The Regional Enlarged Observability for Hilfer Fractional Differential Equations

Abstract

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ${\nu }_0^1$ on an internal subregion $\omega$ from the whole domain $\mathsf{\Omega }$ with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results.

1. Introduction

In recent decades, fractional calculus theory has proven to be a significant tool for the formulation of several problems in science and engineering, where fractional derivatives and integrals can be utilized to describe the characteristics of various real materials in various scientific disciplines; see, e.g., [1,2,3,4,5]. This theory has recently received a large amount of consideration by many academics; we mention Euler, Laplace, Riemann, Liouville, Marchaud, Riesz, and Hilfer; see, e.g., [6,7,8]. Distributed parameter systems can be analysed in terms of controllability, observability, and stability, which lead to numerous applications. However, one of the most basic concerns in system analysis and control is observability, which is concerned with the reconstruction of a system’s initial state that is taken from measurements on a system by means of so-called sensors; see, [9]. Amouroux et al. [10] developed two approaches to investigate regional observability (ReOb) for distributed systems. The first is state-space-based, and the second allows for estimating the state on the considered subregion. El Jai et al. [11] introduced the concept of regional strategic sensors for a class of distributed systems and presented the sensor characterization for various geometrical situations. In [12], Al-Saphory et al. considered and analysed the notion of regional gradient strategic sensors, and the results applied to a two-dimensional linear infinite distributed system in Hilbert space.

In a problem governed by a diffusion system, it is commonly known that the positioning of sensors is restricted by severe practical restrictions. In fact, observation processes are generally restricted to subsets, boundaries, or points [13,14]. This indicates that the operators of the observation can be unbounded in their state spaces.

Recently, the study of ReOb for partial differential equations (PDEs) has received considerable attention in the literature. Zerrik et al. [15] reviewed regional boundary observability for a two-dimensional diffusion system. In [16], Chen investigated infinite time exact observability for the Volterra system in Hilbert spaces. Chen and Yi [17] studied the observability and admissibility of Volterra systems in Hilbert spaces. Zouiten et al. [18] studied the following ReEnOb for a linear parabolic system.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\frac{\partial }{\partial t}\nu (y,t)=A\nu (y,t)\hfill & \mathrm{in}\mathsf{\Omega }\times [0,T],\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \nu (y,0)={\nu }_0\left(y\right)\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \mathfrak{M}\left(t\right)=C\nu \left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(1)

where A is an infinitesimal operator and generates a strongly continuous semigroup ${\left\{Q\left(t\right)\right\}}_{t\ge 0}$ on the state space $L^2\left(\mathsf{\Omega }\right)$, $\mathsf{\Omega }$ is an open bound of $L^2\left(\mathsf{\Omega }\right)$, and $\mathfrak{M}$ is the output function (OuPuFu), which represents the measurements. The authors used the HUM approach to reconstruct the initial state between two profiles in an internal subregion.

More recently, many researchers have investigated the ReOb for fractional differential equations (FDEs). In [19], Zguaid and El Alaoui investigated the notion of the regional boundary observability of Caputo fractional systems. Zguaid et al. [20] studied ReOb for a class of linear time-fractional systems using the HUM approach and proved that the considered approach allows to transform the ReOb problem into a solvability one. Regional gradient observability for Caputo fractional diffusion systems is considered in [21]. In [22], Ge et al. presented the notion of the regional gradient observability for Riemann–Liouville (R-L) diffusion systems for the first time. Cai et al. [23] investigated the concept of exact and approximate ReOb of Hadamard–Caputo diffusion systems using the HUM approach. Zguaid and El Alaoui [24] investigated the notion of regional boundary observability of R-L linear diffusion systems by using an extension of HUM.

On the other hand, some works concerning the concept of ReEnOb-FDEs have recently been conducted. In [25], Zouiten et al. studied the ReEnOb for R-L fractional evolution equations with R-L derivatives:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^{RL}\mathrm{D}_t^{\eta }\nu (y,t)=A\nu (y,t)\hfill & \mathrm{in}\mathsf{\Omega }\times [0,T],\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\eta }\nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \mathfrak{M}\left(t\right)=C\nu \left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(2)

where $\mathsf{\Omega }$ is an open bound of ${\mathbb{R}}^n(n=1,2,3)$, with the regular boundary $\partial \mathsf{\Omega }$ and ${}_0{}^{RL}\mathrm{D}_t^{\eta }$ and ${}_0{I}_t^{1-\eta }$ are R-L fractional derivatives and R-L fractional integrals of orders $0<\eta \le 1$ and $1-\eta$, respectively. The authors developed an approach based on HUM allowing them to reconstruct the initial state between two given functions in an internal subregion of the whole domain. In [26], Zouiten and Boutoulout investigated the ReEnOb for the following Caputo fractional diffusion system in a Hilbert space

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^C\mathrm{D}_t^{\eta }\nu (y,t)=A\nu (y,t)\hfill & \mathrm{in}\mathsf{\Omega }\times [0,T],\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \mathfrak{M}\left(t\right)=C\nu \left(t\right),\hfill & t\in [0,T].\hfill \end{array}\right.\end{array}$$

(3)

The HUM approach for fractional differential systems is used for the process of reconstructing the initial state between two profiles in a considered subregion of the whole domain.

Inspired and motivated by the above discussion, in this manuscript we extend the investigation of the notion of the ReEnOb for sub-diffusion systems with fractional derivatives, augmented and restricted by some measurements given by the so-called OuPuFu. We note that FDEs have been widely used for modelling in various science and engineering fields due to their well-described systems and high accuracy, as well as yielding better results compared with systems with integer differentiation. Therefore, the results obtained from Systems (2) and (3) are better than those of System (1). Moreover, use the Hilfer fractional derivative as we know it has two parameters and contains Caputo and R-L derivatives in its definition. Thus, our findings can be seen as a generalization of the mentioned results.

This paper is interested in the concept of ReEnOb for the following sub-diffusion system via Hilfer FDs of order $\eta$, type $\kappa$ and augmented with the OuPuFu (5). We first characterize the ReEnOb of a diffusion system augmented with the OuPuFu in an internal subregion $\omega$ of $\mathsf{\Omega }$. Moreover, we recognize two types of sensors based on the boundness issue of the observation operator C. Then, we reconstruct the initial state ${\nu }_0^1$ of the addressed system using an approach that relies on the HUM approach introduced by Lions [27]. The investigation of the addressed problem shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Let $\mathsf{\Omega }$ be an open bound of ${\mathbb{R}}^n(n=1,2,3)$ with the regular boundary $\partial \mathsf{\Omega }$, and let $\mathfrak{J}=[0,T]$. The space ${\mathfrak{S}}_T=\mathsf{\Omega }\times \mathfrak{J}$ and ${\Sigma }_T=\partial \mathsf{\Omega }\times \mathfrak{J}$. We consider the following diffusion sub-system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\nu (y,t)=A\nu (y,t)\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }\nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{in}\mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(4)

where ${}_0{}^H\mathrm{D}_t^{\eta ,\kappa }$ stands for the Hilfer fractional derivative (left-sided) of order $0<\eta <1$, type $0\le \kappa \le 1$ with respect to time t, the integral ${}_0{I}_t^{1-\zeta }$, $\zeta =\eta +\kappa -\eta \kappa$, $0<\zeta \le 1$ is the left-sided R-L fractional integral operator (1), and the operator A is linear and has a dense domain, so the coefficients are independent of time t. Moreover, operator A is infinitesimal and generates a strongly continuous semigroup ${\left\{Q\left(t\right)\right\}}_{t\ge 0}$ on the state space $L^2\left(\mathsf{\Omega }\right)$, which is a Hilbert space. Here, the initial state ${\nu }_0\in L^2\left(\mathsf{\Omega }\right)$ is assumed to be unknown. The measurements and information of System (4) are obtained by the OuPuFu below:

$$\mathfrak{M}\left(t\right)=C\nu \left(t\right),t\in \mathfrak{J},$$

(5)

where C is the observation operator, and it is a linear, not necessary a bounded, operator determined by the number of sensors or their structure, with a dense domain $\mathcal{D}\left(C\right)\subseteq L^2\left(\mathsf{\Omega }\right)$ with range in the observation space $\mathcal{O}=L^2(\mathfrak{J};{\mathbb{R}}^q)$$(q\in \mathbb{N}$ is the number of considered sensors), and $\mathcal{O}$ is a Hilbert space.

This paper is arranged as follows: In Section 2, we review the definitions, basic concepts, and lemmas utilized throughout this paper. In Section 3, we characterize the ReEnOb. Moreover, we present some remarks, then introduce and prove the main theorem of the ReOb of the Hilfer diffusion System (4). In Section 4, the HUM approach is introduced and applied in the reconstruction process of the initial state of System (4). In addition, two theoretical illustrative examples are given to support our results. In Section 5, we give some conclusions.

2. Preliminaries

In this section, we review the essential definitions, notations, and basic facts utilized throughout this paper.

Definition 1.

(See [7]) The R-L fractional integral (left-sided) of order η for a function $f:\mathfrak{J}\to \mathbb{R}$ is defined as

$${}_0{I}_t^{\eta }f\left(t\right)=\frac{1}{\Gamma \left(\eta \right)}{\int }_0^t{(t-s)}^{\eta -1}f\left(s\right)\mathrm{d}s,0<\eta <1.$$

Definition 2.

(See [7]) The R-L fractional integral (right-sided) of order η for a function $f:\mathfrak{J}\to \mathbb{R}$ is defined as

$${}_t{I}_T^{\eta }f\left(t\right)=\frac{1}{\Gamma \left(\eta \right)}{\int }_t^T{(s-t)}^{\eta -1}f\left(s\right)\mathrm{d}s,0<\eta <1.$$

Definition 3.

(See [1,28]) The R-L fractional derivative (left-sided) and R-L fractional derivative (right-sided) of order $0<\eta <1$ with respect to t for a function f are defined as

$$\begin{array}{cc}\hfill {}_0{}^{RL}\mathrm{D}_t^{\eta }f\left(t\right)=& {\left({}_0{I}_t^{1-\eta }f\left(t\right)\right)}^{\prime }\hfill \\ \hfill =& \frac{1}{\Gamma (1-\eta )}{\left({\int }_0^t{(t-s)}^{-\eta }f\left(s\right)\mathrm{d}s\right)}^{\prime }\mathit{for}a.e.t\in \mathfrak{J},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {}_t{}^{RL}\mathrm{D}_T^{\eta }f\left(t\right)=& -{\left({}_t{I}_T^{1-\eta }f\left(t\right)\right)}^{\prime }\hfill \\ \hfill =& -\frac{1}{\Gamma (1-\eta )}{\left({\int }_t^T{(s-t)}^{-\eta }f\left(s\right)\mathrm{d}s\right)}^{\prime }\mathit{for}a.e.t\in \mathfrak{J},\hfill \end{array}$$

respectively, where the notation ${}^{\prime }$ stands for differentiation.

Definition 4.

(See [1,28]) The Hilfer fractional derivative (left-sided) and the Hilfer fractional derivative (right-sided) of order $0<\eta <1$, type $0\le \kappa \le 1$ with respect to t for a function f are respectively defined by

$$\begin{array}{cc}\hfill {}_0{}^H\mathrm{D}_t^{\eta }f\left(t\right)=& \left({}_0{I}_t^{\kappa (1-\eta )}{\left({}_0{I}_t^{1-\zeta }f\right)}^{\prime }\right)\left(t\right)\hfill \\ \hfill =& {}_0{I}_t^{\zeta -\eta }{}_0{}^{RL}\mathrm{D}_t^{\zeta }f\left(t\right)\hfill \\ \hfill =& \frac{1}{\Gamma (\zeta -\eta )\Gamma (1-\zeta )}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s,\hfill \end{array}$$

for almost everywhere $t\in \mathfrak{J}$, where $\zeta =\eta +\kappa -\eta \kappa$, $0<\zeta \le 1$, $\zeta \le \eta$, and $\zeta >\kappa$.

$$\begin{array}{cc}\hfill {}_t{}^H\mathrm{D}_T^{\eta ,\kappa }f\left(t\right)=& -\left({}_t{I}_T^{\kappa (1-\eta )}{\left({}_t{I}_T^{1-\zeta }f\right)}^{\prime }\right)\left(t\right)\hfill \\ \hfill =& -{}_t{I}_T^{\zeta -\eta }{}_t{}^{RL}\mathrm{D}_T^{\zeta }f\left(t\right)\hfill \\ \hfill =& -\frac{1}{\Gamma (\zeta -\eta )\Gamma (1-\zeta )}{\int }_t^T{(s-t)}^{(\zeta -\eta )-1}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s,\hfill \\ \mathit{for}a.e.t\in \mathfrak{J}.\end{array}$$

Next, we recall a mild solution for the following Hilfer fractional evolution equation; see [29].

Lemma 1.

Let $\mathcal{X}=L^2\left(\mathsf{\Omega }\right)$ be a Hilbert space, for any $u_0\in \mathcal{X}$, $0<\eta <1$, $0\le \kappa \le 1$ and $f\in \mathfrak{J}\times \mathcal{X}⟶\mathcal{X}$, the function $u\in L^2(\mathfrak{J};\mathcal{X})$ is said to be a mild solution of the following system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_t{}^H\mathrm{D}_t^{\eta ,\kappa }u\left(t\right)=Au\left(t\right)+f(t,u),\hfill & t\in \mathfrak{J},\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }u\left(t\right)=u_0,\hfill \end{array}\right.\end{array}$$

(6)

if u fulfils

$$\begin{array}{cc}\hfill u\left(t\right)=& \frac{1}{\Gamma (\zeta -\eta )}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}{s}^{\eta -1}{\int }_0^{\infty }\eta \theta M_{\eta }\left(\theta \right)Q\left(s^{\eta }\theta \right))u_0\mathrm{d}\theta \mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{\int }_0^{\infty }\eta \theta M_{\eta }\left(\theta \right)Q\left({(t-s)}^{\eta }\theta \right){(t-s)}^{\eta -1}f(s,u\left(s\right))\mathrm{d}\theta \mathrm{d}s,\hfill \end{array}$$

(7)

where $P_{\eta }\left(t\right)={\int }_0^{\infty }\eta \theta M_{\eta }\left(\theta \right)Q\left(t^{\eta }\theta \right)\mathrm{d}\theta$, and the function $M_{\eta }\left(\theta \right)={\sum }_{n=1}^{\infty }\frac{{(-\theta )}^{n-1}}{(n-1)\Gamma (1-\rho n)},$ where $0<\rho <1$, $\theta \in \mathbb{C}$ is the Wright function, which fulfils the following equality:

$${\int }_0^{\infty }{\theta }^{\iota }{M}_{\eta }\left(\theta \right)d\theta =\frac{\Gamma (1+\iota )}{\Gamma (1+\eta \iota )}\mathit{for}\iota \ge 0,\theta \ge 0.$$

Remark 1.

(See Remark 2.14 in [29]) Let $0<\eta <1$, $0\le \kappa \le 1$, $0<\zeta \le 1$ and $t\in \mathfrak{J}$; thus, we have

$${}_0{}^{RL}\mathrm{D}_t^{\zeta -\eta }{\mathcal{S}}_{\eta ,\kappa }\left(t\right)={\mathcal{R}}_{\eta }\left(t\right),t\in (0,T],$$

(8)

where

$${\mathcal{R}}_{\eta }\left(t\right)=t^{\eta -1}{P}_{\eta }\left(t\right),$$

(9)

and

$${\mathcal{S}}_{\eta ,\kappa }\left(t\right)={}_0{I}_t^{\zeta -\eta }{\mathcal{R}}_{\eta }\left(t\right).$$

(10)

We can rewrite the equality in (7) as follows:

$$u\left(t\right)={\mathcal{S}}_{\eta ,\kappa }\left(t\right)u_0+{\int }_0^t{\mathcal{R}}_{\eta }(t-s)f(s,u\left(s\right))\mathrm{d}s.$$

(11)

Note that if the non-linear term of System (6) is zero, then the mild solution (11) becomes $u(·)={\mathcal{S}}_{\eta ,\kappa }(·)u_0$. Consequently, the mild solution of (4) may alternatively be expressed as

$$\nu \left(t\right)={\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0,t\in \mathfrak{J}.$$

(12)

We give the following lemma, which will be utilized afterwards to prove our results.

Lemma 2.

(See [30]) Let a function g be defined on interval $[S,T]$, $(S<T)$ and $S,T\in \mathbb{R}$, then the reflection operator $\mathfrak{Q}$ acting on g is

$$\mathfrak{Q}\left[g\right(t\left)\right]=g(S+T-t).$$

Lemma 3.

Let f be a function defined on the interval $\mathfrak{J}$ and let f be differentiable and integrable in the Hilfer derivative sense. We now introduce the reflection operator $\mathfrak{Q}$ when acting on f as follows:

$$\mathfrak{Q}\left[f\right(t\left)\right]=f(T-t),$$

(13)

Moreover, the following assertions hold,

(i) 

${}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{I}_T^{\eta }f\left(t\right)\right]$.

(ii) 

$\mathfrak{Q}\left[{}_0{I}_t^{\eta }f\left(t\right)\right]={}_t{I}_T^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]$.

(iii) 

$-{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }f\left(t\right)\right]$.

(iv) 

$\mathfrak{Q}\left[{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }f\left(t\right)\right]=-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]$.

Note that, assertions (i) and (ii) are given in [25,26]. Here, we state their proof due to the demonstration of assertions (iii) and (iv).

Proof. 

Our proof is obtained by virtue of Equation (13) and by utilizing changes in the variables, specifically, changes in the role of time.

(i): We show that ${}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{I}_T^{\eta }f\left(t\right)\right]$. Since

$$\begin{array}{cc}\hfill {}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]& =\frac{1}{\Gamma \left(\eta \right)}{\int }_0^t{(t-s)}^{\eta -1}\mathfrak{Q}f\left(s\right)\mathrm{d}s\hfill \\ \hfill & =\frac{1}{\Gamma \left(\eta \right)}{\int }_0^t{(t-s)}^{\eta -1}f(T-s)\mathrm{d}s.\hfill \end{array}$$

(14)

Using the change in the variables, let $\tilde{s}=T-s$, then $-\mathrm{d}\tilde{s}=\mathrm{d}s$. Now, for $s=0$ and $s=t$, we obtain $\tilde{s}=T$ and $\tilde{s}=T-t$, respectively. Let us fix $\mathcal{M}=\frac{1}{\Gamma \left(\eta \right)}$. Substituting these values into (14), we obtain

$$\begin{array}{cc}\hfill {}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]& =-\mathcal{M}{\int }_T^{T-t}{(t-T+\tilde{s})}^{\eta -1}f\left(\tilde{s}\right)\mathrm{d}\tilde{s},\hfill \end{array}$$

Let $\tilde{s}:=s$, we obtain

$$\begin{array}{cc}\hfill {}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]& =\mathcal{M}{\int }_{T-t}^T{(s-T+t)}^{\eta -1}f\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(15)

We now consider the right-hand side:

$$\begin{array}{cc}\hfill \mathfrak{Q}\left[{}_t{I}_T^{\eta }f\left(t\right)\right]& =\mathfrak{Q}\left[\mathcal{M}{\int }_t^T{(s-t)}^{\eta -1}f\left(s\right)\mathrm{d}s\right]\hfill \\ \hfill & =\mathcal{M}{\int }_{T-t}^T{(s-T+t)}^{\eta -1}f\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(16)

Consequently, from (15) and (16), we can see that ${}_0{I}_t^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{I}_T^{\eta }f\left(t\right)\right]$.

(ii): The proof follows the same way as (i). Considering the left-hand side:

$$\begin{array}{cc}\hfill \mathfrak{Q}\left[{}_0{I}_t^{\eta }f\left(t\right)\right]=& \mathfrak{Q}\left[\mathcal{M}{\int }_0^t{(t-s)}^{\eta -1}f\left(s\right)\mathrm{d}s\right]\hfill \\ \hfill =& \mathcal{M}{\int }_0^{T-t}{(T-t-s)}^{\eta -1}f\left(s\right)\mathrm{d}s,\hfill \end{array}$$

and the right-hand side:

$$\begin{array}{cc}\hfill {}_t{I}_T^{\eta }\mathfrak{Q}\left[f\left(t\right)\right]& =\mathcal{M}{\int }_t^T{(s-t)}^{\eta -1}\mathfrak{Q}f\left(s\right)\mathrm{d}s\hfill \\ \hfill & =-\mathcal{M}{\int }_{T-t}^0{(T-s-t)}^{\eta -1}f\left(s\right)\mathrm{d}s\hfill \\ \hfill & =\mathcal{M}{\int }_0^{T-t}{(T-t-s)}^{\eta -1}f\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(iii): We demonstrate $-{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }f\left(t\right)\right]$. Let us fix $\tilde{\mathcal{M}}=\frac{1}{\Gamma (\zeta -\eta )\Gamma (1-\zeta )}$, which will be used in the remainder of the proof of this lemma. We first consider the left-hand side:

$$\begin{array}{cc}\hfill -{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]& =-\frac{1}{\Gamma (\zeta -\eta )\Gamma (1-\zeta )}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}\mathfrak{Q}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\hfill \\ \hfill & =-\tilde{\mathcal{M}}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f(T-\tau )\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s,\hfill \end{array}$$

(17)

Let $\tilde{\tau }=T-\tau$, then $-\mathrm{d}\tilde{\tau }=\mathrm{d}\tau$. Now, for $\tau =0$ and $\tau =s$, we obtain $\tilde{\tau }=T$ and $\tilde{\tau }=T-s$, respectively. Substituting these values into (17), we obtain

$$\begin{array}{cc}\hfill -{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]& =-\tilde{\mathcal{M}}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}{\left(-{\int }_T^{T-s}{(s-T+\tilde{\tau })}^{-\zeta }f\left(\tilde{\tau }\right)\mathrm{d}\tilde{\tau }\right)}^{\prime }\mathrm{d}s,\hfill \end{array}$$

(18)

Let $s=T-\tilde{s}$, then $-\mathrm{d}\tilde{s}=\mathrm{d}s$. Now for $s=0$ and $\tau =s$, we obtain $\tilde{s}=T$ and $\tilde{s}=T-t$, respectively. Substituting these values into (18), we obtain

$$\begin{array}{cc}\hfill -{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]& =\tilde{\mathcal{M}}{\int }_T^{T-t}{(t-T+\tilde{s})}^{(\zeta -\eta )-1}{\left({\int }_{\tilde{s}}^T{(\tilde{\tau }-\tilde{s})}^{-\zeta }f\left(\tilde{\tau }\right)\mathrm{d}\tilde{\tau }\right)}^{\prime }\mathrm{d}\tilde{s},\hfill \end{array}$$

Let $\tau :=\tilde{\tau }$ and $s:=\tilde{s}$, we obtain

$$\begin{array}{c}\hfill -{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]=\tilde{\mathcal{M}}{\int }_T^{T-t}{(t-T+s)}^{(\zeta -\eta )-1}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s.\end{array}$$

On the other hand, we proceed with the right-hand side as follows:

$$\begin{array}{cc}\hfill \mathfrak{Q}\left[{}_t{\mathrm{D}}_T^{\eta ,\kappa }f\left(t\right)\right]& =\mathfrak{Q}\left[-\tilde{\mathcal{M}}{\int }_t^T{(s-t)}^{(\zeta -\eta )-1}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\right]\hfill \\ \hfill & =-\tilde{\mathcal{M}}{\int }_{T-t}^T{(s-T+t)}^{(\zeta -\eta )-1}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\hfill \\ \hfill & =\tilde{\mathcal{M}}{\int }_T^{T-t}{(s-T+t)}^{(\zeta -\eta )-1}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s,\hfill \end{array}$$

Hence, $-{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]=\mathfrak{Q}\left[{}_t{\mathrm{D}}_T^{\eta ,\kappa }f\left(t\right)\right]$.

(iv): The proof follows the same way as (iii). We first consider the left-hand side:

$$\begin{array}{cc}\hfill \mathfrak{Q}\left[{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }f\left(t\right)\right]& =\mathfrak{Q}\left[\tilde{\mathcal{M}}{\int }_0^t{(t-s)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\right]\hfill \\ \hfill & =\tilde{\mathcal{M}}{\int }_0^{T-t}{(T-t-s)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s,\hfill \end{array}$$

then the right-hand side:

$$\begin{array}{cc}\hfill -{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]& =-\left[-\tilde{\mathcal{M}}{\int }_t^T{(s-t)}^{(\zeta -\eta )-1}\mathfrak{Q}{\left({\int }_s^T{(\tau -s)}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\right]\hfill \\ \hfill & =\tilde{\mathcal{M}}{\int }_t^T{(s-t)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(\tau -s)}^{-\zeta }f(T-\tau )\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s\hfill \\ \hfill & =\tilde{\mathcal{M}}{\int }_t^T{(s-t)}^{(\zeta -\eta )-1}{\left(-{\int }_{T-s}^0{(T-\tilde{\tau }-s)}^{-\zeta }f\left(\tilde{\tau }\right)\mathrm{d}\tilde{\tau }\right)}^{\prime }\mathrm{d}s\hfill \\ \hfill & =-\tilde{\mathcal{M}}{\int }_{T-t}^0{(T-\tilde{s}-t)}^{(\zeta -\eta )-1}{\left({\int }_0^{T-s}{(\tilde{s}-\tilde{\tau })}^{-\zeta }f\left(\tilde{\tau }\right)\mathrm{d}\tilde{\tau }\right)}^{\prime }\mathrm{d}\tilde{s}\hfill \\ \hfill & =\tilde{\mathcal{M}}{\int }_0^{T-t}{(T-s-t)}^{(\zeta -\eta )-1}{\left({\int }_0^s{(s-\tau )}^{-\zeta }f\left(\tau \right)\mathrm{d}\tau \right)}^{\prime }\mathrm{d}s.\hfill \end{array}$$

Consequently, $\mathfrak{Q}\left[{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }f\left(t\right)\right]=-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathfrak{Q}\left[f\left(t\right)\right]$.

Thus, this completes the proof of the lemma. □

Since C is an admissible operator, as we will see later, then the OuPuFu of System (4) is given by

$$\mathfrak{M}\left(t\right)=C{\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0={\mathcal{K}}_{\eta ,\kappa }\left(t\right){\nu }_0,t\in \mathfrak{J},$$

(19)

where ${\mathcal{K}}_{\eta ,\kappa }:L^2\left(\mathsf{\Omega }\right)⟶\mathcal{O}$ is a fractional linear operator. Let us recall the observation space $\mathcal{O}=L^2(\mathfrak{J};{\mathbb{R}}^q)(q\in \mathbb{N})$. Two cases arise for obtaining the adjoint operator of ${\mathcal{K}}_{\eta ,\kappa }$.

• Case 1. C is bounded. In this case, we can define zonal sensors. Let operator C be from $L^2\left(\mathsf{\Omega }\right)$ to $\mathcal{O}$. Then, if $C^{*}$ is adjoint on the other hand, the adjoint of operator ${\mathcal{K}}_{\eta ,\kappa }$ can be obtained by

  $$\begin{array}{cc}\hfill {\mathcal{K}}_{\eta ,\kappa }^{*}:\mathcal{O}& ⟶L^2\left(\mathsf{\Omega }\right)\hfill \\ \hfill {\mathfrak{M}}^{*}& ⟼{\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}\left(s\right)C^{*}{\mathfrak{M}}^{*}\left(s\right)\mathrm{d}s.\hfill \end{array}$$
• Case 2. C is unbounded. We can define pointwise sensors. However, in this case, the operator C can be introduced from $\mathcal{D}\left(C\right)\subseteq L^2\left(\mathsf{\Omega }\right)$ to the observation space $\mathcal{O}$. Then, $C^{*}$ is adjoint. However, in order to give this case a sense of (5), we make an assumption on C in the following definition, namely, C is an admissible observation operator, as we will see in Definition 5 below.

Definition 5.

(See [18]) The observation operator C is an admissible of (4) and (5), if for any ${\nu }_0\in \mathcal{D}\left(C\right)$ there is a constant $\mathcal{L}>0$, such that

$${\int }_0^T\parallel C{\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0{\parallel }^2\mathrm{d}s\le \mathcal{L}\Vert {\nu }_0\Vert .$$

Note that operator C being admissible assures that the map

$${\nu }_0⟼C{\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0={\mathcal{K}}_{\eta ,\kappa }\left(t\right){\nu }_0$$

can be extended to a bounded linear operator from $L^2\left(\mathsf{\Omega }\right)$ to the space $\mathcal{O}$. Thus, we can introduce ${\mathcal{K}}_{\eta ,\kappa }^{*}$ as the adjoint of operator ${\mathcal{K}}_{\eta ,\kappa }$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{K}}_{\eta ,\kappa }^{*}:\mathcal{D}\left({\mathcal{K}}_{\eta ,\kappa }^{*}\right)\subseteq & \mathcal{O}⟶L^2\left(\mathsf{\Omega }\right)\hfill \\ \hfill & {\mathfrak{M}}^{*}⟼{\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}\left(s\right)C^{*}{\mathfrak{M}}^{*}\left(s\right)\mathrm{d}s\hfill \end{array}$$

3. Characterization of Enlarged Observability

In this section, we will characterize the ReEnOb of System (4) with the output function (5) in the subregion $\omega$ of $\mathsf{\Omega }$. Let $\omega$ be a positive Lebesgue measure, and let us define the restriction mapping (projection mapping) $p_{\omega }$, as follows:

$$\begin{array}{cc}\hfill p_{\omega }:L^2\left(\mathsf{\Omega }\right)& ⟶L^2\left(\omega \right)\hfill \\ \hfill \nu & ⟼p_{\omega }\nu ={\nu }_{{|}_{\omega }}.\hfill \end{array}$$

We can now define the adjoint $p_{\omega }^{*}$ of $p_{\omega }$ as follows: $\left(p_{\omega }^{*}\nu \right)\left(y\right):=\nu (y,·)$ when $y\in \omega$, and $\left(p_{\omega }^{*}\nu \right)\left(y\right):=0$ when $y\in \mathsf{\Omega }\setminus \omega$. In addition, we note that the regional exact observability of System (4) with (5) can be achieved at time t in the subregion $\omega$, if $\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)=L^2\left(\omega \right)$, see, e.g., [25,26,31,32,33]. Now, let ${\gamma }_1(·)$ and ${\gamma }_2(·)$, ${\gamma }_1(·)\le {\gamma }_2(·)$ almost everywhere in the subregion $\omega$, be two functions defined in $L^2\left(\mathsf{\Omega }\right)$. We thus define the following set

$$\mathfrak{Z}=\{\nu \in L^2\left(\omega \right)|{\gamma }_1(·)\le \nu (·)\le {\gamma }_2(·)\mathrm{almost}\mathrm{everywhere}\mathrm{in}\mathrm{the}\mathrm{subregion}\omega \},$$

where ${\gamma }_1(·)$ and ${\gamma }_2(·)$ are given functions in $\omega$. We assume that the initial state is given by

$$\begin{array}{c}\hfill {\nu }_0=\left\{\begin{array}{cc}{\nu }_0^1\hfill & \mathrm{in}\mathfrak{Z},\hfill \\ {\nu }_0^2\hfill & \mathrm{in}{L}^2\left(\mathsf{\Omega }\right)\setminus \mathfrak{Z}.\hfill \end{array}\right.\end{array}$$

The main objective of the investigation proposed in this paper is to demonstrate ReEnOb for Hilfer time fractional-order diffusion systems, that is, we will answer the following question: Given the Hilfer fractional diffusion System (4) with (5) in the subregion $\omega$ at time $t\in \mathfrak{J}$, can we reconstruct ${\nu }_0^1$ between ${\gamma }_1(·)$ and ${\gamma }_2(·)$?

The following definition will be used in the following.

Definition 6.

If $\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}=\left\{0\right\}$, then System (4) with (5) is exactly $\mathcal{E}$-observable in the subregion ω.

Definition 7.

A sensor is exactly $\mathfrak{Z}$-strategic in the subregion ω if the observed system is exactly $\mathfrak{Z}$-observable in subregion ω.

The following three remarks show that the results obtained in [18,25,26] are particular cases of our results.

Remark 2.

If $\kappa =0$ and $\eta =1$, then the Hilfer fractional diffusion (4) corresponds to the normal diffusion process, which is investigated in [18].

Remark 3.

If $\kappa =0$ and $0<\eta <1$, then the Hilfer fractional diffusion System (4) corresponds to the R-L fractional diffusion process, which is investigated in [25].

Remark 4.

If $\kappa =1$ and $0<\eta <1$, then the Hilfer fractional diffusion (4) corresponds to the Caputo fractional diffusion process, which is considered in [26].

The following result can be obtained directly from Definition 7.

Remark 5.

If System (4) with the OuPuFu (5) is exactly $\mathfrak{Z}$-observable in ${\omega }_1$, then for any subregion ${\omega }_2$ of ${\omega }_1$ it is also exactly $\mathfrak{Z}$-observable in ${\omega }_2$.

The following remark will be used in the proof of the theorem presented below.

Remark 6.

Let X be a Hilbert space and F a linear subspace of X, then $F\cap F^{\perp }=\left\{0\right\}$, where $F^{\perp }$ is the orthogonal complement of F.

Theorem 1.

The following assertions are equivalent:

1. 

System (4) with the OuPuFu (5) is exactly $\mathfrak{Z}$-observable in the subregion ω.

2. 

$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing$.

Proof. 

We show that Statement 1 implies Statement 2, and Statement 2 implies Statement 1. The following two facts play a key role in the proof.

$$\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)=\mathrm{Im}{\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)}^{\perp },$$

(20)

it follows from Remark 6 that

$$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathrm{Im}{\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)}^{\perp }=\left\{0\right\}.$$

(21)

We demonstrate that the left-hand side implies the right-hand side, and vice versa:

$$\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}=\left\{0\right\}⟺\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing .$$

We first show that

$$\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}=\left\{0\right\}⟹\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing .$$

Let $y\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}$, then $y=0$. From (20), one can see that $y\in \mathrm{Im}{\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)}^{\perp }\cap \mathfrak{Z}$. Therefore, it follows from (21) that, $\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)$ has at least one element, which is zero. Thus, $\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing$.

We now prove that statement 2 implies statement 1, that is,

$$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing ⟹\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}=\left\{0\right\}.$$

Suppose

$$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}\ne \varnothing ,$$

(22)

and

$$\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}\ne \left\{0\right\}.$$

(23)

Now, let $y\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}$, then $y\ne 0$, $y\in \mathfrak{Z}$ and $y\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)$. From (20) and (21), we have $y\in \mathrm{Im}{\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)}^{\perp }$ and $y\notin \mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)$, respectively. Consequently, one can see that

$$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \left(\mathrm{Ker}\right({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*})\cap \mathfrak{Z})=\varnothing ;$$

therefore,

$$\mathrm{Im}\left(p_{\omega }{\mathcal{K}}_{\eta ,\kappa }^{*}\right)\cap \mathfrak{Z}=\varnothing ,$$

which contradicts (22). Thus, (23) is not true. Consequently,

$$\mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}=\left\{0\right\}.$$

Therefore, System (4) with (5) is exactly $\mathfrak{Z}$-observable in the subregion $\omega$. This completes the proof. □

4. The Hilbert Uniqueness Method

In this section, we provide an approach for reconstructing the initial state of the system between ${\gamma }_1(·)$ and ${\gamma }_2(·)$ in subregion $\omega$. Let $\mathfrak{P}$ be a space defined as

$$\mathfrak{P}=\left\{g\in L^2\left(\mathsf{\Omega }\right)|g=0\mathrm{in}{L}^2\left(\mathsf{\Omega }\right)\setminus \mathfrak{Z}\right\}.$$

(24)

4.1. Pointwise Sensors

Let System (4) be observed by a pointwise sensor $(l,\delta (l-·\left)\right)$, where $l\in \overline{\mathsf{\Omega }}$ is the location of a sensor and $\delta$ is the Dirac mass (delta function), which is concentrated in l. Here, the OuPuFu is introduced as

$$\mathfrak{M}\left(t\right)=\psi (b,T-t),t\in \mathfrak{J}.$$

(25)

Let ${\psi }_0$ be in $\mathfrak{P}$; thus, we examine the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\psi (y,t)=A\psi (y,t)\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \psi (\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }\psi (y,t)={\psi }_0\left(y\right)\hfill & \mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(26)

For simplicity of notation, we denote $\psi (y,t):=\psi \left(t\right)$. We note that System (26) admits a unique solution $\psi \in L^2(\mathfrak{J};\mathcal{D}\left(A\right))\cap C(\mathsf{\Omega }\times \mathfrak{J})$ given by $\psi \left(t\right)={\mathcal{S}}_{\eta ,\kappa }\left(t\right){\phi }_0$, if ${\psi }_0\left(x\right)\in \mathcal{D}\left(A\right)$. Let us denote a semi-norm on $\mathfrak{P}$ by

$${\psi }_0⟼\parallel {\psi }_0{\parallel }_{\mathfrak{P}}^2={\int }_0^T{\parallel C\psi (T-t)\parallel }^2\mathrm{d}t.$$

(27)

In the following lemma, we will see that a norm can be defined.

Lemma 4.

If System (4) with OuPuFu (25) is exactly $\mathfrak{Z}$-observable in the subregion ω; consequently, Equation (27) defines a norm in the space $\mathfrak{P}$.

Proof. 

Firstly, in light of Theorem 1 and Definition 6, we suppose that System (4) with the OuPuFu (25) is exactly $\mathfrak{Z}$-observable in the space $\mathfrak{P}$. Now, for ${\psi }_0\in \mathfrak{P}$ and a semi-norm in $\mathfrak{P}$, we have

$$\parallel {\psi }_0{\parallel }_{\mathfrak{P}}=0⟹C\psi (T-t)=0\mathrm{for}\mathrm{all}t\in \mathfrak{J}.$$

Let

$${\psi }_0\in L^2\left(\mathsf{\Omega }\right)⟹p_{\omega }{\psi }_0\in L^2\left(\omega \right),$$

then,

$${\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}{p}_{\omega }{\psi }_0=C{\mathcal{S}}_{\eta ,\kappa }\left(t\right)p_{\omega }^{*}{p}_{\omega }{\psi }_0=0.$$

Hence,

$$p_{\omega }{\psi }_0\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right).$$

and for $p_{\omega }{\psi }_0\in \mathfrak{Z}$, one has $p_{\omega }{\psi }_0\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}\right)\cap \mathfrak{Z}$ and $p_{\omega }{\psi }_0=0$, since the system is exactly $\mathfrak{Z}$-observable in the subregion $\omega$. Consequently, ${\psi }_0=0$ and (27) is a norm. □

We now consider the following system, which is controlled by the solution to System (26), that is,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{{\rm Y}}(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{{\rm Y}}(y,t)\right]+C^{*}C\mathfrak{Q}\left[\psi (y,t)\right]\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \mathsf{{\rm Y}}(\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{{\rm Y}}(y,t)\right]=0\hfill & \mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(28)

Next, for ${\psi }_0\in \mathfrak{P}$, we define the operator $\mathsf{\Lambda }:\mathfrak{P}⟶{\mathfrak{P}}^{*}$ by

$$\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)\right),$$

(29)

where $\mathcal{P}=p_{\omega }^{*}{p}_{\omega }$ and $\mathsf{{\rm Y}}\left(0\right)=\mathsf{{\rm Y}}(y,0)$.

Next, let us consider the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{\Phi }(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{\Phi }(y,t)\right]+C^{*}\mathfrak{Q}\left[\mathfrak{M}\left(t\right)\right]\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \mathsf{\Phi }(\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{\Phi }(y,t)\right]=0\hfill & \mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(30)

If we choose the initial state ${\psi }_0$ of System (26) such that $\mathsf{\Phi }\left(0\right)=\mathsf{{\rm Y}}\left(0\right)$ in the subregion $\omega$, then one can see that System (30) stands for the adjoint of System (4). Thus, our problem of ReEnOb can be simplified solved in Equation (29), since following Equation (31) is equivalent to Equation (29).

$$\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{\Phi }\left(0\right)\right).$$

(31)

Theorem 2.

System (4) augmented by (25) is exactly $\mathfrak{Z}$-observable in ω, if Equation (29) has a unique solution ${\psi }_0\in \mathfrak{P}$, that coincides with the state ${\nu }_0^1$ observed between functions ${\gamma }_1(·)$ and ${\gamma }_2(·)$ in the subregion ω. In addition, ${\nu }_0^1=p_{\omega }{\phi }_0$.

Proof. 

We note that, System (4) with (25) is exactly $\mathfrak{Z}$-observable in $\omega$, then the norm ${\parallel ·\parallel }_{\mathfrak{P}}$ can be defined on $\mathfrak{P}$ by Lemma 4. Next, we prove that, if $\mathsf{\Lambda }$ is an isomorphism (see [18]), then (29) admits a unique solution in the set $\mathfrak{P}$. For this, we have

$$\begin{array}{cc}\hfill {⟨\mathsf{\Lambda }{\psi }_0,{\psi }_0⟩}_{L^2\left(\mathsf{\Omega }\right)}& ={⟨\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)\right),{\psi }_0⟩}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill & ={⟨p_{\omega }^{*}{p}_{\omega }\left({}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)\right),{\psi }_0⟩}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill & ={⟨{}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right),{\psi }_0⟩}_{L^2\left(\omega \right)}\hfill \end{array}$$

We note that the following propositions are important in the following proof.

Proposition 1.

Let $0<\eta <1$, $0\le \kappa \le 1$, $0<\zeta \le 1$ and $t\in \mathfrak{J}$. Since System (30) is adjoint of (4), then from (9) and (10), we have

$${\mathcal{R}}_{\eta }^{*}\left(t\right)=t^{\eta -1}{P}_{\eta }^{*}\left(t\right),$$

and

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\eta ,\kappa }^{*}\left(t\right)=& {}_0{I}_t^{\zeta -\eta }{\mathcal{R}}_{\eta }^{*}\left(t\right).\hfill \end{array}$$

Therefore, the solution to System (28) is given by

$$\mathsf{{\rm Y}}\left(t\right)={\int }_0^{T-t}{\mathcal{R}}_{\eta }^{*}(T-t-s)C^{*}C\psi (T-s)\mathrm{d}s.$$

(32)

Proposition 2.

Let $0<\eta <1$, $0\le \kappa \le 1$, $0<\zeta \le 1$ and $t\in \mathfrak{J}$, we have

$${}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)={\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}(T-s)C^{*}C\psi (T-s)\mathrm{d}s.$$

(33)

Proof. 

In view of Fubini’s theorem and Equation (32), and for any $\tau \in \mathfrak{J}$, we have

$$\begin{array}{cc}\hfill {}_{\tau }{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(\tau \right)=& \frac{1}{\Gamma (\zeta -\eta )}{\int }_{\tau }^T{(t-\tau )}^{\zeta -\eta -1}\mathsf{{\rm Y}}\left(t\right)\mathrm{d}t\hfill \\ \hfill =& \frac{1}{\Gamma (\zeta -\eta )}{\int }_{\tau }^T{(t-\tau )}^{\zeta -\eta -1}{\int }_0^{T-t}{\mathcal{R}}_{\eta }^{*}(T-t-s)C^{*}C\psi (T-s)\mathrm{d}s\mathrm{d}t\hfill \\ \hfill =& \frac{1}{\Gamma (\zeta -\eta )}{\int }_0^T\left({\int }_{\tau }^{T-s}{(t-\tau )}^{\zeta -\eta -1}{\mathcal{R}}_{\eta }^{*}(T-t-s)\mathrm{d}t\right)C^{*}C\psi (T-s)\mathrm{d}s\hfill \end{array}$$

Let $u=T-t-s$, then $\mathrm{d}u=-\mathrm{d}t$. Now, for $t=\tau$ and $t=T-s$, we obtain $u=T-\tau -s$ and $u=0$, respectively. Thus, we obtain

$$\begin{array}{cc}\hfill {}_{\tau }{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(\tau \right)=& \frac{1}{\Gamma (\zeta -\eta )}{\int }_0^T\left({\int }_0^{T-\tau -s}{(T-s-\tau -u)}^{\zeta -\eta -1}{\mathcal{R}}_{\eta }^{*}\left(u\right)\mathrm{d}u\right)C^{*}C\psi (T-s)\mathrm{d}s\hfill \\ \hfill =& {\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}(T-s-\tau )C^{*}C\psi (T-s)\mathrm{d}s.\hfill \end{array}$$

We now let $\tau =0$, we obtain

$${}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)={\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}(T-s)C^{*}C\psi (T-s)\mathrm{d}s.$$

□

Now, we continue the proof of our theorem

$$\begin{array}{cc}\hfill {⟨\mathsf{\Lambda }{\psi }_0,{\psi }_0⟩}_{L^2\left(\mathsf{\Omega }\right)}& ={⟨{}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right),{\psi }_0⟩}_{L^2\left(\omega \right)}\hfill \\ \hfill & =⟨{\int }_0^T{\mathcal{S}}_{\eta ,\kappa }^{*}(T-s)C^{*}C\psi (T-s)\mathrm{d}s,{\psi }_0⟩\hfill \\ \hfill & ={\int }_0^T⟨C\psi (T-s),C{\mathcal{S}}_{\eta ,\kappa }(T-s){\psi }_0⟩\mathrm{d}s\hfill \\ \hfill & ={\int }_0^T⟨C\psi (T-s),C\psi (T-s)⟩\mathrm{d}s\hfill \\ \hfill & ={\int }_0^T{\parallel C\psi (T-s)\parallel }^2\mathrm{d}s\hfill \\ \hfill & =\parallel {\psi }_0{\parallel }_{\mathcal{G}}^2.\hfill \end{array}$$

Thus, the operator $\mathsf{\Lambda }$ is an isomorphism. Therefore, we establish that Equation (29) has a unique solution, which corresponds to the desired initial state ${\nu }_0^1=p_{\omega }{\psi }_0$. This completes the proof. □

4.2. Zone Sensors

Here we suppose the measurements of System (4) are given by an internal zone sensor defined by $(\mathcal{A},h)$ with $\mathcal{A}\subset \mathsf{\Omega }$ and $h\in L^2\left(\mathcal{A}\right)$. The system is augmented with the OuPuFu

$$\mathfrak{M}\left(t\right)={\int }_{\mathcal{A}}\nu (y,T-t)h\left(y\right)\mathrm{d}y.$$

(34)

In this case, we consider System (26), and we assume $\mathfrak{P}$ is given by Equation (24). Then, a semi-norm can be introduced by

$$\parallel {\phi }_0{\parallel }_{\mathfrak{P}}^2={\int }_0^T{⟨\psi (T-t),h⟩}_{L^2\left(\mathcal{A}\right)}^2\mathrm{d}t,$$

(35)

and if System (26) with (25) is exactly $\mathfrak{Z}$-observable in a subregion $\omega$ of $\mathsf{\Omega }$, then a norm can be defined.

In this case, we can introduce the adjoint System of (26) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{{\rm Y}}(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{{\rm Y}}(y,t)\right]+{⟨\mathfrak{Q}\left[\psi \left(t\right)\right],h⟩}_{L^2\left(\mathcal{A}\right)}h\left(y\right)\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \mathsf{{\rm Y}}(\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{{\rm Y}}(y,t)\right]=0\hfill & \mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(36)

Thus, the operator $\mathsf{\Lambda }$ can be defined by

$$\begin{array}{cc}\hfill \mathsf{\Lambda }:\mathfrak{P}& ⟶{\mathfrak{P}}^{*}\hfill \\ \hfill {\psi }_0& ⟼\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{{\rm Y}}\left(0\right)\right),\hfill \end{array}$$

(37)

where $\mathcal{P}=p_{\omega }^{*}{p}_{\omega }$ is a projection operator. For simplicity, let us write $\mathsf{{\rm Y}}\left(0\right)=\mathsf{{\rm Y}}(y,0)$.

We introduce the following system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{\Phi }(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{\Phi }(y,t)\right]+{⟨\mathfrak{Q}\left[\mathfrak{M}\left(t\right)\right],h⟩}_{L^2\left(\mathcal{A}\right)}{p}_{\mathcal{A}}h\left(y\right)\hfill & \mathrm{in}{\mathfrak{S}}_T,\hfill \\ \mathsf{\Phi }(\xi ,t)=0\hfill & \mathrm{on}{\Sigma }_T,\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{\Phi }(y,t)\right]=0\hfill & \mathrm{in}\mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(38)

If the initial state ${\psi }_0$ of System (26) is chosen such that $\mathsf{\Phi }\left(0\right)=\mathsf{{\rm Y}}\left(0\right)$ in the subregion $\omega$, then one can see that System (38) is the adjoint of System (4); thus, our ReEnOb problem can be simplified and solved by the following equation

$$\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{\Phi }\left(0\right)\right),$$

(39)

Theorem 3.

If System (4) with OuPuFu (34) is exactly $\mathfrak{Z}$-observable in the subregion ω, then Equation (39) has a unique solution ${\psi }_0\in \mathfrak{P}$ that corresponds with the observed initial state ${\nu }_0^1$ between functions ${\gamma }_1(·)$ and ${\gamma }_2(·)$ in the subregion ω.

Proof. 

The procedures of the proof are remarkably similar to those of Theorem 2. □

4.3. Examples

Example 1.

In this subsection, we will consider the case where C is unbounded (pointwise sensors). The following time fractional diffusion system can be use to describe a chemical reaction or a heat conduction.

Let ${\mathsf{\Omega }}_1=[0,l]$ and ${\overline{\mathfrak{S}}}_T={\mathsf{\Omega }}_1\times \mathfrak{J}$, we thus consider

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\nu (y,t)={\aleph }_{\eta ,\kappa }\frac{{\partial }^2}{\partial y^2}\nu (y,t)+f(y,t)\hfill & \mathrm{in}{\overline{\mathfrak{S}}}_T,\hfill \\ \nu (0,t)=h_1\left(y\right),\nu (l,t)=h_2\left(y\right)\hfill & \mathrm{in}\mathfrak{J},\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }\nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{in}{\mathsf{\Omega }}_1,\hfill \end{array}\right.\end{array}$$

(40)

where $f(y,t)$ is the density of the sources that transmits the substance in/out the system, $A={\aleph }_{\eta ,\kappa }\frac{{\partial }^2}{\partial x^2}$ and ${\aleph }_{\eta ,\kappa }$ represents a constant of physical dimension $\left[{\aleph }_{\eta ,\kappa }\right]=c{m}^2{s}^{\eta }$, which only depends on η and is independent of κ.

For simplicity, we assume ${\aleph }_{\eta ,\kappa }=1$, $f(y,t)=0$, $h_1\left(y\right)=h_2\left(y\right)=0$, and $l=1$, obtaining ${\mathsf{\Omega }}_1=[0,1]$ and ${\overline{\overline{\mathfrak{S}}}}_T={\mathsf{\Omega }}_1\times \mathfrak{J}$. Hence, System (40) can be written as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\nu (y,t)=\frac{{\partial }^2}{\partial y^2}\nu (y,t)\hfill & \mathrm{in}{\overline{\overline{\mathfrak{S}}}}_T,\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{in}[0,T],\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }\nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{in}[0,1],\hfill \end{array}\right.\end{array}$$

(41)

augmented with the OuPuFu

$$\mathfrak{M}\left(t\right)=C\nu (y,t)=\nu (b,t),$$

(42)

where $\frac{1}{4}=b\in [0,1]$, and System (44) has a mild solution $\nu (y,t)$, $t\in \mathfrak{J}$ given by

$$\begin{array}{cc}\hfill \nu (y,t)=& 2\sum _{n=1}^{\infty }{t}^{\eta +\kappa (1-\eta )-1}{E}_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{t}^{\eta }\right)sin\left(n\pi y\right)\hfill \\ \hfill & \times {\int }_0^1{\nu }_0\left(y\right)sin\left(n\pi y\right)\mathrm{d}x,\hfill \end{array}$$

(43)

where $E_{\eta ,\kappa }(·)$ stands for the two-parameter Mittag–Leffler function [4], and one can easily see that the operator $\frac{{\partial }^2}{\partial y^2}$ has a complete set of eigenfunctions ${\varphi }_n=sin\left(n\pi y\right)$ in the Hilbert space $L^2\left({\mathsf{\Omega }}_1\right)$ associated with the eigenvalues ${\lambda }_n=-n^2\pi$. Let us assume the initial state that needs to be observed in System (44) is given by ${\nu }_0\left(y\right)=sin\left(2\pi y\right)$, $\eta =0.2$, and $\kappa =0.4$. Now, for the subregion ${\omega }_1=\left[\frac{1}{2},\frac{2}{3}\right]\subset [0,1]$, the following results hold.

Proposition 3.

There exists a state for which System (44) with the OuPuFu (42) is not weakly observable in ${\mathsf{\Omega }}_1$, but is ${\mathfrak{Z}}_1$-observable in the subregion ${\omega }_1$.

Proof. 

To show that System (44) with the OuPuFu (42) is not weakly observable in ${\mathsf{\Omega }}_1$, it sufficient to verify that ${\nu }_0\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }\right)$. From Equation (43) and the assumptions above we can now calculate

$$\begin{array}{cc}\hfill {\mathcal{K}}_{0.2,0.4}{\nu }_0=& 2\sum _{n=1}^{\infty }{t}^{-0.48}{E}_{0.2,-0.52}\left(-n^2{\pi }^2{t}^{0.2}\right)sin\left(\frac{n\pi }{4}\right)\hfill \\ \hfill & \times {\int }_0^{1}sin\left(2\pi y\right)sin\left(n\pi y\right)\mathrm{d}y\hfill \\ \hfill & =0.\hfill \end{array}$$

Hence, ${\nu }_0\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }\right)$. As a result, System (44) and (42) is not weakly observable in ${\mathsf{\Omega }}_1$,

$$\begin{array}{cc}\hfill {\mathcal{K}}_{0.2,0.4}{p}_{{\omega }_1}^{*}{p}_{{\omega }_1}{\nu }_0=& 2\sum _{n=1}^{\infty }{t}^{-0.48}{E}_{0.2,-0.52}\left(-n^2{\pi }^2{t}^{0.2}\right)sin\left(0.25n\pi \right)\hfill \\ \hfill & \times {\int }_0^1{p}_{{\omega }_1}^{*}{p}_{{\omega }_1}sin\left(2\pi y\right)sin\left(n\pi y\right)\mathrm{d}y\hfill \\ \hfill =& 2\sum _{n=1}^{\infty }{t}^{-0.48}{E}_{0.2,-0.52}\left(-n^2{\pi }^2{t}^{0.2}\right)sin\left(\frac{n\pi }{4}\right)\hfill \\ \hfill & \times {\int }_{\frac{1}{2}}^{\frac{2}{3}}sin\left(2\pi y\right)sin\left(n\pi y\right)\mathrm{d}y\hfill \\ \hfill =& 2t^{-0.48}{E}_{0.2,-0.52}\left(-{\pi }^2{t}^{0.2}\right)sin\left(\frac{\pi }{4}\right)\hfill \\ \hfill & \times {\int }_{\frac{1}{2}}^{\frac{2}{3}}sin\left(2\pi y\right)sin\left(\pi y\right)\mathrm{d}y\hfill \\ \hfill =& \frac{(3\sqrt{3}-8)t^{-0.48}}{6\sqrt{2}\pi }{E}_{0.2,-0.52}\left(-{\pi }^2{t}^{0.2}\right)\hfill \\ \hfill \ne & 0.\hfill \end{array}$$

While on the other hand, this leads us to observe that the initial state ${\nu }_0$ is weakly observable in the subregion ${\omega }_1$. In addition, for all $y\in {\omega }_1$, we have

$${\tilde{\gamma }}_1=\left|{\nu }_{{|}_{{\omega }_1}}^0\left(y\right)\right|-\frac{2}{3}<{\nu }_{{|}_{{\omega }_1}}^0$$

and

$${\tilde{\gamma }}_2=\left|{\nu }_{{|}_{{\omega }_1}}^0\left(y\right)\right|+\frac{2}{3}>{\nu }_{{|}_{{\omega }_1}}^0.$$

Thus, $p_{{\omega }_1}{\nu }_0\in {\mathfrak{Z}}_1$ and (44) together with (42) is ${\mathfrak{Z}}_1$-observable in ${\omega }_1$. This completes the proof. □

Let the space ${\mathfrak{P}}_1$ be given by

$${\mathfrak{P}}_1=\left\{g\in L^2\left({\mathsf{\Omega }}_1\right)|g=0\mathrm{in}{L}^2\left({\mathsf{\Omega }}_1\right)\setminus {\mathfrak{Z}}_1\right\}.$$

From Lemma 4, we have

$${\psi }_0⟼\parallel {\psi }_0{\parallel }_{{\mathfrak{P}}_1}^2={\int }_0^T{\parallel C\psi (T-t)\parallel }^2\mathrm{d}t.$$

which defines a norm on ${\mathfrak{P}}_1$, and thus we can introduce the adjoint System of (44) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{\Phi }(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{\Phi }(y,t)\right]+\mathfrak{M}(b,T-t)\hfill & \mathrm{in}{\overline{\mathfrak{S}}}_T,\hfill \\ \mathsf{\Phi }(\xi ,t)=0\hfill & \mathrm{on}\partial {\mathsf{\Omega }}_1\times [0,T],\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{\Phi }(y,t)\right]=0\hfill & \mathrm{in}{\mathsf{\Omega }}_1,\hfill \end{array}\right.\end{array}$$

then, in view of Theorem 2, we can now conclude that $\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{\Phi }\left(0\right)\right)$ has a unique solution in ${\mathfrak{P}}_1$, and the initial state ${\nu }_0$ is observed between functions ${\tilde{\gamma }}_1$ and ${\tilde{\gamma }}_2$ in the subregion ${\omega }_1$.

Example 2.

In this example, we consider C as bounded (zone sensors). Considering the following diffusion system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_0{}^H\mathrm{D}_t^{\eta ,\kappa }\nu (y,t)=\frac{{\partial }^2}{\partial y^2}\nu (y,t)\hfill & \mathrm{in}[0,1]\times [0,T],\hfill \\ \nu (\xi ,t)=0\hfill & \mathrm{in}[0,T],\hfill \\ \underset{t\to 0^{+}}{lim}{}_0{I}_t^{1-\zeta }\nu (y,t)={\nu }_0\left(y\right)\hfill & \mathrm{is}\mathrm{unknown}\mathrm{in}[0,1],\hfill \end{array}\right.\end{array}$$

(44)

augmented with the OuPuFu

$$\mathfrak{M}\left(t\right)={\int }_0^1\nu (y,T-t)h\left(y\right)\mathrm{d}y=\nu (b,t),$$

(45)

where $A=\frac{{\partial }^2}{\partial y^2}$ with eigenvalues ${\lambda }_n=-n^2{\pi }^2$ and the corresponding eigenfunctions ${\varphi }_n\left(y\right)=sin\left(n\pi y\right)$. Let us fix $\frac{1}{3}=b\in [0,1]={\mathsf{\Omega }}_2$ and take any internal subregion ${\omega }_2=[\frac{1}{6},\frac{1}{2}]$ of the whole domain. We note that System (44) has a unique mild solution $\nu (y,t)$ in $L^2([0,T];\mathcal{D}\left(A\right))\cap C([0,1]\times [0,T])$.

Proposition 4.

There exists a state for which System (44) with the OuPuFu (45) is not weakly observable in ${\mathsf{\Omega }}_2$, but is ${\mathfrak{Z}}_2$-observable in the subregion ${\omega }_2$.

Proof. 

To show that System (44) with the OuPuFu (45) is not weakly observable in ${\mathsf{\Omega }}_2$, it is sufficient to verify that ${\nu }_0\in \mathrm{Ker}\left({\mathcal{K}}_{\eta ,\kappa }\right)$. Thus, we can now derive

$$\begin{array}{cc}\hfill C{\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0={\mathcal{K}}_{\eta ,\kappa }\left(t\right){\nu }_0=& 2\sum _{n=1}^{\infty }{t}^{\eta +\kappa (1-\eta )-1}{E}_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{t}^{\eta }\right)⟨{\nu }_0,{\varphi }_n⟩{\varphi }_n\left(\frac{1}{3}\right),\hfill \end{array}$$

where $E_{\eta ,\kappa }(·)$ stands for the two-parameter Mittag–Leffler function. Now, for all $y\in [0,1]$, $|{\varphi }_n|\le \sqrt{2}$, the Mittag–Leffler function $E_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{t}^{\eta }\right)$ is continuous with $|E_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{t}^{\eta }\right)|\le \frac{K}{1+|-n^2{\pi }^2|t^{\eta }}$ for $t\ge 0$ with $K>0$. Hence,

$$|C{\mathcal{S}}_{\eta ,\kappa }\left(t\right){\nu }_0|=2\sum _{n=1}^{\infty }\frac{K\sqrt{2}\parallel {\nu }_0\parallel t^{\eta +\kappa (1-\eta )-1}}{1+|-n^2{\pi }^2|t^{\eta }}$$

and

$$\begin{array}{cc}\hfill {\mathcal{K}}_{\eta ,\kappa }^{*}\mathfrak{M}\left(t\right)=& 2\sum _{n=1}^{\infty }{\varphi }_n\left(y\right){\int }_0^{\frac{1}{3}}{\sigma }^{\eta +\kappa (1-\eta )-1}{E}_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{\sigma }^{\eta }\right)⟨C^{*}\mathfrak{M}\left(\sigma \right),{\varphi }_n⟩\mathrm{d}\sigma \hfill \\ \hfill & ={\mathcal{S}}_{\eta ,\kappa }^{*}\left(t\right)C^{*}\mathfrak{M}\left(t\right).\hfill \end{array}$$

Thus, the observation operator C is admissible. From the above, we can see that $Ker{\mathcal{K}}_{\eta ,\kappa }\left(t\right)\ne 0$, which means System (44) is not observable in the whole domain $[0,1]$. Next, we investigate the observability of the addressed system in the internal subregion ${\omega }_2$.

$$\begin{array}{cc}\hfill {\mathcal{K}}_{\eta ,\kappa }{p}_{\omega }^{*}{p}_{\omega }{\nu }_0=& 2\sum _{n=1}^{\infty }{\varphi }_n\left(y\right)t^{\eta +\kappa (1-\eta )-1}{E}_{\eta ,\kappa (\eta -1)-\eta }\left(-n^2{\pi }^2{t}^{\eta }\right){⟨p_{\omega }^{*}{p}_{\omega }{\nu }_0,{\varphi }_n⟩}_{L^2\left({\omega }_2\right)}\ne 0.\hfill \end{array}$$

Thus, the initial state ${\nu }_0$ is weakly observable in the subregion ${\omega }_2$. In addition, for all $y\in {\omega }_2$, we have

$${\overline{\tilde{\gamma }}}_1=|{\nu }_{{|}_{{\omega }_2}}^0\left(y\right)|-\frac{2}{3}<{\nu }_{{|}_{{\omega }_2}}^0$$

and

$${\overline{\tilde{\gamma }}}_2=|{\nu }_{{|}_{{\omega }_2}}^0\left(y\right)|+\frac{2}{3}>{\nu }_{{|}_{{\omega }_2}}^0.$$

Thus, $p_{{\omega }_2}{\nu }_0\in {\mathfrak{Z}}_2$ and (44) together with (42) is ${\mathfrak{Z}}_2$-observable in ${\omega }_2$. This completes the proof. □

Let the space ${\mathfrak{P}}_2$ be given by

$${\mathfrak{P}}_2=\left\{g\in L^2\left({\mathsf{\Omega }}_2\right)|g=0\mathrm{in}{L}^2\left({\mathsf{\Omega }}_2\right)\setminus {\mathfrak{Z}}_2\right\}.$$

From Lemma 4, we have

$${\psi }_0⟼\parallel {\psi }_0{\parallel }_{{\mathfrak{P}}_2}^2={\int }_0^T{\parallel C\psi (T-t)\parallel }^2\mathrm{d}t,$$

which defines a norm on ${\mathfrak{P}}_2$, and we can introduce the adjoint system of (44) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathfrak{Q}\left[-{}_t{}^H\mathrm{D}_T^{\eta ,\kappa }\mathsf{\Phi }(y,t)\right]=A^{*}\mathfrak{Q}\left[\mathsf{\Phi }(y,t)\right]+{⟨C^{*}\mathfrak{M}\left(t\right),h⟩}_{L^2\left({\omega }_2\right)}h\left(y\right)\hfill & \mathrm{in}{\mathsf{\Omega }}_2\times [0,T],\hfill \\ \mathsf{\Phi }(\xi ,t)=0\hfill & \mathrm{on}\partial {\mathsf{\Omega }}_2\times [0,T],\hfill \\ \underset{t\to T^{-}}{lim}\mathfrak{Q}\left[{}_t{I}_T^{1-\zeta }\mathsf{\Phi }(y,t)\right]=0\hfill & \mathrm{in}{\mathsf{\Omega }}_2,\hfill \end{array}\right.\end{array}$$

Then, in view of Theorem 3, we can now conclude that $\mathsf{\Lambda }{\psi }_0=\mathcal{P}\left({}_0{I}_T^{\zeta -\eta }\mathsf{\Phi }\left(0\right)\right)$ has a unique solution in ${\mathfrak{P}}_2$, and the initial state ${\nu }_0$ can be observed between functions ${\overline{\tilde{\gamma }}}_1$ and ${\overline{\tilde{\gamma }}}_2$ in the subregion ${\omega }_2$.

5. Conclusions

In this manuscript we studied the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with Hilfer derivatives. We developed an approach based on the Hilbert uniqueness method (HUM). Based on this approach and with the knowledge of the initial information of the system and some given measurements, we reconstructed the initial state ${\nu }_0^1$ on an internal subregion $\omega$ from the whole domain $\mathsf{\Omega }$. Our findings show that it is possible to obtain the desired state between two profiles in some selective internal subregions. Finally, we gave two illustrative examples to support our theoretical results. It is of great interest for future works to investigate the ReOb of sub-diffusion systems with the Hilfer derivative in cases where the reconstructed initial state is in a subregion on the boundary of the whole domain. Furthermore, our paper motivates the study of the ReEnOb of sub-diffusion systems via $\psi$-Hilfer or $(k,\psi )$-Hilfer fractional derivatives.