the lower and upper limitations of the control. Schmitendorf and Barmish [11] researched controllability with compact constraints for ordinary differential equations, and Respondek [12] considered controllability with compact constraints for partial differential equations. Work done in [13], [14]showed numerical analysis on controllability. Otherwise, there was also much interest in approximate controllability problems for different kinds of systems described by differential equations, such as work done in [17], [18], [19], [20], [21]. The researchers in Refs. [17], [18], [19] studied the controllability of differential equations of an arbitrary order. However, to the best of our knowledge, there were not any results dealing with controllability or approximate controllability for abstract measure differential systems before. This is a problem of difficulty and challenge.
In this paper, we investigate the approximate controllability for abstract measure differential systems based on extending notions and results of approximate controllability for ordinary differential systems. We first introduce new concepts of reachable set and approximate controllability for abstract measure differential systems. Then based on the nonlinear alternative for -condensing mapping in Banach space, the sufficient conditions of approximate controllability for our system is derived. Additionally, applying our results of dealing with approximate controllability to study systems considered in [20], the corresponding results for approximate controllability are less conservative than those given in [20]. The work in the previous papers are improved.
The content of this paper is organized as follows. In Section 2, some preliminary fact is recalled; the abstract measure differential system, as well as two new concepts, based on extending knowledge for ordinary differential systems are proposed. In Section 3, the results of approximate controllability for abstract measure differential systems are given; an application of our method to deal with the neutral differential system in [20] is shown by two theorems, and a remark is given to analyze that our results are less conservative. In Section 4, an example is used to illustrate the effectiveness of our results for approximate controllability. In Section 5, conclusion is drawn for this paper.
Section snippets
Preliminaries
Definition 1.1 is a metric space, a mapping is called -condensing, if for any bounded subset of , is bounded and , .[22]
Remark 1.2 The in Definition 1.1 is called the Kuratowski measure of non-compactness, which is defined as : there exist finitely many sets of diameter at most which cover . Measures of non-compactness are useful in the study of infinite-dimensional Banach spaces. Any ball of diameter has in infinite-dimensional Banach space.
Lemma 1.3 Let and be, [22]
Main results
In order to study the approximate controllability of system (1), we consider the approximate controllability of its corresponding linear system as follows:
We give the following notations for convenience:
And denote the relevant operators as:
Consider the following assumptions.
- (A0)
System (2) is an
Example
Now we consider the following controlled neutral differential system: where is continuous.
Let and define as: with the domain and are absolutely continuous, . has the eigenvalues , and the corresponding eigenvectors are , with .
The following properties can be known.
(i) The operator generates a strong
Conclusions
In this paper, we have investigated approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems. We have first introduced the new concepts of the reachable set and approximate controllability for abstract measure differential systems. Then based on the nonlinear alternative for -condensing mapping in the Banach space, we have given sufficient conditions of approximate controllability for a class of abstract measure
Acknowledgments
This work is supported by the National Science Foundation of China under Grants 60874027 and 61174039. The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper.
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