Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type☆
Introduction
Let X be a separable reflexive Banach space and be its dual space. The purpose of this paper is to provide some sufficient conditions for the existence and controllability for the fractional evolution inclusions of Clarke’s subdifferential type. More precisely, we will study the problem having the following form:where denotes the Caputo fractional derivative of order q with the lower limit zero. is the infinitesimal generator of a uniformly bounded -semigroup on X. The control function u belongs to the space with U being a Banach space. Further, B is a bounded linear operator from U to X. is a given function to be specified later. denotes the Clarke’s subdifferential of . H is a bounded linear operator from to X.
As we know, the concept of controllability plays an important role in the analysis and design of control systems. Controllability of the deterministic and stochastic dynamical control systems in infinite-dimensional spaces is well developed by using different kinds of approaches, and the details can be found in various papers (see [2], [8], [18], [19], [24]). In recent years, the study of fractional differential equations and inclusions has acquired a great development (see [9], [10], [11], [13], [14], [23], [28], [29]). Many authors are devoted to the study of this field because it is useful to solve a lot of practical problems in applications. Now, many authors are devoted to the study of the controllability for fractional differential equations and inclusions. For more details, we refer to the works [25], [26], [27] and references therein.
However, the study for the controllability of the control systems described by the fractional evolution inclusions of Clarke’s subdifferential type is still untreated topic in the literature and this fact is the motivation of the present work. In fact, the Clarke’s subdifferential has important applications in mechanics and engineering, especially in nonsmooth analysis and optimization (see [4], [21]). The evolution inclusions with Clarke’s subdifferential type have been studied in many papers (see [7], [15], [16], [17], [20]).
In the study of the existence and controllability for fractional differential equations and inclusions, it is general to assume that the semigroup is compact. In this paper, by using the measure of noncompactness, we shall assume that the semigroup is strongly continuous instead of compact. Furthermore, by applying the knowledge of multivalued maps and the properties of Clarke’s subdifferential, we give the controllability results of the inclusions (1.1).
The rest of this paper is organized as follows. In the next section, we will introduce some useful preliminaries. In Section 3, some sufficient conditions are established for the existence result of mild solutions. In Section 4, we give the controllability of the system (1.1).
Section snippets
Preliminaries
Now, we introduce some basic preliminaries.
The norm of the Banach space X will be denoted by . stands for the set of nonnegative numbers. Denote be the Banach space of all continuous functions from J into X with the norm and be the Banach space of all Bochner integrable functions from J into X with the norm . is the infinitesimal generator of a uniformly bounded -semigroup on the Banach space X. Without
Existence result
In this section, we consider the existence of mild solutions to the problem with the following form:where denotes the Caputo fractional derivative of order q with the lower limit zero. is the infinitesimal generator of a uniformly bounded -semigroup on X. is a given function to be specified later. denotes the Clarke’s subdifferential of . H is a bounded linear operator from to X. Definition 3.1 A function
Controllability result
In this section we study the controllability of problem (1.1). Definition 4.1 A function is said to be a mild solution of problem (1.1) on the interval J ifwhere on .
To the readers’ convenience, we give the definition of controllability as follows. Definition 4.2 System (1.1) is said to be controllable on the interval J, if for every , there exists a control such that a mild solution x of system (1.1) satisfies
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Project supported by NNSF of China Grant Nos. 11271087, 61263006 and NSF of Guangxi Grant No. 2014GXNSFDA118002; the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118 and the National Science Center of Poland under Maestro Advanced Project No. UMO-2012/06/A/ST1/00262, the Innovation Project of Guangxi University for Nationalities No. gxun-chx2014098 and Special Funds of Guangxi Distinguished Experts Construction Engineering.