A discussion on controllability of nonlocal fractional semilinear equations of order $1<r<2$ with monotonic nonlinearity

Definition 2.1

(Kilbas et al., 2006) “Provided that $p\left(\varrho \right)\in L_1\left(\right[0,b\left]\text{;}\mathbb{P}\right)$ , then the Riemann–Liouville (R-L) fractional integral of order $r>0$ is presented as $$J_{\varrho }^{r}p\left(\varrho \right)=\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }{(\varrho -s)}^{r-1}p\left(s\right)\mathit{ds}\text{.}$$ In the above, $\mathrm{\Gamma }\left(r\right)={\int }_0^{\infty }{e}^{-\varrho }{\varrho }^{r-1}d\varrho$ .”

Definition 2.2

(Kilbas et al., 2006) “The R-L fractional derivative for $p\left(\varrho \right)\in L_1\left(\right[0,b\left]\text{;}\mathbb{P}\right)$ of order $r\in (1,2)$ is presented as $$D_{\varrho }^{r}p\left(\varrho \right)=D^2{J}_{\varrho }^{2-r}p\left(\varrho \right)=\frac{1}{\mathrm{\Gamma }(2-r)}\frac{d^2}{d{\varrho }^2}{\int }_0^{\varrho }{(\varrho -s)}^{1-r}p\left(s\right)\mathit{ds}.\mathrm{”}$$

Definition 2.3

(Kilbas et al., 2006) “The Caputo fractional derivative of order $r\in (1,2)$ is presented as $${}^c{D}_{\varrho }^{r}p\left(\varrho \right)=J_{\varrho }^{2-r}{D}^{2}p\left(\varrho \right)=\frac{1}{\mathrm{\Gamma }(2-r)}{\int }_0^{\varrho }{(\varrho -s)}^{1-r}\left[\frac{d^2}{{\mathit{ds}}^2}p\left(s\right)\right]\mathit{ds},$$ where $p\left(\varrho \right)\in L_1\left(\right[0,b\left]\text{;}\mathbb{P}\right)\cap C^1\left(\right[0,b\left]\text{;}\mathbb{P}\right)$ .”

Definition 2.4

(Bajlekova et al., 2001) “Assume that $r\in (1,2)$ . A family ${\left\{{\mathrm{\Upsilon }}_r\right(\varrho \left)\right\}}_{\varrho ⩾0}\subset \mathbb{L}\left(\mathbb{P}\right)$ is said to be a solution operator (or a strongly continuous r-order fractional CF) for (2.1) provided that the subsequent characteristics are fulfilled:

1. ${\mathrm{\Upsilon }}_r\left(\varrho \right)$ is strongly continuous for $\varrho ⩾0$ and ${\mathrm{\Upsilon }}_r\left(0\right)=I$ .

2. $\exists$ $L⩾1$ such that $\Vert {\mathrm{\Upsilon }}_r\left(\varrho \right)\Vert ⩽L$ .

3. ${\mathrm{\Upsilon }}_r\left(\varrho \right)D\left(A\right)\subset D\left(A\right)$ and $A{\mathrm{\Upsilon }}_r\left(\varrho \right)\eta ={\mathrm{\Upsilon }}_r\left(\varrho \right)A\eta ,\forall \eta \in D\left(A\right),\varrho ⩾0$ .

4. ${\mathrm{\Upsilon }}_r\left(\varrho \right)\eta$ is a solution for (2.1), $\forall \eta \in D\left(A\right),t⩾0$ .”

Definition 2.5

The fractional SF ${\mathrm{\Psi }}_r:[0,\infty )\to \mathbb{L}\left(\mathbb{P}\right)$ connected with ${\mathrm{\Upsilon }}_r$ is presented as $${\mathrm{\Psi }}_r\left(\varrho \right)={\int }_0^{\varrho }{\mathrm{\Upsilon }}_r\left(s\right)\mathit{ds},\varrho ⩾0\text{.}$$

Definition 2.6

The fractional R-L family ${\chi }_r:[0,\infty )\to \mathbb{L}\left(\mathbb{P}\right)$ connected with ${\mathrm{\Upsilon }}_r$ is presented as $${\chi }_r\left(\varrho \right)=J_{\varrho }^{r-1}{\mathrm{\Upsilon }}_r\left(\varrho \right)\text{.}$$

Definition 2.7

The mild solution of (1.1)–(1.3) is defined by $p(·)\in \mathbb{P}$ which satisfy the following integral equation. $$\begin{array}{cc}\hfill p\left(\varrho \right)=& {\mathrm{\Upsilon }}_r\left(\varrho \right)(p_0+h(p\left)\right)+{\mathrm{\Psi }}_r\left(\varrho \right)(p_1+h_1(p\left)\right)+{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\left\{\mathit{Bv}\right(\tau )+\kappa (s,p\left(\tau \right)\left)\right\}d\tau ,\varrho \in (0,b],\hfill \\ \hfill p\left(0\right)=& p_0+h\left(p\right),\hfill \\ \hfill p\prime \left(0\right)=& p_1+h_1\left(p\right),\hfill \end{array}$$ and the mild solution for (1.4)–(1.6) is described by the following integral equation $$\begin{array}{cc}\hfill q\left(\varrho \right)=& {\mathrm{\Upsilon }}_r\left(\varrho \right)(p_0+h(p\left)\right)+{\mathrm{\Psi }}_r\left(\varrho \right)(p_1+h_1(p\left)\right)+{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\mathit{Bu}\left(\tau \right)d\tau ,\varrho \in (0,b],\hfill \\ \hfill q\left(0\right)=& p_0+h\left(p\right),\hfill \\ \hfill q\prime \left(0\right)=& p_1+h_1\left(p\right)\text{.}\hfill \end{array}$$

Definition 2.8

(Curtain et al., 1995) “The system (1.1)–(1.3) is said to be approximately controllable in $[0,b]$ , provided that for given starting position and required final position $p_F$ and $∊>0,\exists$ a control function $v\in V$ such that the solution of (1.1)–(1.3) fulfills $$\Vert p\left(b\right)-p_F\Vert <∊,$$ where $p\left(b\right)$ is the state value of (1.1)–(1.3) at time $\varrho =b$ .”

Remark 2.1

Suppose $p\left(b\right)=p_F$ , then the system is called exactly controllable.

Definition 2.9

(Curtain et al., 1995) “Reachable set is the collection of all the possible final positions corresponding to the control $v\in V$ . The set defined by $$\begin{array}{cc}\hfill {\mathfrak{R}}_b\left(\kappa \right)=& \left\{p\right(b)\in \mathbb{P}:p(\varrho )\text{is a mild solution of the semilinear}\hfill \\ \hfill & \text{system corresponding to control}v\in V\},\hfill \end{array}$$ is the reachable set for (1.1)–(1.3) and ${\mathfrak{R}}_b\left(0\right)$ is the reachable set for (1.4)–(1.6).”

Definition 2.10

(Curtain et al., 1995) Definition of Approximate Controllability in terms of Reachable Set: “A control system is said to be approximately controllable on an interval $[0,b]$ , iff the corresponding reachable set is dense in $\mathbb{P}$ . Therefore the given system (1.1)–(1.3) is approximately controllable iff $$\overline{{\mathfrak{R}}_b\left(\kappa \right)}=\mathbb{P},$$ where $\overline{{\mathfrak{R}}_b\left(\kappa \right)}$ denotes the closure of ${\mathfrak{R}}_b\left(\kappa \right)$ .

Remark 2.2

If ${\mathfrak{R}}_b\left(\kappa \right)=\mathbb{P}$ , then the given system is said to exactly controllable.”

Lemma 1

The solution $p\left(\varrho \right)$ of the nonlocal fractional system (1.1)–(1.3) connected with the control $v=u-w$ fulfills the subsequent $$\begin{array}{cc}\hfill \Vert p\left(\varrho \right){\Vert }_{\mathbb{P}}⩽& \left[\left(1+b_2\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\right)L(\Vert p_0\Vert +\Vert p_1\Vert +M_h+M_{h_1})+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left\{\left(1+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}{b}_2}{\mathrm{\Gamma }\left(r\right)}\right)\Vert \mathit{Bu}{\Vert }_H\right.\right.\hfill \\ \hfill & \left.\left.+b_1+{∊}_1\right\}+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}\right]e^{\frac{{\mathit{Lb}}^r{b}_2}{\mathrm{\Gamma }\left(r\right)}},\hfill \end{array}$$ where $\Vert {\mathrm{\Upsilon }}_r\left(\varrho \right)\Vert ⩽L,\Vert {\mathrm{\Psi }}_r\left(\varrho \right)\Vert ⩽L,\Vert {\chi }_r\left(\varrho \right)\Vert ⩽\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}$ for each $\varrho \in [0,b]$ .

Proof

Let $p\left(\varrho \right)$ be the mild solution of (1.1)–(1.3) associated with the control $v=u-w$ . Therefore, one can consider $p\left(\varrho \right)$ in the following way: $$p\left(\varrho \right)={\mathrm{\Upsilon }}_r\left(\varrho \right)(p_0+h(p\left)\right)+{\mathrm{\Psi }}_r\left(\varrho \right)(p_1+h_1(p\left)\right)+{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )B(u-w)\left(\tau \right)d\tau +{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\kappa (\tau ,p(\tau \left)\right)d\tau \text{.}$$ Taking norm on both sides, we get $$\begin{array}{cc}\hfill \Vert p\left(\varrho \right){\Vert }_{\mathbb{P}}⩽& L\Vert p_0\Vert +L\Vert h\left(p\right)\Vert +L\Vert p_1\Vert +L\Vert h_1\left(p\right)\Vert +\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert B(u-w)\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill & +\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert \kappa (\tau ,p(\tau \left)\right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}\sqrt{b}(\Vert \mathit{Bu}{\Vert }_H+\Vert \mathit{Bw}{\Vert }_H)\hfill \\ \hfill & +\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }(b_1+b_2\Vert p(\tau \left){\Vert }_{\mathbb{P}}\right)d\tau \hfill \\ \hfill ⩽& L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}(\Vert \mathit{Bu}{\Vert }_H+\Vert \mathit{Kq}{\Vert }_H+{∊}_1)+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}\hfill \\ \hfill & +\frac{{\mathit{Lb}}^{r-1}{b}_2}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert p\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}(\Vert \mathit{Bu}{\Vert }_H+b_1+b_2\Vert q{\Vert }_H+{∊}_1)+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}\hfill \\ \hfill & +\frac{{\mathit{Lb}}^{r-1}{b}_2}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert p\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left(\Vert \mathit{Bu}{\Vert }_H+b_1+b_2(L\Vert p_0\Vert +{\mathit{LM}}_h\right.\hfill \\ \hfill & \left.+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\Vert \mathit{Bu}{\Vert }_H)+{∊}_1\right)+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}+\frac{{\mathit{Lb}}^{r-1}{b}_2}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert p\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left\{\left(1+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}{b}_2}{\mathrm{\Gamma }\left(r\right)}\right)\Vert \mathit{Bu}{\Vert }_H+b_1+b_{2}L\Vert p_0\Vert \right.\hfill \\ \hfill & \left.+b_2{\mathit{LM}}_h+b_{2}L\Vert p_1\Vert +b_2{\mathit{LM}}_{h_1}+{∊}_1\right\}+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}+\frac{{\mathit{Lb}}^{r-1}{b}_2}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert p\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& \left(1+b_2\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\right)L(\Vert p_0\Vert +\Vert p_1\Vert )+L\left(1+b_2\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\right)(M_h+M_{h_1})\hfill \\ \hfill & +\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left\{(1+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}{b}_2}{\mathrm{\Gamma }\left(r\right)})\Vert \mathit{Bu}{\Vert }_H+b_1+{∊}_1\right\}+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}+\frac{{\mathit{Lb}}^{r-1}{b}_2}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert p\left(\tau \right){\Vert }_{\mathbb{P}}d\tau \text{.}\hfill \end{array}$$ Applying Gronwall’s inequality, we attain $$\begin{array}{cc}\hfill \Vert p\left(\varrho \right){\Vert }_{\mathbb{P}}⩽& \left[\left(1+b_2\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\right)L(\Vert p_0\Vert +\Vert p_1\Vert +M_h+M_{h_1})\right.\hfill \\ \hfill & \left.+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left\{\left(1+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}{b}_2}{\mathrm{\Gamma }\left(r\right)}\right)\Vert \mathit{Bu}{\Vert }_H+b_1+{∊}_1\right\}+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}\right]e^{\frac{{\mathit{Lb}}^r{b}_2}{\mathrm{\Gamma }\left(r\right)}}\text{.}\hfill \end{array}$$ Hence, the result is proved.

Theorem 3.1

Suppose that $\left(T_1\right)$ - $\left(T_6\right)$ are fulfilled, then the fractional semilinear control system with nonlocal conditions (1.1)–(1.3) is approximately controllable on $[0,b]$ .

Proof

Assume that $q\left(\varrho \right)$ be the mild solution of the fractional linear system with nonlocal conditions (1.4)–(1.6) associated with the control u. Then we can write $q\left(\varrho \right)$ as

$$q\left(\varrho \right)={\mathrm{\Upsilon }}_r\left(\varrho \right)(p_0+h(p\left)\right)+{\mathrm{\Psi }}_r\left(\varrho \right)(p_1+h_1(p\left)\right)+{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\left(\mathit{Bu}\right)\left(\tau \right)d\tau \text{.}$$ Let $p\left(\varrho \right)$ be the mild solution of (1.1)–(1.3) associated with the control $v=u-w$ . Then, we can express $p\left(\varrho \right)$ as

(2.3)

$$\begin{array}{cc}\hfill p\left(\varrho \right)=& {\mathrm{\Upsilon }}_r\left(\varrho \right)(p_0+h(p\left)\right)+{\mathrm{\Psi }}_r\left(\varrho \right)(p_1+h_1(p\left)\right)+{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )B(u-w)\left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\kappa (\tau ,p(\tau \left)\right)d\tau \text{.}\hfill \end{array}$$ Using (2.2) and (2.3), one can obtain

(2.4)

$$q\left(\varrho \right)-p\left(\varrho \right)={\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\mathit{Bw}\left(\tau \right)d\tau -{\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\kappa (\tau ,p(\tau \left)\right)d\tau \text{.}$$ Writing the above equation in operator theoretic form, we get $$\begin{array}{cc}\hfill q-p=& \mathit{NBw}-\mathit{NKp}\hfill \\ \hfill =& N(\mathit{Bw}-\mathit{Kq})+(\mathit{NKq}-\mathit{NKp})\text{.}\hfill \end{array}$$ Applying the inner product on both sides with $\mathit{Kq}-\mathit{Kp}$ , one can obtain

(2.5)

$$\begin{array}{cc}\hfill \langle q-p,\mathit{Kq}-\mathit{Kp}{\rangle }_H=& \langle N(\mathit{Bw}-\mathit{Kq})+(\mathit{NKq}-\mathit{NKp}),\mathit{Kq}-\mathit{Kp}{\rangle }_H\hfill \\ \hfill =& \langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H\hfill \\ \hfill & +\langle (\mathit{NKq}-\mathit{NKp}),\mathit{Kq}-\mathit{Kp}{\rangle }_H\text{.}\hfill \end{array}$$ Using condition $\left(T_3\right)$ , we get that the left hand side of the Eq. (2.5) is $⩽-\gamma \Vert q-p{\Vert }^2$ and the second term of the R.H.S is nonnegative from assumption $\left(T_1\right)$ .

If we can show that $\langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H$ is negligbly small, then it will indicate that $\Vert q-p{\Vert }_H$ is also arbitrary small from the Eq. (2.5).

Therefore, we show that $\langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H$ is arbitrarily small.

Using Cauchy Schwartz Inequality, we have

$$\begin{array}{cc}\hfill |\langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H|⩽& \Vert N(\mathit{Bw}-\mathit{Kq}){\Vert }_H\Vert \mathit{Kq}-\mathit{Kp}{\Vert }_H\hfill \\ \hfill & ⩽\frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}.b\Vert \mathit{Bw}-\mathit{Kq}{\Vert }_H\{\Vert \mathit{Kq}{\Vert }_H+\Vert \mathit{Kp}{\Vert }_H\}\hfill \\ \hfill & ⩽\frac{{\mathit{Lb}}^r{∊}_1}{\mathrm{\Gamma }\left(r\right)}\{b_1+b_2\Vert q{\Vert }_H+b_1+b_2\Vert p{\Vert }_H\}\text{.}\hfill \end{array}$$ Using Lemma 1, we get

(2.7)

$$|\langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H|⩽\frac{{\mathit{LGb}}^r{∊}_1}{\mathrm{\Gamma }\left(r\right)}\text{.}$$ where $$\begin{array}{cc}\hfill G=& 2b_1+b_2\left[L\Vert p_0\Vert +{\mathit{LM}}_h+L\Vert p_1\Vert +{\mathit{LM}}_{h_1}+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\Vert \mathit{Bu}{\Vert }_H\right]\hfill \\ \hfill & +b_2\left[\left\{\left(1+b_2\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\right)L(\Vert p_0\Vert +\Vert p_1\Vert +{\mathit{LM}}_h+{\mathit{LM}}_{h_1})\right.\right.\hfill \\ \hfill & \left.\left.+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\left\{\left(1+\frac{{\mathit{Lb}}^{r-\frac{1}{2}}{b}_2}{\mathrm{\Gamma }\left(r\right)}\right)\Vert \mathit{Bu}{\Vert }_H+b_1+{∊}_1\right\}+\frac{{\mathit{Lb}}^r{b}_1}{\mathrm{\Gamma }\left(r\right)}\right\}{e}^{\frac{{\mathit{Lb}}^r{b}_2}{\mathrm{\Gamma }\left(r\right)}}\right]\text{.}\hfill \end{array}$$ Therefore, G is finite for given u and ${∊}_1$ . It indicates that $\langle N(\mathit{Bw}-\mathit{Kq}),\mathit{Kq}-\mathit{Kp}{\rangle }_H$ is arbitrary small as ${∊}_1$ is arbitrarily small.

Now, using Eqs. (2.5), (2.6)condition $\left(T_4\right)$ , we get that $\Vert q-p\Vert <{∊}_2$ , for some ${∊}_2>0$ .

Next, we determine that $\Vert q\left(\varrho \right)-p\left(\varrho \right){\Vert }_{\mathbb{P}}$ is arbitrary small.

Consider,

$$\begin{array}{cc}\hfill q\left(\varrho \right)-p\left(\varrho \right)=& \mathit{NBw}\left(\varrho \right)-N\left(\mathit{Kp}\right)\left(\varrho \right)\hfill \\ \hfill =& N\left\{\mathit{Bw}\right(\varrho )-(\mathit{Kq}\left)\right(\varrho \left)\right\}+N\left\{\right(\mathit{Kq}\left)\right(\varrho )-(\mathit{Kp}\left)\right(\varrho \left)\right\}\text{.}\hfill \end{array}$$ Taking norm on both sides, we get $$\begin{array}{cc}\hfill \Vert q\left(\varrho \right)-p\left(\varrho \right){\Vert }_{\mathbb{P}}=& \Vert N\left\{\mathit{Bw}\right(\varrho )-(\mathit{Kq}\left)\right(\varrho \left)\right\}+N\left\{\right(\mathit{Kq}\left)\right(\varrho )-(\mathit{Kp}\left)\right(\varrho \left)\right\}{\Vert }_{\mathbb{P}}\hfill \\ \hfill ⩽& \Vert N\left\{\mathit{Bw}\right(\varrho )-(\mathit{Kq}\left)\right(\varrho \left)\right\}{\Vert }_{\mathbb{P}}+\Vert N\left\{\right(\mathit{Kq}\left)\right(\varrho )-(\mathit{Kp}\left)\right(\varrho \left)\right\}{\Vert }_{\mathbb{P}}\text{.}\hfill \end{array}$$ Now, $$\begin{array}{cc}\hfill \Vert N\left\{\mathit{Bw}\right(\varrho )-(\mathit{Kq}\left)\right(\varrho \left)\right\}{\Vert }_{\mathbb{P}}=& \Vert {\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\left\{\mathit{Bw}\right(\tau )-(\mathit{Kq}\left)\right(\tau \left)\right\}d\tau {\Vert }_{\mathbb{P}}\hfill \\ \hfill ⩽& \frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\varrho }\Vert \left\{\mathit{Bw}\right(\tau )-(\mathit{Kq}\left)\right(\tau \left)\right\}{\Vert }_{\mathbb{P}}d\tau \hfill \\ \hfill ⩽& \frac{{\mathit{Lb}}^{r-1}}{\mathrm{\Gamma }\left(r\right)}\sqrt{b}\Vert \mathit{Bw}-\mathit{Kq}{\Vert }_H\hfill \\ \hfill <& \frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}{∊}_1,\hfill \end{array}$$ and

(2.9)

$$\begin{array}{cc}\hfill \Vert N\left\{\right(\mathit{Kq}\left)\right(\varrho )-(\mathit{Kp}\left)\right(\varrho \left)\right\}{\Vert }_{\mathbb{P}}=& \Vert {\int }_0^{\varrho }{\chi }_r(\varrho -\tau )\left\{\right(\mathit{Kq}\left)\right(\tau )-(\mathit{Kp}\left)\right(\tau \left)\right\}d\tau {\Vert }_{\mathbb{P}}\hfill \\ \hfill ⩽& \frac{{\mathit{Lb}}^{r-\frac{1}{2}}}{\mathrm{\Gamma }\left(r\right)}\sqrt{b}\Vert \mathit{Kq}-\mathit{Kp}{\Vert }_H\text{.}\hfill \end{array}$$ Since K is continuous on H and $\Vert q-p\Vert <{∊}_2$ , therefore, the right hand side of the above defined equation can be made arbitrarily small.

Thus, we get

$$\Vert q\left(\varrho \right)-p\left(\varrho \right)\Vert <∊\forall \varrho \in [0,b]\mathrm{andforgiven}∊>0\text{.}$$ Thus, $\Vert q\left(\varrho \right)-p\left(\varrho \right)\Vert$ may be formed arbitrarily small by selecting appropriate w.

$\Rightarrow$ ${\mathfrak{R}}_b\left(\kappa \right)$ (reachable set for the fractional semilinear control system with nonlocal conditions (1.1)–(1.3) is dense in ${\mathfrak{R}}_b\left(0\right)$ (corresponding fractional linear system with nonlocal conditions (1.4)–(1.6).

But ${\mathfrak{R}}_b\left(0\right)$ is dense in $\mathbb{P}$ due to the condition $\left(T_2\right)$ as the corresponding fractional linear system with nonlocal conditions is approximately controllable. Therefore, ${\mathfrak{R}}_b\left(\kappa \right)$ is dense in $\mathbb{P}$ , which implies that (1.1)–(1.3) is approximately controllable.