Finite-Approximate Controllability of Impulsive Fractional Functional Evolution Equations of Order $1<\alpha <2$

Abstract

In this manuscript, we study the finite-approximate controllability of impulsive fractional functional evolution equations of order $1<\alpha <2$ in Hilbert spaces. We first discuss a useful characterization of the finite-approximate controllability for linear fractional evolution equations of order $1<\alpha <2$ in terms of a resolvent-like operator. We also find a suitable control to obtain the approximate controllability of the linear system, which also ensures the finite-approximate controllability of the system. Next, we establish sufficient conditions for the finite-approximate controllability of the semilinear impulsive fractional evolution equations, whenever the corresponding linear system is approximately controllable. Moreover, we provide an example of fractional wave equation to illustrate the efficiency of the developed results. Finally, we discuss the finite-approximate controllability of semilinear fractional evolution equations of order $1<\alpha <2$ with finite delay by using a variational method.

Appendix A: A Variational Method

In this section, we discuss the variational method (cf. [27,28,29,30], etc.) for finite-approximate controllability of a semilinear fractional evolution equation with finite delay. For easiness of demonstration, we are not considering impulses in the system. Let us consider the following fractional evolution equation with finite delay:

$$\begin{aligned} \begin{aligned} ^C\textrm{D}_{0,t}^{\alpha }x(t)&=\textrm{A}x(t)+\textrm{B}u(t)+f(t,x_t), \ t\in J =[0,T],\\ x(t)&=\phi (t),\ t\in [-c,0], \ c>0,\\ x'(0)&=\zeta , \end{aligned} \end{aligned}$$

(A.1)

where $^C\textrm{D}_{0,t}^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha \in (1,2)$, the operators $\textrm{A}:\mathrm {D(A)}\subset {\mathbb {H}}\rightarrow {\mathbb {H}}$ and $\textrm{B}:{\mathbb {U}}\rightarrow {\mathbb {H}}$ are the same as defined in (1.1). The control function $u\in \textrm{L}^{2}(J;{\mathbb {U}})$ and the given function $f:J\times \textrm{C}(I_c;{\mathbb {H}})\rightarrow {\mathbb {H}}$, where $I_c=[-c,0]$. For every $t \in J,\ x_t \in \textrm{C}(I_c;{\mathbb {H}})$ and the function $x_t(\cdot )$ is defined by $x_t(s)= x(t + s),\ s\in I_c$.

Definition A.1

A function $x\in \textrm{C}(J_c;{\mathbb {H}}),$ where $J_c=[-c,T]$, is called a mild solution of (A.1), if $x_0=\phi$ and

$$\begin{aligned} x(t)=\textrm{C}_{q}(t)\phi (0)+\textrm{T}_{q}(t)\zeta +\int _{0}^{t}(t-s)^{q-1}\textrm{S}_{q}(t-s)\left[ \textrm{B}u(s)+f(s,x_s)\right] \textrm{d}s,\ t\in J. \end{aligned}$$

Let us impose the following assumptions on the nonlinear term $f(\cdot ,\cdot )$.

$\textit{(A1)}$

• (i) the mapping $t\mapsto f(t, \psi )$ is strongly measurable on J for each $\psi \in \textrm{C}(I_c;{\mathbb {H}})$, and the function $f(t,\cdot ): \textrm{C}(I_c;{\mathbb {H}})\rightarrow {\mathbb {H}}$ is continuous for a.e. $t\in J$.

• (ii) For each positive integer r, there exists a constant $\beta \in [0,q]$ and a function $\varGamma _r\in \textrm{L}^{\frac{1}{\beta }}(J;\mathbb {R^{+}})$ such that

  $$\begin{aligned} \sup _{\left\| \psi \right\| _{\textrm{C}(I_c;{\mathbb {H}})}\le r} \left\| f(t, \psi )\right\| _{{\mathbb {H}}}\le \varGamma _{r}(t), \text{ for } \text{ a.e. } \ t \in J \ \text{ and } \ \psi \in \textrm{C}(I_c;{\mathbb {H}}), \end{aligned}$$

  with

  $$\begin{aligned} \liminf _{r \rightarrow \infty } \frac{\left\| \varGamma _r\right\| _{\textrm{L}^{\frac{1}{\beta }}(J;\mathbb {R^+})}}{r} = l< \infty . \end{aligned}$$

Here, we obtain the finite-approximate controllability of the semilinear system (A.1) by using the technique introduced in the works [27, 29, 30, 55], etc. For this, we consider the following functional:

$$\begin{aligned} {\mathcal {J}}_{\uplambda }(\varphi ,z)&=\frac{1}{2}\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\varphi \right\| ^2_{{\mathbb {U}}}\textrm{d}s+\frac{\uplambda }{2}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| ^2_{{\mathbb {H}}}-\langle \varphi , g(z)\rangle , \end{aligned}$$

(A.2)

for any $\uplambda >0$, $z\in \textrm{C}(J_c;{\mathbb {H}})$ and

$$\begin{aligned} g(z)=h-\textrm{C}_{q}(T)\phi (0)-\textrm{T}_{q}(T)\zeta -\int _{0}^{T} (T-s)^{q-1}\textrm{S}_{q}(T-s)f(s,z_s)\textrm{d}s, \end{aligned}$$

where $h\in {\mathbb {H}}$.

Lemma A.2

The set $\textrm{V}=\{g(z):z\in {\mathcal {B}}_r\}$ is relatively compact in ${\mathbb {H}}$, where ${\mathcal {B}}_r=\{y\in \textrm{C}(J_c;{\mathbb {H}}):\left\| y\right\| _{\textrm{C}(J_c;{\mathbb {H}})}\le r\}$.

Proof

The relative compactness of the set $\textrm{V}$ is immediately follows by the facts that the operator $\textrm{S}_{q}(t)$ is compact for $t\ge 0$ and also the operator $(\textrm{G}f)(\cdot ) =\int _{0}^{\cdot }(\cdot -s)^{q-1}\textrm{S}_{q}(\cdot -s)f(s)\textrm{d}s$ is compact (see Lemma 3.4, [3]). $\square$

Lemma A.3

The functional ${\mathcal {J}}_{\uplambda }(\cdot ,\cdot )$ satisfies the following properties.

• (i) The mapping $\varphi \mapsto {\mathcal {J}}_{\uplambda }(\varphi ,z)$ is strictly convex and Gâteaux differentiable.

• (ii) For any $r>0$

  $$\begin{aligned} \liminf _{\varphi \rightarrow \infty }\inf _{z\in {\mathcal {B}}_r}\frac{{\mathcal {J}}_{\uplambda }(\varphi ,z)}{\left\| \varphi \right\| _{{\mathbb {H}}}}\ge \uplambda . \end{aligned}$$

  (A.3)

Proof

(i) From the definition of ${\mathcal {J}}_{\uplambda }(\cdot ,\cdot )$, we see immediately that the mapping $\varphi \mapsto {\mathcal {J}}_{\uplambda }(\varphi ,z)$ is strictly convex and Gâteaux differentiable.

(ii) In order to prove the estimate (A.3), let us consider two sequences $\{\varphi ^m\}_{m=1}^\infty \subset {\mathbb {H}}$ and $\{z^m\}_{m=1}^\infty \subset {\mathcal {B}}_r$ with $\left\| \varphi ^m\right\| _{{\mathbb {H}}}\rightarrow \infty$ as $m\rightarrow \infty$. From Lemma A.2, we know that the set $\textrm{V}=\{g(z):z\in {\mathcal {B}}_r\}$ is relatively compact. So without loss generality, taking a subsequence $g(z^m)$ (still denoted by $g(z^m)$), such that

$$\begin{aligned} g(z^m)\rightarrow g \ \text{ in } \ {\mathbb {H}}, \end{aligned}$$

for some $g\in {\mathbb {H}}$. Next, we define

$$\begin{aligned} \psi ^m=\frac{\varphi ^m}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}, \end{aligned}$$

it is clear that $\left\| \psi ^m\right\| _{{\mathbb {H}}}=1$. Hence, by applying the Banach–Alaoglu theorem, we can find a subsequence relabeled as $\{\psi ^m\} _{m=1}^\infty$ such that

Using the compactness of the operator $\textrm{S}_q(t)$ for $t\ge 0$, we have the following convergence:

$$\begin{aligned} \textrm{B}^*\textrm{S}_q(T-\cdot )^*\psi ^m\rightarrow \textrm{B}^*\textrm{S}_q(T-\cdot )^*\psi \ \ \text{ in } \ \ \textrm{C}(J;{\mathbb {U}}). \end{aligned}$$

(A.4)

By the expression (A.2), it follows that

$$\begin{aligned} \frac{{\mathcal {J}}_{\uplambda }(\varphi ^m,z^m)}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}&=\frac{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}{2}\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s\\&\quad +\frac{\uplambda \left\| \varphi ^m\right\| _{{\mathbb {H}}}}{2}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\psi ^m\right\| ^2_{{\mathbb {H}}}-\langle \psi ^m, g(z^m)\rangle . \end{aligned}$$

Case I: If either

$$\begin{aligned} \liminf _{m\rightarrow \infty }\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s>0\ \text{ or } \ \lim _{m\rightarrow \infty }\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\psi ^m\right\| _{{\mathbb {H}}}>0. \end{aligned}$$

Then, it is immediate that

$$\begin{aligned} \frac{{\mathcal {J}}_{\uplambda }(\varphi ^m,z^m)}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}\rightarrow \infty \ \text{ as } \ m\rightarrow \infty , \end{aligned}$$

Case II:If

$$\begin{aligned} \liminf _{m\rightarrow \infty }\int _{0}^{T}(T\!-\!s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T\!-\!s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s>0\ \! \text{ or } \ \! \lim _{m\rightarrow \infty }\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\psi ^m\right\| _{{\mathbb {H}}}=0. \end{aligned}$$

In this case, it is again follows that

$$\begin{aligned} \frac{{\mathcal {J}}_{\uplambda }(\varphi ^m,z^m)}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}\rightarrow \infty \ \ \text{ as } \ \ m\rightarrow \infty . \end{aligned}$$

Case III: If $\liminf \limits _{m\rightarrow \infty }\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s=0$. Using the convergence (A.4) and the Fatou lemma, we obtain

$$\begin{aligned}&\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*{\bar{\varphi }}\right\| ^2_{{\mathbb {U}}}\textrm{d}s\\&\le \liminf _{m\rightarrow \infty }\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s=0. \end{aligned}$$

By using the above fact, Assumption 4.4 (H5) and Theorem 3.5, we have ${\bar{\varphi }}=0$, which implies that

(A.5)

Since the space ${\mathcal {D}}$ is finite-dimensional and $\pi _{{\mathcal {D}}}$ is compact, we get that

$$\begin{aligned} \pi _{{\mathcal {D}}}\varphi ^m\rightarrow 0\ \ \text{ as } \ \ m\rightarrow \infty , \end{aligned}$$

and therefore

$$\begin{aligned} \lim _{m\rightarrow \infty }\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\psi ^m\right\| _{{\mathbb {H}}}=1. \end{aligned}$$

(A.6)

From the convergence (A.5) and the estimate (A.6), further we obtain

$$\begin{aligned} \frac{{\mathcal {J}}_{\uplambda }(\varphi ^m,z^m)}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}\rightarrow \infty \ \ \text{ as } \ \ m\rightarrow \infty . \end{aligned}$$

Therefor, in all the cases we get the following:

$$\begin{aligned} \frac{{\mathcal {J}}_{\uplambda }(\varphi ^m,z^m)}{\left\| \varphi ^m\right\| _{{\mathbb {H}}}}\rightarrow \infty \ \ \text{ as } \ \ m\rightarrow \infty , \end{aligned}$$

which ensures that the inequality (A.3) follows. $\square$

The estimate (A.3) implies that the functional ${\mathcal {J}}_{\uplambda }(\cdot ,z)$ is coercive for all $z\in {\mathcal {B}}_r$. Thus, ${\mathcal {J}}_{\uplambda }(\cdot ,z)$ has a minimizer. By the strict convexity of ${\mathcal {J}}_{\uplambda }(\cdot ,z)$, the minimum is unique which can be found as

$$\begin{aligned} {\mathcal {J}}_{\uplambda }'(\varphi ,z)=\Phi _0^T\varphi +\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})\varphi -g(z)=0, \end{aligned}$$

so that

$$\begin{aligned} \varphi _{\min }=(\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(z)). \end{aligned}$$

By using the above minimum, we construct the feedback control as

$$\begin{aligned} u^q_{\uplambda }(t)&=\textrm{B}^*\textrm{S}_q(T-t)^*\varphi _{\min }=\textrm{B}^*\textrm{S}_q(T-t)^*(\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(z)). \end{aligned}$$

In order to prove the existence of a mild solution of the system (A.1), we define a set

$$\begin{aligned} {\mathcal {E}}:=\{x\in \textrm{C}(J;{\mathbb {H}}): x(0)=\phi (0)\}, \end{aligned}$$

with the norm $\left\| \cdot \right\| _{\textrm{C}(J;{\mathbb {H}})}$. Next, for any $\uplambda >0$, we define an operator ${\mathcal {P}}_\uplambda :{\mathcal {E}}\rightarrow {\mathcal {E}}$ as

$$\begin{aligned} ({\mathcal {P}}_{\uplambda }x)(t)&=\textrm{C}_{q}(t)\phi (0)+\textrm{T}_{q}(t)\zeta \nonumber \\&+\int _{0}^{t}(t-s)^{q-1}\textrm{S}_{q}(t-s)\left[ \textrm{B}u_{\uplambda }^{q}(s)+f(s,{\tilde{x}}_s)\right] \textrm{d}s, \ t\in J, \end{aligned}$$

(A.7)

with the control

$$\begin{aligned} u^q_{\uplambda }(t)&=\textrm{B}^*\textrm{S}_q(T-t)^*(\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(x)), \end{aligned}$$

where

$$\begin{aligned} g(x)=h-\textrm{C}_{q}(T)\phi (0)-\textrm{T}_{q}(T)\zeta -\int _{0}^{T} (T-s)^{q-1}\textrm{S}_{q}(T-s)f(s,{\tilde{x}}_s)\textrm{d}s, \end{aligned}$$

and the function ${\tilde{x}}:[-c,T]\rightarrow {\mathbb {H}}$ such that ${\tilde{x}}(t)=\phi (t), \ t\in [-c,0], \ {\tilde{x}}(t)=x(t),\ t\in J$.

In view of the definition of ${\mathcal {P}}_{\uplambda }$, it is clear that the problem of existence of a mild solution for the system (A.1) is equivalent to the operator ${\mathcal {P}}_{\uplambda }$ has a fixed point. In the next theorem, we show that the operator ${\mathcal {P}}_{\uplambda }$ has a fixed point.

Theorem A.4

If Assumptions (H1) and (A1) hold true, then for arbitrary $\uplambda >0$, the operator ${\mathcal {P}}_{\uplambda }$ given in (A.7) has a fixed point, provided

$$\begin{aligned} \frac{KlT^{2q-\beta }}{\Gamma (2q)\nu ^{1-\beta }}\left\{ 1+\left( \frac{KN}{\Gamma (2q)}\right) ^2\frac{T^{3q}}{3q\uplambda (1-\delta _{\uplambda })}\right\} <1, \end{aligned}$$

where $\nu =\frac{2q-\beta }{1-\beta }$ and $\delta _{\uplambda }=\left\| \uplambda (\uplambda \textrm{I}+\Phi _{0}^{T})^{-1}\pi _{{\mathcal {D}}}\right\| _{{\mathcal {L}}({\mathbb {H}})}<1$.

Proof

Proceeding similarly as in the proof of Theorem 4.2, we can obtain that the operator ${\mathcal {P}}_{\uplambda }$ defined in (A.7) satisfies the following properties:

• (i) There exists $r>0$ such that ${\mathcal {P}}_{\uplambda }(\textrm{E}_r)\subset \textrm{E}_r$, where $\textrm{E}_r=\{x\in {\mathcal {E}}: \left\| x\right\| _{\textrm{C}(J;{\mathbb {H}})}\le r\}$.

• (ii) ${\mathcal {P}}_{\uplambda }$ is continuous.

• (iii) ${\mathcal {P}}_{\uplambda }$ is compact.

In view of these three properties and an application of Schauder’s fixed point theorem, the existence of a fixed point of the operator ${\mathcal {P}}_{\uplambda }$ follows immediately. In other words, for any $\uplambda >0$, there exists a function $x^\uplambda \in \textrm{C}(J_c;{\mathbb {H}}),$ such that $x^\uplambda _0=\phi$ and

$$\begin{aligned} x^\uplambda (t)= & {} \textrm{C}_{q}(t)\phi (0)+\textrm{T}_{q}(t)\zeta \\{} & {} \quad +\int _{0}^{t}(t-s)^{q-1}\textrm{S}_{q}(t-s)\left[ \textrm{B}u_\uplambda ^{q}(s)+f(s,x^\uplambda _s)\right] \textrm{d}s,\ t\in J, \end{aligned}$$

with the control

$$\begin{aligned} u^q_{\uplambda }(t)&=\textrm{B}^*\textrm{S}_q(T-t)^*(\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(x^\uplambda )), \end{aligned}$$

where

$$\begin{aligned} g(x^\uplambda )=h-\textrm{C}_{q}(T)\phi (0)-\textrm{T}_{q}(T)\zeta -\int _{0}^{T} (T-s)^{q-1}\textrm{S}_{q}(T-s)f(s,x^\uplambda _s)\textrm{d}s, \end{aligned}$$

(A.8)

and the proof can be completed. $\square$

In order to investigate the finite-approximate controllability of the semilinear system (A.1), we assume the following:

(A2):

The function $f: J \times \textrm{C}(I_c;{\mathbb {H}}) \rightarrow {\mathbb {H}}$ satisfies Assumption (A1)(i) and there exists a function $\varGamma \in \textrm{L}^{\frac{1}{\beta }}(J;{\mathbb {R}}^+)$ with $\beta \in [0,q]$ such that

$$\begin{aligned} \Vert f(t,\psi )\Vert _{{\mathbb {H}}}\le \varGamma (t),\ \text { for all }\ (t,\psi ) \in J \times \textrm{C}(I_c;{\mathbb {H}}). \end{aligned}$$

Theorem A.5

If Assumptions (H1), (H5), (A2) are satisfied, then the system (A.1) is finite-approximately controllable.

Proof

Since we know that the functional ${\mathcal {J}}_{\uplambda }(\varphi ,x^\uplambda )$ has a unique minima, say $\hat{\varphi _{\uplambda }}\in {\mathbb {H}}$ of the form

$$\begin{aligned} \hat{\varphi _{\uplambda }}=(\uplambda (\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(x^\uplambda )), \end{aligned}$$

(A.9)

where $g(x^\uplambda )$ given in (A.8). For any given $\varphi \in {\mathbb {H}}$ and $\mu \in {\mathbb {R}},$ we have

$$\begin{aligned} {\mathcal {J}}_{\uplambda }(\hat{\varphi _{\uplambda }},x^\uplambda )\le {\mathcal {J}}_{\uplambda }(\hat{\varphi _{\uplambda }}+\mu \varphi ,x^\uplambda ), \end{aligned}$$

or in other words

$$\begin{aligned} \frac{\uplambda }{2}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| ^2_{{\mathbb {H}}}&\le \frac{\mu ^2}{2}\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\varphi \right\| ^2_{{\mathbb {U}}}\textrm{d}s\\&\quad +\mu \int _{0}^{T}(T-s)^{q-1}\langle \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \rangle \textrm{d}s\\&\quad +\frac{\uplambda }{2}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\hat{\varphi _{\uplambda }}+\mu \varphi \right\| ^2_{{\mathbb {H}}}-\mu \langle \varphi , g(x^\uplambda )\rangle . \end{aligned}$$

Dividing the above inequality by $\mu >0$ and passing $\mu \rightarrow 0^+$, we deduce that

$$\begin{aligned}&\langle \varphi , g(x^\uplambda )\rangle \\&\le \int _{0}^{T}(T-s)^{q-1}\langle \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \rangle \textrm{d}s\\ {}&\quad +\frac{\uplambda }{2}\lim _{\mu \rightarrow 0}\frac{\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\hat{\varphi _{\uplambda }}+\mu \varphi \right\| ^2_{{\mathbb {H}}}-\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| ^2_{{\mathbb {H}}}}{\mu }\\&\le \int _{0}^{T}(T-s)^{q-1}\langle \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \rangle \textrm{d}s+\uplambda \langle (\textrm{I}-\pi _{{\mathcal {D}}})\hat{\varphi _{\uplambda }},(\textrm{I}-\pi _{{\mathcal {D}}})\varphi \rangle \\&\le \int _{0}^{T}(T-s)^{q-1}\langle \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \rangle \textrm{d}s+\uplambda \left\| (\textrm{I}-\pi _{{\mathcal {D}}})\hat{\varphi _{\uplambda }}\right\| _{{\mathbb {H}}}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| _{{\mathbb {H}}}\\&\le \int _{0}^{T}(T-s)^{q-1}\langle \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \rangle \textrm{d}s+\uplambda \left\| \hat{\varphi _{\uplambda }}\right\| _{{\mathbb {H}}}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| _{{\mathbb {H}}}. \end{aligned}$$

Repeating this argument with $\mu <0,$ we finally obtain that

$$\begin{aligned} \left| \int _{0}^{T}\left\langle (T-s)^{q-1} \textrm{B}^*\textrm{S}_q(T-s)^*\hat{\varphi _{\uplambda }},\textrm{B}^*\textrm{S}_q(T-s)^*\varphi \right\rangle \textrm{d}s-\left\langle \varphi ,g(x^\uplambda )\right\rangle \right| \\ \le \uplambda \left\| \hat{\varphi _{\uplambda }}\right\| _{{\mathbb {H}}}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| _{{\mathbb {H}}}. \end{aligned}$$

Using the above estimate, we easily find

$$\begin{aligned} \left| \langle x^\uplambda (T)-h, \varphi \rangle \right|&\le \uplambda \left\| \hat{\varphi _{\uplambda }}\right\| _{{\mathbb {H}}}\left\| (\textrm{I}-\pi _{{\mathcal {D}}})\varphi \right\| _{{\mathbb {H}}}, \end{aligned}$$

(A.10)

holds for any $\varphi \in {\mathbb {H}}$, which implies that

$$\begin{aligned} \left\| x^\uplambda (T)-h\right\| _{{\mathbb {H}}}\le \uplambda \left\| \hat{\varphi _{\uplambda }}\right\| _{{\mathbb {H}}}. \end{aligned}$$

Next, by using Assumption (A2), we obtain for $0<\beta \le q$

$$\begin{aligned} \int _{0}^{T}\left\| f(s,x^{\uplambda }_s)\right\| _{{\mathbb {H}}}^{2}\textrm{d}s&\le \int _{0}^{T}\varGamma ^2(s)\textrm{d} s\le \left( \int _{0}^{T}\varGamma ^{\frac{1}{\beta }}(s)\textrm{d}s\right) ^{2\beta }T^{1-2\beta }<+\infty . \end{aligned}$$

The case $\beta =0$ can be obtained in a similar way. The above relation ensures that the set $\{f(\cdot , x^{\uplambda }_{(\cdot )}): \uplambda >0\}$ in $\textrm{L}^2([0,T]; {\mathbb {H}})$ is bounded. By applying the Banach–Alaoglu theorem, we can find a subsequence $\{f(\cdot , x^{\uplambda _i}_{(\cdot )})\}_{i=1}^{\infty }$ such that

Using the above weak convergence together with the compactness of the operator $(\textrm{G}f)(\cdot ) =\int _{0}^{\cdot }(\cdot -s)^{q-1}\textrm{S}_{q}(\cdot -s)f(s)\textrm{d}s:\textrm{L}^2(J;{\mathbb {H}})\rightarrow \textrm{C}(J;{\mathbb {H}})$ (see Lemma 3.4 [3]), we get

$$\begin{aligned} \left\| g(x^{\uplambda _i})-w\right\| _{{\mathbb {H}}}&\le \left\| \int _{0}^{T}(T-s)^{q-1}\textrm{S}_q(T-s)\left[ f(s,x^{\uplambda _i}_s)-f(s)\right] \textrm{d}s\right\| _{{\mathbb {H}}}\nonumber \\&\rightarrow 0\ \text{ as } \ \uplambda \rightarrow 0^+\ \ (i\rightarrow \infty ), \end{aligned}$$

(A.11)

where

$$\begin{aligned} w=h-\textrm{C}_{q}(T)\phi (0)-\textrm{T}_{q}(T)\zeta -\int _{0}^{T} (T-s)^{q-1}\textrm{S}_{q}(T-s)f(s)\textrm{d}s. \end{aligned}$$

Combining the estimates (A.9) and (A.10), we evaluate

$$\begin{aligned} \left\| x^{\uplambda _i}(T)-h\right\| _{{\mathbb {H}}}&\le \uplambda _i\left\| (\uplambda _i(\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _0^T)^{-1}(g(x^{\uplambda _i}))\right\| _{{\mathbb {H}}}\nonumber \\&\le \left\| \uplambda _i(\uplambda _i(\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _{0}^{T})^{-1}\right\| _{{\mathcal {L}}({\mathbb {H}})}\left\| g(x^{\uplambda _i})-w\right\| _{{\mathbb {H}}}\nonumber \\&+\left\| \uplambda _i(\uplambda _i(\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _{0}^{T})^{-1}w\right\| _{{\mathbb {H}}}\nonumber \\&\le \frac{\uplambda _i}{\min (\uplambda _i,\varsigma )}\left\| g(x^{\uplambda _i})-w\right\| _{{\mathbb {H}}}+\left\| \uplambda _i(\uplambda _i(\textrm{I}-\pi _{{\mathcal {D}}})+\Phi _{0}^{T})^{-1}w\right\| _{{\mathbb {H}}}, \end{aligned}$$

(A.12)

where $\varsigma =\min \{\langle \pi _{{\mathcal {D}}}\Phi _{0}^T\pi _{{\mathcal {D}}}\varphi ,\varphi \rangle : \left\| \pi _{{\mathcal {D}}}\varphi \right\| _{{\mathbb {H}}}=1\}>0$. Using the convergence (A.11), Assumption 4.4 (H5) and Theorem 3.5, we obtain

$$\begin{aligned} \left\| x^{\uplambda _i}(T)-h\right\| _{{\mathbb {H}}}\rightarrow 0\ \text{ as } \ \uplambda _i\rightarrow 0^+\ \ (i\rightarrow \infty ). \end{aligned}$$

By taking $\varphi \in {\mathcal {D}}$ in the estimate (A.10), we get

$$\begin{aligned} \pi _{{\mathcal {D}}}x^{\uplambda _i}(T)=\pi _{{\mathcal {D}}}h. \end{aligned}$$

Thus, the system (A.1) is finite-approximately controllable. $\square$