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Approximate controllability of fractional nonlocal evolution equations with multiple delays
- He Yang^{1}Email authorView ORCID ID profile and
- Elyasa Ibrahim^{1}
https://doi.org/10.1186/s13662-017-1334-8
© The Author(s) 2017
Received: 24 April 2017
Accepted: 26 August 2017
Published: 7 September 2017
Abstract
This paper deals with the existence and approximate controllability for a class of fractional nonlocal control systems governed by abstract fractional evolution equations with multiple delays. Under some weaker assumptions, the existence as well as the approximate controllability is established by using fixed point theory. An example is given to illustrate the applicability of the abstract results.
Keywords
- fractional evolution equations
- nonlocal initial condition
- multiple delays
- compact analytic semigroup
- approximate controllability
MSC
- 34A08
- 34K35
1 Introduction
Since the fractional differential equations have extensive physical background and realistic mathematical model, the theory has considerably developed in recent years; see [1–17] and the references therein. And the nonlocal initial conditions have better effects in applications than the classical ones; see [3, 8, 10, 16, 18–21] and the references therein. Therefore, the theory of fractional nonlocal differential equations has been a research field of focus in recent years.
Controllability of deterministic and stochastic dynamical control systems in infinite-dimensional spaces is well-developed in which the details can be found in various papers; see [3, 7–11, 13, 14, 22]. Several authors [3, 10, 14] investigated the exact controllability of control systems represented by nonlinear fractional evolution equations using fixed point approach. Debbouche and Baleanu [3] established the exact null controllability result for a class of fractional integro-differential control systems governed by nonlinear fractional evolution equations with nonlocal initial conditions in Banach spaces. Liang and Yang [10] investigated the exact controllability for a class of fractional integro-differential control systems represented by nonlinear fractional evolution equations involving specific nonlocal functions. Sakthivel et al. [14] studied the exact controllability for a class of fractional neutral control systems governed by abstract nonlinear fractional neutral evolution equations.
However, in infinite-dimensional spaces, the concept of exact controllability is usually too strong [22]. Therefore, it is necessary to present a weaker concept of controllability, namely approximate controllability for nonlinear control systems. In the recent literature, the approximate controllability of nonlinear fractional evolution systems has not yet been sufficiently studied. More precisely, there are limited papers regarding the approximate controllability of abstract nonlinear fractional evolution systems under different conditions [8, 9, 11, 13]. Kumar and Sukavanam [9] obtained a new set of sufficient conditions of approximate controllability for a class of semilinear fractional control systems involving delay. Mahmudov and Zorlu [11] established the sufficient conditions of approximate controllability for certain classes of abstract fractional evolution control systems. Sakthival et al. [13] investigated the approximate controllability for a class of nonlinear fractional dynamical systems governed by abstract fractional evolution equations with nonlocal conditions.
The rest of this paper is organized as follows. In Section 2, some preliminaries are given on fractional calculus and fractional power of generator of compact analytic semigroup. In Section 3, we study the existence of mild solutions for fractional nonlocal control system (1.1). The approximate controllability of fractional nonlocal control system (1.1) is discussed in Section 4. In Section 5, an example is given to illustrate the applicability of the abstract results.
2 Preliminaries
Lemma 2
[23]
\(A^{\alpha}S(t)\) is bounded in X for any \(t>0\) and there exists a constant \(M_{\alpha}>0\) such that \(\| A^{\alpha}S(t)\|\leq M_{\alpha}t^{-\alpha}\).
Lemma 3
[21]
\(S_{\alpha}(t)\) (\(t\geq0\)) is a compact semigroup in \(X_{\alpha}\), and hence it is norm-continuous.
Denote by \(C([-r,T],X_{\alpha})\) the Banach space of all continuous \(X_{\alpha}\)-valued functions on the interval \([-r,T]\) with norm \(\|x\| _{C}=\max_{t\in[-r,T]}\|x(t)\|_{\alpha}\) for any \(x\in C([-r,T],X_{\alpha})\). Similarly, denote by \(C([-r,0],X_{\alpha})\) the Banach space of all continuous \(X_{\alpha}\)-valued functions on the interval \([-r,0]\) with norm \(\|x\|_{C[-r,0]}=\max_{t\in [-r,0]}\|x(t)\|_{\alpha}\) for any \(x\in C([-r,0],X_{\alpha})\). In this paper, we adopt the following definition of the mild solution of fractional nonlocal evolution equation (1.1).
Definition 1
Lemma 4
- (i)
For fixed \(t\geq0\) and any \(x\in X_{\alpha}\), \(\|U(t)x\| _{\alpha}\leq M\|x\|_{\alpha}, \|V(t)x\|_{\alpha}\leq\frac {M}{\Gamma(q)}\|x\|_{\alpha}\).
- (ii)
For fixed \(t>0\) and any \(x\in X\), \(\|V(t)x\|_{\alpha}\leq C_{\alpha}t^{-q\alpha}\|x\|\), where \(C_{\alpha}=\frac{M_{\alpha}q\Gamma(2-\alpha)}{\Gamma (1+q(1-\alpha))}\).
- (iii)
\(U(t)\) and \(V(t)\) are strongly continuous for all \(t\geq0\).
- (iv)
\(U(t)\) and \(V(t)\) are norm-continuous in X for \(t>0\).
- (v)
\(U(t)\) and \(V(t)\) are compact operators in X for \(t>0\).
- (vi)
For every \(t>0\), the restriction of \(U(t)\) to \(X_{\alpha}\) and the restriction of \(V(t)\) to \(X_{\alpha}\) are norm-continuous.
- (vii)
For every \(t>0\). the restriction of \(U(t)\) to \(X_{\alpha}\) and the restriction of \(V(t)\) to \(X_{\alpha}\) are compact operators.
Let \(x(T; u)\) be the state value of fractional nonlocal evolution equation (1.1) at terminal time T corresponding to control u. Introduce the set \(\mathcal{R}(T)\) by \(\mathcal{R}(T):=\{x(T; u): u\in L^{2}(J, Y)\}\). \(\overline{\mathcal{R}(T)}\) denotes its closure in \(X_{\alpha}\).
Definition 2
Approximate controllability
The fractional nonlocal control system (1.1) is called approximately controllable on the interval \([-r,T]\) if \(\overline{\mathcal {R}(T)}=X_{\alpha}\).
Lemma 5
[11]
The linear fractional differential system (2.1) is approximately controllable on the interval [0, T] if and only if \(\epsilon R(\epsilon , \Gamma_{0}^{T})\rightarrow0\) as \(\epsilon\rightarrow0^{+}\) in the strong operator topology.
Definition 3
Remark 1
It is clear if \(\phi(\tau)\equiv k\tau\) for some \(k\in(0,1)\), the nonlinear contraction mapping degenerates into contraction mapping.
Lemma 6
[24]
- (a)
\(Q_{1}\) is a nonlinear contraction, and
- (b)
\(Q_{2}\) is completely continuous.
- (i)
the operator equation \(x=Q_{1}x+Q_{2}x\) has a solution, or
- (ii)
the set \(\Sigma:=\{x\in E: \lambda(Q_{1}x+Q_{2}x)=x, 0<\lambda<1\} \) is unbounded.
Lemma 7
[15]
3 Existence of mild solutions
We make the following assumptions:
Remark 2
The condition \((H_{1})\) can be replaced by the condition
Lemma 8
If the assumption \((H_{2})\) holds, \(Q_{1}\) is a nonlinear contraction.
Proof
Lemma 9
If the assumptions \((H_{1})\) and \((H_{2})\) hold, \(Q_{2}\) is completely continuous.
Proof
Theorem 1
Assume that the conditions \((H_{1})\) and \((H_{2})\) hold. Then the fractional nonlocal control system (1.1) has at least one mild solution.
Proof
Define two operators \(Q_{1}, Q_{2}: C([-r,T], X_{\alpha })\rightarrow C([-r,T], X_{\alpha})\) as in (3.1) and (3.3). By Lemma 8 and 9, it follows that all the conditions of Lemma 6 are satisfied and a direct application of Lemma 6 shows that either the conclusion (i) or the conclusion (ii) holds. We next show that the conclusion (ii) is not possible. Equivalently, we prove that the set \(\Sigma:=\{x\in C([-r,T], X_{\alpha}): \lambda(Q_{1}x+Q_{2}x)=x, 0<\lambda<1\}\) is bounded.
Remark 3
Even if \(g(x)\equiv0\) and without control u in the fractional nonlocal control system (1.1), Theorem 1 is still new.
The condition \((H_{2})\) can be replaced by the following condition:
Theorem 2
Let the conditions \((H_{1})\) and \((H_{2})'\) hold. Then the fractional nonlocal control system (1.1) has at least one mild solution.
Proof
Remark 4
In some existing literature, see [2, 14, 16, 17], the authors always assume that \(f(t,x)\leq m(t)\) with some functions \(m\in L^{1}(J, {\mathbb{R}}^{+})\) independent of x, and g is either completely continuous or Lipschitz continuous and the coefficients satisfy some inequality conditions. But in Theorems 1 and 2, we only assume that the conditions \((H_{1})\) and \((H_{2})\) (or \((H_{2})'\)) hold. Hence, Theorems 1 and 2 greatly extend the main results of [2, 13, 14, 16, 17].
4 Approximate controllability
To prove the approximate controllability of (1.1), the following assumption is required:
\((H_{5})\) The function \(f: J\times X_{\alpha}^{n+1}\rightarrow X\) is bounded.
\((H_{6})\) The linear fractional control system (2.1) is approximately controllable.
Theorem 3
In addition to the assumptions of Theorem 1, suppose that the conditions \((H_{5})\) and \((H_{6})\) hold. Then the fractional nonlocal control system (1.1) is approximately controllable.
Proof
Theorem 4
In addition to the assumptions of Theorem 2, suppose that the conditions \((H_{5})\) and \((H_{6})\) hold. Then the fractional nonlocal control system (1.1) is approximately controllable.
Proof
The proof is similar to the proof of Theorem 3, and we omit it here. □
5 Existence and approximate controllability of delay parabolic equations
(A2) \(\phi_{0}\in C^{\mu}(\overline{\Omega})\) for some \(\mu\in (0,1)\), and \(\phi_{0}(z)\geq0\) on Ω̅.
By Theorem 1, we have the following existence result.
Theorem 5
Assume that the following conditions are satisfied:
Then the delay parabolic boundary value problem (5.2) has at least one mild solution.
By Theorem 3, we have the following approximate controllability result.
Theorem 6
In addition to the assumptions of Theorem 5, suppose that the function F is bounded and the following condition holds:
\((P_{3})\) The linear fractional control system corresponding to (5.2) is approximately controllable.
Then the delay parabolic boundary value problem (5.2) is approximately controllable.
By Theorem 4, we obtain the following theorem.
Theorem 7
Assume that the conditions \((P_{1})\), \((P_{3})\) and
\((P_{4})\) The function \(g: C([-r,T], X_{\alpha})\rightarrow X_{\alpha }\) is Lipschitz continuous with constant \(L_{1}\in(0,\frac{1}{M})\).
Then the delay parabolic boundary value problem (5.2) is approximately controllable.
Example 1
Hence, by Theorem 7, if the linear fractional control system corresponding to (5.3) is approximately controllable, then the delay parabolic boundary value problem (5.3) is approximately controllable.
6 Conclusion
In this paper, the approximate controllability of a nonlocal control system governed by fractional multi-delay evolution equation is investigated. By using fixed point theory and also the Gronwall-Bellman inequality of fractional order, sufficient conditions of approximate controllability are obtained. As an example, the approximate controllability of fractional multi-delay parabolic boundary value problem is discussed and the abstract result is applied to a special fractional multi-delay parabolic function.
Declarations
Acknowledgements
The research is supported by the NSF (No. 11661071).
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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