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Mathematical Problems in Engineering
Volume 2017, Article ID 4715861, 15 pages
https://doi.org/10.1155/2017/4715861
Review Article

Controllability Problem of Fractional Neutral Systems: A Survey

Institute of Automatic Control, Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland

Correspondence should be addressed to Michał Niezabitowski; lp.lslop@ikswotibazein.lahcim

Received 4 August 2016; Revised 25 October 2016; Accepted 30 October 2016; Published 18 January 2017

Academic Editor: Leonid Shaikhet

Copyright © 2017 Artur Babiarz and Michał Niezabitowski. 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.

Abstract

The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces, controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, controllability for a class of fractional neutral integrodifferential equations with unbounded delay, controllability of neutral fractional functional equations with impulses and infinite delay, and controllability for a class of fractional order neutral evolution control systems.

1. Introduction

Controllability plays a very important role in various areas of engineering and science. In particular in control systems many fundamental problems of control theory, such as optimal control, stabilizability, or pole placement can be solved with assumption that the system is controllable [1, 2]. Controllability in general means that there exists a control function which steers the solution of the system from its initial state to a final state using a set of admissible controls, where the initial and final states may vary over the entire space. A standard approach is to transform the controllability problem into a fixed point problem for an appropriate operator in a functional space. There are many papers devoted to the controllability problem, in which authors used the theory of fractional calculus [313] and a fixed point approach [1423].

The subject of fractional calculus and its applications has gained a lot of importance during the past four decades. This was mainly because it has become a powerful tool in modeling several complex phenomena in numerous seemingly diverse and widespread fields such as engineering, chemistry, mechanics, aerodynamics, and physics [2432].

For infinite-dimensional systems two basic concepts of controllability can be distinguished: approximate and exact controllability, as in infinite-dimensional spaces there exist linear subspaces which are not closed. Approximate controllability enables steering the system to an arbitrarily small neighbourhood of final state. The second one, that is, exact controllability, means that system can be steered to arbitrary final state. From these definitions it is obvious that approximate controllability is essentially weaker notion than exact controllability. In the case of finite-dimensional systems notions of approximate and exact controllability coincide.

Many control systems arising from realistic models can be described as partial fractional differential or integrodifferential inclusions [3336]. In [37] authors present a new approach to obtain the existence of mild solutions and controllability results. For this purpose they avoid hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Author of [38] focuses on fractional evolution equations and inclusions. Moreover author presents their applications to control theory. The existence of solutions for fractional semilinear differential or integrodifferential equations has been studied by many authors [3943].

The impulsive differential systems can be used to model processes which are subject to sudden changes and which cannot be described by classical differential systems [44]. The controllability problem for impulsive differential and integrodifferential systems in Banach spaces has been discussed in [45]. Papers [46, 47] are devoted to the controllability of fractional evolution systems. The problem of controllability and optimal controls for functional differential systems has been extensively studied in many papers [4850].

1.1. Motivation

Controllability is one of the properties of dynamical systems that is continuously studied by control theory scientists. In case of infinite-dimensional systems there are many articles tackling this problem, in particular for approximate controllability, exact controllability, and relative controllability. This field can be divided based on the nature of controllability, but also on the basis of main equations describing a system of interest as well as the space in which the mathematical model is described. Additionally researchers frequently use different fixed point theorems for finding controllability conditions. That introduces high intricacy of problems which one can encounter during an analysis of a particular problem. The main purpose of this work is to perform a survey on the main types of equations describing dynamical systems based on a definition of a fractional order derivative. Additionally, as a result this work performs a systematization of knowledge in the field of controllability fractional systems, which by itself becomes a major discipline in the realm of control theory. This work shows schematics present in the analysis of controllability problems as well as points out which fixed point theorems are particularly useful.

2. Basic Notations

Let us introduce the following necessary notations.(i) is a Banach space.(ii) is a Hilbert space.(iii) is a bounded and closed interval.(iv) is a measurable function and Bochner integrable [51].(v) is the Hilbert space of all continuous functions from into with the norm .(vi) denotes the Hilbert space of bounded linear operators from to .(vii) is a Hilbert space.(viii) denotes the Hilbert space of measurable functions which are Bochner integrable normed by for all .(ix) is a space of all strongly measurable functions .(x) is the closed ball with centre at and radius in .(xi) denotes the class of all nonempty subsets of .(xii), , , and denote, respectively, the families of all nonempty bounded-closed, compact-convex, bounded-closed-convex, and compact-acyclic [52] subsets of .(xiii) is completely continuous.(xiv) is measurable multivalued map.(xv) is a measurable function on .(xvi) is a bounded linear operator from to .(xvii).(xviii)If is a uniformly bounded and analytic semigroup with infinitesimal generator such that then it is possible to define the fractional power , for , as a closed linear operator on its domain . Furthermore, the subspace is dense in and the expression defines a norm on . Hereafter we represent by the space endowed with the norm .(xix) is constant number such that .(xx) represents the Caputo derivative of order defined by where is the smallest integer greater than or equal to , is the gamma function, and .(xxi) and are the operator families defined by (xxii) are fixed points.(xxiii) and represent the right and left limits of at , respectively.(xxiv), , and are the operators defined by where denotes the adjoint of . Below we present definition of phase space.

Definition 1 (see [53]). Suppose that is a continuous function with . For all , one definesand equips the space with the norm , . Let us define the phase space If is endowed with the norm , then it is clear that is a Banach space.

Now we consider the spacewhere is the restriction of to . Set be a seminorm in defined by

Definition 2 (see [54]). Let be a metric space and . One will say that operator is a contraction if there exists some such that

Theorem 3 (Krasnoselskii’s fixed point theorem). Let be a bounded, closed, and convex subset of . Let be two mappings such that for every pair . If is a contraction and is completely continuous, then the operator equation has a solution on .

Then, the Banach fixed point theorem has the following form.

Theorem 4 ((Banach fixed point theorem) [54]). Let be a contraction on . Then, there exists a unique such that

3. Selected Problems of Controllability of Fractional Order Systems

In this section, we describe recent results of controllability problem of semilinear systems in infinite-dimensional spaces. The dynamical systems are expressed by different types of semilinear fractional order equations.

3.1. Approximate Controllability of Fractional Impulsive Partial Neutral Integrodifferential Inclusions with Infinite Delay in Hilbert Spaces

The authors of paper [55] derived a new set of sufficient conditions for the approximate controllability of fractional impulsive evolution system under the assumption that the corresponding linear system is approximately controllable. To do this they considered the approximate controllability of a class of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces of the formwhere(i) takes values in the Hilbert space ;(ii) is an initial condition;(iii);(iv), , are closed linear operators defined on a common domain which is dense in ;(v) is admissible control functions;(vi)the function defined by , belongs to some abstract phase space ;(vii) is a bounded, closed, convex-valued, multivalued map;(viii) is the family of all nonempty subsets of ;(ix), , and are functions subject to some additional conditions which will be given later.

In order to obtain theorem about existing of solutions and a new set of sufficient conditions for the approximate controllability of system (11) we recall few important definitions and present necessary conditions.

Definition 5. The set is called the reachable set of system (11) at terminal time . Its closure in is denoted by .

Definition 6. System (11) is said to be approximately controllable on the interval if .

Condition 1. The operator families and are compact for all , and there exist constants and such that and for every .

Condition 2. The function is continuous and there exists a such that

Condition 3. (i) For each the function is continuous and for each , the function is strongly measurable.
(ii) There exists a continuous function , such that for a.e. and , where is a continuous nondecreasing function.

Condition 4. The multivalued map ; for each , the function is upper semicontinuous and for each , the function is measurable; for each fixed , the set is nonempty.

Condition 5. There exists a continuous function and a continuous nondecreasing function such that for a.e. and each and with

Condition 6. The functions are continuous and there exist constants such that for every , .

Lemma 7 (see [56]). Let be a compact interval and be a Hilbert space. Let be a multivalued map satisfying Condition 4 and let be a linear continuous operator from to . Then the operator is a closed graph in .

Theorem 8 (see [55]). Suppose that Conditions 16 are satisfied and that, for all , system (11) has at least one mild solution on , provided that where , , , , and

Now we present the main result of paper [55] on the approximate controllability of system (11).

Theorem 9 (see [55]). Assume that assumptions of Theorem 8 hold and, in addition, there exists a positive constant such that and the linear system corresponding to system (11) is approximately controllable on . Then system (11) is approximately controllable on .

The proofs of the Theorems 8 and 9 presented in [55] are obtained with nonlinear alternative of Leray-Schauder type for multivalued maps [57].

3.2. Controllability of Nonlinear Neutral Fractional Impulsive Differential Inclusions in Banach Space

Controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space was investigated in paper [53]:where(i);(ii);(iii) is the infinitesimal generator of an analytic semigroup of the bounded linear operator in ;(iv) is a bounded, closed, convex-valued multivalued map;(v) are given functions;(vi) are bound functions.

The author of [53] used the following fixed point theorem.

Theorem 10 (see [58]). Let be a Banach space. and are two multivalued operators satisfying the following.(a) is a contraction.(b) is completely continuous.Then either(i)the operator inclusion has a solution for , or(ii)the set is unbounded.

Definition 11. A function is called a mild solution of system (22) if the following holds: on , ; the restriction of to the interval is continuous and the integral equation is satisfied, where where is probability density function defined on ; that is, , and .

The properties of the operators and can be found in [53].

In order to study the exact controllability of system (22), the following definition and conditions were made [53].

Definition 12 (see [53]). System (22) is said to be exactly controllable on the interval if for every continuous initial function, , , there exists a control such that the mild solution of (22) satisfies .

Condition 7. is the infinitesimal generator of an analytic semigroup of bounded linear operators and ; for , there exist constants such that .

Condition 8. The linear operator defined by has an induced inverse operator , which takes values in and there exist positive constants and such that and .

Condition 9. There exist constants such that is -valued and is continuous, and(i), ;(ii) , , with

Condition 10. There exists a constant such that , for each .

Condition 11. There exist an integrable function and a nondecreasing function such that for almost all and all .

Condition 12. There exists a positive constant such that where

Next theorem includes the condition for exact controllability of system (22) on the interval .

Theorem 13 (see [53]). If the Conditions 712 hold, then system (22) is controllable on the interval .

Based on a fixed point theorem (Theorem 10), sufficient conditions for the exact controllability of the fractional impulsive neutral functional differential inclusions have been obtained.

3.3. Approximate Controllability of Nonlocal Neutral Fractional Integrodifferential Equations with Finite Delay

In paper [59], authors obtain a set of sufficient conditions to prove the approximate controllability for a class of nonlocal neutral fractional integrodifferential equations, with time varying delays, considered in a Hilbert space.

They consider the following equation:where(i);(ii);(iii), and are continuous functions;(iv), , and are continuous and nonlinear functions; here , .Let be the state value of (29) at terminal time corresponding to the initial value and the control function . Define the set , which is called reachable set of the system (29) at time , and its closure in is denoted by .

Definition 14 (see [59]). The dynamical system (29) is called approximately controllable on if ; that is, for given , however small, it is possible to steer from the point to within a distance from all points in the state space at time .

Now, we introduce some conditions which will be used in presented results.

Condition 13.

Condition 14. The function satisfies that, for each , the function is strongly measurable in over the interval and there exists a positive constant such that for each

Condition 15. The function satisfies the following. (i)For any , the function is continuous, and for all , the function is strongly measurable.(ii)For each , there exist and , such that

Condition 16. is a continuous function and there exists a positive constant such that

For any and , we define a control aswhere satisfies the condition of a probability density function defined on ; that is, , , and ; and denote the adjoint of and , respectively.

For the sake of convenience, we introduce the following denotations: where , , , , and .

Theorem 15 (see [59]). Assume that the Conditions 1416 hold. System (29) corresponding to the control has a mild solution for each provided that

Theorem 16 (see [59]). Suppose that the Conditions 13, 14, 15, and 16 hold. Besides, one assumes additionally that the functions , , and are bounded and . Then the nonlocal neutral fractional integrodifferential equations with finite delay (29) are approximately controllable on .

Theorem 16 is proved by Krasnoselskii’s fixed point theorem.

3.4. Exact Controllability of Fractional Neutral Integrodifferential Systems with State-Dependent Delay in Banach Spaces

In paper [60] the authors execute Banach contraction fixed point theorem combined with resolvent operator to analyze the exact controllability results for fractional neutral integrodifferential systems with state-dependent delay in Banach spaces. Motivation to do it implies from their papers [6163]. In article [60] they study the controllability of mild solutions for a fractional neutral integrodifferential system with state-dependent delay of the modelwhere(i) is unknown and needs values in the Banach space having norm ;(ii);(iii) and are closed linear operators described on a regular domain which is dense in ;(iv) is a bounded linear operator from to ;(v), , , ; , and are apposite functions.If , , is continuous on and , then for every the accompanying conditions hold.(1) is .(2).(3), where is a constant and is continuous, is locally bounded, and and are independent of .(4)The function is well described and continuous from the set into and there is a continuous and bounded function to ensure that for every .

Recognize the space where is the constraint of to the real compact interval on . The function to be a seminorm in is described by

Definition 17. Let be the state value of model (37) at terminal time corresponding to the control and the initial value . Present the set , which is known as the reachable set of model (37) at terminal time .

Definition 18. Model (37) is said to be exactly controllable on if .

Now, according to the article [60] we will present the exact controllability results for the structure (37) under Banach fixed point theorem. First of all, we present the mild solution for model (37).

Definition 19 ([64], Definition ). A function is called a mild solution of (37) on , if ; ; the function is integrable on for all and for ;

Presently, we itemizing the subsequent conditions.

Condition 17. The operator families and are compact for all , and there exists a constant in a way that and for every and where symbolizes the Banach space of all bounded linear operators from into endowed with the uniform operator topology, having its norm recognized as .

Condition 18. The subsequent conditions are fulfilled.(a) for every .(b)There is a function , to ensure that

Condition 19. The function is continuous and one can find positive constants , , and in ways that, for all and , ,

Condition 20. is continuous and one can find constants and to ensure that, for all and , ;

Condition 21. The function is -values; is continuous and there exist positive constants , and such that, for all , ; , where

Condition 22. The following inequalities hold.(i)Let for some .(ii)Let be such that .

Theorem 20 (see [60]). Assume that the Conditions 1722 hold. Then, control system (37) is exactly controllable on .

Proof of the Theorem 20 is based on contraction mapping principle [60].

3.5. Controllability for a Class of Fractional Neutral Integrodifferential Equations with Unbounded Delay

The paper [65] focuses on establishing the sufficient conditions for the exact controllability for a class of fractional neutral integrodifferential equations with infinite delay in Banach spaces formulated as follows:where(i);(ii), for , are closed linear operators defined on a common domain which is dense in ;(iii) are appropriate functions. Some necessary notations for the above-mentioned system were presented in Basic Notations Section. The other ones are as follows.(i) is the domain of endowed with the graph norm.(ii) and are Banach spaces.(iii) stands for the Banach space of bounded linear operators from into endowed with the uniform operator topology. When then we will write .(iv) denotes the Laplace transform of for appropriate functions .(v) for a bounded function and ; shortly we will write when no confusion about the space arises.

In [65] the contraction mapping principle is used to formulate and prove conditions for exact controllability for the system (51). To obtain the exact controllability result the following lemmas and conditions were made [65].

Lemma 21. One can assume there exists such that and for all . Additionally, and are the constants. Moreover , , represent the supreme of the functions , and on , respectively.

Lemma 22 (see [66]). There exists a constant such that

Condition 23. The given conditions hold. (i).(ii)There is function , such that ,

Condition 24. The function is -valued, , the function is defined on , and there exist positive constants and such that for all the following inequalities are satisfied

Condition 25. The linear fractional control system defined asis exactly controllable.

In the next theorem we present conditions for exact controllability for the system (51).

Theorem 23 (see [65]). If Conditions 2325 and are satisfied, then control system (51) is exactly controllable on .

Theorem 23 is proved in [65] by using the contraction mapping.

Additionally, the authors of paper [65] study the exact controllability of the fractional neutral integrodifferential system with nonlocal condition of the following form: