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Complete controllability of nonlinear fractional neutral functional differential equations
Advances in Continuous and Discrete Models volumeÂ 2022, ArticleÂ number:Â 33 (2022)
Abstract
This paper is concerned with the complete controllability of a nonlinear fractional neutral functional differential equation. Some sufficient conditions are established for the complete controllability of the nonlinear fractional system. The conditions are established based on the fractional power of operators and the fixedpoint theorem under the assumption that the associated linear system is completely controllable. Finally, an example is presented to illustrate our main result.
1 Introduction
In this paper, we assume that X is a Hilbert space with the norm \(\cdot \). Let \(r>0\) and C be the Banach space of all continuous functions from an interval \([r,0]\) into X with the norm \(\x\=\sup_{t\in [r,0]}x(t)\). The purpose of this paper is to study the complete controllability for the following nonlinear fractional neutral functional differential system
where \({}^{C}D_{t}^{q}\) is the Caputo fractional derivative of order \(0< q<1\), the state variable \(x(\cdot )\) takes values in the Hilbert space X, \(h: I\times C \rightarrow X\) is a given function, \(A:D(A)\subset X \rightarrow X\) is the infinitesimal generator of an analytic semigroup \(\{T(t)\}_{t\geq 0}\). \(B: U\rightarrow X\) is a bounded linear operator, U is a Hilbert space, the control function \(u(\cdot ) \in L^{2}(I,U)\) and \(f:I\times C \times U\rightarrow X\) is a given function satisfying some assumptions. If \(x:[r,T]\rightarrow X\) is a continuous function, then \(x_{t}\) is an element in C defined by \(x_{t}(\theta )=x(t+\theta )\), \(\theta \in [r,0]\) and \(\phi \in C\).
In mathematical control theory, the controllability is one of the important concepts that has been studied by many authors (see [1â€“7] and the references therein). By means of semigroup theory and the fixedpoint approach, various types of controllability problems have been investigated, for instance, approximate controllability [8â€“15] and complete controllability [16â€“25]. There are several papers devoted to the approximate or complete controllability for fractional differential systems, when the nonlinear term is independent of the control function. Wang and Zhou [22] studied the complete controllability of fractional evolution systems in infinitedimensional spaces by using fractional calculus, the properties of characteristic solution operators and fixedpoint theorems. Meanwhile, Sakthivel et al. [23] established a new set of sufficient conditions for the complete controllability of a fractional nonlinear neutral functional differential equation. Du and Jiang [24] investigated the approximate controllability of impulsive Hilfer fractional differential inclusions.
In [25, 26], the approximate controllability of firstorder delay control systems has been proved when the nonlinear term is a function of both the state function and the control function by assuming that the corresponding linear system is approximately controllable. As far as we know, the fractional functional differential systems have been proved to be abstract formulations in many problems arising in engineering, physics, automatic control, etc. The delay differential equations had shown their efficiency in the study of the behavior of real populations. Thus, the study of controllability for such systems is important for many applications. In 2011, Sukavanam [27] investigated the approximate controllability of the following fractionalorder semilinear delay system in which the nonlinear term is a function of both the state function and the control function.
where \(\frac{1}{2}< q<1\), the state \(x(\cdot )\) takes values in the Banach space X, the control function \(u(\cdot ) \) takes values in the Banach space Y, \(A:D(A)\subset X \rightarrow X\) is a closed linear operator with dense domain \(D(A) \) generating a \(C_{0}\)semigroup \(S(t)\); B is a bounded linear operator from \(L^{2}([0, T]; Y )\) to \(L^{2}([0, T];X )\); the operator \(f:[0,T]\times C([r,0];X) \times Y\rightarrow X\) is nonlinear. If \(x:[r,T]\rightarrow X\) is a continuous function, then \(x_{t}: [r,0]\rightarrow X\) is defined as \(x_{t}(\theta )=x(t+\theta )\), \(\theta \in [r,0]\) and \(\phi \in C([r,0];X)\).
Motivated by the above work, we consider the functional differential equations in which the delay terms also occur in the derivative of the unknown solution. That is, we study the complete controllability of the nonlinear fractional neutral functional differential system (1) when the nonlinear term is a function of both the state function and the control function.
In the past decades, with the development of theories of fractional differential equations, there has been a great deal of interest in the study of solutions of fractional differential systems; see Byszewski [28], Podlubny [29], Kilbas et al. [30] and Lakshmikantham et al. [31]. Moreover, there are different types of mild solutions that have been investigated; see Byszewski and Lakshmikantham [32], Pazy [33], Zhou and Jiao [34, 35] and Wang and Zhou [36, 37]. In particular, Zhou and Jiao [34] obtained the existence and uniqueness of mild solutions for fractional neutral evolution equations by using the fractional power of operators and some fixedpoint theorems. In addition, Wang and Zhou [36] introduced a new mild solution for semilinear fractional evolution equations and the existence and uniqueness of Î±mild solutions are proved.
In order to derive the complete controllability of the nonlinear fractional neutral functional differential system (1), in this paper, we first give the concept of mild solutions of the system (1) in the light of [34, 36]. Then, we establish sufficient conditions for the complete controllability of the nonlinear fractional neutral functional differential system (1) in which the nonlinear term depends on the control function. To obtain the complete controllability of the system (1), we impose some necessary hypotheses on A, B, h, f and the assumption that the corresponding linear system of the system (1) is completely controllable.
The rest of our paper is organized as follows. SectionÂ 2 is devoted to some necessary preliminaries. In Sect.Â 3, the complete controllability for the system (1) is given. Finally, an example is presented to demonstrate our complete controllability result.
2 Preliminaries
In this section, we introduce some notations, definitions and lemmas that will be used throughout the paper.
Definition 2.1
([29])
The fractional integral of order Î± with the lower limit 0 for a function f is defined as
provided the righthand side is pointwise defined on \([0,\infty )\), where Î“ is the gamma function.
Definition 2.2
([29])
The Caputo derivative of order Î± with the lower limit 0 for a function f can be written as
Remark 2.1
If f is an abstract function with values in X, then integrals that appear in Definitions 2.1 and 2.2 are taken in Bochnerâ€™s sense.
Throughout this paper, we assume that A is the infinitesimal generator of an analytic semigroup \(\{T(t)\}_{t\geq 0}\) of bounded operators on X. Let \(0\in \rho (A)\), where \(\rho (A)\) is the resolvent set of A, then for \(\eta \in (0,1]\), we define the fractional power \(A^{\eta }\) as a closed linear operator on its domain \(D(A^{\eta })\). Moreover, \(T(t)\) and \(A^{\eta }\) have the following basic properties.
(i) There is a \(M\geq 1\) such that
(ii) For any \(\eta \in (0,1]\), there exists a constant \(c_{\eta }> 0\) such that
For more details, see [33].
Lemma 2.1
([33])
There exists a constant C such that
where \(A^{\alpha }=\frac{1}{\Gamma (\alpha )} \int _{0}^{\infty }t^{\alpha 1} T(t)\,dt\).
By comparison with the fractional differential equation given in [34], we give the following definition of the mild solution of the system (1).
Definition 2.3
A function \(x \in C([r,T];X)\) is called a mild solution of the system (1) if on \([r,T]\) it satisfies
where \(S_{q}(t)=\int _{0}^{\infty }\phi _{q}(\theta )T(t^{q} \theta )\,d\theta \), \(T_{q}(t)=q \int _{0}^{\infty }\theta \phi _{q}(\theta )T(t^{q} \theta )\,d\theta \) and for \(\theta \in (0,\infty )\)
In addition, \(\phi _{q}(\theta )\) is the probability density function defined as
Lemma 2.2
The operators \(S_{q}(t)\) and \(T_{q}(t)\) have the following properties:

(i)
For any \(t\geq 0\), the operators \(S_{q}(t)\) and \(T_{q}(t)\) are linear and bounded operators, that is, for any \(x\in X\),
$$\bigl\Vert S_{q}(t)x \bigr\Vert \leq M \Vert x \Vert \quad \textit{and}\quad \bigl\Vert T_{q}(t)x \bigr\Vert \leq \frac{Mq}{\Gamma (1+q)} \Vert x \Vert .$$ 
(ii)
\(\{S_{q}(t)\}_{t\geq 0}\) and \(\{T_{q}(t)\}_{t\geq 0}\) are strongly continuous.
Lemma 2.3
([34])
For any \(x\in E\), E is a Banach space, \(\beta \in (0,1)\) and \(\eta \in (0,1]\), we have
and
Consider the following linear fractional differential system
and it is convenient to introduce the controllability operator associated with (11) as
where \(B^{*}\), \(T_{q}^{*}(t)\) denote the adjoints of B and \(T_{q}(t)\), respectively.
By [23, 38], the definition of complete controllability for the linear fractional differential system (11) is as follows.
Lemma 2.4
([23, 38]) The linear fractional control system (11) is completely controllable on I if and only if there exists a \(\gamma >0\) such that
3 Complete controllability
We present our main results of the paper in this section. We need the definition of complete controllability of the system (1).
Definition 3.1
The system (1) is said to be completely controllable on the interval I if \(\mathcal{R}(T,\phi ) =X\), where \(\mathcal{R}(T,\phi )=\{x_{T}(\phi ,u)(0):u(\cdot )\in L^{2}(I,U)\}\).
To prove the main results, we impose the following hypotheses:
\((P_{1})\) The function \(h:[0,T]\times C \rightarrow X\) is continuous and there exists a constant \(\beta \in (0,1)\) and L, \(L_{1}\), for any \(x,y\in C\), \(A^{\beta }h(\cdot , x)\) is strongly measurable and \(A^{\beta }h(t,\cdot )\) satisfies the Lipchitz condition \(\A^{\beta }h(t, x)A^{\beta }h(t, y)\\leq L\xy\\) and the inequality \(\A^{\beta }h(t, x)\\leq L_{1}(\x\+1)\).
\((P_{2})\) The nonlinear function \(f:I\times C\times U \rightarrow X\) is continuous and there exists a constant \(L_{2}>0\) such that
\((P_{3})\) The linear fractional control system (11) is completely controllable.
\((P_{4})\) The nonlinear function \(f(t,x_{t}, u(t))\) satisfies the Lipschitz condition, that is, there exists a constant \(L_{3}\) such that
Define an operator Î¦ on \(C(I,C)\times C(I,U)\) as
with the norm \(\(x,u)\=\x_{t} \_{C}+\u \\), \((x,u)\in C(I,C)\times C(I,U)\), \(t\in I\), where
\(p(x,u)= x_{T}S_{q}(T)[ \phi (0) h(0,x_{0})]h(T,x_{T})\int _{0}^{T} (Ts)^{q1} AT_{q}(Ts)h(s,x_{s})\,ds  \int _{0}^{T} (Ts)^{q1} T_{q}(Ts) f(s,x_{s},u(s))\,ds\).
It will be shown that the system (1) is completely controllable on I if the operator Î¦ has a fixed point in \(C(I,C)\times C(I,U)\).
Theorem 3.1
Assume that the hypotheses \((P_{1})\)â€“\((P_{4})\) are satisfied. Then, the problem (1) has a unique mild solution in \(C([r,T];X)\) provided that
where \(\beta \in (0,1)\), \(M_{B}=B\), \(d=\frac{M_{B} Mq}{\gamma \Gamma (1+q)}\).
Proof
Obviously, \(x \in C([r,T];X)\) is a mild solution of the system (1) if and only if the operator Î¦ has a fixed point in \(C(I,C)\times C(I,U)\). Therefore, it is sufficient to prove that Î¦ has a fixed point in \(C(I,C)\times C(I,U)\). We first show that Î¦ maps \(C(I,C)\times C(I,U)\) into itself. Based on LemmaÂ 2.3 and the condition \((P_{1})\), we have
According to LemmaÂ 2.2(i) and the hypothesis \((P_{2})\), we have
By using (17) and (18), Lemmas 2.1 and 2.4, and hypothesis \((P_{3})\), it can be shown that there exist two constants \(C_{1},C_{2}>0\) such that
and
where \(\beta \in (0,1)\). It follows from (15), (19) and (20) that there exists a constant \(C_{3}\) such that
which means that Î¦ maps \(C(I,C)\times C(I,U)\) into itself.
We next prove that the operator Î¦ is a contraction mapping on \(C(I,C)\times C(I,U)\). For any \((x,u),(y,w)\in C(I,C)\times C(I,U)\), it holds that
By hypotheses \((P_{1})\)â€“\((P_{4})\), LemmaÂ 2.2(i), and (17) and (18), we have
where \(d=\frac{M_{B} Mq}{\gamma \Gamma (1+q)}\). The condition \((P_{1})\) implies
Based on LemmaÂ 2.2(i) and (23), one can obtain
Similar to the discussion of \(I_{1}\), we obtain
In view of (16), we obtain that Î¦ is a contraction. Consequently, Î¦ has a fixed point in \(C(I,C)\times C(I,U)\) by the Banach fixedpoint theorem, which is a mild solution of the system (1). This completes the proof.â€ƒâ–¡
Theorem 3.2
If all the assumptions of TheoremÂ 3.1hold, then the system (1) is completely controllable on I.
Proof
Let \((\bar{x}(\cdot ),\bar{u})\) be a fixed point of the operator Î¦ in (13), that is
where
and the control function
here \(p(\bar{x},\bar{u})=x_{T}S_{q}(T)[\phi (0)h(0,x_{0})] + h(T, x_{T})+ \int _{0}^{T} (Ts)^{q1} AT_{q}(Ts)h(s,\bar{x}_{s})\,ds  \int _{0}^{T} (Ts)^{q1}T_{q}(Ts) f(s,\bar{x}_{s},\bar{u}(s))\,ds\).
According to TheoremÂ 3.1, any fixed point of Î¦ is a mild solution of the system (1). Then, by (12), (29) and (30), we have
Thus, the system (1) is approximately controllable on I by DefinitionÂ 3.1. The proof is completed.â€ƒâ–¡
4 Example
As an application of our complete controllability result, we consider the following fractional partial differential equation
where \(\partial _{t}^{q}\) is the Caputo fractional partial derivative of order \(0< q<1\), g, f are given continuous functions, \(\mu :I \times [0,\pi ] \rightarrow [0,\pi ]\) is continuous in t, \(x_{t}(\theta ,z)=x(t+\theta ,z)\) and \(\phi (\theta ,z) \) is continuous.
Taking \(X=U=L^{2}[0,\pi ]\). Let \(A:D(A)\subset X\rightarrow X\) be an operator defined by \(Aw=w^{\prime \prime }\) with the domain
Then,
where \(e_{n}(z)=\sqrt{\frac{2}{\pi }} \sin (nz)\), \(0\leq z\leq \pi \), \(n=1,2,\ldots \)â€‰. It is known that \(\{e_{n}\}\), \(n=1,2,\ldots \) is an orthonormal base for U and A generates a compact semigroup \(T(t)\), \(t>0\) in X that is given by
For more details, please refer to [33].
Put \(x_{t}=x_{t}(\theta ,\cdot )\), i.e., \((x(t+\theta ))(z)=x(t+\theta ,z)\), \(\theta \in [r,0]\). Define the functions \(h:[0,1]\times C\rightarrow X\) by \((h(t,x_{t}))(z)=\int _{0}^{\pi }g(z,y)x_{t}(\theta ,y)\,dy\) and \(f: [0,1]\times C \times U \rightarrow X\) as \(f(t,x_{t},u )(z)= f(t,x_{t}(\theta ,z),u(t))\). Moreover, define \(B: U\rightarrow X\) by \((Bu(t))(z)=\mu (t,z)\), \(z\in [0,\pi ]\). Let us take \(f(t,x_{t},u )=\x_{t}\_{C} e_{3}+\u\e_{4}\), then the conditions \((P_{2})\) and \((P_{4})\) are satisfied.
Consequently, the system (32) can be written in the abstract form (1) with the appropriate choices of A, B, h and f, and its associated linear system
is completely controllable according to LemmaÂ 2.4, which means that the condition \((P_{4})\) is satisfied. Meanwhile, the inequality (3.4) is also satisfied with the appropriate choices of A, B, h and f. Therefore, all the conditions of TheoremÂ 3.2 are satisfied. Hence, the system (32) is completely controllable on \([0,1]\).
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Acknowledgements
The authors would like to thank the reviewers very much for their valuable comments for improvement of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which contributed to an improvement in the presentation of this paper.
Funding
This work is supported by the National Natural Science Foundation of China (11471015 and 11601003); the Natural Science Foundation of Anhui Province (1708085MA15 and 2008085QA12); the Natural Science Fund of Colleges and Universities in Anhui Province (KJ2021A0967) and the Scientific Research Project of Anhui Jianzhu University (2019QDZ37).
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Wen, Y., Xi, XX. Complete controllability of nonlinear fractional neutral functional differential equations. Adv Cont Discr Mod 2022, 33 (2022). https://doi.org/10.1186/s13662022037068
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DOI: https://doi.org/10.1186/s13662022037068