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 \(\|A^{\beta }h(t, x)-A^{\beta }h(t, y)\|\leq L\|x-y\|\) 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
$$\bigl\Vert f(t,\varphi ,u) \bigr\Vert \leq L_{2}\bigl(1+ \Vert \varphi \Vert _{C}+ \Vert u \Vert \bigr),\quad (t,\varphi ,u) \in I\times C\times U. $$
\((P_{3})\) 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
$$\bigl\Vert f(t,\varphi _{1},u_{1})-f(t,\varphi _{2},u_{2}) \bigr\Vert \leq L_{3} \bigl( \Vert \varphi _{1}-\varphi _{2} \Vert _{C}+ \Vert u_{1}-u_{2} \Vert \bigr),\quad (\varphi _{1},u_{1}), (\varphi _{2},u_{2}) \in C \times U. $$
Define an operator Φ on \(C(I,C)\times C(I,U)\) as
$$ \Phi (x,u)=(z,v) $$
(13)
with the norm \(\|(x,u)\|=\|x_{t} \|_{C}+\|u \|\), \((x,u)\in C(I,C)\times C(I,U)\), \(t\in I\), where
$$\begin{aligned}& v(t)=B^{*}T_{q}^{*}(T-t) \bigl( \Gamma _{0}^{T} \bigr)^{-1}p(x,u), \end{aligned}$$
(14)
$$\begin{aligned}& \begin{aligned} &z(t)= S_{q}(t) \bigl[\phi (0)-h(0,x_{0}) \bigr] + h(t,x_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \\ &\hphantom{z(t)=}{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(Bv(s)+f \bigl(s,x_{s},u(s) \bigr) \bigr)\,ds, \quad t\in [0,T], \\ & z_{0}(\theta )= \phi (\theta ), \quad -r\leq \theta \leq 0, \end{aligned} \end{aligned}$$
(15)
\(p(x,u)= x_{T}-S_{q}(T)[ \phi (0) -h(0,x_{0})]-h(T,x_{T})-\int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,x_{s})\,ds - \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f(s,x_{s},u(s))\,ds\).
It will be shown that the system (1) is completely controllable on I if the operator Φ has a fixed point in \(C(I,C)\times C(I,U)\).
Theorem 3.1
Assume that the hypotheses \((P_{1})\)–\((P_{4})\) are satisfied. Then, the problem (1) has a unique mild solution in \(C([-r,T];X)\) provided that
$$\begin{aligned} \bigl\vert A^{-\beta } \bigr\vert L+ \biggl(1+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggr) \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) + \frac{M T^{q} L_{3} }{\Gamma (1+q)} < 1, \end{aligned}$$
(16)
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 Φ has a fixed point in \(C(I,C)\times C(I,U)\). Therefore, it is sufficient to prove that Φ has a fixed point in \(C(I,C)\times C(I,U)\). We first show that Φ maps \(C(I,C)\times C(I,U)\) into itself. Based on Lemma 2.3 and the condition \((P_{1})\), we have
$$\begin{aligned} & \biggl\Vert \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \biggr\Vert \\ &\quad \leq \int _{0}^{t} (t-s)^{q-1} A^{1-\beta }T_{q}(t-s) A^{\beta }h(s,x_{s})\,ds \\ &\quad \leq \int _{0}^{t} (t-s)^{q-1} \frac{q\Gamma (1+\beta )c_{1-\beta }}{(t-s)^{q(1-\beta )} \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{s} \Vert _{C} +1 \bigr)\,ds \\ &\quad \leq \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta },\quad \beta \in (0,1). \end{aligned}$$
(17)
According to Lemma 2.2(i) and the hypothesis \((P_{2})\), we have
$$ \biggl\Vert \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f \bigl(s,x_{s},u(s) \bigr)\,ds \biggr\Vert \leq \frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr). $$
(18)
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
$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert ={}& \biggl\Vert B^{*}T_{q}^{*}(T-t) \bigl(\Gamma _{0}^{T} \bigr)^{-1} \biggl( x_{T}-S_{q}(T) \bigl[ \phi (0) -h(0,x_{0}) \bigr]-h(T,x_{T}) \\ & {}- \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,x_{s})\,ds \\ &{} - \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) f \bigl(s,x_{s},u(s) \bigr)\,ds \biggr) \biggr\Vert \\ \leq{} & \frac{M_{B} Mq}{\gamma \Gamma (1+q)} \biggl[ \vert x_{T} \vert +M \Vert \phi \Vert +M \bigl\vert A^{- \beta } \bigr\vert L_{1} \bigl( \Vert \phi \Vert +1 \bigr) + \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert x_{T} \Vert +1 \bigr) \\ &{}+ \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta }+\frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \biggr] \\ \leq {}& C_{1} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \end{aligned}$$
(19)
and
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert ={}& \biggl\Vert S_{q}(t) \bigl[\phi (0)-h(0,x_{0}) \bigr] + h(t,x_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,x_{s})\,ds \\ &{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(Bv(s)+f \bigl(s,x_{s},u(s) \bigr) \bigr)\,ds \biggr\Vert \\ \leq{} & M \Vert \phi \Vert +M \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert \phi \Vert +1 \bigr) + \bigl\vert A^{-\beta } \bigr\vert L_{1} \bigl( \Vert x_{t} \Vert _{C}+1 \bigr) \\ &{}+ \frac{\Gamma (1+\beta )c_{1-\beta }}{\beta \Gamma (1+q\beta )} L_{1} \bigl( \Vert x_{t} \Vert _{C} +1 \bigr) T^{q\beta } \\ &{}+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} C_{1} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr)+ \frac{MT^{q}}{\Gamma (1+q)}L_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr) \\ \leq {}& C_{2} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr), \end{aligned}$$
(20)
where \(\beta \in (0,1)\). It follows from (15), (19) and (20) that there exists a constant \(C_{3}\) such that
$$ \bigl\Vert \Phi (x,u) \bigr\Vert = \Vert z \Vert _{C([-r,T];X)} + \Vert v \Vert \leq C_{3} \bigl(1+ \Vert x_{t} \Vert _{C}+ \Vert u \Vert \bigr), $$
(21)
which means that Φ maps \(C(I,C)\times C(I,U)\) into itself.
We next prove that the operator Φ 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
$$\begin{aligned} & \bigl\Vert \Phi (x,u)-\Phi (y,w) \bigr\Vert \\ &\quad = \Vert v_{1}-v_{2} \Vert + \Vert z_{1}-z_{2} \Vert _{C([-r,T];X)} \\ &\quad \leq \Vert v_{1}-v_{2} \Vert + \bigl\Vert h(t,x_{t})-h(t,y_{t}) \bigr\Vert + \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) B \bigl(v_{1}(s)-v_{2}(s) \bigr)\,ds \biggr\Vert \\ &\qquad {}+ \bigg\| \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) ( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr) \,ds\bigg\| \\ &\quad = I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$
(22)
By hypotheses \((P_{1})\)–\((P_{4})\), Lemma 2.2(i), and (17) and (18), we have
$$\begin{aligned} I_{1}=& \Vert v_{1}-v_{2} \Vert \\ =& \biggl\Vert B^{*}T_{q}^{*}(T-t) \bigl( \Gamma _{0}^{T} \bigr)^{-1} \bigg( \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s) \bigl(h(s,x_{s})-h(s,y_{s}) \bigr) \,ds \\ &+ \int _{0}^{T} (T-s)^{q-1}T_{q}(T-s) \bigl( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr)\,ds \bigr) \biggr\Vert \\ \leq & \frac{M_{B} Mq}{\gamma \Gamma (1+q)} \biggl( \frac{\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} \Vert x_{t}-y_{t} \Vert _{C} + \frac{M T^{q} L_{3} }{\Gamma (1+q)} \bigl( \Vert x_{t}-y_{t} \Vert _{C}+ \Vert u-w \Vert \bigr) \biggr) \\ \leq & \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr), \end{aligned}$$
(23)
where \(d=\frac{M_{B} Mq}{\gamma \Gamma (1+q)}\). The condition \((P_{1})\) implies
$$\begin{aligned} I_{2}=& \bigl\Vert h(t,x_{t})-h(t,y_{t}) \bigr\Vert \leq \bigl\vert A^{-\beta } \bigr\vert L \Vert x-y \Vert _{C([-r,T];X)}. \end{aligned}$$
(24)
Based on Lemma 2.2(i) and (23), one can obtain
$$\begin{aligned} I_{3}={}& \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) B \bigl(v_{1}(s)-v_{2}(s) \bigr)\,ds \biggr\Vert \\ \leq{} & \frac{M M_{B} T^{q}}{\Gamma (1+q)} \Vert v_{1}-v_{2} \Vert \\ \leq {}& \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} + \frac{dM T^{q} L_{3} }{\Gamma (1+q)} \biggr) \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$
(25)
Similar to the discussion of \(I_{1}\), we obtain
$$\begin{aligned} I_{4}={}& \biggl\Vert \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl( f \bigl(s,x_{s},u(s)- f \bigl(s,y_{s},w(s) \bigr) \bigr)\bigr)\,ds \biggr\Vert \\ \leq{} & \frac{M T^{q} L_{3} }{\Gamma (1+q)} \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$
(26)
Then, (22)–(26) imply
$$\begin{aligned} & \bigl\Vert \Phi (x,u)-\Phi (y,w) \bigr\Vert _{C([-r,T];X)} \\ &\quad \leq \biggl[ \bigl\vert A^{-\beta } \bigr\vert L+ \biggl(1+ \frac{M M_{B} T^{q}}{\Gamma (1+q)} \biggr) \biggl( \frac{ d\Gamma (1+\beta )c_{1-\beta }L T^{q\beta }}{\beta \Gamma (1+q\beta )} \biggr) \\ &\qquad {} +\frac{M T^{q} L_{3} }{\Gamma (1+q)} \biggr] \bigl( \Vert x-y \Vert _{C([-r,T];X)}+ \Vert u-w \Vert \bigr). \end{aligned}$$
(27)
In view of (16), we obtain that Φ is a contraction. Consequently, Φ has a fixed point in \(C(I,C)\times C(I,U)\) by the Banach fixed-point theorem, which is a mild solution of the system (1). This completes the proof. □
Theorem 3.2
If all the assumptions of Theorem 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 Φ in (13), that is
$$ \Phi \bigl(\bar{x} (\cdot ),\bar{u} \bigr)= \bigl(\bar{x} (\cdot ),\bar{u} \bigr), $$
(28)
where
$$ \begin{aligned} &\bar{x}(t)= S_{q}(t) \bigl[ \phi (0)-h(0,x_{0}) \bigr] + h(t,\bar{x}_{t})+ \int _{0}^{t} (t-s)^{q-1} AT_{q}(t-s)h(s,\bar{x}_{s})\,ds \\ &\hphantom{\bar{x}(t)=}{}+ \int _{0}^{t} (t-s)^{q-1}T_{q}(t-s) \bigl(B\bar{u}(s)+f \bigl(s, \bar{x}_{s},\bar{u}(s) \bigr) \bigr)\,ds, \quad t \in [0,T], \\ & \bar{x}_{0}(\theta )= \phi (\theta ), \quad -r\leq \theta \leq 0 \end{aligned} $$
(29)
and the control function
$$ \bar{u}(t)=B^{*}T_{q}^{*}(T-t) \bigl(\Gamma _{0}^{T} \bigr)^{-1} p(\bar{x}, \bar{u} ), $$
(30)
here \(p(\bar{x},\bar{u})=x_{T}-S_{q}(T)[\phi (0)-h(0,x_{0})] + h(T, x_{T})+ \int _{0}^{T} (T-s)^{q-1} AT_{q}(T-s)h(s,\bar{x}_{s})\,ds - \int _{0}^{T} (T-s)^{q-1}T_{q}(T-s) f(s,\bar{x}_{s},\bar{u}(s))\,ds\).
According to Theorem 3.1, any fixed point of Φ is a mild solution of the system (1). Then, by (12), (29) and (30), we have
$$\begin{aligned} \bar{x}(T) ={}& x_{T}- p(\bar{x},\bar{u})+ \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s) B\bar{u}(s)\,ds \\ ={}& x_{T}- p(\bar{x},\bar{u})+ \int _{0}^{T} (T-s)^{q-1} T_{q}(T-s)B B^{*}T_{q}^{*}(T-s) \bigl( \Gamma _{0}^{T} \bigr)^{-1} p(\bar{x},\bar{u})\,ds \\ ={}& x_{T}- p(\bar{x},\bar{u} ) + p(\bar{x},\bar{u}) \\ ={}& x_{T}. \end{aligned}$$
(31)
Thus, the system (1) is approximately controllable on I by Definition 3.1. The proof is completed. □