**4. Dual System**

In order to establish approximate controllability, we also need to consider the *dual system* for (3), a similar strategy for partial differential equations of integer order (see Section 8 in [21] or Chapters 2 and 3 in [22] for example). The dual system for (3), which runs backward in time, is given by;

$$\begin{cases} -\partial\_t v + D\_t^{1-\alpha} Av = 0 & \text{in} \quad \Omega \times (0, T), \\ \Delta v(\cdot, t\_i) = I\_i^\*(v(\cdot, t\_i)), & i = 1, 2, 3, \cdots, P\_\prime \\ v = 0 & \text{on} \quad \Gamma \times (0, T), \\ v(\cdot, T) = v\_0 & \text{in} \quad \Omega. \end{cases} \tag{19}$$

## *4.1. Solution of Dual System*

**Proposition 1.** *Let v*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω). *Then, there exists a unique solution of (19) and the solution is given by*

$$\begin{split} v(\mathbf{x},t) &= \sum\_{n=1}^{\infty} (T-t)^{n-1} E\_{\mathbf{z},\mathbf{z}} (-\lambda\_{\text{il}} (T-t)^{n}) (\upsilon\_{0}, \varrho\_{\text{il}}) \varrho\_{\text{n}}(\mathbf{x}) \\ &+ \sum\_{t$$

*and has the following estimate:*

$$\|\boldsymbol{v}(\cdot,t)\|\_{L^{2}(\Omega)} \leq \mathbb{C}\left((T-t)^{a-1}||\boldsymbol{v}\_{0}||\_{L^{2}(\Omega)} + P\|I\_{i\_{\text{w}}}^{\*}\|\_{L^{2}(\Omega)}\left(\sum\_{t$$

*where Iim <sup>L</sup>*2(Ω) <sup>=</sup> sup1≤*i*≤*P*{ *Ii <sup>L</sup>*2(Ω)}.

*Moreover, the mapping v* : [0, *<sup>T</sup>*] <sup>→</sup> *<sup>L</sup>*2(Ω) *is analytically extended to ST* :<sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup>; Re *<sup>z</sup>* <sup>&</sup>lt; *<sup>T</sup>*}*.*

**Proof.** Here, we establish existence and uniqueness of solution of (19) for *v*<sup>0</sup> = 0. Multiplying (19) with *ϕ<sup>n</sup>* and setting *vn*(*t*)=(*v*(·, *t*), *ϕn*), we get

$$
\partial\_t \upsilon\_n(t) + \lambda\_n \partial\_t^{1-a} \upsilon\_n(t) + \sum\_{t < T - t\_i} (I\_i^\*(\upsilon(t\_i), \boldsymbol{\varrho}\_n) = 0. \tag{22}
$$

Since

$$|v\_{\mathfrak{n}}(t)|^2 \le \sum |v\_{\mathfrak{n}}(t)|^2 = ||v(\cdot, t)||\_{L^2(\Omega)}^2 \to 0 \text{ as } t \to T\_{\mathfrak{n}}$$

we have

$$
\upsilon\_n(T) = 0.\tag{23}
$$

From existence and uniqueness of the solution of the fractional differential equation (see [12]), we get

$$
\upsilon\_n(t) = 0, \quad n = 1, 2, 3, \dots, \dots
$$

As {*ϕn*} is a complete orthonormal system, we have

$$v = 0 \text{ in } \Omega \times (0, T).$$

Thus, Equation (19) has a unique solution. Now, we show the estimate (21).

By (20), we have

 *<sup>v</sup>*(·, *<sup>t</sup>*) <sup>2</sup> *L*2(Ω) ≤ 4 4 4 4 4 ∞ ∑ *n*=1 (*v*0, *<sup>ϕ</sup>n*)(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)*α*−1*Eα*,*α*(−*λn*(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)*α*)*ϕ<sup>n</sup>* 4 4 4 4 4 2 *L*2(Ω) + 4 4 4 4 4 ∑ ∑*t*<*T*−*ti mk* ∑ *j*=1 (*I* ∗ *<sup>i</sup>* (*v*(·, *ti*)), *ϕn*)*ϕn*(*x*) (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *ti*)*α*−1*Eα*,*α*(−*λn*(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *ti*)*α*) 4 4 4 4 4 2 *L*2(Γ) = ∞ ∑ *n*=1 (*v*0, *<sup>ϕ</sup>n*)(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)*α*−1*Eα*,*α*(−*μk*(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)*α*) 2 + ∞ ∑ *n*=1 ∑ *t*<*T*−*ti* (*I* ∗ *<sup>i</sup>* (*v*(·, *ti*)), *<sup>ϕ</sup>n*)(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *ti*)*α*−1*Eα*,*α*(−*μk*(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *ti*)*α*) 2 = *C*<sup>2</sup> ∞ ∑ *n*=1 |(*v*0, *ϕn*)| 2 (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)2*α*−<sup>2</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> ∑ *t*<*T*−*ti* (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *ti*)2*α*−<sup>2</sup> ∞ ∑ *n*=1 |(*I* ∗ *<sup>i</sup>* (*v*(·, *ti*)), *ϕn*)| 2 

Therefore,

$$\|\|v(\cdot,t)\|\|\_{L^2(\Omega)} \le C \left( (T-t)^{a-1} \|\|v\_0\|\|\_{L^2(\Omega)} + P \|I\_{i\_m}^\*\|\|\_{L^2(\Omega)} \left( \sum\_{t < T-t\_i} (T-t-t\_i)^{2a-2} \right)^{\frac{1}{2}} \right).$$

.

Next, we show the analyticity of *v*(·, *t*) in *t* ∈ *ST*.

We note that *Eα*,*α*(−*λnz*) is an entire function (see [20] for example) and consequently each (*<sup>T</sup>* <sup>−</sup> *<sup>z</sup>*)*α*−1*Eα*,*α*(−*λn*(*<sup>T</sup>* <sup>−</sup> *<sup>z</sup>*)*α*) is analytic in *<sup>z</sup>* <sup>∈</sup> *ST*. Therefore, <sup>∑</sup>*<sup>N</sup> <sup>n</sup>*=1(*v*0, *<sup>ϕ</sup>n*)(*<sup>T</sup>* <sup>−</sup> *<sup>z</sup>*)*α*−1*Eα*,*α*(−*λn*(*<sup>T</sup>* <sup>−</sup> *<sup>z</sup>*)*α*)*ϕ<sup>n</sup>* in *ST*.

If we fix *<sup>δ</sup>* <sup>&</sup>gt; 0 arbitrarily, then, for *<sup>z</sup>* <sup>∈</sup> <sup>C</sup> with Re *<sup>z</sup>* <sup>≤</sup> *<sup>T</sup>* <sup>−</sup> *<sup>δ</sup>*, we have

$$\begin{split} & \left\| \sum\_{n=M}^{N} (v\_{0\prime}, \boldsymbol{\varrho}\_{n}) (T - z)^{\alpha - 1} E\_{\boldsymbol{a}, \alpha} (-\lambda\_{n} (T - z)^{\alpha}) \boldsymbol{\varrho}\_{n} \right\|\_{L^{2}(\Omega)}^{2} \\ & = \sum\_{n=M}^{N} \left| (v\_{0\prime}, \boldsymbol{\varrho}\_{n}) (T - z)^{\alpha - 1} E\_{\boldsymbol{a}, \alpha} (-\lambda\_{n} (T - z)^{\alpha}) \right|^{2} \\ & \leq C \sum\_{n=M}^{N} |(v\_{0\prime}, \boldsymbol{\varrho}\_{n})|^{2} |T - z|^{2\alpha - 2} \\ & \leq C \delta^{2\alpha - 2} \sum\_{n=M}^{N} |(v\_{0\prime}, \boldsymbol{\varrho}\_{n})|^{2} \to 0 \quad \text{as } M, N \to \infty. \end{split}$$

That is, (20) is uniformly convergent in {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup>;Re *<sup>z</sup>* <sup>≤</sup> *<sup>T</sup>* <sup>−</sup> *<sup>δ</sup>*}. Hence, *<sup>v</sup>*(·, *<sup>t</sup>*) is also analytic in *t* ∈ *ST*.

#### *4.2. Unique Continuation Property*

**Proposition 2.** *Let <sup>ω</sup> be open in* <sup>Ω</sup> *and <sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω). *If a solution <sup>v</sup>* <sup>∈</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*2(Ω) <sup>∩</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) *be the solution of* (19) *vanishing in ω* × (0, *T*), *then v* = 0 *in* Ω × (0, *T*)*.*

**Proof.** Since *<sup>v</sup>*(*x*, *<sup>t</sup>*) = 0 in *<sup>ω</sup>* <sup>×</sup> (0, *<sup>T</sup>*) and *<sup>v</sup>* : [0, *<sup>T</sup>*) <sup>→</sup> *<sup>L</sup>*2(Γ) can be analytically extended to *ST* :<sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup>; Re *<sup>z</sup>* <sup>&</sup>lt; *<sup>T</sup>*}, we have

$$\begin{split} \boldsymbol{v}(\mathbf{x},t) &= \sum\_{n=1}^{\infty} (v\_0, q\_n) (T - t)^{n-1} E\_{\mathbf{a}, \mathbf{a}} (-\lambda\_{\mathbf{n}} (T - t)^{\mathbf{a}}) q\_n(\mathbf{x}) \\ &+ \sum\_{t < T - t\_i} \sum\_{n=1}^{\infty} (I\_t^\*(\mathbf{v}(\cdot, t\_i)), q\_n) (T - t - t\_i)^{\mathbf{a} - 1} E\_{\mathbf{a}, \mathbf{a}} (-\lambda\_{\mathbf{n}} (T - t - t\_i)^{\mathbf{a}}) q\_n(\mathbf{x}) \\ &= \boldsymbol{0}, \quad \mathbf{x} \in \omega, \ t \in ( -\infty, T). \end{split} \tag{24}$$

Let {*μk*}*k*∈<sup>N</sup> be all spectra of *L* without multiplicities and we denote by {*ϕkj*}1≤*j*≤*mk* an orthonormal basis of Ker(*μ<sup>k</sup>* − *L*). By using these notations, we can rewrite (24) by

$$\begin{split} \boldsymbol{\nu}(\mathbf{x},t) &= \sum\_{k=1}^{\infty} \left( \sum\_{j=1}^{m\_k} (\boldsymbol{v}\_{0\*} \boldsymbol{q}\_{kj}) \boldsymbol{q}\_{kj}(\mathbf{x}) \right) (T-t)^{a-1} E\_{\mathbf{z},a} (-\mu\_k (T-t)^a) \\ &+ \sum\_{k$$

Then, for any *<sup>z</sup>* <sup>∈</sup> <sup>C</sup> with Re *<sup>z</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>&</sup>gt; 0 and *<sup>N</sup>* <sup>∈</sup> <sup>N</sup>, we have

$$\begin{split} & \left\| \sum\_{k=1}^{N} \left( \sum\_{j=1}^{m\_k} (v\_{0,\prime} \, q\_{kj}) \, q\_{kj}(\mathbf{x}) \right) e^{z(t-T)} (T-t)^{a-1} E\_{a,a} (-\mu\_k (T-t)^a) \right\|\_{L^2(\Gamma)}^2 \\ & = \sum\_{k=1}^{N} \left( \sum\_{j=1}^{m\_k} |(v\_{0,\prime} \, q\_{kj})|^2 \right) e^{2\xi(t-T)} \left| (T-t)^{a-1} E\_{a,a} (-\mu\_k (T-t)^a) \right|^2 \\ & \le \mathcal{C}^2 e^{2\xi(t-T)} (T-t)^{2a-2} ||v\_0||\_{L^2(\Omega)} \end{split}$$

and

$$\begin{split} & \left\| \sum\_{k=1}^{N} \sum\_{t < T - t\_i} \left( \sum\_{j=1}^{m\_k} \langle I\_i^\*(\boldsymbol{\nu}(., t\_i)), \boldsymbol{\rho}\_{kj} \rangle \boldsymbol{\varrho}\_{kj}(\boldsymbol{x}) \right) (T - t - t\_i)^{a-1} E\_{b, \boldsymbol{a}} (-\mu\_k (T - t - t\_i)^a) \right\|\_{L^2(\Gamma)}^2 \\ &= \sum\_{k=1}^{N} \sum\_{t < t\_i} \left( \sum\_{j=1}^{m\_k} |\langle I\_i^\*(\boldsymbol{\nu}(., t\_i)), \boldsymbol{\rho}\_{kj} \rangle|^2 \right) e^{2\xi (t - T)} \left| (T - t - t\_i)^{a-1} E\_{a, \boldsymbol{a}} (-\mu\_k (T - t - t\_i)^a) \right|^2 \\ & \leq \sum\_{k=1}^{N} \sum\_{t < T - t\_i} \left( \sum\_{j=1}^{m\_k} |\langle I\_i^\*(\boldsymbol{\nu}(., t\_i)), \boldsymbol{\rho}\_{kj} \rangle|^2 \right) e^{2\xi (t - T)} \left| (T - t - t)^{a-1} E\_{a, \boldsymbol{a}} (-\mu\_k (T - t - t\_i)^a) \right|^2 \\ & \leq PC^2 \epsilon^{2\tilde{\xi} (t - T)} \| |I\_{i\boldsymbol{a}}^\*| \|\_{L^2(\Omega)}^2 \sum\_{t < T - t\_i} (T - t - t\_i)^{2a - 2}, \end{split}$$

where *Iim <sup>L</sup>*2(Ω) <sup>=</sup> sup1≤*i*≤*P*{ *Ii <sup>L</sup>*2(Ω)}. Therefore,

$$\begin{aligned} & \left\| \sum\_{k=1}^N \left( \sum\_{j=1}^{m\_k} (\upsilon\_{0\prime} \,\,\rho\_{kj}) \,\rho\_{kj}(\mathbf{x}) \right) e^{z(t-T)} (T-t)^{a-1} E\_{a,a} (-\mu\_k (T-t)^a) \right\|\_{L^2(\Gamma)} \\ & \le C e^{\overline{\zeta}(t-T)} (T-t)^{a-1} \|\upsilon\_0\|\_{L^2(\Omega)} \end{aligned}$$

and

$$\begin{aligned} &\left\|\sum\_{k=1}^N \sum\_{t$$

The right-hand sides of the two inequalities above are integrable on (−∞, *T*):

$$\int\_{-\infty}^{T} e^{\xi(t-T)} (T-t)^{\alpha-1} dt = \frac{\Gamma(\alpha)}{\xi^{\alpha}}$$

and

$$\int\_{-\infty}^{T-t\_i} e^{\xi(t\_i+t-T)} (T-t-t\_i)^{a-1} dt = \int\_0^{\infty} e^{-\xi t} t^{a-1} dt = \frac{\Gamma(a)}{\xi^a}.$$

Hence, the Lebesgue theorem yields that

$$\begin{cases} \int\_{-\infty}^{T} \boldsymbol{\varepsilon}^{z(t-T)} \left( \sum\_{k=1}^{\infty} \left( \sum\_{j=1}^{m\_k} (\upsilon\_0, \rho\_{kj}) \boldsymbol{\varrho}\_{kj} (\mathbf{x}) \right) (T-t)^{a-1} \mathbb{E}\_{\mathbf{z},a} (-\boldsymbol{\mu}\_k (T-t)^a) \right) dt + \int\_{-\infty}^{T-t\_i} \boldsymbol{\varepsilon}^{z(t\_i+t-T)} \\ \quad + \left( \sum\_{k 0, \end{cases} \tag{26}$$

where we have used the Laplace transform formula;

$$\int\_0^\infty e^{-zt}t^{\alpha-1}E\_{\alpha,a}(-\mu\_k t^\alpha)dt = \frac{1}{z^\alpha + \mu\_k}, \quad \text{Re}\, z > 0$$

(see (1.80) in p. 21 of [20]). By (25) and (26), we have

$$\sum\_{k=1}^{\infty} \sum\_{j=1}^{m\_k} \frac{(v\_0 + \sum\_{t \le T - t\_i} I\_i^\*(v(\cdot, t\_i)), q\_{kj})}{z^a + \mu\_k} \varphi\_{kj}(\mathbf{x}) = 0, \quad \text{a.e.} \, \mathbf{x} \in \omega, \, \text{Re} \, z > 0,$$

that is, <sup>∞</sup>

$$\sum\_{k=1}^{\infty} \sum\_{j=1}^{m\_k} \frac{(v\_0 + \sum\_{\substack{\ell < T - t\_i}} I\_i^\*(v(\cdot, t\_i)), \,\boldsymbol{\varrho}\_{kj})}{\eta + \mu\_k} \boldsymbol{\varrho}\_{kj}(\mathbf{x}) = 0, \quad \text{a.e.} \,\mathbf{x} \in \omega, \,\text{Re}\,\eta > 0.$$

By using analytic continuation in *η*, we have

$$\sum\_{k=1}^{\infty} \sum\_{j=1}^{m\_k} \frac{(\upsilon\_0 + \sum\_{t < T - t\_i} I\_i^\* \left(\upsilon \left(\cdot, t\_i\right)\right), \rho\_{kj})}{\eta + \mu\_k} \varphi\_{kj}(\mathbf{x}) = 0, \quad \text{a.e.} \ x \in \omega, \ \eta \in \mathbb{C} \ \backslash \{ -\mu\_k \}\_{k \in \mathbb{N}}.\tag{27}$$

Then, we can take a suitable disk which includes −*μ* and does not include {−*μk*}*k*=. By integrating (27) in the disk, we have

$$\sum\_{j=1}^{m\_{\ell}} (\upsilon\_0 + \sum\_{t < T - t\_i} I\_i^\*(\upsilon(\cdot, t\_i)), \,\varrho\_{\ell j}) \,\rho\_{\ell j}(\mathbf{x}) = 0, \quad \text{a.e.} \,\mathbf{x} \in \omega.$$

By setting *<sup>v</sup>*= :<sup>=</sup> <sup>∑</sup>*m <sup>j</sup>*=1(*v*<sup>0</sup> + <sup>∑</sup>*t*<*T*−*ti <sup>I</sup>*<sup>∗</sup> *<sup>i</sup>* (*v*(·, *ti*)), *ϕj*)*ϕj*(*x*), we have

$$(A - \mu\_\ell)\widetilde{v}\_\ell = 0 \quad \text{in } \Omega \quad \text{and} \quad \widetilde{v}\_\ell = 0 \quad \text{on } \omega.$$

Therefore, the unique continuation result for eigenvalue problem of elliptic operator (see [23,24]) implies

$$\tilde{v}\_\ell(\mathbf{x}) = \sum\_{j=1}^{m\_\ell} (v\_0 + \sum\_{t < T - t\_i} I\_i^\*(v(\cdot, t\_i)), \,\varrho\_{\ell j}) \,\varrho\_{\ell j}(\mathbf{x}) = 0, \quad \mathbf{x} \in \Omega$$

for each <sup>∈</sup> <sup>N</sup>. Since {*ϕj*}1≤*j*≤*m* is linearly independent in <sup>Ω</sup>, we see that

$$(\upsilon\_0 + \sum\_{t < T - t\_i} I\_i^\*(\upsilon(\cdot, t\_i)), q\_{\ell \dot{j}}) = 0, \quad 1 \le \dot{j} \le m\_{\ell \prime} \text{ } \ell \in \mathbb{N}.$$

This implies *v* = 0 in Ω × (0, *T*).

#### **5. Approximate Controllability**

In this section, we complete the proof of our main theorems.

**Theorem 3.** *Let* 0 < *α* < 1 *and ω be an open set in* Ω*. Then, Equation* (3) *is approximately controllable for arbitrarily given T* > 0*. That is,*

$$\overline{\left\{\mu(\cdot,T);\,f\in\mathbb{C}\_{0}^{\infty}(\omega\times(0,T))\right\}}=L^{2}(\Omega),\tag{28}$$

*where u is the solution to* (3) *and the closure on the left-hand side is taken in L*2(Ω)*.*

We start the proof with a lemma.

**Lemma 4.** *If the conclusion of Theorem* (3) *is true for u*<sup>0</sup> <sup>≡</sup> <sup>0</sup>*, then it is true for any u*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)*.*

**Proof.** Let *<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω) and *uT* <sup>∈</sup> *<sup>L</sup>*2(Ω). Let <sup>&</sup>gt; 0. Let us introduce *<sup>u</sup>*¯ the (mild) solution of

$$\begin{cases} \begin{aligned} \ddot{\boldsymbol{u}}\_{t} + \dot{\boldsymbol{\theta}}\_{t}^{1-\alpha} \boldsymbol{A} \ddot{\boldsymbol{u}} &= \boldsymbol{0} & (\boldsymbol{x},t) \in \Omega \times (0,T), \\ \Delta \boldsymbol{u}(\cdot, t\_{i}) &= I\_{i}(\boldsymbol{u}(\cdot, t\_{i})), & \dot{\boldsymbol{\iota}} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \cdots, \mathbf{P}, \\ \ddot{\boldsymbol{u}}(\boldsymbol{x},t) &= \mathbf{0}, & \boldsymbol{t} \in \Gamma \times (0,T), \\ \ddot{\boldsymbol{u}}(\boldsymbol{x},0) &= \boldsymbol{u}\_{0}(\boldsymbol{x}), & \boldsymbol{x} \in \Omega. \end{aligned} \end{cases}$$

Then, *<sup>u</sup>*¯(*T*) <sup>∈</sup> *<sup>L</sup>*2(Ω). Therefore, using the assumption of Lemma 4, there exists *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*ω* × (0, *T*)) such that the solution *w* of

$$\begin{cases} \begin{aligned} \partial\_t w + \partial\_t^{1-\alpha} Aw &= f(x,t) & (x,t) \in \Omega \times (0,T)\_{\prime}, \\ \Delta u(\cdot, t\_i) &= I\_i(u(\cdot, t\_i)), & i = 1,2,3,\cdots \end{aligned} & t = 1,2,3,\cdots \end{cases} \\ w(1,t) = 0, & t \in \Gamma \times (0,T)\_{\prime} \\ w(x,0) = 0, & x \in \Omega, \end{cases}$$

satisfies

$$\|w(T) - (\mu\_T - \bar{u}(T))\|\_{L^2(\Omega)} \le \epsilon.$$

One can easily see that *u*(*T*) = *w*(*T*) + *u*¯(*T*), so that the proof of Lemma 4 is achieved.

We now assume that *u*<sup>0</sup> ≡ 0.

In order to complete the proof of Theorem 3, we will see that the unique continuation property for (19) is equivalent to the approximate controllability for (3) stated in Theorem 3.

**Proof.** Let *<sup>u</sup>* be a solution of (3) for *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*<sup>ω</sup>* <sup>×</sup> (0, *<sup>T</sup>*)) and let *<sup>v</sup>* be a solution of (19) for *<sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω). Then, we see that

$$\begin{aligned} 0 &= \int\_0^T \int\_{\Omega} \left( \partial\_t u + \partial\_t^{1-\alpha} A u - f \right) v dx dt \\ &= \int\_0^T \int\_{\Omega} (\partial\_t u) v dx dt + \int\_0^T \int\_{\Omega} (\partial\_t^{1-\alpha} A u) v dx dt \\ &- \int\_0^T \int\_{\Omega} f v dx dt - \int\_0^T \int\_{\Omega} \sum\_{1 \le i \le P} I\_i(u(t\_i)) \delta(t - t\_i) v dx dt. \end{aligned}$$

In the above equation, the first term is calculated as follows:

$$\begin{aligned} \int\_0^{T-\delta} \int\_{\Omega} (\partial\_t u) v dx dt &= \int\_0^{T-\delta} \int\_{\Omega} (\partial\_t u) v dx dt \\ &= \int\_{\Omega} u v \Big|\_{t=0}^{t=T-\delta} dx - \int\_0^{T-\delta} \int\_{\Omega} u (\partial\_t v) dx dt \\ &= \int\_{\Omega} u (\cdot, T-\delta) v\_0 dx + \int\_0^{T-\delta} \int\_{\Omega} u (\partial\_t v) dx dt. \end{aligned}$$

Here, we have used the integration in *t* by parts and the initial conditions in (3) and (19).

In terms of *<sup>u</sup>* <sup>∈</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*2(Ω)) and *<sup>v</sup>* <sup>∈</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*2(Ω) <sup>∩</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)), we apply the Green formula to the second term, we have

$$\begin{split} \int\_{0}^{T-\delta} \int\_{\Omega} (\partial\_{t}^{1-a}Au)\upsilon dxdt &= \int\_{0}^{T-\delta} \int\_{\Omega} (\partial\_{t}^{1-a}u)(Av) dxdt + \int\_{0}^{T-\delta} \int\_{\Gamma} \left( u \frac{\partial \upsilon}{\partial \upsilon\_{A}} - \frac{\partial u}{\partial \upsilon\_{A}} \upsilon \right) d\sigma\_{x} dt \\ &= - \int\_{0}^{T-\delta} \int\_{\Omega} u(D\_{t}^{1-a}Av) dxdt. \end{split}$$

In the above calculation, we have used boundary conditions in (3) and (19). Therefore, we have

$$\begin{split} 0 &= \int\_{0}^{T-\delta} \int\_{\Omega} (\partial\_{t}u)v dx dt + \int\_{0}^{T-\delta} \int\_{\Omega} (\partial\_{t}^{1-a}Au) v dx dt - \int\_{0}^{T-\delta} \int\_{\Omega} f v dx dt \\ &= \int\_{\Omega} u(\cdot, T-\delta)v u dx + \int\_{0}^{T-\delta} \int\_{\Omega} u(D\_{t}v) dx dt - \int\_{0}^{T-\delta} \int\_{\Omega} u(D\_{t}^{1-a}Av) dx dt \\ &- \int\_{0}^{T-\delta} \int\_{\Omega} f v dx dt - \int\_{0}^{T} \int\_{\Omega} \sum\_{1 \le i \le P} I\_{i}(u(t\_{i})) \delta(t - t\_{i}) v dx dt \\ &= \int\_{\Omega} u(\cdot, T-\delta)v\_{0} dx + \int\_{0}^{T-\delta} \int\_{\Omega} u \left(\partial\_{t}v - D\_{t}^{1-a}Av\right) dx dt \\ &- \int\_{0}^{T-\delta} \int\_{\Omega} f v dx dt - \int\_{\Omega} \sum\_{1 \le i \le P} I\_{i}(u(t\_{i})) v dx \\ &= \int\_{\Omega} u(\cdot, T-\delta)v\_{0} dx - \int\_{0}^{T-\delta} \int\_{\Omega} f v dx dt - \sum\_{1 \le i \le P} \int\_{O} I\_{i}(u(t\_{i})) v dx. \end{split}$$

Since *<sup>u</sup>* <sup>∈</sup> *PC*([0, *<sup>T</sup>*], *<sup>L</sup>*2(Ω)) and *<sup>v</sup>*(·, *<sup>T</sup>*) = *<sup>v</sup>*<sup>0</sup> and taking *<sup>δ</sup>* <sup>→</sup> 0, we get

$$\int\_{\Omega} \mathbf{u}(\cdot, T) \mathbf{v}\_0 d\mathbf{x} - \sum\_{1 \le i \le P} \int\_{O} I\_i(\mathbf{u}(t\_i)) v d\mathbf{x} = \int\_0^T \int\_{\Omega} f v d\mathbf{x} dt. \tag{29}$$

In order to prove density of {*u*(·, *<sup>T</sup>*); *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*<sup>ω</sup>* <sup>×</sup> (0, *<sup>T</sup>*))} in *<sup>L</sup>*2(Ω), we have to show that, if *<sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω) satisfies

$$\mu(\mu(\cdot, T), v\_0) = \int\_{\Omega} \mu(\cdot, T) v\_0 dx = 0 \tag{30}$$

for any *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*ω* × (0, *T*)), then *v*<sup>0</sup> ≡ 0. This can be shown as follows: we have

$$\int\_{0}^{T} \int\_{\Omega} fv dx dt = 0$$

for any *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*ω* × (0, *T*)). Then, by the fundamental theorem of the calculus of variations. we have

$$
\upsilon(\mathfrak{x}, t) = 0,\\
(\mathfrak{x}, t) \in \mathfrak{w} \times (0, T).
$$

By proposition (2), we have

$$
\upsilon(\mathbf{x}, t) = \mathbf{0}, (\mathbf{x}, t) \in \Omega \times (0, T).
$$

By uniqueness of the solution of (1),

$$v\_0(\mathbf{x}) = 0, \mathbf{x} \in \Omega,$$

which gives {*u*(·, *<sup>T</sup>*); *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*ω* × (0, *T*))} <sup>⊥</sup> <sup>=</sup> {0}. Hence, {*u*(·, *<sup>T</sup>*); *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (*ω* × (0, *T*))} is dense in *L*2(Ω).

Thus, the proof of Theorem (3) is completed.

#### **6. Example**

**Example 1.** *Consider the following relaxations' oscillation equation with fractional order given by*

$$\begin{cases} \frac{\partial}{\partial t} u(\mathbf{x}, t) = \frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}} \frac{\partial^2}{\partial x^2} u(\mathbf{x}, t) + f(\mathbf{x}, t), \quad t \in I = (0, T), \; \mathbf{x} \in \Omega = (0, \pi), \\ u(0, t) = u(\pi, t) = 0 \quad t \in (0, T), \\ u(\mathbf{x}, 0) = u\_0, \quad \mathbf{x} \in (0, \pi), \\ \Delta u(\mathbf{x}, t\_k) = -u(\mathbf{x}, t\_k) \quad k = 1, 2, \cdots, N. \end{cases} \tag{31}$$

Now, consider the corresponding system Let *u*(*t*)*x* = *u*(*x*, *t*) and assume *f*(*x*(*t*), *t*) to be a continuous function with respect to *t* that satisfies the Lipschitz condition in *x*. Define the operator *Au* = *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> with domain

$$D(A) = \{ \mathbf{x} \in L^2(0, \pi) : \mathbf{x}, \mathbf{x}' \text{ are absolutely continuous and } \mathbf{x}, \mathbf{x}', \mathbf{x}'' \in L^2(0, \pi) \}.$$

It is well known that for *α* = 1, sectorial operator, *A* = *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> generates an analytic semigroup and for *α* = 2, sectorial operator, *A* = *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> generates a cosine family of operators.

Using the above notation, now consider the following system

$$\begin{aligned} \frac{\partial u}{\partial t} &= \frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}} A u, \quad t \in I = (0, T), \ x \in \Omega = (0, \pi), \\ u(0, t) &= u(\pi, t) = 0 \quad t \in (0, T), \\ u(x, 0) &= 0, \quad x \in (0, \pi), \\ \Delta u(x, t\_k) &= -u(x, t\_k) \quad k = 1, 2, \cdots, N. \end{aligned} \tag{32}$$

The above problem can be posed as an abstract problem on *X* = *L*2(0, *π*) = *U*, and hence it has a unique solution. Hence under the assumption of Theorem, the problem is approximately controllable.

**Example 2.** *By choosing the function cos*(*t* <sup>2</sup>)*exp*(−*t*), *we get the following relaxations oscillation equation with fractional order given by*

$$
\partial\_t^{1.8} u(t) + Au(t) = \cos(t^2) \exp(-t), \\
u(0) = 1, \\
u'(0) = 1,\tag{33}
$$

*where A is the operator mentioned above.*

The graphical illustration of Example 2 is depicted in the Figure 1.

**Figure 1.** Comparison of solution of (33) with varied relaxation coefficients, *A* = 1, 2 and 3.

## **7. Discussion**

This paper presents a fractional sub-diffusion equation of an impulsive system (3) and its dual (19). The unique continuation Property 2 of the dual system plays a crucial role in the proof of our main result, approximate controllability Theorem 3 of the primal system with an interior control acts on a sub-domain. As an example, the approximate controllability of a fractional relaxation-oscillation equation is discussed and simulated for different relaxation coefficients.

**Author Contributions:** Conceptualization, L.M. and S.A.; methodology, S.A. and M.H.; software, M.H.; validation, S.A. and H.M.S.; formal analysis, L.M. and H.M.S.; writing—original draft preparation, S.A. and M.H.; writing—review and editing, S.A. and H.M.S.; supervision, S.A. and H.M.S.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors are thankful to the anonymous reviewers for their careful reading of the manuscript and constructive comments and suggestions. Lakshman Mahto would like to thank The Institute of Mathematical Sciences, Chennai, for support and hospitality during the postdoctoral work, where this work was initiated.

**Conflicts of Interest:** The authors declare no conflict of interest.
