 Research
 Open access
 Published:
Approximate controllability of fractional nonlocal evolution equations with multiple delays
Advances in Difference Equations volume 2017, Article number: 272 (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.
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 infinitedimensional spaces is welldeveloped 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 integrodifferential 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 integrodifferential 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 infinitedimensional 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.
However, to the best of our knowledge, the approximate controllability for nonlinear fractional nonlocal control systems governed by abstract fractional evolution equations involving multiple delays and compact analysis semigroup has not been investigated yet and it is also the motivation of this paper. In this paper, we consider the existence and approximate controllability for fractional nonlocal control system
where \(D^{q}\) is Caputo fractional derivative of order \(q\in(0, 1)\), \(T>0\) is a constant, A generates a compact analytic semigroup \(S(t)\) (\(t\geq0\)) of uniformly bounded linear operator, \(u\in L^{2}(J, Y)\) is a control, Y is a Banach space, \(B: Y\rightarrow X_{\alpha}\) is a linear bounded operator, \(\tau_{1}, \tau_{2}, \ldots, \tau_{n}\) are positive constants, \(r=\max\{\tau_{1}, \tau_{2}, \ldots, \tau_{n}\}\), \(\varphi: [r, 0]\rightarrow X_{\alpha}\) is continuous, f and g are given functions and will be specified later. The space \(X_{\alpha }\) will be specified later.
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
Let \((X, \\cdot\)\) be a Banach space. Throughout this paper, we always assume that \(A: D(A)\subset X\rightarrow X\) generates a compact analytic semigroup \(S(t)\) (\(t\geq0\)) of uniformly bounded linear operator in X. Then there exists a constant \(M\geq1\) such that \(\S(t)\\leq M\) for all \(t\geq0\). Let \(0\in\rho(A)\). Then
for some \(0<\alpha<1\). Thus, we define \(A^{\alpha}\) by \(A^{\alpha }=(A^{\alpha})^{1}\) with \(D(A^{\alpha})=A^{\alpha}(X)\). Let \(X_{\alpha}\) be the Banach space of \(D(A^{\alpha})\) with norm \(\x\ _{\alpha}:=\A^{\alpha}x\\) for any \(x\in D(A^{\alpha})\). Denote by \(S_{\alpha}(t)\) the restriction of \(S(t)\) to \(X_{\alpha}\) for all \(t\geq0\). Then \(S_{\alpha}(t)\) (\(t\geq0\)) is a \(C_{0}\)semigroup in \(X_{\alpha}\) and \(\S_{\alpha}(t)\_{\alpha}\leq\S(t)\\) for all \(t\geq0\). To prove our main results, the following lemmas are needed.
Lemma 1
[23]
\(A^{\alpha}\) is a bounded linear operator in X for any \(\alpha>0\).
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 normcontinuous.
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
By the mild solution of fractional nonlocal evolution equation (1.1), we mean a function \(x\in C([r,T],X_{\alpha})\) satisfying initial condition \(x(t)+g(x)=\varphi (t), t\in[r,0]\) and integral equation
where \(F(x)(s)=f(s,x(s),x(s\tau_{1}),\ldots,x(s\tau_{n}))\), the operators \(\{U(t)\}_{t\geq0}\) and \(\{V(t)\}_{t\geq0}\) are defined by
where
It is well known that \(\eta_{q}(\theta)\geq0\) for all \(\theta\in (0,\infty)\) and
Lemma 4
The operators \(\{U(t)\}_{t\geq0}\) and \(\{ V(t)\}_{t\geq0}\) satisfy the following properties:

(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 normcontinuous 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 normcontinuous.

(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 \(\overline{\mathcal {R}(T)}=X_{\alpha}\).
Consider the following linear fractional differential system:
Let
where \(B^{*}\) and \(V^{*}(t)\) denote the adjoint of B and \(V(t)\), respectively. It is well known that \(\Gamma_{0}^{T}\) is a linear bounded operator and \(\R(\epsilon, \Gamma_{0}^{T})\\leq\frac{1}{\epsilon}\).
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
Let E be a Banach space with norm \(\\cdot\_{E}\). A mapping \(Q: E\rightarrow E\) is called a nonlinear contraction if there exists a continuous and nondecreasing function \(\phi: {\mathbb {R}}^{+}\rightarrow{\mathbb{R}}^{+}\) satisfying
for all \(x,y\in E\) with \(\phi(\tau)<\tau\) for \(\tau>0\).
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]
Let E be a Banach space and let \(Q_{1}, Q_{2}: E\rightarrow E\) be two operators satisfying

(a)
\(Q_{1}\) is a nonlinear contraction, and

(b)
\(Q_{2}\) is completely continuous.
Then either

(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]
Let \(\sigma>0\) and let \(a_{1}(t)\) be a nonnegative nondecreasing function locally integrable on \(0\leq t< b\) (some \(b\leq+\infty\)) and \(a_{2}(t)\) be a nonnegative nondecreasing continuous function defined on \(0\leq t< b, a_{2}(t)\leq\widetilde{M}\) (constant). Suppose that \(x(t)\) is nonnegative and local integrable on \(0\leq t< b\) with
Then
3 Existence of mild solutions
We make the following assumptions:
\((H_{1})\) The function \(f: J\times X_{\alpha}^{n+1}\rightarrow X\) is continuous and there exist positive constants \(\beta_{0}, \beta_{1}, \ldots, \beta_{n}\) and \(K\geq0\) such that
\((H_{2})\) The function \(g: C([r,T], X_{\alpha})\rightarrow X_{\alpha }\) is continuous and there exists a constant \(L\geq M\) such that
Remark 2
The condition \((H_{1})\) can be replaced by the condition
\((H_{1})'\) The function \(f: J\times X_{\alpha}^{n+1}\rightarrow X\) is continuous and there exist functions \(\overline{\beta}_{i}\in L^{1}(J, {\mathbb{R}} ^{+}), i=0,1,\ldots,n\) and nondecreasing function \(\Phi: [0,+\infty )\rightarrow(0,+\infty)\) such that
for any \(t\in J, x\in C([r,T],X_{\alpha})\), where \(\tau_{0}=0\).
By \((H_{1})'\), we have
Let \(\Phi(r)=r\) for some \(r>0\). Then \((H_{1})'\Rightarrow(H_{1})\) with \(\beta_{i}=\\overline{\beta}_{i}\_{L^{1}}\) and \(K=0\). Since \((H_{1})'\) would not be an essential generalization, we only consider \((H_{1})\) in the following.
Define an operator \(Q_{1}: C([r,T], X_{\alpha})\rightarrow C([r,T], X_{\alpha})\) by
Lemma 8
If the assumption \((H_{2})\) holds, \(Q_{1}\) is a nonlinear contraction.
Proof
Let \(x,y\in C([r,T], X_{\alpha})\). For \(t\in[r,0]\), we have
For \(t\in[0,T]\), we have
This implies that
Let \(\phi(r)=\frac{Mr}{L+r}\). Then \(\phi: {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\) is continuous and nondecreasing and \(\phi(r)< r\) for \(r>0\). Hence \(Q_{1}\) is a nonlinear contraction in \(C([r,T], X_{\alpha})\). This completes the proof. □
For any \(\epsilon>0\) and \(h_{1}\in X_{\alpha}\), define a control \(u(t):=u(t;x)\) by
Then, from assumptions \((H_{1})\) and \((H_{2})\), we have
where
Define an operator \(Q_{2}: C([r,T], X_{\alpha})\rightarrow C([r,T], X_{\alpha})\) by
Lemma 9
If the assumptions \((H_{1})\) and \((H_{2})\) hold, \(Q_{2}\) is completely continuous.
Proof
By the assumptions \((H_{1})\) and \((H_{2})\), it is easy to prove that \(Q_{2}: C([r,T], X_{\alpha})\rightarrow C([r,T], X_{\alpha})\) is continuous. So, it remains to prove that \(Q_{2}\) is a compact operator on \(C([r,T], X_{\alpha})\). The case \(t\leq0\) is trivial. Thus, let \(t\in (0,T]\) be fixed. For each \(\delta\in(0,t), \rho>0\) and \(x\in B_{r}:=\{x\in C([r,T], X_{\alpha}): \x\_{C}\leq r\}\), \(r>0\), we define \(Q_{2}^{\delta,\rho}\) by
Then the set \(\{(Q_{2}^{\delta,\rho}x)(t): x\in B_{r}\}\) is relatively compact in \(X_{\alpha}\) because of Lemma 3. By \((H_{1})\), \((H_{2})\) and (3.2), we have
This implies that the set \(\{(Q_{2}x)(t): x\in B_{r}\}\) is relatively compact in \(X_{\alpha}\) for all \(t\in(0, T]\). Hence, we obtain the relative compactness of \((Q_{2}B_{r})(t)\) in \(X_{\alpha}\) for all \(t\in[r, T]\). We further show that the operator \(Q_{2}\) is equicontinuous in \(C([r,T], X_{\alpha})\). For \(x\in C([r,T], X_{\alpha})\), if \(\nu\in[0,T)\), we have
as \(\nu\rightarrow0\). Hence, it is only necessary to consider the case \(t>0\). For \(0< t_{1}< t_{2}\leq T\), denote
By assumptions \((H_{1}), (H_{2})\) and (3.2), we can find
This implies that \(I_{i}\rightarrow0, i=2,3\) as \(t_{2}t_{1}\rightarrow0\). For \(t_{1}>0\) and \(\eta\in(0,t_{1})\) small enough, we have
Since compact semigroup is equicontinuous semigroup, it follows that \(I_{1}\rightarrow0\) as \(t_{2}t_{1}\rightarrow0\) and \(\eta\rightarrow0\). Therefore, from the inequality
we see that the operator \(Q_{2}\) is equicontinuous in \(C([r,T], X_{\alpha})\). Hence, by the AscoliArzela theorem, \(Q_{2}\) is a compact operator in \(C([r,T], X_{\alpha})\). This completes the 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.
Let \(x\in C([r,T], X_{\alpha})\) satisfy the operator equation \(x=\lambda(Q_{1}x+Q_{2}x)\) for some \(\lambda\in(0,1)\). Then, for any \(t\in[r, 0]\), by assumption \((H_{2})\), we have
For \(t\geq0\), by assumptions \((H_{1})\) and \((H_{2})\), we have
where \(\tau_{0}=0\) and \(C=M\\varphi\_{C[r,0]}+M(1+\g(0)\_{\alpha })+\frac{KC_{\alpha}T^{q(1\alpha)}}{q(1\alpha)}+\frac {ML_{u}T^{q}}{\Gamma(q+1)}\).
Let
Then \(\psi\in C([r, T], {\mathbb{R}}^{+})\) and \(\x(t)\_{\alpha}\leq \psi(t)\) for \(t\in[r, T]\). For every \(t\geq0\), by the definition of ψ, there exists \(\theta_{t}\in[r, t]\) such that \(\psi(t)=\x(\theta_{t})\ _{\alpha}\).
If \(r\leq\theta_{t}\leq0\), we have
If \(\theta_{t}>0\), we have
Using the wellknown singular version of Gronwall inequality [12], we can deduce that there exists a constant \(\overline{M}_{2}>0\) such that \(\psi(t)\leq\overline{M}_{2}\). Thus, for any \(t\geq0\), we have
Consequently,
This implies that the set Σ is bounded. Therefore, by Lemma 6, the operator equation \(x=Q_{1}x+Q_{2}x\) has at least one fixed point which is the mild solution of the fractional control system (1.1) on \(C([r,T], X_{\alpha})\). This completes the proof. □
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:
\((H_{2})'\) The function \(g: C([r,T], X_{\alpha})\rightarrow X_{\alpha }\) is Lipschitz continuous with constant \(L_{1}\in(0,\frac{1}{M})\), that is, for any \(x,y\in C([r,T], X_{\alpha})\), we have
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
By the condition \((H_{2})'\), similar to the proof as in Lemma 8, we obtain
Let \(\phi(r)=ML_{1}r\). Then \(\phi: {\mathbb{R}}^{+}\rightarrow{\mathbb {R}}^{+}\) is continuous and nondecreasing and \(\phi(r)< r\) for \(r>0\). Hence \(Q_{1}\) is a nonlinear contraction on \(C([r,T], X_{\alpha})\). The remaining proof is similar to the proof of Theorem 1, we omit it here. This completes the 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
Let \(x_{\epsilon}\) be a fixed point of the operator \(Q_{1}+Q_{2}\) on \(C([r,T], X_{\alpha})\). By Theorem 1, \(x_{\epsilon}\) is a mild solution of fractional nonlocal control system (1.1) with control
where
A direct calculation shows that \(x_{\epsilon}\) satisfies
Moreover, by the assumption \((H_{5})\), there exists a constant \(N>0\) such that
Consequently, there is a sequence still denoted by \(\{F(x_{\epsilon })(s)\}\) weakly converging to, say, \(\{F(s)\}\) in \(L^{2}(J, X)\). Denote
Now, we have
By using infinitedimensional version of the AscoliArzela theorem, we see that the operator \(\ell(\cdot)\rightarrow\int_{0}^{\cdot}(\cdot s)^{q(1\alpha)1}\ell(s)\,ds: L^{2}(J,X)\rightarrow C(J,X)\) is compact. Hence, \(\P(x_{\epsilon})\rho\_{\alpha}\rightarrow0\) as \(\epsilon \rightarrow0^{+}\). Moreover, by (4.1), we have
Hence, by Lemma 5 and \((H_{5})\), we have
as \(\epsilon\rightarrow0^{+}\). This proves the approximate controllability of fractional nonlocal control system (1.1). This completes the 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. □
Remark 5
Compared with [11], the present paper studies the fractional control system (1.1) with nonlocal condition and delays, and we suppose that the function f maps \(J\times X_{\alpha}^{n+1}\) to X. Hence, our results extend some existing results.
5 Existence and approximate controllability of delay parabolic equations
Let \(\Omega\in{\mathbb{R}}^{N}\) be a bounded domain, whose boundary ∂Ω is sufficiently smooth. Let
be a uniformly elliptic differential operator of divergence form in Ω̅. We assume that following conditions are satisfied:
(A1) \(\phi_{ij}\in C^{1+\mu}(\overline{\Omega})\) (\(i,j=1,2,\ldots ,N\)) for some \(\mu\in(0,1)\), \([\phi_{ij}(z)]_{N\times N}\) is a positive definite symmetric matrix for \(z\in\overline{\Omega}\) and there exists a constant \(\ell>0\) such that
(A2) \(\phi_{0}\in C^{\mu}(\overline{\Omega})\) for some \(\mu\in (0,1)\), and \(\phi_{0}(z)\geq0\) on Ω̅.
Let \(X=L^{2}(\Omega)\). Define an operator \(A: D(A)\subset X\rightarrow X\) by
Then, from [25], A generates a compact analysis semigroup \(S(t)\) (\(t\geq 0\)) in X and there exists a constant \(M\geq1\) such that \(\S(t)\ \leq M\). It is wellknown that \(D(A^{\frac{1}{2}})=H_{0}^{1}(\Omega)\).
Under above assumptions we discuss the approximate controllability of delay parabolic boundary value problem
where \(\frac{\partial^{q}}{\partial t^{q}}\) is the Caputo fractional partial derivative of order \(q\in(0,1)\), \(\omega>0\) is a constant, \(\tau_{1},\ldots,\tau_{n}\) are positive constants, \(r=\max\{\tau _{1},\ldots,\tau_{n}\}, \varphi: \Omega\times[r,0]\rightarrow {\mathbb{R}}\).
By Theorem 1, we have the following existence result.
Theorem 5
Assume that the following conditions are satisfied:
\((P_{1})\) The function \(F: \overline{\Omega}\times[0,T]\times{\mathbb{R}} ^{n+1}\rightarrow{\mathbb{R}}\) is continuous and there exist positive constants \(\beta_{0}, \beta_{1}, \ldots, \beta_{n}\) and K such that
\((P_{2})\) The function g is continuous and there exists a constant \(L\geq M\) such that
for all \(x, y\in{\mathbb{R}}\).
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
Consider the approximate controllability of delay parabolic boundary value problem
where \(m\leq T\) and \(\gamma_{k}\in{\mathbb{R}}, k=1,2,\ldots, m\) with \(\sum_{k=1}^{m}\gamma_{k}\leq\frac{1}{M}\).
Let
Then
where \(\tau_{0}=0\).
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 multidelay evolution equation is investigated. By using fixed point theory and also the GronwallBellman inequality of fractional order, sufficient conditions of approximate controllability are obtained. As an example, the approximate controllability of fractional multidelay parabolic boundary value problem is discussed and the abstract result is applied to a special fractional multidelay parabolic function.
References
Ahmad, B, Nieto, J: Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions. Comput. Math. Appl. 58, 18381843 (2009)
Agarwal, R, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 10951100 (2010)
Debbouche, A, Baleanu, D: Controllability of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems. Comput. Math. Appl. 62, 14421450 (2011)
ElBorai, M: Exact solutions for some nonlinear fractional parabolic partial differential equations. Appl. Math. Comput. 206, 150153 (2008)
Glockle, W, Nonnenmacher, T: A fractional calculus approach of selfsimilar protein dynamics. J. Biophys. 68, 4653 (1995)
Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Kailasavalli, S, Baleanu, D, Suganya, S, Arjunan, M: Exact controllability of fractional neutral integrodifferential systems with statedependent delay in Banach spaces. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa 24, 2955 (2016)
Kerboua, M, Debbouche, A, Baleanu, D: Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces. Electron. J. Qual. Theory Differ. Equ. 2014, 58 (2014)
Kumar, S, Sukavanam, N: Approximate controllability of fractional onder semilinear systems with bounded deday. J. Differ. Equ. 251, 61636174 (2012)
Liang, J, Yang, H: Controllability of fractional integrodifferential evolution equations with nonlocal conditions. Appl. Math. Comput. 254, 2029 (2015)
Mahmudov, N, Zorlu, S: On the approximate controllability of fractional evolution equations with compact analytic semigroup. J. Comput. Appl. Math. 259, 194204 (2014)
Ravichandran, C, Baleanu, D: On the controllability of fractional functional integrodifferential systems with an infinite delay in Banach spaces. Adv. Differ. Equ. 2013, 291 (2013)
Sakthivel, R, Ganesh, R, Ren, Y, Anthoni, S: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18, 34983508 (2013)
Sakthivel, R, Mahmudov, N, Nieto, J: Controllability for a class of fractionalorder neutral evolution control systems. Appl. Math. Comput. 218, 1033410340 (2012)
Ye, HP, Gao, JM, Ding, YS: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 10751081 (2007)
Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 44654475 (2010)
Zhou, Y, Jiao, F: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 10631077 (2010)
Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 419437 (1991)
Byszewski, L, Lakshmikantham, V: Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 1119 (1991)
Liang, J, Van Casteren, J, Xiao, TJ: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Anal. 50, 173189 (2002)
Liu, H, Chang, JC: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70, 30763083 (2009)
Mahmudov, N: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42, 16041622 (2003)
Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Dhage, B: A nonlinear alternative with applications to nonlinear perturbed differential equations. Nonlinear Stud. 13, 343354 (2006)
Li, YX: Existence and asymptotic stability of periodic solution for evolution equations with delays. J. Funct. Anal. 261, 13091324 (2011)
Acknowledgements
The research is supported by the NSF (No. 11661071).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
None of the authors have any competing interests in the manuscript.
Authors’ contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Yang, H., Ibrahim, E. Approximate controllability of fractional nonlocal evolution equations with multiple delays. Adv Differ Equ 2017, 272 (2017). https://doi.org/10.1186/s1366201713348
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201713348