Allergic responses are defined by hypersensitive reactions against foreign antigens that activate immune cell interactions, leading B lymphocytes to produce IgE. This results in tissue mast cells and circulating basophil cells releasing leukotrienes and histamine, which causes an early inflammatory response; in the late phase, immune cells release chemokines and cytokines. Ferroptosis is an iron-dependently regulated cell death process in which excessive production of reactive oxygen species (ROS) causes massive lipid peroxidation–mediated membrane damage. Allergens, IgE regulation, inflammatory cytokines, and lipid metabolism from an allergic response can induce ferroptosis, which can then enhance the allergic response. This review summarizes the mechanism of ferroptosis and its key regulators. We particularly focus on the potential roles of allergen triggers, IgE regulation, inflammatory cytokines, and lipid metabolism in ferroptosis. We also describe recent research progress regarding ferroptosis in allergic asthma, allergic rhinitis, and allergic dermatitis. Further research on the process of ferroptosis in allergic responses and diseases can aid potential novel therapeutic tools.
Citation: Haiyan Guo, Mingsheng Lei, Jinan Ma, Hongchun Du, Youming Zhang. Allergy and ferroptosis[J]. AIMS Allergy and Immunology, 2025, 9(1): 8-26. doi: 10.3934/Allergy.2025002
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Allergic responses are defined by hypersensitive reactions against foreign antigens that activate immune cell interactions, leading B lymphocytes to produce IgE. This results in tissue mast cells and circulating basophil cells releasing leukotrienes and histamine, which causes an early inflammatory response; in the late phase, immune cells release chemokines and cytokines. Ferroptosis is an iron-dependently regulated cell death process in which excessive production of reactive oxygen species (ROS) causes massive lipid peroxidation–mediated membrane damage. Allergens, IgE regulation, inflammatory cytokines, and lipid metabolism from an allergic response can induce ferroptosis, which can then enhance the allergic response. This review summarizes the mechanism of ferroptosis and its key regulators. We particularly focus on the potential roles of allergen triggers, IgE regulation, inflammatory cytokines, and lipid metabolism in ferroptosis. We also describe recent research progress regarding ferroptosis in allergic asthma, allergic rhinitis, and allergic dermatitis. Further research on the process of ferroptosis in allergic responses and diseases can aid potential novel therapeutic tools.
The theory of fractional calculus has a long-standing history, and it can be traced back to nearly four centuries ago, which firstly appeared in the correspondence between mathematicians Leibniz and L'Hospital about the definition of fractional derivative in the 17th century. After a long and tortuous development, fractional calculus has received considerable attention due mainly to its potential and wide applications in various kinds of scientific fields such as chemical physics, pure mathematics, signal processing, mechanics and engineering, viscoelasticity, biology, neural network model, fractal theory, etc. See [22,55,62,75,96,102,118,120,121] and references therein for further details. Actually, fractional calculus can describe mathematical models involving practical background with less parameters, and present a more vivid and accurate description over things than integral order ones [10,23,38,41,57,91,101,117,130]. Recent research trends and achievements in science and technology show that fractional differential equations including both ordinary and partial ones have more extensive applications than integral order differential equations [37,49,57,63,76,77,97,103,104,124].
As we all know, control theory is an interdisciplinary branch of economics, engineering and mathematics that investigates and analyses some dynamical behaviors of various systems [78,92,115,130,131]. In addition, control theory of dynamical systems with impulse [7,35,47,74] or time delay [39,48,109,110] have been studied in the last decates. Some other excellent results for such systems are obtained on stability [13,33,42,43,44,61,70,105,106,126,127] and stabilization [8,34,36,46,52,82,112,113], optimal control [1,4,11,17,27,30], etc. As an important component of technology and mathematical control theory, controllability has already gained considerable attention. Among the methods to investigate the controllability of diverse nonlinear systems, fixed-point theory has been used widely and effectively, which was initiated by Tarnove in 1967 [94]. For the sake of using fixed-point theorem, the controllability problem of nonlinear systems is transformed to a fixed-point problem of corresponding nonlinear operator in a appropriate function space. Frequently used fixed-point theorems include Banach's fixed-point theorem [12], Schauder's fixed-point theorem [21,29,129], Darbo's fixed-point theorem [16,26], Schaefer's fixed-point theorem [16], Krasnoselskii's fixed-point theorem [59,100,116], Sadovskii's fixed-point theorem [40,68,122], Mönch's fixed-point theorem [14,32,54,123,125], etc. It should be particularly noted that the controllability of fractional evolution systems (FESs) is an important issue for lots of practical problems since the fractional calculus can derive better results than integeral order one. The two most extensively studied subjects of controllability for FESs are exact controllability and approximate controllability.
On the other hand, time delay is a widespread phenomenon in the fields of science and engineering. So it can not be ignored in many practical situations [9,51,58,65]. This is especially true for the evolution processes which are closely related to time. For example, the time delay is often inevitable in the process of pregnancy, maturation, and hatching at the different stages of population development. Sometimes, a minor delay may restrict the system seriously, or affect the structure and performance of the system to a great extent, and even lead to instability of the system. Therefore, various dynamic behaviors of FESs with time delay have been investigated during the last few decades, such as optimal controllability, stability and stabilization problems. With the continuous development of fractional calculus, more and more attention has been paid to the controllability of various kinds of FESs with time delay in recent years [12,14,24,29,123]. In addition, there is another inevitable factor in practice, that is, the phenomenon of impulse which widely exists in the real world. For example, it can be used to describe the sudden fluctuation of population caused by disease, famine, etc, in population model. The impulse phenomena also appear in the timing fishing or replenishment of population ecosystem, the external stimulation in nervous system, the closing of switch in circuit system and so on. The development of impulsive differential equations and related theories has been growing rapidly in recent decades [56,107,108,119]. As we know, there are many processes in nature that are influenced by impulses as well as memory and heredity such as neural network and population dynamics model, etc. Obviously, they are most appropriately described by using fractional impulsive differential systems. Therefore, the investigation on controllability of FESs with impulsive effect has important theoretical significance and practical value. Nevertheless, analyzing controllability for FESs with impulse is more complicated than that for integral order ones. In the study of the fractional (impulsive) controllability, how to represent mild solutions of FESs is the first important step. Up to now, many divergences and academic controversies on the mild solutions to fractional impulsive systems still exist due to the fact that fractional derivatives have heredity, nonlocal behavior and memory property. Fortunately, some relevant literatures [88] have corrected these issues and given their correct expressions.
By the analysis and comparison of the relevant literatures, this work will present a comprehensive overview for the controllability on some classes of FESs, such as some basic systems with classical initial and nonlocal conditions, FESs with time delay or impulse. An outline of the rest of this work is arranged as follows. In Section 2, some notations, definitions and lemmas are listed. Section 3 discusses the controllability of some basic systems. Section 4 considers the controllability of some classes of systems with time delay. In Section 5, controllability of systems with impulsive effects is presented. Controllability results via resolvent operator theory is discussed in Section 6. At last, Section 7 provides a conclusion of the present paper and the research prospect in this area.
This section mainly lists some necessary notations, definitions and lemmas, which will be used throughout the present paper. The preliminaries here can be found in, for example, [40,72,73].
Denote by
Definition 2.1. The Riemann-Liouville standard fractional integral with fractional order
Iqtx(t)=D−qtx(t)=1Γ(q)∫t0(t−s)q−1x(s)ds, |
provided that the right side integral is pointwise defined on
Definition 2.2. The Riemann-Liouville fractional derivative with fractional order
RLDqtx(t)=1Γ(n−q)dndtn∫t0(t−s)n−q−1x(s)ds=dndtnD−(n−q)t, |
where
As an example, for
As early as the 19th century, Riemann-Liouville fractional derivative has been well developed in the works of Abel, Riemann and Liouville. Up to now, a complete theoretical system has been established. However, such theory had led to many difficulties in the application of some practical problems. In order to avoid these difficulties and to facilitate modeling the problems of engineering and physics, a new definition of fractional derivative was proposed by Caputo in 1967:
Definition 2.3. The Caputo fractional derivative with fractional order
CDqtx(t)=1Γ(n−q)∫t0(t−s)n−q−1x(n)(s)ds=D−(n−q)tdndtnx(t), |
where
It is not difficult to see that the Caputo derivative of a constant is equal to zero. Especially, if
CDqtx(t)=1Γ(1−q)∫t0(t−s)−qx′(s)ds, |
which can be used to describe the state of the system at a given time depends on past events. Apart from that, the solutions of Caputo-type equations can approximate any given smooth function arbitrarily [5].
Definition 2.4. The Hilfer fractional derivative with fractional order
Dν,μtx(t)=Iν(1−μ)tddtI(1−ν)(1−μ)tx(t), |
for functions such that the expression on the right hand side exists.
Obviously, when
Lemma 2.5. Suppose that
IqtRLDtqx(t)=x(t)+c1tq−1+c2tq−2+⋅⋅⋅+cntq−n, |
where
Lemma 2.6. Let
IqtCDtqx(t)=x(t)+c0+c1t+c2t2+⋅⋅⋅+cn−1tn−1, |
where
Since the classical solution of a fractional evolution system satisfies a convolution equation on the halfline, it is natural to employ the theory of Laplace transform for its study. Therefore, mild solutions of the considered FESs can be obtained mainly by applying the Laplace transform technique. For this reason, we present the following properties:
Lemma 2.7. Let
(i)
gq(t)={1Γ(q)tq−1, t>0,0, t≤0. |
(ii)
(iii)
(iv)
The definitions of mild solutions for various kinds of
Definition 2.8. The Mainardi's Wright function
Ψq(θ)=∞∑n=0(−θ)nn!Γ(−qn+1−q)=1π∞∑n=1(−θ)n(n−1)!Γ(nq)sin(nπq), θ∈C. |
Remark 1. If
Ψq(θ)=1πq∞∑n=1(−θ)n−1Γ(1+qn)n!sin(nπq), q∈(0,1). |
Lemma 2.9. Let
(i)
(ii)
(iii)
(iv)
During the past four decades, controllability problems of various dynamical systems, including integeral order and fractional order derivatives, have been widely investigated in finite-dimensional and infinite-dimensional spaces [3,25,26,57,60,69,122]. Among such controllability problems, the two most fundamental types are exact controllability and approximate controllability. The exact controllability enables to steer the considered systems to arbitrary final state. If there exists a control such that the systems can be steered to zero point, we call it null controllability [84], which can be regarded as a special case of exact controllability in some way. Under the assumption that the invertible controllability operator can be constructed, the controllability problem can be transformed into a fixed point problem.
In this section, we consider the controllability of some basic nonlinear fractional control systems with local conditions and nonlocal conditions. First, consider the following systems:
CDqtx(t)=Ax(t)+f(t,x(t))+Bu(t), t∈J=[0,b], | (3.1) |
with classical initial condition
x(0)=x0, | (3.2) |
where
x(t)∈D, ∀t∈[0,b], |
which indicates that the systems will not be steered to arbitrary final state of
Definition 3.1. The fractional control system (3.1)-(3.2) is said to be exactly controllable on
It is worth mentioning that the introduction of mild solution in the investigation of fractional controllability problems is the first step. However, Hernández et al. [18] pointed out that the definition of mild solutions used in some papers was not appropriate for FESs as it is just a simple extension of mild solutions for integral order evolution equations. In 2002, El-Borai [15] firstly introduced some probability density functions and gave the fundamental solutions of fractional evolution equations, which is a significant contribution to the construction of mild solutions for FESs. Since then, mild solutions for most of fractional controllability problems are constructed in terms of such probability density functions. For example, with regard to (3.1)-(3.2), the mild solution can be given as follows:
Definition 3.2. For each
x(t)=T(t)x0+∫t0(t−s)q−1S(t−s)f(s,x(s))ds+∫t0(t−s)q−1S(t−s)Bu(s)ds, |
where
T(t)=∫∞0ξq(θ)T(tqθ)dθ, S(t)=q∫∞0θξq(θ)T(tqθ)dθ |
and for
ξq(θ)=1qθ−1−1qΨq(θ−1q)≥0. |
∫∞0ξq(θ)dθ=1, ∫∞0θνξq(θ)dθ=Γ(1+ν)Γ(1+qν), ν∈[0,1]. |
We are now in the position to describe the exact controllability of (3.1)-(3.2) based on the above definition of mild solution. Suppose that the following hypothesis holds:
(C) The linear operator
Wu=∫b0(b−s)q−1S(b−s)Bu(s)ds |
has a bounded inverse operator
Then, for
ux(t)=W−1(x1−T(b)x0−∫b0(b−s)q−1S(b−s)f(s,x(s))ds)(t), t∈J, | (3.3) |
and condition (C) infers
Note that the formula of Definition 3.2 contains characteristic solution operators associated with semigroup
Theorem 3.3. ([100]) Suppose that the following assumptions hold:
(i)
(ii) The linear operator
Wu=∫b0(b−s)q−1S(b−s)Bu(s)ds |
has an inverse operator
‖W−1‖Lb(X,L2(J,U)/kerW)≤M3. |
(iii)
‖f(t,x1)−f(t,x2))‖≤Lf(t)‖x1−x2‖, t∈J, xi∈X, i=1,2. |
(iv)
(v) For all bounded subsets
Πh,δ={q∫t−h0∫∞δθ(t−s)q−1ξq(θ)T((t−s)qθ)f(s,x(s))dθds:x∈B} |
is relatively compact in
Then system (3.1)-(3.2) is exactly controllable on
c[1+M3M2M1bqΓ(1+q)]<1, |
where
Using the similar approach in [100], J. Du et al. [14], studied the exact controllability of some fractional delay control systems, and A. Kumar et al. [28], A. Debbouche et al. [12], investigated the fractional impulsive control problems or such problems with delay. Some fixed point theorems, such as Banach's fixed-point theorem, Dhage's fixed-point theorem and Krasnoselskii's fixed-point theorem, are used widely in establishing the controllability results for the aforementioned papers in which the Lipschitz conditions are all required. The premise of these results is that the semigroups generated by infinitesimal generators are noncompact. This can be illustrated by the following simple example:
Lemma 3.4. ([100]) Consider the following fractional differential systems
{∂23∂t23x(t,y)=xy(t,y)+ωμ(t,y)+e−tk+etx(t,y), t∈J=[0,1],x(t,0)=x(t,1)=0,x(0,y)=0, 0<y<1, |
where
Obviously,
‖f(t,x1−f(t,x2))‖≤11+k‖x1−x2‖, x1,x2∈X, t∈J. |
It is easy to see that all conditions of Theorem 3.3 are satisfied. Hence, the given system is exactly controllable on
Authors in [100] used the equicontinuity of semigroup
Definition 3.5. ([122]) The fractional control system (3.1)-(3.2) is called exactly controllable on interval
Theorem 3.6. ([122]) Suppose that the following hypotheses are satisfied:
(i)
(ii) The linear operator
W(t)u=Bu(t)+∫t0˙S(t−s)Bu(s)ds, t∈J, |
where
supt∈J‖W−1(t)‖L(X,L2(J,U)/kerW(t))≤M2. |
(iii) There exists a positive constant
Then the fractional control system (3.1)-(3.2) is exactly controllable on
It is shown that in Definition 3.5, the time when objective system is steered to
(i)
(ii) The linear operator
W(t)u=S(t)Bu(t)+∫t0˙S(t−s)(Bu(s)−Bu(t))ds, t∈J |
has an induced inverse operator
∥B∥L(U,X)≤M1, supt∈J‖W−1(t)‖L(X,L2(J,U)/kerW(t))≤M2. |
Next, we consider the system (3.1) with nonlocal conditions. In practice, nonlocal conditions are usually more accurate than classical initial conditions for physical estimation. For example, when observing the diffusion of a small amount of gas in the tube, there is more than one initial observation value, that is, the initial value can be determined by multiple observation points. The powerful tool to describe such phenomenon is the differential equations with nonlocal conditions. Therefore, more and more studies on the exact controllability of FESs with nonlocal conditions emerge in large numbers. Generally, nonlocal conditions for many practical problems can be described by the following formula:
x(0)+g(x)=x0, | (3.4) |
Definition 3.7. ([99]) The fractional control system (3.1) and (3.4) is said to be nonlocally exactly controllable on
Different from (3.3), using hypothesis (C), for an arbitrary function
ux(t)=W−1(x1−g(x)−T(b)(x0−g(x))−∫b0(b−s)q−1S(b−s)f(s,x(s))ds)(t). |
It is not difficult to deduce that, by using this control, the mild solution of (3.1) and (3.4) satisfies
CDqtx(t)+Ax(t)=f(t,x(t),Gx(t))+Bu(t), t∈J=[0,b] | (3.5) |
with nonlocal condition
x(0)=m∑k=1ckx(tk), 0<t1<t2<⋅⋅⋅<tm≤b, | (3.6) |
which covers condition (3.2) as a special case. Here
Thirdly, consider the following fractional evolution equations of mixed type
CDqtx(t)+Ax(t)=f(t,x(t),(Sx)(t),(Tx)(t))+Bu(t), t∈J=[0,b] | (3.7) |
with nonlocal condition (3.4), where
(i) For
‖f(t,x,Sx,Tx)‖≤φr(t), a.e. t∈J, ∀x∈X satisfying ‖x‖≤r. |
Moreover, suppose
(ii) The nonlocal term
It should be pointed out that, different from [122], [80] used the methods including a new estimation of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. Obviously, condition (3.4) reduces to condition (3.6) when
Generally speaking, the controllability of the objective system is usually transformed into a fixed point problem for some nonlinear integral operator in an suitable function space. It is worth noting that the compactness conditions play a crucial role in the proof of their results. Sometimes, the compactness of the operator semigroup is required to obtain the controllability of the systems. Unfortunately, this is the case only in finite-dimensional spaces [95] since the inverse of control operator may not exist if the state space is infinite-dimensional. Similar technical errors due to the compactness of semigroup
Compared with the exact controllability, approximate controllability can steer the considered system to arbitrary small neighborhood of final state. It is stressed here that exact controllability and approximate controllability coincide when the space is finite-dimensional. On the premise that the corresponding linear systems are approximate controllable, many scholars have studied the approximate controllability of various fractional semilinear evolution systems such as the time-delay systems [24,29], impulsive systems [116], stochastic equations [84,87], neutral equations [68] and so on. Generally speaking, the major techniques to investigate approximate controllability can be classified into three categories. The first one is to use iterative and approximate techniques, that is called sequential approach also [45,89,90]. The second one is the range conditions of the operator
Definition 3.8. The fractional system (3.1)-(3.2) is said to be approximately controllable on
Kb(f)={x(b,u)∈X:u∈L2(J,U), x(⋅,u) is the mild solution of (3.1)−(3.2)}. |
We can also use the following equivalent expression:
Definition 3.9. The fractional system (3.1)-(3.2) is said to be approximately controllable on
With regard to the method to use resolvent conditions, it is particularly important to note that the approximate controllability results are based on the supposition that the corresponding linear dynamical systems are approximately controllable. For this reason, consider the following linear fractional systems corresponding to (3.1) and (3.2):
{CDqtx(t)=Ax(t)+Bu(t), t∈J=[0,b],x(0)=x0. | (3.8) |
Note that approximate controllability of (3.8) is an extension of approximate controllability of linear first-order control systems. Then it is natural to introduce the control operator associated with (3.8) as
Γb0=∫b0(b−s)q−1S(b−s)BB∗S∗(b−s)ds, |
where
Lemma 3.10. ([86]) The linear fractional system (3.8) is approximately controllable on
For fractional systems (3.1) and (3.2), R. Sakthivel et al. in [86] established a new set of sufficient conditions guaranteeing the approximate controllability by using the semigroup theory and fixed point strategy, which is given as follows:
Theorem 3.11. Assume that the following hypotheses hold:
(i)
(ii) For each
(iii) There exists a constant
(iv) The function
Then the semilinear fractional system (3.1)-(3.2) is approximately controllable on
Notice that Theorem 3.11 requires the compactness of operator
Theorem 3.12. Assume that all the assumptions of Theorem 3.11 hold and, in addition, the following hypothesis holds:
(v)
Then the semilinear fractional system (3.1) with nonlocal condition (3.4) is approximately controllable on
On the basis of the ideas presented in [86], S. Ji [21] further investigated the approximate controllability of the same nonlocal fractional systems by resolvent conditions and approximation method:
Theorem 3.13. ([21]) Assume that the following hypotheses are satisfied:
(i)
(ii)
(iii)
(iv)
Then the fractional control system (3.1) with nonlocal condition (3.4) is approximately controllable on
Here, the nonlinear term
As we all know, the current states of some practical systems clearly depend on the past history, which is the so-called phenomenon of time delay. It occurs frequently and is inevitable in numerous practical systems of the real world. In fact, time delay is closely related to various evolution equations. Hence, the effect of time delay must be considered if we intend to describe and analyze evolution systems accurately. With the development of the applications for fractional calculus, the research on the controllability of FESs with time delay is more and more extensive [29,54,90].
According to that the time delay is finite or not, such systems in the existing literatures on fractional controllability can be classified into two types. One is FES with finite time delay, and the other is FES with infinite time delay. There are many excellent results on the controllability of FESs with finite time delay [12,84,85]. Compared with the spaces chosen to investigate infinite time delay systems, the Banach spaces selected to study finite time delay systems are regular and relatively simple.
Consider the following fractional finite time delay system:
CDqtx(t)=Ax(t)+A1x(t−h)+Bu(t)+f(t,x(t−h)), t∈J=[0,b], | (4.1) |
with initial condition
x(t)=ϕ(t), t∈[−h,0], | (4.2) |
where
Based on the ideas of [89,128], authors in [45] investigated the approximate controllability of system (4.1)-(4.2) by using iterative and approximate techniques, i.e. sequential approach. Their main result is given as below:
Theorem 4.1. ([45]) Suppose that the following hypotheses hold:
(i) The semigroup
(ii) The nonlinear function
‖f(t,x)−f(t,y)‖≤L‖x−y‖V, ∀x,y∈V. |
(iii) For any
Gh=∫b0(b−s)q−1S(b−s)h(s)ds, h∈Z, |
and
M(‖A1‖+L)γΓ(α)√b2α−12α−1Eα(M(‖A1‖+L)bα)<1. |
Then, system (4.1)-(4.2) is approximately controllable.
Different from the method of resolvent conditions mentioned in Section 3, the sequential approach utilized in [45] has many advantages. Hypothesis (i) in Theorem 4.1 indicates that
Another interesting work on this subject is to study the relationship of approximate controllability between FES with delay and FES without delay. A. Shukla et al. [85] studied the approximate controllability of the following two classes of FESs of order
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t−h)), t∈J=[0,b],x(t)=ϕ(t), t∈[−h,0],x′(0)=x0, | (4.3) |
and the corresponding semilinear system without delay
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t)), t∈J=[0,b],x(0)=ϕ(0),x′(0)=x0, | (4.4) |
where
Different from [45], [85] took advantage of relationships of the approximate controllability between system (4.3) and its corresponding semilinear system (4.4). The approximate controllability of system (4.3) can be obtained from that of the approximate controllability of system (4.4) under some suitable assumptions. The two above-mentioned controllability results in [85] can be characterized by the following two theorems:
Theorem 4.2. ([85]) If the following assumptions hold:
(i)
(ii)
‖f(t,x(t))−g(t,w(t))‖X≤Lg‖x(t)−w(t)‖X. |
(iii)
(iv) The system (4.4) is approximately controllable.
Then, the system (4.3) is approximately controllable.
Theorem 4.3. ([85]) Suppose that the assumptions (i), (ii) and (iv) in Theorem 4.2 hold. In addition, assumption (ⅲ) in Theorem 4.2 is replaced by (iii)
Then, the system (4.3) is approximately controllable.
Besides using Lipschitz continuity of nonlinearity to obtain the existence and uniqueness of mild solutions, some other types of fixed-point theorems are also used to study the controllability of fractional finite time delay systems. For instance, authors in [25] established some necessary and sufficient conditions for the exact controllability of certain linear fractional finite delay systems by using the Laplace transformation techniques and Mittag-Leffler function, and also presented a sufficient condition for the exact controllability of nonlinear fractional finite delay system via Schauder's fixed-point theorem. [26] investigated a implicit fractional finite delay system with multiple delays in control. Under the hypothesis that control Gramian matrix
As we know, the theory of noncompactness measures is also a powerful tool to study the controllability of fractional time delay systems in infinite-dimensional Banach spaces. Consider the following fractional nonlocal semilinear evolution systems with finite time delay:
{Dqx(t)=Ax(t)+f(t,x(t),xt)+Bu(t), a.e. t∈J=[0,b],x(t)+M(t,x)=ϕ(t), t∈[−h,0], | (4.5) |
where
Inspired by [54,122], authors in [123] established a new result of exact controllability for system (4.5) by utilizing the theory of noncompactness measures and resolvent operators. Due to the existence of time delay, sometimes we need to choose or construct some appropriate spaces. Therefore, different from [122], [123] selects the complete space
Generally speaking, for the sake of investigating approximate controllability of nonlinear FESs with delay or without delay, some restrictive assumptions need to be imposed on the system components, such as continuity or Lipschitz continuity of nonlinearity, control interval, compactness of semigroup, and range conditions of operator
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t−h)), t∈J=[0,b],x(t)=ϕ(t), t∈[−h,0]. | (4.6) |
Denote the range of the operator
(HB) for each ξ∈Z, there exists a function η∈¯R(B) such that Lξ=Lη. |
It should be pointed out that hypothesis (HB) is essential to obtain the approximate controllability for system (4.6), since it implies that the corresponding linear system of (4.6) is approximately controllable. Also, from hypothesis (HB), they deduced that there is a geometrical relation in
Z=N0(L)⊕N⊥0(L)=N0(L)⊕¯R(B), |
where
As we know, the theory of differential equations with finite time delay has been developed extensively. However, there are many complex dynamic systems in practice which can not be described and analyzed accurately by using finite time delay. In recent years, the theory of differential equations with infinite delay has received a great deal of concerns and gained rapid progress due to its applications in science and engineering. The choice of phase space is of vital importance. How to choose a suitable phase space is essential to solve such problems with infinite time delay. The most effective phase spaces used so far are
In addition, [31] studied the following fractional systems with infinite delay:
Dξ,η0+[x(t)−g(t,xt)]=Ax(t)+f(t,xt)+Bu(t), t∈(0,b], | (4.7) |
I(1−ξ)(1−η)0+x(t)|t=0=ϕ(0)∈Ch, t∈(−∞,0], | (4.8) |
I(1−ξ)(1−η)0+x(t)|t=0=ϕ(0)+q(zt1,zt2,zt3,⋅⋅⋅,ztn)∈Ch, t∈(−∞,0], | (4.9) |
where
As we know, the general method to study the approximate controllability on FESs with infinite time delay is resolvent condition technique [83]. Differently, A. Shukla et al. in [90] studied the approximate controllability for the following semilinear FES of order
{CDqtx(t)=Ax(t)+Bu(t)+f(t,xt), t∈[0,b],x(t)=x0=ϕ∈B, t∈(−∞,0].x′(0)=ψ∈Y, | (4.10) |
where
Theorem 4.4. ([90]) Assume the following conditions hold:
(i)
‖f(t,x)−f(t,y)‖Y≤L‖x−y‖B, ∀x,y∈B, t∈[0,b]. |
(ii) For any given
(iii)
(iv) The constant
Then, the fractional infinite time delay system (4.10) is approximately controllable.
It should be pointed out that hypothesis (ii) in Theorem 4.4 plays an important role in ensuring the approximate controllability for correponding linear system. Authors in [90] defined a solution mapping
In real applications, a system (such as signal processing systems, computer networks, automatic control systems, and telecommunications) is often affected by abrupt changes and instantaneous disturbances at a certain moment. These systems are often described by impulsive differential systems which contain some continuous-time differential equations and some jumping operators. Since impulsive differential systems involved in piecewise continuous function spaces, some properties of continuous function spaces may not be applicable if they are extended to piecewise continuous spaces, such as Ascoli-Arzela theorem and the properties of noncompactness measure. For the sake of describing the evolution processes more reasonably and accurately, the influence of abrupt changes and instantaneous disturbances must be fully considered.
In many cases, impulse phenomenon and memory effect cross each other in evolution processes. For instance, impulsive system is suitable to describe the population model which is suddenly affected by disease and famine. However, the current number of population is closely related to the previous population base, gender ratio and age structure. These phenomena with memory effect can be described more precisely by fractional impulsive differential systems. Hence, it is believed that fractional differential equations with impulse and fractional impulsive control systems have great research significance and wide application background.
With regard to the study on controllability of fractional impulsive evolution equations, the most important step is to obtain the existence of mild solutions to the considered systems. The first systematic investigation for the mild solutions of fractional impulsive evolution equations was made by Mophou [67]. Although this type of mild solution has been quoted by many scholars, it is not suitable for their considered systems. The main reason is that these definitions based on that in [67] do not consider its memory and heredity. In recent years, some mathematicians have pointed out this error in the comments of [88].
Based on this, Z. Liu et al. [59] considered the following nonlinear fractional impulsive evolution systems:
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t)), t∈J=[0,b]∖{t1,t2,⋅⋅⋅,tk},Δx(ti)=Ii(x(ti)), i=1,2,⋅⋅⋅,k,x(0)=x0, | (5.1) |
where
x(t)={T(t)x0+∫t0(t−s)q−1S(t−s)(f(s,x(s))+Bu(s))ds, t∈[0,t1],T(t)x0+∫t0(t−s)q−1S(t−s)(f(s,x(s))+Bu(s))ds+T(t−t1)I1(x(t−1)), t∈(t1,t2], ⋮T(t)x0+∫t0(t−s)q−1S(t−s)(f(s,x(s))+Bu(s))ds+∑ki=1T(t−ti)Ii(x(t−i)), t∈(tk,b]. | (5.2) |
In addition to the hypotheses of Theorem 3.2 in [100], authors in [59] supposed that impulsive functions
Also by using the similar techniques as in [59] and with the Lipschitz continuity imposed on nonlocal terms, [81] and [114] further studied the exact controllability of the following fractional impulsive differential and integro-differential evolution equations with nonlocal conditions, respectively:
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t)), t∈J=[0,b]∖{t1,t2,⋅⋅⋅,tk},Δx(ti)=Ii(x(ti)), i=1,2,⋅⋅⋅,k,x(0)+g(x)=x0, | (5.3) |
and
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t),(Hx)(t)), t∈J=[0,b]∖{t1,t2,⋅⋅⋅,tk},Δx(ti)=Ii(x(ti)), i=1,2,⋅⋅⋅,k,x(0)+g(x)=x0. | (5.4) |
As for the case
Recently, authors in [16] considered the following similar FESs with nonlocal initial condition and impulsive effects:
{CDqtx(t)=Ax(t)+Bu(t)+f(t,x(t)), t∈J=[0,b]∖{t1,t2,⋅⋅⋅,tk},Δx(ti)=Ii(x(ti)), i=1,2,⋅⋅⋅,k,x(0)=g(x), | (5.5) |
The approximate controllability was established by using Schaefer's fixed-point theorem, Darbo's fixed-point theorem and the theory of noncompactness measure. It should be further pointed out that [16] adopted the approximate technique to deal with system (5.5), which is different from the conventional method that certain fixed point theorem is applied directly to the corresponding integral operator. This method can overcome the difficulty caused by nonlocal conditions and reduce the assumptions on the impulsive terms effectively. In addition, unlike the hypotheses of [12,81,86,114] in which the nonlinearity
At present, most of the fractional impulsive controllability results are involved in Caputo fractional derivative. However, there are still a few literatures on the controllability of FESs with Riemann-Liouville derivatives [53,60]. In [53], Z. Liu et al. studied the approximate controllability of a class of fractional impulsive neutral evolution equations with Riemann-Liouville derivatives under the assumption
Lag‖A−1‖+LagMbαΓ(α+1)+k∑i=1Mdi(t−ti)α−1+LfMbαΓ(1+α)<1. | (5.6) |
Unfortunately, this assumption is unreasonable due to the possibility that operator
In addition, Z. Liu et al. in [60] considered the following fractional impulsive evolution systems with Riemann-Liouville derivatives:
{RLDqtx(t)=Ax(t)+Bu(t)+f(t,x(t)), t∈J=[0,b]∖{t1,t2,⋅⋅⋅,tk},△I1−qtix(ti)=Gi(t−i,x(t−i)), i=1,2,⋅⋅⋅,k,I1−qtx(t)|t=0=x0∈X, | (5.7) |
where
In order to avoid the error (5.6), authors in [60] established some new sufficient conditions of approximate controllability for system (5.7). By introducing Banach space
‖x‖PC1−q=max{supt∈(ti,ti+1](t−ti)1−q‖x(t)‖X: i=0,1,2,⋅⋅⋅,k}, |
they derived the existence and uniqueness of
Theorem 5.1. ([60]) Suppose that the following hypotheses hold:
(i) The function
‖f(t,x)−f(t,y)‖≤L(t−ti)1−q‖x−y‖X |
for a.e.
(ii) There exist constants
(iii) For any
‖Gφ−GBu‖X<ε, ‖Bu‖Lp(J,X)<N‖φ‖Lp(J,X), |
where
Gh=∫b0(b−s)q−1S(b−s)h(s)ds, h(⋅)∈Lp(J,X), |
N(1−D∗)Eq(MLb)Mb1−1pΓ(q)(p−1pq−1)1−1p<1, |
and
Then, system (5.7) is approximately controllable on
Generally speaking, the approach of resolvent operator can be applied to the inhomogeneous equations to derive various variation of parameters formulas, and also it can lead to improved perturbation results and stronger properties of the variation of parameter formulas [73]. For equations with unbounded operators in infinite-dimensional space, the resolvent operator is more appropriate because it is a direct generalization of
Definition 6.1. ([6]) Let
λα−β(λαI−A)−1u=∫∞0e−λtSα,β(t)udt, Reλ>ω. |
In this case,
Definition 6.2. ([73]) The bounded linear operator
u(t)=∫t0σ(t−s)Au(s)ds, t≥0, |
where the scalar kernel
(i)
(ii)
(iii)
For the special cases
Based on the subordination principle, if
Theoretically, for the evolution equations with order of fractional derivative
Up to now, there are not many papers on controllability problems of FESs with order
Recently, K. Li et al. in [40] considered the fractional systems (3.1), (3.4) together with the initial condition
x′(0)=y0, | (6.1) |
An appropriate definition of mild solutions via Laplace transformation was introduced. Else, they derived some sufficient conditions to ensure the exact controllability for nonlocal problem (3.1), (3.4) and (6.1) by using Sadovskii's fixed-point theorem and vector-valued operator theory under the following hypotheses on nonlinearity together with some conditions imposed on operators
(i)
(ii)
(iii) There exists a function
∥f(t,x)−f(t,z)∥≤lf(t)∥x−z∥, x,z∈X. |
Following [40], authors in [90] and [85] addressed the semilinear fractional control system of order
For most practical problems, it is especially effective to solve the abstract differential equations of second order directly rather than transforming them into first-order systems. Travis and Webb [98] established the theory of strongly continuous sine and cosine operator families, in which they analyzed the advantages of such method. But for fractional controllability problems of order
Definition 6.3. ([129]) For each
x(t)=Cq(t)x0+Kq(t)x1+∫t0(t−s)q−1Pq(t−s)f(s,x(s))ds+∫t0(t−s)q−1Pq(t−s)Bu(s)ds, |
for each
Cq(t)=∫∞0Mq(θ)C(tqθ)dθ, Kq(t)=∫t0Cq(s)ds, Pq(t)=∫∞0qθMq(θ)S(tqθ)dθ, |
Their main results are as follows:
Theorem 6.4. ([129]) Assume the following conditions holds:
(i)
(ii) There exists a function
∣f(t,x)∣≤kf(t)φ(∣x∣), |
for each
lim infr→∞φ(r)r=0. |
(iii) The linear operator
Wu=∫b0(b−s)q−1Pq(b−s)Bu(s)ds |
has an inverse operator
‖W−1‖Lb(X,L2(J,U)/kerW)≤M2. |
(iv) For every
Then, the evolution system (3.1), (3.2) and (6.1) is exactly controllable on
It is observed that the infinitesimal generator
The controllability is one of the fundamental problems for FESs. This work has provided a comprehensive survey on the elementary results and some recent progress of the controllability for FESs. Exact controllability and approximate controllability of some kinds of FESs are reviewed. Firstly, several basic FESs with classical initial and nonlocal conditions are considered. Some fundamental theory and sufficient conditions are presented to ensure the exact controllability and approximate controllability for such systems. Secondly, FESs with finite time delay and infinite time delay are discussed, respectively. We talked over three different methods for investigating approximate controllability of these time-delay systems. Thirdly, controllability results on some types of FESs with impulsive effects are discussed. At last, exact controllability and approximate controllability for FESs obtained by using resolvent operator theory have been carefully analyzed.
The interested directions of the theoretical study in the future on the controllability for FESs may be as follows:
(i) The systems studied on this topic will be more and more extensive and complicated. For instance, it is valuable to investigate controllability for hybrid FESs with delay, impulse or stochastic factors.
(ii) Besides fixed point theorem, more effective tools for studying controllability of FESs should be further developed.
(iii) Application area of controllability for FESs in reality need to be investigated in depth.
For our subsequent work, the following issues will continue to be focused on:
(1) It is noted that Lipschitz continuity and compact conditions are required in most of existing work on this area. How to relax such conditions imposed on nonlinear terms still need to be well investigated.
(2) Relationship of approximate controllability between FES with impulse and FES without impulse has not been fully described. Further research progress in this field is expected to be well achieved.
(3) When FES is affected by impulse and time delay simultaneously, the investigation of controllability on its nonlocal problem is still insufficient. In particular, to deal with such problems via resolvent operator should be further explored.
(4) Numerical simulation or other applications about the theoretical results, such as digital filters in Digital Signal Processing (DSP), should be paid more attention in the future work.
Research supported by the National Natural Science Foundation of China under grant 62073202, the Natural Science Foundation of Shandong Province under grant ZR2020MA007 and a project of Shandong Province Higher Educational Science and Technology Program of China under grant J18KA233.
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