Controllability Problem of Fractional Neutral Systems : A Survey

The following article presents recent results of controllability problemof 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.


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 [3][4][5][6][7][8][9][10][11][12][13] and a fixed point approach [14][15][16][17][18][19][20][21][22][23].
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 [24][25][26][27][28][29][30][31][32].
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 [33][34][35][36].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 [39][40][41][42][43].
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 2 Mathematical Problems in Engineering 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 [48][49][50].
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.

Basic Notations
Let us introduce the following necessary notations.
(iii)  is a bounded and closed interval.
(vi) () denotes the Hilbert space of bounded linear operators from  to .
(vii)  is a Hilbert space.
(x)   (, ) is the closed ball with centre at  and radius  > 0 in .
(xi) P() denotes the class of all nonempty subsets of .
Below we present definition of phase space.

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.

Mathematical Problems in Engineering
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) for a.e. ∈  and each  ∈  and  ∈  with Condition 6.The functions   : B ℎ →  are continuous and there exist constants   such that lim sup for every  ∈ B ℎ ,  = 1, . . ., .
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  1 (, ) to (, ).Then the operator is a closed graph in (, ) × (, ).
Theorem 8 (see [55]).Suppose that Conditions 1-6 are satisfied and that, for all  > 0, system (11) has at least one mild solution on , provided that where 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 C 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].
The author of [53] used the following fixed point theorem.
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,  ∈ B ℎ ,  1 ∈ , there exists a control  ∈  2 (, ) such that the mild solution () of ( 22 Condition 9.There exist constants 0 ⩽  < 1,  0 ,  1 ,  2 ,   such that  is   -valued and (−)   is continuous, and where Next theorem includes the condition for exact controllability of system (22) 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.

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: Let (, , ) be the state value of (29) at terminal time  corresponding to the initial value and the control function .Define the set (, ) = {(, , ) :  ∈  2 (, )}, 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  > 0, 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.

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 [61][62][63].
In article [60] they study the controllability of mild solutions for a fractional neutral integrodifferential system with statedependent delay of the model where (i) (⋅) is unknown and needs values in the Banach space  having norm ‖ ⋅ ‖; (ii)  ∈ (1, 2); (iii)  and (()) ≥0 are closed linear operators described on a regular domain which is dense in (, ‖ ⋅ ‖); (iv)  is a bounded linear operator from  to ; If  : (−∞, ] → ,  > 0, is continuous on  and  0 ∈ B ℎ , then for every  ∈  the accompanying conditions hold. (1)   is B ℎ .
Recognize the space where |  is the constraint of  to the real compact interval on .The function ‖ ⋅ ‖ B  to be a seminorm in B  is described by Definition 17.Let   (; ) be the state value of model (37) at terminal time  corresponding to the control  and the initial value  ∈ B ℎ .Present the set R(, ) = {  (; )(0) : (⋅) ∈  2 (, )}, which is known as the reachable set of model (37) at terminal time .
Definition 18. Model ( 37) is said to be exactly controllable on J if R(; ) = .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).
where L() symbolizes the Banach space of all bounded linear operators from  into  endowed with the uniform operator topology, having its norm recognized as ‖ ⋅ ‖ L() .
Condition 18.The subsequent conditions are fulfilled.(b) There is a function (⋅) ∈  1 (, R + ), to ensure that where Condition 22.The following inequalities hold.
Proof of the Theorem 20 is based on contraction mapping principle [60].

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: (ii) , (), for  ⩾ 0, are closed linear operators defined on a common domain D = () which is dense in ; (iii) ,  : [0, ] × B ℎ →  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.
(iii) L(, ) stands for the Banach space of bounded linear operators from  into  endowed with the uniform operator topology.When  =  then we will write L().
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 22 (see [66]).There exists a constant  such that      (−)       ⩽  for 0 ⩽  ⩽ 1. (52) Condition 23.The given conditions hold.
(ii) There is function ( Condition 25.The linear fractional control system defined as is exactly controllable. In the next theorem we present conditions for exact controllability for the system (51).
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: where 0 <  The next theorem includes the required conditions for system (57) to be exactly controllable.
Theorem 24 (see [65]).Assume that the conditions of Theorem 23 As before, the proof of Theorem 24 is led by contraction mapping.

Controllability of Neutral Fractional Functional Equations
with Impulses and Infinite Delay.Authors of [67] investigate the exact controllability of a class of fractional order neutral integrodifferential equations with impulses and infinite delay in the following form: (iv)   (; ) is the state value.
To formulate a set of sufficient conditions for exact controllability of system (61) Conditions for exact controllability of the fractional impulsive system (61) on  are the content of the next theorem.
Theorem 25 (see [67]).If the Conditions 25 and 27-30 are satisfied and there exists  > 0, then fractional impulsive system (61) is exactly controllable on  provided that where Moreover, in paper [67], the approximate controllability of system (61) was discussed too and the results are presented below.
Theorem 26 (see [67]).Assume that Conditions 27-30 hold and that the family {  () :  > 0} is compact.In addition, assume that the function  is uniformly bounded and the linear system (54) associated with the system ( 61) is approximately controllable; then the nonlinear fractional control system with infinite delay ( 61) is approximately controllable on [0, ].
Theorems 25 and 26 are proved in [67] by contraction mapping theorem.

Controllability for a Class of Fractional Order Neutral
Evolution Control Systems.In [68], authors study the exact controllability of fractional control systems with states and controls in Hilbert spaces.Their investigations were started from fractional nonlinear neutral functional differential equation described as follows: Some necessary notations for the above-mentioned system were presented in Basic Notations Section.The other ones are as follows.
Next conditions [68] are necessary to present conditions for exact controllability for the nonlinear fractional control system (67) by using the contraction mapping principle.
Then, the following theorem is true.
Additionally the authors assumed that function  satisfies the below-presented conditions.(75) Necessary conditions for the controllability of nonlinear systems are established in the following theorem.
Theorem 28 (see [68]).If the conditions of Theorem 27 and Conditions 35 and 36 are satisfied, then fractional system (67) with nonlocal condition ( 73) is exactly controllable on .
Theorems 27 and 28 are proved by contraction mapping theorem.

Conclusions
The presented paper focuses on the controllability problem of different types of dynamical systems described with fractional order equation.Precisely, the paper presents the results for the selected works from the scope of the investigated controllability of fractional semilinear dynamical systems.Generally speaking, at the beginning, we prove that the semilinear system is controllable if the associated linear system is controllable, too.Next, we pose some conditions for the semilinear dynamical system.The main role is the assumption about Lipschitz continuity.After scrutinizing we observed a research methodology, which is used to solve the controllability problem, not only approximately but also exactly.Below is presented the methodology resulting from indepth analysis of the papers concerning the controllability of nonlinear systems: The controllability problems for dynamical systems require the application of various mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, and matrix analysis.It should be noticed that there are many unsolved problems for controllability concepts for different types of dynamical systems.The methodology presented in this paper may well be used in a research on controllability of stochastic dynamical systems [69], in a search of optimal control [70,71], for systems with constraints on control signal [11], and for dynamical systems with delay in state and control [12,72].

( 1 )( 3 ) 6 )
Showing a mathematical model of dynamical system (2) Formulation of the assumptions concerning dynamical systems Proof of solution existence of state space equation using the fixed point theorem or generally fixed point technique (4) Proposition of a control transferring the initial state to some neighbourhood of final state (5) Formulation theorem containing necessary conditions of controllability (Proof of the above-mentioned theorem (xiv)  :  → P ,,V () 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 semigroupwith infinitesimal generator  such that 0 ∈ () then it is possible to define the fractional power (−)  , for 0 <  ⩽ 1, as a closed linear operator on its domain ((−)  ).Furthermore, the subspace ((−)  ) is dense in  and the expression  ).Hereafter we represent by   the space ((−)  ) endowed with the norm ‖ ⋅ ‖  .