Approximate controllability of fractional differential equations via resolvent operators
 Zhenbin Fan^{1}Email author
https://doi.org/10.1186/16871847201454
© Fan; licensee Springer. 2014
Received: 30 December 2013
Accepted: 16 January 2014
Published: 3 February 2014
Abstract
Of concern are the existence and approximate controllability of fractional differential equations governed by a linear closed operator which generates a resolvent. Using the analytic resolvent method and the continuity of a resolvent in the uniform operator topology, we derive the existence and approximate controllability results of a fractional control system.
MSC:34K37, 47A10, 49J15.
Keywords
1 Introduction
where ${D}^{\alpha}$ is the Caputo fractional derivative of order α with $0<\alpha <1$, $A:D(A)\subset X\to X$ is the infinitesimal generator of a resolvent ${S}_{\alpha}(t)$, $t\ge 0$, $B:U\to X$ is a bounded linear operator, $u\in {L}^{2}([0,b],U)$, X and U are two real Hilbert spaces, ${J}_{t}^{1\alpha}h$ denotes the $1\alpha $ order fractional integral of $h\in {L}^{1}([0,b],X)$.
The controllability problem has attracted a lot of mathematicians and engineers’ attention since it plays a key role in control theory and engineering and has very important applications in these fields. Many contributions on exact and approximate controllability have been made in recent years. We refer the reader to the recent papers [1–12] and the references therein.
However, there are few articles to study fractional control system (1.1) governed by a linear closed operator which generates a resolvent. The main difficulty is that the resolvent does not have the semigroup property, even the continuity in the uniform operator topology. Fortunately, we can prove the continuity of a resolvent in the uniform operator topology and the compactness of the solution operator in the case of an analytic resolvent. For more details, we refer the reader to the papers [13, 14] by Fan and Mophou. A similar idea on the uniform continuity of operators can be found in [15] by Liang, Liu and Xiao. In the present paper, we study approximate controllability of fractional control system (1.1) by using the analytic resolvent method and the uniform continuity of the resolvent.
This paper has three sections. In Section 2, we recall some definitions of Caputo fractional derivatives, analytic resolvent, mild solutions to equation (1.1) and the concept of approximate controllability of fractional control systems. In Section 3, we prove the existence and approximate controllability of fractional control system (1.1).
2 Preliminaries
Throughout this paper, let $b>0$ be fixed, ℕ be the set of positive integers. We denote by $(X,\parallel \cdot \parallel )$ and $(U,\parallel \cdot \parallel )$ two Hilbert spaces, by $C([0,b],X)$ the space of all Xvalued continuous functions on $[0,b]$ with the norm $\parallel u\parallel =sup\{\parallel u(t)\parallel ,t\in [0,b]\}$, by ${L}^{p}([0,b],X)$ the space of Xvalued Bochner integrable functions on $[0,b]$ with the norm ${\parallel f\parallel}_{{L}^{p}}={({\int}_{0}^{b}{\parallel f(t)\parallel}^{p}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t)}^{1/p}$, where $1\le p<\mathrm{\infty}$. Also, we denote by $\mathcal{L}(X)$ the space of bounded linear operators from X into X endowed with the norm of operators.
Now, let us recall some basic definitions and results on fractional derivative, resolvent and approximate controllability.
Definition 2.1 ([16])
where Γ is the gamma function.
Definition 2.2 ([16])
where $\alpha \in (n1,n]$, $n\in \mathbb{N}$.
Definition 2.3 ([16])
where $\alpha \in (n1,n]$, $n\in \mathbb{N}$.
In the remainder of this paper, we always suppose that $0<\alpha <1$ and A is a closed and densely defined linear operator on X.
Definition 2.4 ([17])

(S1) ${S}_{\alpha}(t)$ is strong continuous on ${\mathbb{R}}_{+}$ and ${S}_{\alpha}(0)=I$;

(S2) ${S}_{\alpha}(t)D(A)\subseteq D(A)$ and $A{S}_{\alpha}(t)x={S}_{\alpha}(t)Ax$ for all $x\in D(A)$ and $t\ge 0$;

(S3) the resolvent equation holds${S}_{\alpha}(t)x=x+{\int}_{0}^{t}{g}_{\alpha}(ts)A{S}_{\alpha}(s)x\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\phantom{\rule{1em}{0ex}}\text{for all}x\in D(A),t\ge 0.$
Since A is a closed and densely defined operator on X, it is easy to show that the resolvent equation holds for all $x\in X$ (see [17]).
Definition 2.5 ([17])
A resolvent ${S}_{\alpha}(t)$ is called analytic if the function ${S}_{\alpha}(\cdot ):{\mathbb{R}}_{+}\to \mathcal{L}(X)$ admits analytic extension to a sector $\sum (0,{\theta}_{0})$ for some $0<{\theta}_{0}\le \pi /2$. An analytic resolvent ${S}_{\alpha}(t)$ is said to be of analyticity type $({\omega}_{0},{\theta}_{0})$ if for each $\theta <{\theta}_{0}$ and $\omega >{\omega}_{0}$, there is ${M}_{1}={M}_{1}(\omega ,\theta )$ such that $\parallel S(z)\parallel \le {M}_{1}{e}^{\omega Rez}$ for $z\in \sum (0,\theta )$, where Rez denotes the real part of z.
Definition 2.6 A resolvent ${S}_{\alpha}(t)$ is called compact for $t>0$ if for every $t>0$, ${S}_{\alpha}(t)$ is a compact operator.
A function $x\in C([0,b],X)$ is called a strong solution of (2.1) if $x(t)\in D(A)$ for all $t\in [0,b]$, ${g}_{1\alpha}\ast x\in {C}^{1}([0,b],X)$ and (2.1) holds, where ${C}^{1}([0,b],X)=\{x:{x}^{\prime}\in C([0,b],X)\}$, $({g}_{1\alpha}\ast x)(t)=\frac{1}{\mathrm{\Gamma}(1\alpha )}{\int}_{0}^{t}{(ts)}^{\alpha}x(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$.
A function $x\in C([0,b],X)$ is called an integral solution of (2.1) if $({g}_{\alpha}\ast x)(t)\in D(A)$ and $x(t)={x}_{0}+A({g}_{\alpha}\ast x)(t)+{\int}_{0}^{t}f(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$ for all $t\in [0,b]$.
which implies that $x(t)={S}_{\alpha}(t){x}_{0}+{\int}_{0}^{t}{S}_{\alpha}(ts)f(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$, $0\le t\le b$. That is, the variation of a constant formula is satisfied.
So, we can give the following definition of mild solutions for (1.1).
for ${x}_{0}\in X$ and $u\in {L}^{2}([0,b],U)$.
is called the reachable set of system (1.1).
Definition 2.8 The fractional system is said to be approximately controllable on $[0,b]$ if $\overline{{K}_{b}(f)}=X$, where $\overline{{K}_{b}(f)}$ denotes the closure of ${K}_{b}(f)$.
where ${B}^{\ast}$, ${S}_{\alpha}^{\ast}(bs)$ denote the adjoint of operators B and ${S}_{\alpha}(bs)$, respectively.
where x is the solution of (1.1) with control $u,{x}_{b}\in X$, $\lambda >0$.
It is known that the control u concerned with approximate controllability of integer order differential equation is just the unique solution of the above optimal problem. Following this idea, we have the following lemma, which can be used to explain the following construction of control function u in (2.4).
□
In the next section, we will prove the approximate controllability of fractional order system (1.1) by using this integral system. More precisely, we will approximate any fixed point ${x}_{b}\in X$ under appropriate conditions by using the final state of solution x with the control u given in system (2.4).
3 Approximate controllability
In this section, we first show that for every $\lambda >0$ and ${x}_{b}\in X$, integral system (2.4) has at least one mild solution. That is, there exists at least one function ${x}_{\lambda}\in C([0,b],X)$ which satisfies (2.4). Then, we can approximate any point ${x}_{b}$ in X by using these solutions $\{{x}_{\lambda}:\lambda >0\}$. For this purpose, we need two important lemmas.
If ${S}_{\alpha}(t)$ is a compact ${C}_{0}$semigroup, it is well known that G is compact. However, it is unknown in the case of a compact resolvent. The main difficulty is that the resolvent does not have the property of semigroups. Thus, it seems to be more complicated to prove the compactness of the Cauchy operator. However, we can prove the continuity of a resolvent in the uniform operator topology in the case of an analytic resolvent, thus the compactness of the Cauchy operator. Moreover, the continuity of a resolvent in the uniform operator topology plays a key role in the proof of the next existence theorem.
Lemma 3.1 ([[13], Lemma 10])
 (i)
${lim}_{h\to 0}\parallel {S}_{\alpha}(t+h){S}_{\alpha}(t)\parallel =0$ for $t>0$;
 (ii)
${lim}_{h\to {0}^{+}}\parallel {S}_{\alpha}(t+h){S}_{\alpha}(h){S}_{\alpha}(t)\parallel =0$ for $t>0$;
 (iii)
${lim}_{h\to {0}^{+}}\parallel {S}_{\alpha}(t){S}_{\alpha}(h){S}_{\alpha}(th)\parallel =0$ for $t>0$.
Lemma 3.2 ([[13], Lemma 11])
Suppose that ${S}_{\alpha}(t)$ is a compact analytic resolvent of analyticity type $({\omega}_{0},{\theta}_{0})$. Then the Cauchy operator G defined by (3.1) is a compact operator.

(H1) ${S}_{\alpha}(t)$ is a compact analytic resolvent of analyticity type $({\omega}_{0},{\theta}_{0})$ and $M={sup}_{t\in [0,b]}\parallel {S}_{\alpha}(t)\parallel <+\mathrm{\infty}$.

(H2) $f:[0,b]\times X\to X$ is continuous and there exists a positive constant K such that $\parallel f(t,x)\parallel \le K$ for all $(t,x)\in [0,b]\times X$.

(H3) $B:U\to X$ is a linear bounded operator and there exists $N>0$ such that $\parallel B\parallel =N$.
Under these assumptions, we can prove the first main result in this paper. We hereafter always suppose that $\parallel R(\lambda ,{\mathrm{\Lambda}}_{b})\parallel \le \frac{1}{\lambda}$ for all $\lambda >0$.
Theorem 3.3 Assume that conditions (H1)(H3) are satisfied. Then integral system (2.4) has at least one mild solution on $[0,b]$ for every $\lambda >0$ and ${x}_{b}\in X$.
It is easy to see that the fixed point of Q is a mild solution of integral system (2.4). Subsequently, we will prove that Q has a fixed point by using Schauder’s fixed point theorem.
as $n\to \mathrm{\infty}$, which implies that Q is continuous on $C([0,b],X)$.
Next, we will show that ${Q}_{1}$ is compact by using the AscoliArzela theorem.
where $L=\parallel {x}_{b}\parallel +M\parallel {x}_{0}\parallel +MKb$, and K comes from condition (H2).
Thus, ${Q}_{1}{W}_{r}$ is equicontinuous on $C([0,b],X)$.
which implies that $\{({Q}_{1}x)(t):x\in {W}_{r}\}$ is precompact in X by using the total boundedness. Thus, ${Q}_{1}$ is compact in view of the ArzelaAscoli theorem. Therefore, the solution operator Q is compact.
Then we obtain that for large enough ${r}_{0}>0$, the inequality $\parallel (Qx)\parallel \le {r}_{0}$ holds for all $x\in C([0,b],X)$. Thus $Q{W}_{{r}_{0}}\subseteq {W}_{{r}_{0}}$.
Therefore, by Schauder’s fixed point theorem, the operator Q has a fixed point in ${W}_{{r}_{0}}$, which is just the mild solution of integral system (2.4). □
Next, we present the approximate controllability of fractional control system (1.1). We make the following hypothesis:
(H4) $\lambda R(\lambda ,{\mathrm{\Lambda}}_{b})\to 0$ as $\lambda \to {0}^{+}$ in the strong operator topology.
Theorem 3.4 Assume that conditions (H1)(H4) are satisfied. Then fractional control system (1.1) is approximately controllable on $[0,b]$.
which implies that fractional control system (1.1) is approximately controllable on $[0,b]$. □
Remark 3.5 In the case of a ${C}_{0}$semigroup and an integer order derivative, condition (H4) is equivalent to the approximate controllability of the corresponding homogenous linear system. However, due to the complexity of fractional derivatives, one should be more careful to deal with this equivalence. Further discussions on this equivalence and concrete examples will be presented in our consequent papers.
Declarations
Acknowledgements
The work was supported by the NSF of China (11001034, 11171210) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middleaged Teachers and Presidents.
Authors’ Affiliations
References
 Ahmed H: Controllability for Sobolev type fractional integrodifferential systems in a Banach spaces. Adv. Differ. Equ. 2012., 2012: Article ID 167Google Scholar
 Balachandran K, Park JY, Trujillo JJ: Controllability of nonlinear fractional dynamical systems. Nonlinear Anal. 2012, 75: 19191926. 10.1016/j.na.2011.09.042MathSciNetView ArticleMATHGoogle Scholar
 Fec̆kan M, Wang J, Zhou Y: Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory Appl. 2013, 156(1):7995. 10.1007/s1095701201747MathSciNetView ArticleGoogle Scholar
 Ganesh R, Sakthivel R, Mahmudov NI, Anthoni SM: Approximate controllability of fractional integrodifferential evolution equations. J. Appl. Math. 2013., 2013: Article ID 291816Google Scholar
 Ji S, Li G, Wang M: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 2011, 217(16):69816989. 10.1016/j.amc.2011.01.107MathSciNetView ArticleMATHGoogle Scholar
 Kumar S, Sukavanam N: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 2012, 252: 61636174. 10.1016/j.jde.2012.02.014MathSciNetView ArticleMATHGoogle Scholar
 Mahmudov NI: Approximate controllability of fractional Sobolevtype evolution equations in Banach spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 502839Google Scholar
 Mahmudov NI: Approximate controllability of fractional neutral evolution equations in Banach spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 531894Google Scholar
 Mophou GM, N’Guérékata GM: Optimal control of a fractional diffusion equation with state constraints. Comput. Math. Appl. 2011, 62: 14131426. 10.1016/j.camwa.2011.04.044MathSciNetView ArticleMATHGoogle Scholar
 Rykaczewski K: Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal. 2012, 75: 27012712. 10.1016/j.na.2011.10.049MathSciNetView ArticleMATHGoogle Scholar
 Sakthivel R, Mahmudov NI, Nieto JJ: Controllability for a class of fractionalorder neutral evolution control systems. Appl. Math. Comput. 2012, 218: 1033410340. 10.1016/j.amc.2012.03.093MathSciNetView ArticleMATHGoogle Scholar
 Wang J, Zhou Y, Medved̆ M: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 2012, 389: 261274.MathSciNetGoogle Scholar
 Fan Z, Mophou G: Nonlocal problem for fractional differential equations via resolvent operators. Int. J. Differ. Equ. 2013., 2013: Article ID 490673Google Scholar
 Fan Z, Mophou G: Existence and optimal controls for fractional evolution equations. Nonlinear Stud. 2013, 20(2):163172.MathSciNetMATHGoogle Scholar
 Liang J, Liu JH, Xiao TJ: Nonlocal problems for integrodifferential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2008, 15: 815824.MathSciNetMATHGoogle Scholar
 Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
 Prüss J: Evolutionary Integral Equations and Applications. Birkhäuser, Basel; 1993.View ArticleMATHGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.