# Controllability Problem of Fractional Neutral Systems: A Survey

Authors: Artur Babiarz; Michał Niezabitowski
Journal: Mathematical Problems in Engineering
 (2017)

---

## 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.


---

## Body

## 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 [3–13] and a fixed point approach [14–23].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 [24–32].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 [33–36]. 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 [39–43].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 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 [48–50].


### 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.


## 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)
(X,·) is a Banach space.(ii)
H,· is a Hilbert space.(iii)
J is a bounded and closed interval.(iv)
x:J→H is a measurable function and Bochner integrable [51].(v)
CJ,H is the Hilbert space of all continuous functions from J into H with the norm x∞=sup⁡xt:t∈J.(vi)
LH denotes the Hilbert space of bounded linear operators from H to H.(vii)
U is a Hilbert space.(viii)
L1J,H denotes the Hilbert space of measurable functions x:J→H which are Bochner integrable normed by xL1=∫Jxtdt for all x∈L1J,H.(ix)
L2(J,U) is a space of all strongly measurable functions u:J→U.(x)
Brx,H is the closed ball with centre at x and radius r>0 in H.(xi)
PH denotes the class of all nonempty subsets of H.(xii)
Pbd,clH, Pcp,cvH, Pbd,cl,cvH, and PcdH denote, respectively, the families of all nonempty bounded-closed, compact-convex, bounded-closed-convex, and compact-acyclic [52] subsets of H.(xiii)
F is completely continuous.(xiv)
G:J→Pbd,cl,cvH is measurable multivalued map.(xv)
t↦Dx,Gt is a measurable function on J.(xvi)
B is a bounded linear operator from U to H.(xvii)
MB=B.(xviii)
IfT is a uniformly bounded and analytic semigroup with infinitesimal generator A such that 0∈ρ(A) then it is possible to define the fractional power (-A)α, for 0<α⩽1, as a closed linear operator on its domain D((-A)α). Furthermore, the subspace D((-A)α) is dense in X and the expression (1)xα≔-Aαx,x∈D-Aαdefines a norm onD((-A)α). Hereafter we represent by Xα the space D((-A)α) endowed with the norm ·α.(xix)
M is constant number such that Tt⩽M.(xx)
cDtαξt=∫0t‍gn-αt-s represents the Caputo derivative of order α>0 defined by (2)Dtαcξt=∫0tgn-αt-sdndsnξt-sds,wheren is the smallest integer greater than or equal to α, Γ· is the gamma function, and gβt≔tβ-1/Γβ,t>0,β≥0.(xxi)
Rαt and Sαt are the operator families defined by (3)Rαt≔12πi∫Γr,θeλtGαλdλfort>0,Ifort=0,Sαtx≔∫0tgα-1t-sRαsdsforα∈1,2andeacht≥0.(xxii)
0<t1<⋯<tm<b are fixed points.(xxiii)
xtk- and xtk+ represent the right and left limits of xt at t=tk, respectively.(xxiv)
Γτb, Γtk-1tk, and Ra,Γtk-1tk are the operators defined by (4)Γτb=∫τbSαb-sBB∗Sα∗b-sds,0≤τ≤b,Γtk-1tk=∫tk-1tkSαtk-sBB∗Sα∗tk-sds,k=1,2,…,m,m+1,Ra,Γtk-1tk=aI+Γtk-1tk-1fora>0,k=1,2,…,m,m+1,whereB∗ denotes the adjoint of B. Below we present definition of phase space.Definition 1 (see [53]).
Suppose thath:(-∞,0]→(0,∞) is a continuous function with l=∫-∞0‍h(t)dt<∞. For all a>0, one defines(5)B=ψ:-a,0⟶Xsuchthatψtis  bounded  and  measurableand equips the space B with the norm ψ[-a,0]=sups∈[-a,0]ψ(s), ∀ψ∈B. Let us define the phase space (6)Bh=ψ:-∞,0⟶Xsuchthat,foranyc>0,ψ-c,0∈B,∫-∞0hsψs,0ds<∞.If Bh is endowed with the norm ψBh=∫-∞0‍hsψ[s,0]ds,∀ψ∈B, then it is clear that (Bh,·Bh) is a Banach space.Now we consider the space(7)Bb=ψ:-∞,b⟶Xsuchthatxk∈CJk,Xand  there  existxtk+,xtk-withxtk=xtk-,x0=ϕ∈Bh,k=0,1,…,m,where xk is the restriction of x to Jk=(tk,tk+1],k=0,1,…,m. Set ·b be a seminorm in Bb defined by (8)xb=x0Bh+sup⁡xs:s∈0,b,x∈Bb.Definition 2 (see [54]).
LetX,d be a metric space and F:X→X. One will say that operator F is a contraction if there exists some k∈0,1 such that(9)⋀x,y∈XdFx,Fy≤kdx,y.Theorem 3 (Krasnoselskii’s fixed point theorem).
LetΩ be a bounded, closed, and convex subset of X. Let F1,F2:Ω→X be two mappings such that F1x+F2y∈Ω for every pair x,y∈Ω. If F1 is a contraction and F2 is completely continuous, then the operator equation F1x+F2x=x has a solution on Ω.Then, the Banach fixed point theorem has the following form.Theorem 4 ((Banach fixed point theorem) [54]).
LetF be a contraction on X. Then, there exists a unique x0∈X such that(10)Fx0=x0.


## 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 form(11)DαcNxt∈ANxt+∫0tQt-sNxsds+Bus+Ft,xt,∫0tht,x,xsds,t∈J=0,b,t≠tk,Δxtk=Ikxtk,k=1,…,m,x0∈φ∈Bh,x′0=0,where(i)
x· takes values in the Hilbert space H;(ii)
φ is an initial condition;(iii)
α∈1,2;(iv)
A, Qtt≥0, are closed linear operators defined on a common domain which is dense in (H,·);(v)
u∈L2(J,U) is admissible control functions;(vi)
the functionxt:-∞,0→H defined by xtθ=xt+θ, θ∈(-∞,0] belongs to some abstract phase space Bh;(vii)
F:J×Bh×H→PH is a bounded, closed, convex-valued, multivalued map;(viii)
PH is the family of all nonempty subsets of H;(ix)
G:J×Bh→H, Nψ=ψ0+G(t,ψ),ψ∈Bh, and Ik:Bh→H(k=1,…,m) 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(12)Bhb,x0=xbx0;u0:u·∈L2J,Uis called the reachable set of system (11) at terminal time b. Its closure in H is denoted by Bhb,x0¯.Definition 6.
System (11) is said to be approximately controllable on the interval 0,b if Bhb,x0¯=H.Condition 1.
The operator familiesRαt and Sαt are compact for all t>0, and there exist constants M and δ such that RαtLH≤Meδt and SαtLH≤Meδt for every t∈J.Condition 2.
The functionG:J×Bh→H is continuous and there exists a L>0 such that (13)Gt,ψ1-Gt,ψ2≤Lt1-t2+ψ1-ψ2Bh,t1,t2∈J,ψ1,ψ2∈Bh,Gt,ψ≤LψBh+1,t∈J,ψ∈Bh.Condition 3.
(i) For eacht,s∈Λ the function ht,s,·:Bh→H is continuous and for each x∈Bh, the function h·,·,x:Λ→H is strongly measurable.
(ii) There exists a continuous functionp:Λ→0,∞, such that (14)ht,s,ψ≤pt,sΘ0ψBhfor a.e. t,s∈J and ψ∈Bh, where Θ0:0,∞→0,∞ is a continuous nondecreasing function.Condition 4.
The multivalued mapF:J×Bh×H→Pbd,cl,cvH; for each t∈J, the function Ft,·,·:Bh×H→Pbd,cl,cvH is upper semicontinuous and for each ψ,y∈Bh×H, the function F·,ψ,y is measurable; for each fixed ψ,y∈Bh×H, the set (15)SF,ψ=f∈L1J,H:ft∈Ft,ψ,∫0tht,s,ψdsfor  a.e.t∈Jis nonempty.Condition 5.
There exists a continuous functionm:J→0,∞ and a continuous nondecreasing function Θ:0,∞→0,∞ such that (16)Ft,ψ,y=sup⁡f:f∈Ft,ψ,y≤mtΘψBh+y,for a.e. t∈J and each ψ∈B and y∈H with (17)∫1∞1s+Θs+Θ0sds=∞.Condition 6.
The functionsIk:Bh→H are continuous and there exist constants ck such that (18)limsupψBh→∞⁡IkψψBh=ckfor every ψ∈Bh, k=1,…,m.Lemma 7 (see [56]).
LetJ be a compact interval and H be a Hilbert space. Let F be a multivalued map satisfying Condition 4 and let P be a linear continuous operator from L1J,H to CJ,H. Then the operator (19)P∘SF:CJ,H⟶Pcp,cvH,x⟶P∘SFx≔PSF,xis a closed graph in CJ,H×CJ,H.Theorem 8 (see [55]).
Suppose that Conditions1–6 are satisfied and that, for all a>0, system (11) has at least one mild solution on J, provided that (20)max1≤k≤m⁡M21+KbMck+ML+M3KbML<1,where M2=MN∗1+(1/a)M∗2N∗2M12b, M3=1+(1/a)MM∗N∗M12bN∗, M∗=Mmax⁡1,eδb, N∗=max⁡1,e-δb, and M1=B.Now we present the main result of paper [55] on the approximate controllability of system (11).Theorem 9 (see [55]).
Assume that assumptions of Theorem8 hold and, in addition, there exists a positive constant C~ such that (21)Ft,ψ,y=sup⁡f:f∈Ft,ψ,y≤C~,t,ψ,y∈J×Bh×Hand the linear system corresponding to system (11) is approximately controllable on J. Then system (11) is approximately controllable on J.The proofs of the Theorems8 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]:(22)Dtαcxt-gt,xt∈Axt+Ft,xt+But,t∈J=0,b,t≠tk,k=1,2,…,m,Δxt=tk=Ikxtk-,k=1,2,…,m,x0=ϕ∈Bh,t∈J0=-∞,0,where(i)
α∈(0,1);(ii)
x(·)∈X;(iii)
A is the infinitesimal generator of an analytic semigroup of the bounded linear operator {T(t),t⩾0} in X;(iv)
F:J×Bh→P(X) is a bounded, closed, convex-valued multivalued map;(v)
g:J×Bh→X are given functions;(vi)
Ik∈C(X,X)(k=1,2,…,m) are bound functions.The author of [53] used the following fixed point theorem.Theorem 10 (see [58]).
LetX be a Banach space. Φ1:X→Pcl,cv,bd(X) and Φ2:X→Pcp,cv(X) are two multivalued operators satisfying the following.(a)
Φ1 is a contraction.(b)
Φ2 is completely continuous.
Then either(i)
the operator inclusionλx∈Φ1x+Φ2x has a solution for λ=1, or(ii)
the setG={x∈X:λx∈Φ1x+Φ2x,λ>1} is unbounded.Definition 11.
A functionx:(-∞,b]→X is called a mild solution of system (22) if the following holds: x0=ϕ∈Bh on (-∞,0], Δx|t=tk=Ik(x(tk-)),k=1,2,…,m; the restriction of x(·) to the interval [0,b)-{t1,t2,…,tm} is continuous and the integral equation (23)xt=Sαtϕ0-g0,ϕ+gt,xt+∫0tt-sα-1ATαt-sgs,xsds+∫0tt-sα-1Tαt-sfsds+∫0tt-sα-1Tαt-sBusds+∑0<tk<tSαt-tkIkxtk-,t∈J,x0=ϕ∈Bh,t∈J0is satisfied, where (24)f∈SF,x=f∈L1J,X:ft∈Ft,xt,for  a.e.t∈J,Sαt=∫0∞ξαθTtαθdθ,Tαt=α∫0∞θξαθTtαθdθ,ξαθ=1αθ-1-1/αϖαθ-1/α⩾0,ϖαθ=1π∑n=1∞-1n-1θ-nα-1Γnα+1nsin⁡nπα,θ∈0,∞,where ξα is probability density function defined on (0,∞); that is, ξα(θ)⩾0,θ∈(0,∞), and ∫0∞‍ξα(θ)dθ=1.The properties of the operatorsSα(t) and Tα(t) 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 J if for every continuous initial function, ϕ∈Bh, x1∈X, there exists a control u∈L2(J,U) such that the mild solution x(t) of (22) satisfies x(b)=x1.Condition 7.
A is the infinitesimal generator of an analytic semigroup of bounded linear operators T(t) and 0∈ρ(A); for t⩾0, there exist constants M such that Tt⩽M.Condition 8.
The linear operatorW:L2(J,U)→X defined by (25)Wu=∫0bb-sα-1Tb-sBusdshas an induced inverse operator W-1, which takes values in L2(J,U)/kerW and there exist positive constants M2 and M3 such that B⩽M2 and W-1⩽M3.Condition 9.
There exist constants0⩽β<1,c0,c1,c2,Lg such that g is Xβ-valued and (-A)βg is continuous, and(i)
-Aβg(t,x)⩽c1xBh+c2, (t,x)∈J×Bh;(ii)
-Aβg(t,x1)--Aβg(t,x2)⩽Lgx1-x2Bh,(t,xi)∈J×Bh, i=1,2, with (26)C0=Lgl-A-β+c1-βΓ1+βbαββΓ1+αβ<1.Condition 10.
There exists a constantdk such that Ikx⩽dk, k=1,2,…,m for each x∈X.Condition 11.
There exist an integrable functionp:J→[0,∞) and a nondecreasing function ψ:R+→(0,∞) such that F(t,x)=sup⁡f:ft∈Ft,x⩽p(t)ψ(xBh) for almost all t∈J and all x∈Bh.Condition 12.
There exists a positive constantr such that (27)rF1+F2r+F3ψlr+ϕBh+lMϕ0>1,where (28)F1=K1+MM2M3bαΓ1+α·x1+Mϕ0+K1,F2=K2+MM2M3bαΓ1+α·K2,F3=1+MM2M3bαΓ1+αbαMΓ1+αsups∈J⁡ps,K1=M-A-βc1ϕBh+c2+-A-β+c1-βΓ1+βΓ1+αβ·bαββc1ϕBh+c1lMϕ0+c2+M∑k=1mdk,K2=-A-βc1l+c1-βΓ1+βΓ1+αβ·bαββc1l.Next theorem includes the condition for exact controllability of system (22) on the interval J.Theorem 13 (see [53]).
If the Conditions7–12 hold, then system (22) is controllable on the interval J.Based on a fixed point theorem (Theorem10), 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:(29)Dcαxt+ht,xt-kt=-Axt+It1-αft,xt-υt+But,t∈J=0,b,xt=ϕt+gxt,t∈-a,0,where(i)
a>0;(ii)
α∈(0,1);(iii)
k,υ:[0,+∞)→(0,a], and (a>0) are continuous functions;(iv)
f:[0,+∞)×X→X, h:[0,+∞)×X→Xα, and g:C→C([a,0],X) are continuous and nonlinear functions; here 0<q≤1, C≔C([a,b],X).Let x(b,ϕ,u) be the state value of (29) at terminal time b corresponding to the initial value and the control function u. Define the set R(b,ϕ)={x(b,ϕ,u):u∈L2(J,U)}, which is called reachable set of the system (29) at time b, and its closure in X is denoted by R(b,ϕ)¯.Definition 14 (see [59]).
The dynamical system (29) is called approximately controllable on J if R(b,ϕ)¯=X; that is, for given ε>0, however small, it is possible to steer from the point to within a distance ε from all points in the state space X at time b.Now, we introduce some conditions which will be used in presented results.Condition 13.
(30) ε R ε , Γ 0 b ⟶ 0   a s ε ⟶ 0 +   i n   s t r o n g   o p e r a t o r   t o p o l o g y .Condition 14.
The functionh:J×X→Xq satisfies that, for each x→X, the function h(·,x) is strongly measurable in Xα over the interval J and there exists a positive constant Lh such that for each t∈J(31)ht,x-ht,yα≤Lhx-y,∀x,y∈X,ht,xα≤Lh1+x,∀x∈X.Condition 15.
The functionf:J×X→X satisfies the following. (i)
For anyt∈J, the function f(t,·):X→X is continuous, and for all x∈X, the function f(·,x) is strongly measurable.(ii)
For eachr>0, there exist ar(·)∈L1([0,t],R+) and t∈J, such that (32)sup⁡ft,x:x≤r≤art,for  a.e.t∈J,limr→∞inf⁡1r∫0tarsds=σ<+∞.Condition 16.
g : C → C ( [ - a , 0 ] , X ) is a continuous function and there exists a positive constant Lg such that (33)gx-gyC-a,0,X≤Lgx-yC,∀x,y∈C,gxC-a,0,X≤Lg1+xC,∀x∈C.For anyε>0 and z∈X, we define a control uε(t,x) as(34)uεt,x=B∗V∗b-tRε,Γ0bz-Ubϕ0+gx0+h0,ϕ-k0+gx-k0+hb,xb-kb-∫0bb-sq-1AVb-s×hs,xs-ksds-∫0bUb-sfs,xs-υsds,Vt=q∫0∞ϑψqϑTtqϑdϑ,where ψq(ϑ) satisfies the condition of a probability density function defined on (0,∞); that is, ψq(ϑ)≥0, ∫0∞ψq(ϑ)dϑ=1, and ∫0∞ϑψq(ϑ)=1/Γ(1+q); B∗ and V∗ denote the adjoint of B and V, respectively.For the sake of convenience, we introduce the following denotations:(35)K=1εMbqΓq+1Bsup0≤t≤b⁡B∗V∗b-t,Nα=qMαΓ1+αΓ1+qα,l=1εBsup0≤t≤b⁡B∗V∗b-tz+Mϕ0+Lg1+xC+CαLh1+ϕ-k0+Lg1+xC+CαLh1+xb-kb+NαbqαqαLh1+xC+M∫0barsds,where M≥1, Cα>0, M1-α>0, α>0, and Mα>0.Theorem 15 (see [59]).
Assume that the Conditions14–16 hold. System (29) corresponding to the control uε(t,x) has a mild solution for each ε>0 provided that(36)K+1MLg+σ+LhCαMLg+1+Nαbqαqα<1.Theorem 16 (see [59]).
Suppose that the Conditions13, 14, 15, and 16 hold. Besides, one assumes additionally that the functions f:J×X→X, h:J×X→Xq, and g:C([a,b],X)→C([a,0],X) are bounded and MLg+Lh(MLg+1)Cα<1. Then the nonlocal neutral fractional integrodifferential equations with finite delay (29) are approximately controllable on J.Theorem16 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 [61–63]. In article [60] they study the controllability of mild solutions for a fractional neutral integrodifferential system with state-dependent delay of the model(37)Dtαxt+Gt,xϱt,xt,∫0te1t,s,xϱs,xsds=Axt+∫0tBt-sxsds+Ft,xϱt,xt,∫0te2t,s,xϱs,xsds+Cut,t∈J=0,T,x0=ςt∈Bh,x′0=0,t∈-∞,0,where(i)
x· is unknown and needs values in the Banach space X having norm ·;(ii)
α∈1,2;(iii)
A and Btt≥0 are closed linear operators described on a regular domain which is dense in X,·;(iv)
C is a bounded linear operator from U to X;(v)
G, :J×Bh×X→X, ei:D×Bh→X, i=1,2; D=t,s∈J×J:0≤s≤t≤T, and ϱ:J×Bh→-∞,T are apposite functions.If x:-∞,T→X, T>0, is continuous on J and x0∈Bh, then for every t∈J the accompanying conditions hold.(1)
xt is Bh.(2)
xtX≤HxtBh.(3)
xtBh≤D1tsup⁡xsX:0≤s≤t+D2tx0Bh, where H>0 is a constant and D1·:0,+∞→0,+∞ is continuous, D2·:0,+∞→0,+∞ is locally bounded, and D1 and D2 are independent of x·.(4)
The functiont→ςt is well described and continuous from the set (38)Rϱ-=ϱs,ς:s,ς∈0,T×Bh,intoBh and there is a continuous and bounded function Jς:Rϱ-→0,∞ to ensure that ςtBh≤JςtςBh for every t∈Rϱ-.Recognize the space(39)BT=x:-∞,T⟶X:xJiscontinuousandx0∈Bh,where xJ is the constraint of x to the real compact interval on J. The function ·BT to be a seminorm in BT is described by (40)xBT=ςBh+sup⁡xsX:s∈0,T,x∈BT.Definition 17.
LetxTς;u be the state value of model (37) at terminal time T corresponding to the control u and the initial value ς∈Bh. Present the set RT,ς=xTς;u0:u·∈L2J,U, which is known as the reachable set of model (37) at terminal time T.Definition 18.
Model (37) is said to be exactly controllable on J if R(T;ς)=X.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 3.4).
A functionx:-∞,T→X is called a mild solution of (37) on 0,T, if x0=ς; x|0,T∈C0,T:X; the function (41)s⟶ASαt-sGs,xϱs,xs,∫0se1s,τ,xϱτ,xτdτ,s⟶∫0sBs-τSαt-sGτ,xϱτ,xτ,∫0τe1τ,ε,xϱε,xτdεdτis integrable on 0,t for all t∈0,T and for t∈0,T; (42)xt=Rαtς0+G0,ς0,0-Gt,xϱt,xt,∫0te1t,s,xϱs,xsds-∫0tASαt-sGs,xϱs,xs,∫0se1s,τ,xϱτ,xτdτds-∫0t∫0sBs-τSαt-sGτ,xϱτ,xτ,∫0τe1τ,ε,xϱε,xεdεdτds+∫0tSαt-sFs,xϱs,xs,∫0se2s,τ,xϱτ,xτdτds+∫0tSαt-sCusds.Presently, we itemizing the subsequent conditions.Condition 17.
The operator familiesRαt and Sαt are compact for all t>0, and there exists a constant M in a way that RαtLX≤M and SαtLX≤M for every t∈J and (43)-AϑSαtX≤Mtα1-ϑ-1,0<t≤T,where L(X) symbolizes the Banach space of all bounded linear operators from X into X endowed with the uniform operator topology, having its norm recognized as ·L(X).Condition 18.
The subsequent conditions are fulfilled.(a)
B·x∈CJ,X for every x∈D-A1-ϑ.(b)
There is a functionμ·∈L1J,R+, to ensure that (44)BsSαtLD-Aϑ,X≤Mμstαϑ-1,0≤s<t≤T.Condition 19.
The functionF:J×Bh×X→X is continuous and one can find positive constants LF, L~F, and LF∗>0 in ways that, for all t∈J and x, y∈X, (45)Ft,ψ1,x-Ft,ψ2,yX≤LFψ1-ψ2Bh+L~Fx-yX,LF∗=maxt∈J⁡Ft,0,0X.Condition 20.
e i : D × B h → X is continuous and one can find constants Lei>0 and Lei∗>0 to ensure that, for all t,s∈D and ς,ψ∈Bh2, i=1,2; (46)eit,s,ς-eit,s,ψX≤Leiς-ψBh,Lei∗=maxt∈J⁡eit,s,0X,i=1,2.Condition 21.
The functionG· is -Aϑ-values; G:J×Bh×X→D-A-ϑ is continuous and there exist positive constants LG, L~G>0 and LG∗>0 such that, for all t,ςj∈J×Bh, j=1,2; x,y∈X, (47)-AϑGt,ς1,x--AϑGt,ς2,yX≤LGς1-ς2Bh+L~Gx-yX,-AϑGt,ς,0X≤LGςBh+LG∗,where (48)LG∗=maxt∈J⁡-AϑGt,0,0X.Condition 22.
The following inequalities hold.(i)
Let(49)1γM2MC2TxT+1+1γM2MC2TMM0LGςBh+MM0LG∗+LG∗+L~GTLe1∗M0+MTαϑαϑ1+∫0Tμτdτ+MTLF∗+L~FTLe2∗+D1∗r+cnMTLF+L~FTLe2+M0+MTαϑαϑ1+∫0TμτdτLG+L~GTLe1≤r,for some r,γ>0.(ii)
Let(50)Λ=1+1γM2MC2TD1∗MTLF+L~FTLe2+M0+MTαϑαϑ1+∫0TμτdτLG+L~GTLe1<1be such that 0≤Λ<1.Theorem 20 (see [60]).
Assume that the Conditions17–22 hold. Then, control system (37) is exactly controllable on J.Proof of the Theorem20 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:(51)Dtαcxt+ft,xt=Axt+∫0tGt-sxsds+But+gt,xt,t∈I=0,b,x0=ϕ∈Bh,x′0=x1,where(i)
α∈(1,2);(ii)
A,G(t), for t⩾0, are closed linear operators defined on a common domain D=D(A) which is dense in X;(iii)
f,g:[0,b]×Bh→X are appropriate functions. Some necessary notations for the above-mentioned system were presented in Basic Notations Section. The other ones are as follows.(i)
[D(A)] is the domain of A endowed with the graph norm.(ii)
(Z,·Z) and (W,·W) are Banach spaces.(iii)
L(Z,W) stands for the Banach space of bounded linear operators from Z into W endowed with the uniform operator topology. When Z=W then we will write L(Z).(iv)
K^ denotes the Laplace transform of K for appropriate functions K:[0,∞)→Z.(v)
xZ,b=sup⁡{x(s)Z:s∈[0,b]} for a bounded function x:[0,a]→Z and b∈[0,a]; shortly we will write xb when no confusion about the space Z 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 existsM>0 such that Rα(t)⩽M and Sα(t)⩽M for all t∈[0,b]. Additionally, Mb=sups∈[0,b]M(s) and Kb=sups∈[0,b]K(s) are the constants. Moreover N(-A)ϑf, Nf, Ng represent the supreme of the functions (-A)ϑf, f and g on [0,b]×Br[0,Bh], respectively.Lemma 22 (see [66]).
There exists a constantC such that (52)-Aϑ⩽Cfor0⩽ϑ⩽1.Condition 23.
The given conditions hold.(i)
G(·)x∈C(I,X)foreveryx∈[D((-A)1-ϑ)].(ii)
There is functionμ(·)∈L1(I;R+), such that G(s)Sα(t)L([D((-A)ϑ)],X)⩽Mμ(s)tαϑ-1, 0⩽s<t⩽b.Condition 24.
The functionf(·) is (-A)ϑ-valued, f:I×Bh→[D((-A)-ϑ)], the function g(·) is defined on g:I×Bh→X, and there exist positive constants Lf and Lg such that for all (ti,ψj)∈I×Bh the following inequalities are satisfied (53)-Aϑft1,ψ1--Aϑft2,ψ2⩽Lft1-t2+ψ1-ψ2Bh,gt1,ψ1-gt2,ψ2⩽Lgt1-t2+ψ1-ψ2Bh.Condition 25.
The linear fractional control system defined as(54)Dtαcxt=Axt+But,(55)x0=x0,x′0=0is exactly controllable.In the next theorem we present conditions for exact controllability for the system (51).Theorem 23 (see [65]).
If Conditions23–25 and (56)1+1γM2MB2bKb·MHϕBh+Nf+Nf+N-AϑfMbαϑαϑ+N-AϑfMbαϑαϑ∫0bμξdξ+NgMb<r,1+1γM2MB2Kb·Lf-A-ϑ+Mbαϑαϑ+Mbαϑαϑ∫0bμξdξ+MLgb<1are satisfied, then control system (51) is exactly controllable on I.Theorem23 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:(57)Dtαcxt+ft,xt=Axt+∫0tGt-sxsds+But+gt,xt,t∈I=0,b,x0=φ+qxt1,xt2,…,xtn∈Bh,x′0=x1,where 0<t1<t2<⋯<tn⩽b; q:Bhn→Bh is given function such that the next condition holds.Condition 26.
The functionq:Bhn→Bh is continuous and there exist positive constants Li(q) such that (58)qψ1,ψ2,…,ψn-qφ1,φ2,…,φn⩽∑i=1nLiqψi-φiBhfor every ψi,φi∈Br[0,Bh] and assume Nq=sup⁡{q(ψt1,ψt2,…,ψtn):ψi∈Br[0,Bh]}.The next theorem includes the required conditions for system (57) to be exactly controllable.Theorem 24 (see [65]).
Assume that the conditions of Theorem23 are satisfied. Further, if Condition 26 is satisfied, then fractional system (57) is exactly controllable on I provided that (59)1+1γM2MB2bMb+KbMHφBh+Mb+KbMNq+KbM+1Nf+KbN-AϑfMbαϑαϑ1+∫0bμξdξ+KbNgMb<r,Λ=max⁡MbMb∑i=1nLiq+Kbθ,KbMb∑i=1nLiq+Kbθ<1,where (60)θ=1+1γM2MB2bM∑i=1nLiq+Lf-A-ϑ+Mbαϑαϑ+Mbαϑαϑ∫0bμξdξ+MLgb.As before, the proof of Theorem24 is led by contraction mapping.


### 3.6. 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:(61)Dtαcxt+gt,xt=Axt+gt,xt+Jt1-αBut+ft,xt,Hxt,t∈J=0,b,t≠tk,Δxtk=Ikxtk-,k=1,2,…,m,x0=ϕ∈Bh,where(i)
0<α<1;(ii)
g:J×Bh×X→X are given functions;(iii)
Hx(t)=∫0t‍G(t,s)x(s)ds, where G∈C(D,R+) is the set of all positive continuous functions on D={(t,s)∈R2:0⩽s⩽t⩽b};(iv)
xb(ϕ;u) is the state value.To formulate a set of sufficient conditions for exact controllability of system (61) next conditions are necessary [67].Condition 27.
There exists a constantM>0 such that (62)SαtLX⩽M,∀t∈0,b.Condition 28.
The functiong:J×Bh→X is continuous, and there exists a constant Lg>0 such that (63)gt1,ψ1-gt2,ψ2X⩽Lgt1-t2+ψ1-ψ2Bh,ti∈J,ψi∈Bh,i=1,2.Condition 29.
There exist constantsμ1>0 and μ2>0 such that (64)ft,φ,x-ft,ψ,yX⩽μ1φ-ψBh+μ2x-yX,t∈J,φ,ψ∈Bh,x,y∈X.Condition 30.
I k ∈ C ( X , X ), and there exist constants ρ>0 such that (65)Ikx-IkyX⩽ρx-yX,x,y∈X,for  eachk=1,…,m.Conditions for exact controllability of the fractional impulsive system (61) on J are the content of the next theorem.Theorem 25 (see [67]).
If the Conditions25 and 27–30 are satisfied and there exists γ>0, then fractional impulsive system (61) is exactly controllable on J provided that (66)L^=1+1γMB2M2mMρ+mMLgC12+ρ+LgC1+Mbμ1C1+μ2H<1,where C1=sup0<τ<bC1(τ) and H=supt∈[0,b]∫0t‍G(t,s)ds<∞.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 Conditions27–30 hold and that the family {Sα(t):t>0} is compact. In addition, assume that the function f 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,b].Theorems25 and 26 are proved in [67] by contraction mapping theorem.


### 3.7. 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:(67)DtαCxt-ht,xt=Axt+But+ft,xt,t∈J=0,b,x0ϑ=ϕϑ,ϑ∈-r,0.Some necessary notations for the above-mentioned system were presented in Basic Notations Section. The other ones are as follows.(i)
f, h:[0,∞)×C→H are given functions satisfying certain assumptions.(ii)
ϕ∈C.(iii)
xt(ϑ)=x(t+ϑ), for ϑ∈[-r,0].Next conditions [68] are necessary to present conditions for exact controllability for the nonlinear fractional control system (67) by using the contraction mapping principle.Condition 31.
For eacht∈[0,b], the function f(t,·):C→H is continuous and for each x∈C, the function f(·,x):[0,b]→H is strongly measurable.Condition 32.
There exists a constantα1∈[0,α] and m∈L1/α1([0,b],R+) such that f(t,x)⩽m(t) for all x∈C and almost all t∈[0,b].Condition 33.
The functionh:[0,b]×C→H is continuous and there exists a constant β∈(0,1) and H,H1>0 such that h∈D(Aβ) and for any x,y∈C, the function Aβh(·,x) is strongly measurable and Aβh(t,x) satisfies the Lipschitz condition (68)Aβht,x-Aβht,y⩽Hx-yand the inequality (69)Aβht,x⩽H1x+1.Condition 34.
There exists a constantα2∈[0,α] and ρ∈L1/α2([0,b],R+) such that (70)ft,x-ft,y⩽ρtx-yfor any x,y∈C([-r,b],H).Then, the following theorem is true.Theorem 27 (see [68]).
If Conditions25 and 31–34 are satisfied, then control system (67) is exactly controllable on J provided that (71)N+1A-βH+Γ1+βC1-βHbαββΓ1+αβ+αNN2b1+a′1-α2Γ1+α1+a′1-α2N2MB2N2αb1+a′1-α2γΓ1+α1+a′1-α2+1<1,where(72)N=supt∈0,∞⁡Tt≥1,AηTt≤Cηtη,Cη>0,η∈0,1,0<t≤b,N1=mL1/α1∈0,b,N2=ρL1/α2∈0,b,a′=α-11-α2∈-1,0.Tt is an analytic semigroup.Moreover, the authors of [68] investigated the exact controllability of system (67) with nonlocal condition defined in the following way:(73)x0ϑ+gxt1,…,xtnϑ=ϕϑ,ϑ∈-r,0,where g:Cn→C are given functions.Additionally the authors assumed that functiong satisfies the below-presented conditions.Condition 35.
There exists a constantL>0 such that (74)gxt1,…,xtn-gyt1,…,ytn⩽Lx-yfor x,y∈C([-r,b],H).Condition 36.
(75) N L + N + 1 A - β H + Γ 1 + β C 1 - β H b α β β Γ 1 + α β + α N N 2 b 1 + a ′ 1 - α 2 Γ 1 + α 1 + a ′ 1 - α 2 · N 2 M B 2 N 2 α b 1 + a ′ 1 - α 2 γ Γ 1 + α 1 + a ′ 1 - α 2 + 1 < 1 .Necessary conditions for the controllability of nonlinear systems are established in the following theorem.Theorem 28 (see [68]).
If the conditions of Theorem27 and Conditions 35 and 36 are satisfied, then fractional system (67) with nonlocal condition (73) is exactly controllable on J.Theorems27 and 28 are proved by contraction mapping theorem.


## 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 form(11)DαcNxt∈ANxt+∫0tQt-sNxsds+Bus+Ft,xt,∫0tht,x,xsds,t∈J=0,b,t≠tk,Δxtk=Ikxtk,k=1,…,m,x0∈φ∈Bh,x′0=0,where(i)
x· takes values in the Hilbert space H;(ii)
φ is an initial condition;(iii)
α∈1,2;(iv)
A, Qtt≥0, are closed linear operators defined on a common domain which is dense in (H,·);(v)
u∈L2(J,U) is admissible control functions;(vi)
the functionxt:-∞,0→H defined by xtθ=xt+θ, θ∈(-∞,0] belongs to some abstract phase space Bh;(vii)
F:J×Bh×H→PH is a bounded, closed, convex-valued, multivalued map;(viii)
PH is the family of all nonempty subsets of H;(ix)
G:J×Bh→H, Nψ=ψ0+G(t,ψ),ψ∈Bh, and Ik:Bh→H(k=1,…,m) 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(12)Bhb,x0=xbx0;u0:u·∈L2J,Uis called the reachable set of system (11) at terminal time b. Its closure in H is denoted by Bhb,x0¯.Definition 6.
System (11) is said to be approximately controllable on the interval 0,b if Bhb,x0¯=H.Condition 1.
The operator familiesRαt and Sαt are compact for all t>0, and there exist constants M and δ such that RαtLH≤Meδt and SαtLH≤Meδt for every t∈J.Condition 2.
The functionG:J×Bh→H is continuous and there exists a L>0 such that (13)Gt,ψ1-Gt,ψ2≤Lt1-t2+ψ1-ψ2Bh,t1,t2∈J,ψ1,ψ2∈Bh,Gt,ψ≤LψBh+1,t∈J,ψ∈Bh.Condition 3.
(i) For eacht,s∈Λ the function ht,s,·:Bh→H is continuous and for each x∈Bh, the function h·,·,x:Λ→H is strongly measurable.
(ii) There exists a continuous functionp:Λ→0,∞, such that (14)ht,s,ψ≤pt,sΘ0ψBhfor a.e. t,s∈J and ψ∈Bh, where Θ0:0,∞→0,∞ is a continuous nondecreasing function.Condition 4.
The multivalued mapF:J×Bh×H→Pbd,cl,cvH; for each t∈J, the function Ft,·,·:Bh×H→Pbd,cl,cvH is upper semicontinuous and for each ψ,y∈Bh×H, the function F·,ψ,y is measurable; for each fixed ψ,y∈Bh×H, the set (15)SF,ψ=f∈L1J,H:ft∈Ft,ψ,∫0tht,s,ψdsfor  a.e.t∈Jis nonempty.Condition 5.
There exists a continuous functionm:J→0,∞ and a continuous nondecreasing function Θ:0,∞→0,∞ such that (16)Ft,ψ,y=sup⁡f:f∈Ft,ψ,y≤mtΘψBh+y,for a.e. t∈J and each ψ∈B and y∈H with (17)∫1∞1s+Θs+Θ0sds=∞.Condition 6.
The functionsIk:Bh→H are continuous and there exist constants ck such that (18)limsupψBh→∞⁡IkψψBh=ckfor every ψ∈Bh, k=1,…,m.Lemma 7 (see [56]).
LetJ be a compact interval and H be a Hilbert space. Let F be a multivalued map satisfying Condition 4 and let P be a linear continuous operator from L1J,H to CJ,H. Then the operator (19)P∘SF:CJ,H⟶Pcp,cvH,x⟶P∘SFx≔PSF,xis a closed graph in CJ,H×CJ,H.Theorem 8 (see [55]).
Suppose that Conditions1–6 are satisfied and that, for all a>0, system (11) has at least one mild solution on J, provided that (20)max1≤k≤m⁡M21+KbMck+ML+M3KbML<1,where M2=MN∗1+(1/a)M∗2N∗2M12b, M3=1+(1/a)MM∗N∗M12bN∗, M∗=Mmax⁡1,eδb, N∗=max⁡1,e-δb, and M1=B.Now we present the main result of paper [55] on the approximate controllability of system (11).Theorem 9 (see [55]).
Assume that assumptions of Theorem8 hold and, in addition, there exists a positive constant C~ such that (21)Ft,ψ,y=sup⁡f:f∈Ft,ψ,y≤C~,t,ψ,y∈J×Bh×Hand the linear system corresponding to system (11) is approximately controllable on J. Then system (11) is approximately controllable on J.The proofs of the Theorems8 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]:(22)Dtαcxt-gt,xt∈Axt+Ft,xt+But,t∈J=0,b,t≠tk,k=1,2,…,m,Δxt=tk=Ikxtk-,k=1,2,…,m,x0=ϕ∈Bh,t∈J0=-∞,0,where(i)
α∈(0,1);(ii)
x(·)∈X;(iii)
A is the infinitesimal generator of an analytic semigroup of the bounded linear operator {T(t),t⩾0} in X;(iv)
F:J×Bh→P(X) is a bounded, closed, convex-valued multivalued map;(v)
g:J×Bh→X are given functions;(vi)
Ik∈C(X,X)(k=1,2,…,m) are bound functions.The author of [53] used the following fixed point theorem.Theorem 10 (see [58]).
LetX be a Banach space. Φ1:X→Pcl,cv,bd(X) and Φ2:X→Pcp,cv(X) are two multivalued operators satisfying the following.(a)
Φ1 is a contraction.(b)
Φ2 is completely continuous.
Then either(i)
the operator inclusionλx∈Φ1x+Φ2x has a solution for λ=1, or(ii)
the setG={x∈X:λx∈Φ1x+Φ2x,λ>1} is unbounded.Definition 11.
A functionx:(-∞,b]→X is called a mild solution of system (22) if the following holds: x0=ϕ∈Bh on (-∞,0], Δx|t=tk=Ik(x(tk-)),k=1,2,…,m; the restriction of x(·) to the interval [0,b)-{t1,t2,…,tm} is continuous and the integral equation (23)xt=Sαtϕ0-g0,ϕ+gt,xt+∫0tt-sα-1ATαt-sgs,xsds+∫0tt-sα-1Tαt-sfsds+∫0tt-sα-1Tαt-sBusds+∑0<tk<tSαt-tkIkxtk-,t∈J,x0=ϕ∈Bh,t∈J0is satisfied, where (24)f∈SF,x=f∈L1J,X:ft∈Ft,xt,for  a.e.t∈J,Sαt=∫0∞ξαθTtαθdθ,Tαt=α∫0∞θξαθTtαθdθ,ξαθ=1αθ-1-1/αϖαθ-1/α⩾0,ϖαθ=1π∑n=1∞-1n-1θ-nα-1Γnα+1nsin⁡nπα,θ∈0,∞,where ξα is probability density function defined on (0,∞); that is, ξα(θ)⩾0,θ∈(0,∞), and ∫0∞‍ξα(θ)dθ=1.The properties of the operatorsSα(t) and Tα(t) 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 J if for every continuous initial function, ϕ∈Bh, x1∈X, there exists a control u∈L2(J,U) such that the mild solution x(t) of (22) satisfies x(b)=x1.Condition 7.
A is the infinitesimal generator of an analytic semigroup of bounded linear operators T(t) and 0∈ρ(A); for t⩾0, there exist constants M such that Tt⩽M.Condition 8.
The linear operatorW:L2(J,U)→X defined by (25)Wu=∫0bb-sα-1Tb-sBusdshas an induced inverse operator W-1, which takes values in L2(J,U)/kerW and there exist positive constants M2 and M3 such that B⩽M2 and W-1⩽M3.Condition 9.
There exist constants0⩽β<1,c0,c1,c2,Lg such that g is Xβ-valued and (-A)βg is continuous, and(i)
-Aβg(t,x)⩽c1xBh+c2, (t,x)∈J×Bh;(ii)
-Aβg(t,x1)--Aβg(t,x2)⩽Lgx1-x2Bh,(t,xi)∈J×Bh, i=1,2, with (26)C0=Lgl-A-β+c1-βΓ1+βbαββΓ1+αβ<1.Condition 10.
There exists a constantdk such that Ikx⩽dk, k=1,2,…,m for each x∈X.Condition 11.
There exist an integrable functionp:J→[0,∞) and a nondecreasing function ψ:R+→(0,∞) such that F(t,x)=sup⁡f:ft∈Ft,x⩽p(t)ψ(xBh) for almost all t∈J and all x∈Bh.Condition 12.
There exists a positive constantr such that (27)rF1+F2r+F3ψlr+ϕBh+lMϕ0>1,where (28)F1=K1+MM2M3bαΓ1+α·x1+Mϕ0+K1,F2=K2+MM2M3bαΓ1+α·K2,F3=1+MM2M3bαΓ1+αbαMΓ1+αsups∈J⁡ps,K1=M-A-βc1ϕBh+c2+-A-β+c1-βΓ1+βΓ1+αβ·bαββc1ϕBh+c1lMϕ0+c2+M∑k=1mdk,K2=-A-βc1l+c1-βΓ1+βΓ1+αβ·bαββc1l.Next theorem includes the condition for exact controllability of system (22) on the interval J.Theorem 13 (see [53]).
If the Conditions7–12 hold, then system (22) is controllable on the interval J.Based on a fixed point theorem (Theorem10), 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:(29)Dcαxt+ht,xt-kt=-Axt+It1-αft,xt-υt+But,t∈J=0,b,xt=ϕt+gxt,t∈-a,0,where(i)
a>0;(ii)
α∈(0,1);(iii)
k,υ:[0,+∞)→(0,a], and (a>0) are continuous functions;(iv)
f:[0,+∞)×X→X, h:[0,+∞)×X→Xα, and g:C→C([a,0],X) are continuous and nonlinear functions; here 0<q≤1, C≔C([a,b],X).Let x(b,ϕ,u) be the state value of (29) at terminal time b corresponding to the initial value and the control function u. Define the set R(b,ϕ)={x(b,ϕ,u):u∈L2(J,U)}, which is called reachable set of the system (29) at time b, and its closure in X is denoted by R(b,ϕ)¯.Definition 14 (see [59]).
The dynamical system (29) is called approximately controllable on J if R(b,ϕ)¯=X; that is, for given ε>0, however small, it is possible to steer from the point to within a distance ε from all points in the state space X at time b.Now, we introduce some conditions which will be used in presented results.Condition 13.
(30) ε R ε , Γ 0 b ⟶ 0   a s ε ⟶ 0 +   i n   s t r o n g   o p e r a t o r   t o p o l o g y .Condition 14.
The functionh:J×X→Xq satisfies that, for each x→X, the function h(·,x) is strongly measurable in Xα over the interval J and there exists a positive constant Lh such that for each t∈J(31)ht,x-ht,yα≤Lhx-y,∀x,y∈X,ht,xα≤Lh1+x,∀x∈X.Condition 15.
The functionf:J×X→X satisfies the following. (i)
For anyt∈J, the function f(t,·):X→X is continuous, and for all x∈X, the function f(·,x) is strongly measurable.(ii)
For eachr>0, there exist ar(·)∈L1([0,t],R+) and t∈J, such that (32)sup⁡ft,x:x≤r≤art,for  a.e.t∈J,limr→∞inf⁡1r∫0tarsds=σ<+∞.Condition 16.
g : C → C ( [ - a , 0 ] , X ) is a continuous function and there exists a positive constant Lg such that (33)gx-gyC-a,0,X≤Lgx-yC,∀x,y∈C,gxC-a,0,X≤Lg1+xC,∀x∈C.For anyε>0 and z∈X, we define a control uε(t,x) as(34)uεt,x=B∗V∗b-tRε,Γ0bz-Ubϕ0+gx0+h0,ϕ-k0+gx-k0+hb,xb-kb-∫0bb-sq-1AVb-s×hs,xs-ksds-∫0bUb-sfs,xs-υsds,Vt=q∫0∞ϑψqϑTtqϑdϑ,where ψq(ϑ) satisfies the condition of a probability density function defined on (0,∞); that is, ψq(ϑ)≥0, ∫0∞ψq(ϑ)dϑ=1, and ∫0∞ϑψq(ϑ)=1/Γ(1+q); B∗ and V∗ denote the adjoint of B and V, respectively.For the sake of convenience, we introduce the following denotations:(35)K=1εMbqΓq+1Bsup0≤t≤b⁡B∗V∗b-t,Nα=qMαΓ1+αΓ1+qα,l=1εBsup0≤t≤b⁡B∗V∗b-tz+Mϕ0+Lg1+xC+CαLh1+ϕ-k0+Lg1+xC+CαLh1+xb-kb+NαbqαqαLh1+xC+M∫0barsds,where M≥1, Cα>0, M1-α>0, α>0, and Mα>0.Theorem 15 (see [59]).
Assume that the Conditions14–16 hold. System (29) corresponding to the control uε(t,x) has a mild solution for each ε>0 provided that(36)K+1MLg+σ+LhCαMLg+1+Nαbqαqα<1.Theorem 16 (see [59]).
Suppose that the Conditions13, 14, 15, and 16 hold. Besides, one assumes additionally that the functions f:J×X→X, h:J×X→Xq, and g:C([a,b],X)→C([a,0],X) are bounded and MLg+Lh(MLg+1)Cα<1. Then the nonlocal neutral fractional integrodifferential equations with finite delay (29) are approximately controllable on J.Theorem16 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 [61–63]. In article [60] they study the controllability of mild solutions for a fractional neutral integrodifferential system with state-dependent delay of the model(37)Dtαxt+Gt,xϱt,xt,∫0te1t,s,xϱs,xsds=Axt+∫0tBt-sxsds+Ft,xϱt,xt,∫0te2t,s,xϱs,xsds+Cut,t∈J=0,T,x0=ςt∈Bh,x′0=0,t∈-∞,0,where(i)
x· is unknown and needs values in the Banach space X having norm ·;(ii)
α∈1,2;(iii)
A and Btt≥0 are closed linear operators described on a regular domain which is dense in X,·;(iv)
C is a bounded linear operator from U to X;(v)
G, :J×Bh×X→X, ei:D×Bh→X, i=1,2; D=t,s∈J×J:0≤s≤t≤T, and ϱ:J×Bh→-∞,T are apposite functions.If x:-∞,T→X, T>0, is continuous on J and x0∈Bh, then for every t∈J the accompanying conditions hold.(1)
xt is Bh.(2)
xtX≤HxtBh.(3)
xtBh≤D1tsup⁡xsX:0≤s≤t+D2tx0Bh, where H>0 is a constant and D1·:0,+∞→0,+∞ is continuous, D2·:0,+∞→0,+∞ is locally bounded, and D1 and D2 are independent of x·.(4)
The functiont→ςt is well described and continuous from the set (38)Rϱ-=ϱs,ς:s,ς∈0,T×Bh,intoBh and there is a continuous and bounded function Jς:Rϱ-→0,∞ to ensure that ςtBh≤JςtςBh for every t∈Rϱ-.Recognize the space(39)BT=x:-∞,T⟶X:xJiscontinuousandx0∈Bh,where xJ is the constraint of x to the real compact interval on J. The function ·BT to be a seminorm in BT is described by (40)xBT=ςBh+sup⁡xsX:s∈0,T,x∈BT.Definition 17.
LetxTς;u be the state value of model (37) at terminal time T corresponding to the control u and the initial value ς∈Bh. Present the set RT,ς=xTς;u0:u·∈L2J,U, which is known as the reachable set of model (37) at terminal time T.Definition 18.
Model (37) is said to be exactly controllable on J if R(T;ς)=X.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 3.4).
A functionx:-∞,T→X is called a mild solution of (37) on 0,T, if x0=ς; x|0,T∈C0,T:X; the function (41)s⟶ASαt-sGs,xϱs,xs,∫0se1s,τ,xϱτ,xτdτ,s⟶∫0sBs-τSαt-sGτ,xϱτ,xτ,∫0τe1τ,ε,xϱε,xτdεdτis integrable on 0,t for all t∈0,T and for t∈0,T; (42)xt=Rαtς0+G0,ς0,0-Gt,xϱt,xt,∫0te1t,s,xϱs,xsds-∫0tASαt-sGs,xϱs,xs,∫0se1s,τ,xϱτ,xτdτds-∫0t∫0sBs-τSαt-sGτ,xϱτ,xτ,∫0τe1τ,ε,xϱε,xεdεdτds+∫0tSαt-sFs,xϱs,xs,∫0se2s,τ,xϱτ,xτdτds+∫0tSαt-sCusds.Presently, we itemizing the subsequent conditions.Condition 17.
The operator familiesRαt and Sαt are compact for all t>0, and there exists a constant M in a way that RαtLX≤M and SαtLX≤M for every t∈J and (43)-AϑSαtX≤Mtα1-ϑ-1,0<t≤T,where L(X) symbolizes the Banach space of all bounded linear operators from X into X endowed with the uniform operator topology, having its norm recognized as ·L(X).Condition 18.
The subsequent conditions are fulfilled.(a)
B·x∈CJ,X for every x∈D-A1-ϑ.(b)
There is a functionμ·∈L1J,R+, to ensure that (44)BsSαtLD-Aϑ,X≤Mμstαϑ-1,0≤s<t≤T.Condition 19.
The functionF:J×Bh×X→X is continuous and one can find positive constants LF, L~F, and LF∗>0 in ways that, for all t∈J and x, y∈X, (45)Ft,ψ1,x-Ft,ψ2,yX≤LFψ1-ψ2Bh+L~Fx-yX,LF∗=maxt∈J⁡Ft,0,0X.Condition 20.
e i : D × B h → X is continuous and one can find constants Lei>0 and Lei∗>0 to ensure that, for all t,s∈D and ς,ψ∈Bh2, i=1,2; (46)eit,s,ς-eit,s,ψX≤Leiς-ψBh,Lei∗=maxt∈J⁡eit,s,0X,i=1,2.Condition 21.
The functionG· is -Aϑ-values; G:J×Bh×X→D-A-ϑ is continuous and there exist positive constants LG, L~G>0 and LG∗>0 such that, for all t,ςj∈J×Bh, j=1,2; x,y∈X, (47)-AϑGt,ς1,x--AϑGt,ς2,yX≤LGς1-ς2Bh+L~Gx-yX,-AϑGt,ς,0X≤LGςBh+LG∗,where (48)LG∗=maxt∈J⁡-AϑGt,0,0X.Condition 22.
The following inequalities hold.(i)
Let(49)1γM2MC2TxT+1+1γM2MC2TMM0LGςBh+MM0LG∗+LG∗+L~GTLe1∗M0+MTαϑαϑ1+∫0Tμτdτ+MTLF∗+L~FTLe2∗+D1∗r+cnMTLF+L~FTLe2+M0+MTαϑαϑ1+∫0TμτdτLG+L~GTLe1≤r,for some r,γ>0.(ii)
Let(50)Λ=1+1γM2MC2TD1∗MTLF+L~FTLe2+M0+MTαϑαϑ1+∫0TμτdτLG+L~GTLe1<1be such that 0≤Λ<1.Theorem 20 (see [60]).
Assume that the Conditions17–22 hold. Then, control system (37) is exactly controllable on J.Proof of the Theorem20 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:(51)Dtαcxt+ft,xt=Axt+∫0tGt-sxsds+But+gt,xt,t∈I=0,b,x0=ϕ∈Bh,x′0=x1,where(i)
α∈(1,2);(ii)
A,G(t), for t⩾0, are closed linear operators defined on a common domain D=D(A) which is dense in X;(iii)
f,g:[0,b]×Bh→X are appropriate functions. Some necessary notations for the above-mentioned system were presented in Basic Notations Section. The other ones are as follows.(i)
[D(A)] is the domain of A endowed with the graph norm.(ii)
(Z,·Z) and (W,·W) are Banach spaces.(iii)
L(Z,W) stands for the Banach space of bounded linear operators from Z into W endowed with the uniform operator topology. When Z=W then we will write L(Z).(iv)
K^ denotes the Laplace transform of K for appropriate functions K:[0,∞)→Z.(v)
xZ,b=sup⁡{x(s)Z:s∈[0,b]} for a bounded function x:[0,a]→Z and b∈[0,a]; shortly we will write xb when no confusion about the space Z 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 existsM>0 such that Rα(t)⩽M and Sα(t)⩽M for all t∈[0,b]. Additionally, Mb=sups∈[0,b]M(s) and Kb=sups∈[0,b]K(s) are the constants. Moreover N(-A)ϑf, Nf, Ng represent the supreme of the functions (-A)ϑf, f and g on [0,b]×Br[0,Bh], respectively.Lemma 22 (see [66]).
There exists a constantC such that (52)-Aϑ⩽Cfor0⩽ϑ⩽1.Condition 23.
The given conditions hold.(i)
G(·)x∈C(I,X)foreveryx∈[D((-A)1-ϑ)].(ii)
There is functionμ(·)∈L1(I;R+), such that G(s)Sα(t)L([D((-A)ϑ)],X)⩽Mμ(s)tαϑ-1, 0⩽s<t⩽b.Condition 24.
The functionf(·) is (-A)ϑ-valued, f:I×Bh→[D((-A)-ϑ)], the function g(·) is defined on g:I×Bh→X, and there exist positive constants Lf and Lg such that for all (ti,ψj)∈I×Bh the following inequalities are satisfied (53)-Aϑft1,ψ1--Aϑft2,ψ2⩽Lft1-t2+ψ1-ψ2Bh,gt1,ψ1-gt2,ψ2⩽Lgt1-t2+ψ1-ψ2Bh.Condition 25.
The linear fractional control system defined as(54)Dtαcxt=Axt+But,(55)x0=x0,x′0=0is exactly controllable.In the next theorem we present conditions for exact controllability for the system (51).Theorem 23 (see [65]).
If Conditions23–25 and (56)1+1γM2MB2bKb·MHϕBh+Nf+Nf+N-AϑfMbαϑαϑ+N-AϑfMbαϑαϑ∫0bμξdξ+NgMb<r,1+1γM2MB2Kb·Lf-A-ϑ+Mbαϑαϑ+Mbαϑαϑ∫0bμξdξ+MLgb<1are satisfied, then control system (51) is exactly controllable on I.Theorem23 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:(57)Dtαcxt+ft,xt=Axt+∫0tGt-sxsds+But+gt,xt,t∈I=0,b,x0=φ+qxt1,xt2,…,xtn∈Bh,x′0=x1,where 0<t1<t2<⋯<tn⩽b; q:Bhn→Bh is given function such that the next condition holds.Condition 26.
The functionq:Bhn→Bh is continuous and there exist positive constants Li(q) such that (58)qψ1,ψ2,…,ψn-qφ1,φ2,…,φn⩽∑i=1nLiqψi-φiBhfor every ψi,φi∈Br[0,Bh] and assume Nq=sup⁡{q(ψt1,ψt2,…,ψtn):ψi∈Br[0,Bh]}.The next theorem includes the required conditions for system (57) to be exactly controllable.Theorem 24 (see [65]).
Assume that the conditions of Theorem23 are satisfied. Further, if Condition 26 is satisfied, then fractional system (57) is exactly controllable on I provided that (59)1+1γM2MB2bMb+KbMHφBh+Mb+KbMNq+KbM+1Nf+KbN-AϑfMbαϑαϑ1+∫0bμξdξ+KbNgMb<r,Λ=max⁡MbMb∑i=1nLiq+Kbθ,KbMb∑i=1nLiq+Kbθ<1,where (60)θ=1+1γM2MB2bM∑i=1nLiq+Lf-A-ϑ+Mbαϑαϑ+Mbαϑαϑ∫0bμξdξ+MLgb.As before, the proof of Theorem24 is led by contraction mapping.


## 3.6. 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:(61)Dtαcxt+gt,xt=Axt+gt,xt+Jt1-αBut+ft,xt,Hxt,t∈J=0,b,t≠tk,Δxtk=Ikxtk-,k=1,2,…,m,x0=ϕ∈Bh,where(i)
0<α<1;(ii)
g:J×Bh×X→X are given functions;(iii)
Hx(t)=∫0t‍G(t,s)x(s)ds, where G∈C(D,R+) is the set of all positive continuous functions on D={(t,s)∈R2:0⩽s⩽t⩽b};(iv)
xb(ϕ;u) is the state value.To formulate a set of sufficient conditions for exact controllability of system (61) next conditions are necessary [67].Condition 27.
There exists a constantM>0 such that (62)SαtLX⩽M,∀t∈0,b.Condition 28.
The functiong:J×Bh→X is continuous, and there exists a constant Lg>0 such that (63)gt1,ψ1-gt2,ψ2X⩽Lgt1-t2+ψ1-ψ2Bh,ti∈J,ψi∈Bh,i=1,2.Condition 29.
There exist constantsμ1>0 and μ2>0 such that (64)ft,φ,x-ft,ψ,yX⩽μ1φ-ψBh+μ2x-yX,t∈J,φ,ψ∈Bh,x,y∈X.Condition 30.
I k ∈ C ( X , X ), and there exist constants ρ>0 such that (65)Ikx-IkyX⩽ρx-yX,x,y∈X,for  eachk=1,…,m.Conditions for exact controllability of the fractional impulsive system (61) on J are the content of the next theorem.Theorem 25 (see [67]).
If the Conditions25 and 27–30 are satisfied and there exists γ>0, then fractional impulsive system (61) is exactly controllable on J provided that (66)L^=1+1γMB2M2mMρ+mMLgC12+ρ+LgC1+Mbμ1C1+μ2H<1,where C1=sup0<τ<bC1(τ) and H=supt∈[0,b]∫0t‍G(t,s)ds<∞.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 Conditions27–30 hold and that the family {Sα(t):t>0} is compact. In addition, assume that the function f 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,b].Theorems25 and 26 are proved in [67] by contraction mapping theorem.


## 3.7. 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:(67)DtαCxt-ht,xt=Axt+But+ft,xt,t∈J=0,b,x0ϑ=ϕϑ,ϑ∈-r,0.Some necessary notations for the above-mentioned system were presented in Basic Notations Section. The other ones are as follows.(i)
f, h:[0,∞)×C→H are given functions satisfying certain assumptions.(ii)
ϕ∈C.(iii)
xt(ϑ)=x(t+ϑ), for ϑ∈[-r,0].Next conditions [68] are necessary to present conditions for exact controllability for the nonlinear fractional control system (67) by using the contraction mapping principle.Condition 31.
For eacht∈[0,b], the function f(t,·):C→H is continuous and for each x∈C, the function f(·,x):[0,b]→H is strongly measurable.Condition 32.
There exists a constantα1∈[0,α] and m∈L1/α1([0,b],R+) such that f(t,x)⩽m(t) for all x∈C and almost all t∈[0,b].Condition 33.
The functionh:[0,b]×C→H is continuous and there exists a constant β∈(0,1) and H,H1>0 such that h∈D(Aβ) and for any x,y∈C, the function Aβh(·,x) is strongly measurable and Aβh(t,x) satisfies the Lipschitz condition (68)Aβht,x-Aβht,y⩽Hx-yand the inequality (69)Aβht,x⩽H1x+1.Condition 34.
There exists a constantα2∈[0,α] and ρ∈L1/α2([0,b],R+) such that (70)ft,x-ft,y⩽ρtx-yfor any x,y∈C([-r,b],H).Then, the following theorem is true.Theorem 27 (see [68]).
If Conditions25 and 31–34 are satisfied, then control system (67) is exactly controllable on J provided that (71)N+1A-βH+Γ1+βC1-βHbαββΓ1+αβ+αNN2b1+a′1-α2Γ1+α1+a′1-α2N2MB2N2αb1+a′1-α2γΓ1+α1+a′1-α2+1<1,where(72)N=supt∈0,∞⁡Tt≥1,AηTt≤Cηtη,Cη>0,η∈0,1,0<t≤b,N1=mL1/α1∈0,b,N2=ρL1/α2∈0,b,a′=α-11-α2∈-1,0.Tt is an analytic semigroup.Moreover, the authors of [68] investigated the exact controllability of system (67) with nonlocal condition defined in the following way:(73)x0ϑ+gxt1,…,xtnϑ=ϕϑ,ϑ∈-r,0,where g:Cn→C are given functions.Additionally the authors assumed that functiong satisfies the below-presented conditions.Condition 35.
There exists a constantL>0 such that (74)gxt1,…,xtn-gyt1,…,ytn⩽Lx-yfor x,y∈C([-r,b],H).Condition 36.
(75) N L + N + 1 A - β H + Γ 1 + β C 1 - β H b α β β Γ 1 + α β + α N N 2 b 1 + a ′ 1 - α 2 Γ 1 + α 1 + a ′ 1 - α 2 · N 2 M B 2 N 2 α b 1 + a ′ 1 - α 2 γ Γ 1 + α 1 + a ′ 1 - α 2 + 1 < 1 .Necessary conditions for the controllability of nonlinear systems are established in the following theorem.Theorem 28 (see [68]).
If the conditions of Theorem27 and Conditions 35 and 36 are satisfied, then fractional system (67) with nonlocal condition (73) is exactly controllable on J.Theorems27 and 28 are proved by contraction mapping theorem.


## 4. 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:(1)
Showing a mathematical model of dynamical system(2)
Formulation of the assumptions concerning dynamical systems(3)
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(6)
Proof of the above-mentioned theorem 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].



---

*Source: 4715861-2017-01-18.xml*