Approximate controllability of fractional impulsive evolution systems involving nonlocal initial conditions
 Xiaozhi Zhang^{1}Email author,
 Chuanxi Zhu^{1} and
 Chenggui Yuan^{2}
https://doi.org/10.1186/s136620150580x
© Zhang et al. 2015
Received: 5 May 2015
Accepted: 19 July 2015
Published: 7 August 2015
Abstract
This work is concerned with the approximate controllability of a nonlinear fractional impulsive evolution system under the assumption that the corresponding linear system is approximate controllable. Using the fractional calculus, the Krasnoselskii fixed point theorem, and the technique of controllability theory, some new sufficient conditions for approximate controllability of fractional impulsive evolution equations are obtained. The results in this paper are generalizations and continuations of the recent results on this issue. At the end, an example is given to illustrate the effectiveness of the main results.
Keywords
MSC
1 Introduction
In recent years, fractional differential systems have provided us with an excellent tool in electrochemistry, physics, porous media, control theory, engineering, etc., due to the descriptions of memory and hereditary properties of various materials and processes. The research as regards the fractional systems has received more and more attention very recently.
Controllability is one of the important concepts both in mathematics and in control theory. Generally speaking, controllability enables one to steer the control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. The controllability problem is a mathematical description of many physical systems such as fluid mechanic systems, quantum systems, and so forth. Controllability of deterministic and stochastic dynamical control systems is well developed by using different kinds of methods which can be found in [1–5]. But the question is that the concept of exact controllability is usually too strong if we consider the problem in the infinite dimensional spaces. Therefore, approximate controllability, the weaker concept of controllability, has gained much attention recently, which steers the system to an arbitrary small neighborhood of a final state (see, for example, [6–10]). Mahmudov and Zorlu [7] researched the approximate controllability of fractional evolution equations involving the Caputo fractional derivative, the sufficient conditions are established under the assumption that the corresponding linear system is approximate controllable, by using the theory of fractional calculus and semigroup and the Schauder fixed point theorem. Ganesh et al. [6] derived a set of sufficient conditions for the approximate controllability of a class of fractional integrodifferential evolution equations.
A strong motivation for investigating the fractional evolution equation comes from the fact that it provides an excellent tool for the modeling of various phenomena in many fields of physics, engineering, economics, etc. The existence and uniqueness of the fractional evolution equation have been studied by several authors (see [11–16] and references therein) with the help of various fixed point theorem and operator theory. Zhou and Jiao [15, 16] discussed the fractional evolution equations and defined the mild solution by means of the probability density function, which has been developed by Wang et al. [13]. On the other hand, there are significant developments in the theory of impulses especially in the area of impulsive differential equations with fixed moments, which provide a natural description of observed evolution processes, regarding these as an important tool for better understanding several realworld phenomena in applied sciences. Very recently, we [17] considered the fractional impulsive differential equations with delay, and the resonance case in [18]. For more details as regards impulsive differential equations, the reader can refer to the monograph of Lakshmikantham et al. [19] and [1, 20–22].
However, there are limited works considering the approximate controllability of the fractional impulsive evolution system with nonlocal conditions [22, 23]. So, in this work, the main objective is to provide the sufficient conditions for the approximate controllability of the control system (1.1). The nonlocal boundary condition, initiated by Byszewski [24], is studied in [8, 13]. It is claimed there that it may be used in some physical problems successfully. The technique we use is the Krasnoselskii fixed point theorem and the semigroup theory. More precisely, by using the constructive control function, we transfer the approximate controllability problem for control system (1.1) into the fixed point problem for operator Λ. Furthermore, the results on the approximate controllability of fractional control systems are derived.
The outline of this paper is as follows. In Section 2, we recall some essential results on the fractional powers of the generator of a compact analytic semigroup and introduce the mild solution for the system (1.1). In Section 3, we study the existence of a mild solution for the system (1.1) under the feedback control \(u_{\epsilon}(x)\) defined in (3.4). We show that the control system (1.1) is approximately controllable on \([0,T]\) provided that the corresponding linear system is approximate controllable. Finally, an example is given to illustrate the effectiveness of the main results.
2 Background materials and preliminaries
Denote \(X_{\alpha}\) by the Hilbert space \(D(A^{\alpha})\) equipped with the norm \(\x\_{\alpha}=\A^{\alpha}x\\) for \(x\in D(A^{\alpha})\), which is equivalent to the graph norm of \(A^{\alpha}\). Moreover, the fractional power \(A^{\alpha}\) has the following basic properties.
Lemma 2.1
([25])
 (1)
\(S(t):X\to X_{\alpha}\) for each \(t>0\) and \(\alpha\ge0\).
 (2)
\(A^{\alpha}S(t)x=S(t)A^{\alpha}x\) for each \(x\in D(A^{\alpha})\) and \(t\ge0\).
 (3)For every \(t>0\), \(A^{\alpha}S(t)\) is bounded in X and there exists \(M_{\alpha}>0\) such that$$\bigl\Vert A^{\alpha}S(t)\bigr\Vert \le M_{\alpha}t^{\alpha}. $$
 (4)
\(A^{\alpha}\) is bounded linear operator for \(0\le\alpha\le1\), there exists \(C_{\alpha}>0\) such that \(\A^{\alpha}\\le C_{\alpha}\).
Next, we present some basic knowledge and definitions as regards fractional calculus theory, which can be found in the monographs of Podlubny [27], Miller and Ross [28], and Kilbas et al. [29].
Definition 2.1
Definition 2.2
Remark 2.1
Definition 2.3
We now state the following lemmas which will be used in the sequel.
Lemma 2.2
([13])
 (i)For any fixed \(t\ge0\) and \(x\in X_{\alpha}\), we find that the operators \(U(t)\) and \(V(t)\) are linear and bounded operators, i.e., for any \(x\in X\),$$\bigl\Vert U(t)x\bigr\Vert _{\alpha}\le M\x\_{\alpha}\quad \textit{and} \quad \bigl\Vert V(t)x\bigr\Vert _{\alpha}\le \frac{M}{\Gamma(q)}\x\_{\alpha}. $$
 (ii)
The operators \(U(t)\) and \(V(t)\) are strongly continuous for all \(t\ge0\).
 (iii)
\(U(t)\) and \(V(t)\) are compact operators in X for all \(t>0\).
 (iv)
For every \(t>0\), the restriction of \(U(t)\) to \(X_{\alpha}\) and the restriction of \(V(t)\) to \(X_{\alpha}\) are compact operators in \(X_{\alpha}\).
 (v)For all \(x\in X\) and \(t\in(0,\infty)\),$$\bigl\Vert A^{\alpha}V(t)x\bigr\Vert \le C_{\alpha}t^{\alpha q}\x\, \quad \textit{where } C_{\alpha}:= \frac{M_{\alpha}q\Gamma(2\alpha)}{\Gamma(1+q(1\alpha))}. $$
Lemma 2.3
(Krasnoselskii fixed point theorem [31, 32])
 (i)
\(Ax+By\in M\), wherever \(x,y\in M\);
 (ii)
A is completely continuous;
 (iii)
B is a contraction mapping.
3 Main results
Let \(x(T;x_{0},u)\) be the state value of (1.1) at terminal time T corresponding to the control u and the initial value \(x_{0}\). Introduce the set \(\mathfrak{R}(T,x_{0})=\{x(T;x_{0},u):u\in L^{2}([0,T],U)\} \), which is called the reachable set of system (1.1) at terminal time T, its closure in \(X_{\alpha}\) is denoted by \(\overline {\mathfrak{R}(T,x_{0})}\).
Definition 3.1
([7])
The system (1.1) is said to be approximately controllable on \([0,T]\) if \(\overline{\mathfrak{R}(T,x_{0})}=X_{\alpha}\), that is, given an arbitrary \(\epsilon>0\) it is possible to steer from the point \(x_{0}\) to within a distance ϵ from all points in the state space \(X_{\alpha}\) at time T.
Lemma 3.1
([8])
The linear fractional control system (3.1) is approximately controllable on \([0,T]\) if and only if \(\epsilon R(\epsilon,\Gamma_{0}^{T})\to0\) as \(\epsilon\to0^{+}\) in the strong operator topology.
 (H_{ f }):

\(f:[0,T]\times X_{\alpha}\times X_{\alpha}\to X\) is continuous, and for any \(r\in\mathbb{R}^{+}\), there exist a constant \(\gamma\in(0,(1\alpha)q)\) and functions \(\varphi_{r}\in L^{\frac{1}{\gamma}}([0,T],\mathbb {R}^{+})\) such that$$\sup\bigl\{ \bigl\Vert f(t,x,Gx)\bigr\Vert :\x\_{\alpha}\le r\bigr\} \le\varphi_{r}(t) \quad \mbox{and} \quad \liminf_{r\to\infty} \frac{\\varphi_{r}\_{L}^{\frac{1}{\gamma }}}{r}=\sigma< \infty. $$
 (H_{ h }):

\(h:\mathit{PC}([0,T],X_{\alpha})\to X_{\alpha}\) is a Lipschitz function with Lipschitz constant \(L_{h}\).
 (H_{ I }):

\(I:\mathit{PC}([0,T],X_{\alpha})\to X_{\alpha}\) is a Lipschitz function with Lipschitz constant \(L_{I}\).
 (H_{ c }):

The linear system associated with (1.1) is approximately controllable on \([0,T]\).
Theorem 3.1
Proof
Obviously, the fractional Cauchy problem (1.1) with the control in (3.4) has a mild solution if and only if the operator Λ has a fixed point on \(B_{r}\). According to the requirements of the Krasnoselskii theorem, our proof will be divided into several steps.
Step 2. We show that Π in (3.7) is completely continuous on \(B_{r(\epsilon)}\); here we divide the argument into two substeps.
Therefore, by means of the Krasnoselskii fixed point theorem, we conclude that Λ in (3.5) has a fixed point, which gives rise to a mild solution of Cauchy problem (1.1) with the control given in (3.4). The proof is completed. □
Theorem 3.2
If the hypothesis (H_{ c }) is satisfied, and the conditions of Theorem 3.1 hold, and, further, f, I, h are bounded in \(X_{\alpha}\), then the fractional Cauchy problem (1.1) is approximately controllable.
Proof
With the help of boundedness of the functions f and I, and the DunfordPettis theorem [34], stating that a class of random variables \(x_{n}\in L^{1}(\mu)\) is uniformly integrable if and only if it is relatively weakly compact, the sequences \(\{f(s,x_{\epsilon}(s),Gx_{\epsilon}(s))\}\) and \(\{ I_{k}(x_{\epsilon})\}\) are relatively weakly compact in \(L^{2}([0,T],X_{\alpha})\). There are subsequences still denoted by \(\{ f(s,x_{\epsilon}(s),Gx_{\epsilon}(s))\}\) and \(\{I_{k}(x_{\epsilon})\}\) weakly converging to \(f(s)\) and \(\tilde{I}_{k}\). Meanwhile, there exists \(\tilde{h}\in X_{\alpha}\) such that \(h(x_{\epsilon})\) converges to \(\tilde{h}\) weakly in \(X_{\alpha}\).
4 An example
Example 4.1
Let \(X_{\frac{1}{2}}=(D(A^{\frac{1}{2}}),\\cdot\_{\frac{1}{2}})\), \(B=E\) (identity operator), and \(U=X_{\frac{1}{2}}\), where \(\x\_{\frac{1}{2}}=\A^{\frac{1}{2}}x\\) for \(x\in D(A^{\frac{1}{2}})\).
With the above choice of f, h, and I, the system (4.1) can be written in the abstract form of system (1.1). Moreover, f, h, and I are bounded linear operators and satisfy the Lipschitz condition. All the conditions of Theorem 3.2 are fulfilled, so we can claim that the system (4.1) is approximately controllable.
Declarations
Acknowledgements
The authors would like to thank the referee for careful reading and valuable comments which led to an improvement of the article. This work has been supported by the National Natural Science Foundation of China (11361042, 11461045, 11071108), the Tian Yuan Special Funds of the National Natural Science Foundation of China (11426129, 11326099), the Provincial Natural Science Foundation of Jiangxi, China (20142BAB211004, 20142BAB201007, 20132BAB201001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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