Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

Abstract

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order $\beta \in (1,2)$, ${}_0^C{D}_t^{\beta }u(t,\zeta )=Au(t,\zeta )+f(t,u(t,\zeta ))d{C}_t+Bx\left(t\right)Cx\left(t\right)dt$ with initial conditions $u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1,$ where $u(t,\zeta )$ takes values from $U(\subset E_N),V(\subset E_N)$ is the other bounded space, and $E_N$ represents the set of all upper semi-continuously convex fuzzy numbers on $\mathbb{R}$. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

1. Introduction

In real-world phenomena, a large number of physical processes can be modeled using dynamical equations containing fractional-order derivatives [1]. The theory of fuzzy sets is continuously drawing the attention of researchers because of its rich applicability in several fields, including mechanics, electrical, engineering, processing signals, thermal systems, robotics and control, and many other fields [2,3]. Therefore, it has been an object of increasing interest for researchers during the past few years.

Until 2010, the concept in terms of Hukuhara differentiability [4] was unable to produce the vast and varied behavior of the crisp solution. However, later in 2012, a Riemann–Liouville H-derivative based on strongly generalized Hukuhara differentiability [5,6] was defined by Allahviranloo and Salahshour [7,8]. They also defined a fuzzy Riemann–Liouville fractional derivative.

Differential equations with fractional derivatives are known as fractional differential equations. Owing to the study of fractional derivatives, it is clear that they arise universally for major mathematical reasons. There are various types of derivatives, such as Caputo and Riemann–Liouville [9,10] derivatives.

Initially, Zadeh presented the concept of the fuzzy set in 1965 via the membership function. The most interesting field is that of fuzzy fractional differential equations. These are useful for analyzing phenomena where there is an inherent impression. Solutions of uniqueness and existence for fuzzy equations have been studied by Kwun et al. [11,12] and Lee et al. [13].

One of the most recent mathematical concepts is the theory of controlled processes in modern engineering to enable significant applications. Furthermore, due to various random factors that affect their behavior, actual systems under control do not allow for a strictly deterministic analysis. The random existence of a system’s actions is taken into account in the theory of controlled processes.

Many scholars have worked on controlled processes. Concerning fuzzy systems, controllability in an n-dimensional fuzzy vector space for an impulsive semi-linear fuzzy differential equation (FDE) was proved by Kwun and Park [14]. Research on controllability with nonlocal conditions of semi-linear fuzzy integro-differential equations was performed by Park et al. [15]. The controllability of impulsive semi-linear fuzzy integro-differential equations was proved by Park et al. [16]. Research on the stability and controllability of fuzzy control set differential equations was performed by Phu and Dung [17]. Lee et al. [18] studied controllability with nonlocal initial conditions in a nonlinear fuzzy control system’s n-dimensional fuzzy space $E_N^n$.

The controllability of a stochastic system of quasi-linear stochastic evolution equations in Hilbert space was studied by Balasubramanian [19] and Yuhu [20], who studied the controllability with time-variant coefficients of stochastic control systems. Arapostathis et al. investigated the controllability properties of stochastic differential systems characterized by linear controlled diffusion perturbed by bounded, smooth, uniformly Lipschitz non-linearity [21]. Brownian-motion-driven stochastic differential equations are a mature branch of modern mathematics and have been studied for a long time. The Liu process [22] was used to drive a new form of FDE, which was described as follows:

$$d{X}_t=f(X_t,t)dt+g(X_t,t)d{C}_t,$$

where $C_t$ is the standard Liu operation, while f and g are assigned functions. A fuzzy method is used to solve this type of equation. The solutions of uniqueness and existence of some special FDEs were discussed by Chen [23] for homogeneous FDEs. An approximate technique was studied by Liu [24] for solving uncertain differential equations. Young et al. [25] worked on exact controllability for abstract FDEs in credibility space by using the results of Liu [24]. In a credibility space, the exact controllability of abstract FDEs is expressed as follows:

$$\begin{array}{cc}\hfill dx(t,\theta )& =Ax(t,\theta )dt+f(t,x(t,\theta ))d{C}_t+Bu\left(t\right),t\in [0,T],\hfill \\ \hfill x\left(0\right)& =x_0.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {}_0^c{D}_t^{\beta }u(t,\zeta )& =Au(t,\zeta )+f(t,u(t,\zeta ))d{C}_t+Bx\left(t\right)Cx\left(t\right)dt,\beta \in (1,2),\hfill \\ \hfill u\left(0\right)& =u_0,\hfill \\ \hfill u^{\prime }\left(0\right)& =u_1,\hfill \end{array}$$

(1)

where the state take values from two bounded spaces $U(\subset E_N)$ and $V(\subset E_N)$. The set of all upper semi-continuously convex fuzzy numbers on $\mathbb{R}$ is $E_N$ and the credibility space is $(\Theta ,\mathcal{P},C_r)$.

The state function $u:[0,T]\times (\Theta ,\mathcal{P},C_r)\to U$ is a fuzzy coefficient. $f:[0,T]\times U\to U$ is a fuzzy function, $x:[0,T]\times (\Theta ,\mathcal{P},C_r)\to V$ is a control function, B and C are V to U linear bounded operators. $u_0\in E_N$ is an initial value and $C_t$ is a standard Liu process.

The aim of this paper is to look into the existence of solutions to FDEs as well as their exact controllability. Some researchers have found results about fuzzy differential equations in the literature, but most of them were for first-order differential equations or fractional orders between $(0,1]$. In our work we have found results for Caputo derivatives of order $(1,2);$ see [5,10,25] for more details. Our results are more complicated than the previous ones, and we require more boundary conditions than previous methods. Due to the change in boundary conditions, using Caputo derivatives, and for order $(1,2)$, almost all the results are original, but for previous results references have already been mentioned. The theory of fuzzy sets is continuously drawing the attention of researchers due to its rich suitability in various fields, including mechanics, engineering, electrical, thermal systems, robotics, control, and signal processing.

We go through some fundamental concepts relevant to Liu processes and fuzzy sets in Section 3. The existence of solutions to free FDEs is shown in Section 4. Finally, we show that the fuzzy differential equation is exactly controllable in Section 5.

2. Preliminaries

Let the family of all nonempty compact convex subsets of $\mathbb{R}$ be denoted by $M_k\left(\mathbb{R}\right)$ and addition and scalar multiplication are also usually defined as $M_k\left(\mathbb{R}\right)$. Let $A_1$ and $B_1$ be two nonempty bounded subsets of $\mathbb{R}$. The Hausdorff metric is used to define the distance between $A_1$ and $B_1$ as

$$d(A_1,B_1)=max\left\{\underset{a_1\in A_1}{sup}\underset{b_1\in B_1}{inf}\parallel a_1-b_1\parallel ,\underset{b_1\in B_1}{sup}\underset{a_1\in A_1}{inf}\parallel a_1-b_1\parallel \right\},$$

where $\parallel ·\parallel$ indicates the usual Euclidean norm in $\mathbb{R}$. Then it is clear that $(M_k\left(\mathbb{R}\right),d)$ becomes a separable and complete metric space [23]. Denote

$$E^n=\left\{x:\mathbb{R}\to [0,1]|u\mathrm{satisfies}\left(\mathrm{i}\right)\sum \left(\mathrm{iv}\right)\mathrm{below}\right\},$$

where

(i)

x is normal, there exists an $x_0\in \mathbb{R}$ such that $x\left(x_0\right)=1$;

(ii)

x is fuzzy convex, that is $x(\lambda t+(1-\lambda \left)s\right)\ge 2$;

(iii)

x is an upper semi-continuous function on $\mathbb{R}$ that is $x\left(t_0\right)\ge \underset{k\to \infty }{lim}\overline{x\left(t_k\right)}$ for any $t_k\in \mathbb{R}(k=0,1,2,\dots ),t_k\to t_0$;

(iv)

${\left[x\right]}^0=cl\{u\in \mathbb{R}|x\left(t\right)>0\}$ is compact.

For $1<\beta <2$, denote ${\left[x\right]}^{\beta }=\{t\in \mathbb{R}|u\left(t\right)\ge \beta \}$ and ${\left[u\right]}^0$ are nonempty compact convex sets in $\mathbb{R}$ [26]. Then from (i)–(iv), it follows that $\beta$-level set ${\left[x\right]}^{\beta }t\in M_k\left(\mathbb{R}\right)$ for all $1<\beta <2$. We can have scalar multiplication and addition in fuzzy number space $E^n$ by using Zadeh’s extension principle as follows:

$${[x\oplus y]}^{\beta }={\left[x\right]}^{\beta }\oplus {\left[y\right]}^{\beta },{\left[kx\right]}^{\beta }=k{\left[y\right]}^{\beta },$$

where $x,y\in E^n,k\in \mathbb{R}$ and $1<\beta <2$.

Suppose that $E_N$ represents the set of all upper semi-continuously convex fuzzy numbers on $\mathbb{R}$.

Definition 1

([27]). Define a complete metric $D_L$ by

$$\begin{array}{cc}\hfill D_L(x,y)& =\underset{1<\beta <2}{sup}{d}_L\left\{{\left[x\right]}^{\beta },{\left[y\right]}^{\beta }\right\}\hfill \\ \hfill & =\underset{1<\beta <2}{sup}max\left\{|x_l^{\beta }-y_l^{\beta }|,|x_l^{\beta }-y_r^{\beta }|\right\},\hfill \end{array}$$

for any $u,v\in E_N$, which satisfies $D_L(x+z,y+z)=D_L(x,y)$ for each $z\in E_N$ and ${\left[x\right]}^{\alpha }=[x_l^{\beta },u_r^{\beta }]$, for each $\beta \in (x,y)$ where $x_l^{\beta },u_r^{\beta }\in \mathbb{R}$ with $x_l^{\beta }\le u_r^{\beta }$.

Definition 2

([28]). The Riemann–Liouville fractional derivative is defined as

$${}_a{D}_t^{p}f\left(t\right)={\left(\frac{d}{dt}\right)}^{n+1}{\int }_a^t{(t-\tau )}^{n-p}f\left(\tau \right)d\tau ,(n\le p\le n+1).$$

Definition 3

([28]). The Caputo fractional derivatives ${}_a^C{D}_t^{\alpha }f\left(t\right)$ of order $\alpha \in {\mathbb{R}}^{+}$ are defined by

$${}_a^C{D}_t^{\alpha }f\left(t\right)={}_a{D}_t^{\alpha }\left(f\left(t\right)-\sum _{k=0}^{n-1}\frac{f^{\left(k\right)}\left(a\right)}{k!}{(t-a)}^k\right),$$

where $n=\left[\alpha \right]+1$ for $\alpha \notin N_0;n=\alpha$ for $\alpha \in N_0$.

In this paper, we consider a Caputo fractional derivative of order $1<\alpha \le 2$, e.g.,

$${}_a^C{D}_t^{3/2}f\left(t\right)={}_a{D}_t^{3/2}\left(f\left(t\right)-\sum _{k=0}^{n-1}\frac{f^{\left(k\right)}\left(a\right)}{k!}{(t-a)}^k\right).$$

Definition 4

([29]). The Wright function ${\psi }_{\alpha }$ is defined by

$$\begin{array}{cc}\hfill {\psi }_{\alpha }(\theta )& =\sum _{n=0}^{\infty }\frac{{(-\theta )}^n}{n!\Gamma (-\alpha n+1-\alpha )}\hfill \\ \hfill & =\frac{1}{\pi }\sum _{n=1}^{\infty }\frac{{(-\theta )}^n}{(n-1)!}\Gamma \left(n\alpha \right)sin\left(n\pi \alpha \right),\hfill \end{array}$$

where $\theta \in \mathbb{C}$ with $0<\alpha <1$.

Definition 5

([30]). For any $x,y\in C\left([0,T],E_N\right)$, metric $H_1(x,y)$ on $C\left([0,T],E_N\right)$ is defined by

$$H_1(x,y)=\underset{0\le t\le T}{sup}{D}_L(x\left(t\right),y\left(t\right)).$$

Allow Θ to be a nonempty set and $\mathcal{P}$ to be Θ ’s power set. Each element of $\mathcal{P}$ is referred to as a case. To offer an axiomatic concept of credibility based on the assumption that A will happen; to ensure that a number $C_r\left\{A_1\right\}$ is assigned to each event $A_1$, indicating the credibility of $A_1$ occurring; and to ensure the number $C_r\left\{A_1\right\}$ has certain mathematical properties that we intuitively predict, we accept the following four axioms:

(i)

(Normality) $C_r\{\Theta \}=2$;

(ii)

(Monotonicity) $C_r\left\{A_1\right\}\le C_r\left\{B_1\right\}$, whenever $A_1\subset B_1$;

(iii)

(Self-Duality) $C_r\left\{A_1\right\}+C_r\left\{A_1^c\right\}=2$ for any event $A_1$;

(iv)

(Maximality) $C_r\left\{{\cup }_i{A}_i\right\}={sup}_i{C}_r\left\{A_1\right\}$ for any events $\left\{A_i\right\}$ with ${sup}_i{C}_r\left\{A_i\right\}<1.5$.

Definition 6

([31]). Let Θ be a nonempty set, $\mathcal{P}$ be Θ ’s power set, and $C_r$ be a credibility measure. The triplet $(\Theta ,\mathcal{P},C_r)$ is then added to a set of real numbers.

Definition 7

([31]). A fuzzy variable is a function from the set of real numbers $(\Theta ,\mathcal{P},C_r)$ to credibility space $(\Theta ,\mathcal{P},C_r)$.

Definition 8

([31]). Let $(\Theta ,\mathcal{P},C_r)$ be a credibility space and $(\Theta ,\mathcal{P},C_r)$ be an index set. A fuzzy process is a function from a set of real numbers to $T\times (\Theta ,\mathcal{P},C_r)$.

That is, it is fuzzy process. $u(t,\zeta )$ is a two-variable function, with $u(t,{\zeta }^{*})$ acting as a fuzzy variable for each $t^{*}$. The function $u(t,\zeta )$ is called the sample path of a fuzzy process for each fixed ${\zeta }^{*}$. If sampling is continuous for almost all ζ, fuzzy process $u(t,\zeta )$ is said to be sample-continuous. We often use the symbol $u_t$ instead of $u(t,\zeta )$.

Definition 9

([31]). A credibility space is known as $(\Theta ,\mathcal{P},C_r)$. For each $\beta \in (1,2)$, the β-level set is used for the fuzzy random variable $u_t$ in credibility space.

$${\left[u_t\right]}^{\beta }=[{\left(u_t\right)}_l^{\beta },{\left(u_t\right)}_r^{\beta }]$$

is defined by

$$\begin{array}{cc}\hfill {\left(u_t\right)}_l^{\beta }& =inf{\left(u_t\right)}^{\beta }=inf\{a\in \mathbb{R};u_t\left(a\right)\ge \beta \},\hfill \\ \hfill {\left(u_t\right)}_r^{\beta }& =sup{\left(u_t\right)}^{\beta }=inf\{a\in \mathbb{R};u_t\left(a\right)\ge \beta \},\hfill \end{array}$$

where ${\left(u_t\right)}_l^{\beta },{\left(u_t\right)}_r^{\beta }\in \mathbb{R}$ with ${\left(u_t\right)}_l^{\beta }\le {\left(u_t\right)}_r^{\beta }$ when $\beta \in (1,2)$.

Definition 10

([32]). Assume that θ is a fuzzy variable and that r is a real number. Then θ’s expected value is defined as

$$E\theta ={\int }_0^{+\infty }{C}_r\{\theta \ge r\}dr-{\int }_{-\infty }^0{C}_r\{\theta \le r\}dr$$

provided that at least one integral is finite.

Lemma 1

([32]). Assume that θ is a fuzzy vector. Below are the properties of the expected value operator E:

(i)

if $f\le g,E\left[f\right(\theta \left)\right]\le E\left[g\right(\theta \left)\right]$;

(ii)

$E[-f(\theta \left)\right]=-E\left[f\right(\theta \left)\right]$;

(iii)

if f and g are comonotonic, we have for any nonnegative real numbers $a_1$ and $b_1$

$$E[a_{1}f(\theta )+b_{1}g(\theta )]=a_{1}E\left[f(\theta )\right]+b_{1}E\left[g(\theta )\right],$$

where $f(\theta )$ and $g(\theta )$ are fuzzy variables.

Definition 11

([32]). A fuzzy process $C_t$ is a Liu process if

(i)

$C_0=0$;

(ii)

the $C_t$ has independent and stationary increments;

(iii)

any increment $C_{t+s}-C_s$ is a normally distributed fuzzy variable with expected value $et$ and variance ${\varphi }^2{t}^2$, with membership function

$$\xi \left(u\right)=2{\left(1+exp\left(\frac{\pi |u-et|}{\sqrt{6}\varphi t}\right)\right)}^{-1},u\in \mathbb{R}.$$

The diffusion and drift coefficients are the parameters ϕ and e, respectively. The Liu process is said to be standard if $e=0$ and $\varphi =1$.

Definition 12

([33]). Suppose $C_t$ to be a standard Liu process and $u_t$ to be a fuzzy process. The mesh is written as $c=t_0<\cdots <t_n=d$ for any partition of the closed interval $[c,d]$ with $c=t_0<\cdots <t_n=d$,

$$\Delta =\underset{1\le i\le n}{max}(t_i-t_{i-1}).$$

The fuzzy integral of $u_t$ with respect to $C_t$ is then determined.

$${\int }_c^d{u}_{t}d{C}_t=\underset{\Delta \to 0}{lim}\sum _{i=1}^n\mu \left(t_{i-1}\right)(C_{t_i}-C_{t_{i-1}})$$

provided that a limit exists almost certainly and is a fuzzy variable.

Lemma 2

([33]). Let $C_t$ be a standard Liu process. The direction $C_t$ is Lipschitz continuous for any given with $C_r\{\zeta \}>0$, which implies that the following inequality holds:

$$|C_{t_1}-C_{t_2}|<K(\zeta )|t_1-t_2|,$$

where $K(\zeta )$ is the Lipschitz constant of a Liu process, which is a fuzzy variable defined by

$$K(\zeta )=\left\{\begin{array}{cc}\underset{0\le s\le t}{sup}\frac{|C_t-C_s|}{t}-s,\hfill & C_r\{\zeta \}>1;\hfill \\ \infty ,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and $E\left[K^p\right]<\infty$ for all $p>1$.

Lemma 3

([33]). Suppose $h(t;c)$ to be a continuously differentiable function and $C_t$ to be standard Liu process. $u_t=h(t;C_t)$ is the function to define. In addition, there is the chain rule that follows:

$$d{u}_t=\frac{\partial h(t;C_t)}{\partial t}dt+\frac{\partial h(t;C_t)}{\partial C}d{C}_t.$$

Lemma 4

([33]). If $f\left(t\right)$ is a continuous fuzzy process, the below fuzzy integral inequality holds:

$$\left|{\int }_c^{d}f\left(t\right)d{C}_t\right|\le K{\int }_c^d\left|f\left(t\right)\right|dt.$$

The expression $K=K(\zeta )$ is defined in Lemma 2.

Definition 13.

The fractional integral for a function f with lower limit $t_0$ and order γ can be defined as

$$I_{t_0^{+}}^{\gamma }f\left(t\right)=\frac{1}{\Gamma \left(\gamma \right)}{\int }_{t_0^{+}}^t\frac{f\left(s\right)}{{(t-s)}^{1-\gamma }}ds,\gamma >0,t>t_0,$$

where the right-hand side of the equality is defined point-wise on ${\mathbb{R}}^{+}$.

Lemma 5

([34]). Let $\gamma >0$. Then

$$I_{{t_0}^{+}}^{\gamma }{{}^{c}D}_{t_0^{+}}^{\gamma }f\left(t\right)=f\left(t\right)+c_0+c_{1}t+c_2{t}^2+\cdots +c_{n-1}{t}^{n-1}$$

for some $c_i\in \mathbb{R},i=0,1,2,\dots ,n-1$, where $n=\left[\gamma \right]+1$.

Lemma 6

([35]). Let ${\left\{C\left(t\right)\right\}}_{t\in \mathbb{R}}$ be a strongly continuous cosine family in X satisfying ${\parallel C\left(t\right)\parallel }_{L_b\left(X\right)}\le M{e}^{\omega \left|t\right|},t\in \mathbb{R}$, and let A be the infinitesimal generator of ${\left\{C\left(t\right)\right\}}_{t\in \mathbb{R}}$, then for $Re\lambda >\omega ,{\lambda }^2\in \rho \left(A\right)$

$$\lambda R({\lambda }^2;A)x={\int }_0^{\infty }{e}^{-\lambda t}C\left(t\right)xdt,\mathrm{and}R({\lambda }^2;A)x={\int }_0^{\infty }{e}^{-\lambda t}S\left(t\right)xdt$$

for $x\in X$.

3. Existence of Solutions for Fuzzy Fractional Evolution Equations

By Definition 8, we use symbol $u_t$ instead of longer notation $u(t,\zeta )$ in this section. The uniqueness and existence of solutions for fuzzy differential Equation (3) $(x\equiv 0)$ are examined.

$$\left\{\begin{array}{c}{}_0^C{D}_t^{\beta }{u}_t=A{u}_t+f(t,u_t)d{C}_t,\beta \in (1,2),\hfill \\ u\left(0\right)=u_0,\hfill \\ u^{\prime }\left(0\right)=u_1\in E_N\hfill \end{array}\right.$$

(2)

where $u_t$ is a state that takes values from $U(\subset E_N)$. The set of all upper semi-continuously convex fuzzy numbers on $\mathbb{R}$ is labeled $E_N$, $(\Theta ,\mathcal{P},C_r)$ is a credibility space, A is a fuzzy coefficient, state function $u:[0,T]\times (\Theta ,\mathcal{P},C_r)\to U$ is a fuzzy process, $f:[0,T]\times U\to U$ is a regular fuzzy function, $C_t$ is a standard Liu process, and the initial value is $u_0\in E_N$.

Lemma 7.

If $u_t$ is a solution of (2) for $u\left(0\right)=u_0$, then $u_t$ is given by

$$u_t=C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\left[f(s,u_s)\right]d{C}_s+{\int }_0^t{(t-s)}^{q-1}Bx\left(s\right)Cx\left(s\right)ds,$$

where B and C are linear bounded operators and

$$\begin{array}{cc}\hfill C_q\left(t\right)& ={\int }_0^{\infty }{M}_{q}C(t^q\zeta )d\zeta ,\hfill \\ \hfill K_q\left(t\right)& ={\int }_0^t{C}_q\left(s\right)ds,\hfill \\ \hfill P_q\left(t\right)& ={\int }_0^{\infty }q\zeta M_{q}C(t^q\zeta )d\zeta ,\hfill \\ \hfill M_q\left({\zeta }^{-q}\right)& ={\psi }_q(\zeta )\frac{{\zeta }^{q+1}}{q}\hfill \end{array}$$

such that $C_q\left(t\right)$ and $K_q\left(t\right)$ are continuous with $S\left(0\right)=I$ and $K\left(0\right)=I$, $|C_q\left(t\right)|\le c,c>1$ and $|K_q\left(t\right)|\le c,c>1$ for all $t\in [0,T]$.

Proof. 

Let $Re\lambda >0$ and $\mathcal{L}$ be the Laplace transform

$$\mu (\lambda )=\mathcal{L}\left[u\left(t\right)\right](\lambda )={\int }_0^{\infty }{e}^{-\lambda s}u\left(s\right)ds,\nu (\lambda )=\mathcal{L}\left[f\left(t\right)\right](\lambda )={\int }_0^{\infty }{e}^{-\lambda s}f\left(s\right)ds.$$

According to Lemma 5, the Laplace transform is now being applied to Equation

$$u\left(t\right)=u_0+u_{1}t+\frac{1}{\Gamma (\beta )}{\int }_0^t{(t-s)}^{\beta -1}[Au\left(s\right)+f(s,u(s,\zeta ))]d{C}_s+{\int }_0^t{(t-s)}^{\beta -1}Bx\left(s\right)Cx\left(s\right)ds.$$

(3)

By Lemma 6, it follows that for $t\in [0,\infty )$.

Taking the Laplace transform on both sides of the above equation, we have

$$\begin{array}{cc}\hfill \mathcal{L}\left\{u_t\right\}& =\mathcal{L}\left\{u_0+u_{1}t\right\}+\mathcal{L}\left\{\frac{1}{\Gamma (\beta )}{\int }_0^t{(t-s)}^{\beta -1}[Au\left(s\right)+f(s,u(s,\zeta ))]d{C}_s\right\}\hfill \\ \hfill & +\mathcal{L}\left\{{\int }_0^t{(t-s)}^{\beta -1}Bx\left(s\right)Cx\left(s\right)ds\right\},\hfill \\ \hfill \mu (\lambda )& =\frac{1}{\lambda }{u}_0+\frac{1}{{\lambda }^2}{u}_1+\frac{1}{{\lambda }^{\beta }}A\mu (\lambda )+\frac{1}{{\lambda }^{\beta }}\nu (\lambda ),\hfill \\ \hfill \mu (\lambda )-\frac{1}{{\lambda }^{\beta }}A\mu (\lambda )& =\frac{1}{\lambda }{u}_0+\frac{1}{{\lambda }^2}{u}_1+\frac{1}{{\lambda }^{\beta }}\nu (\lambda ),\hfill \\ \hfill \mu (\lambda )\left(1-\frac{1}{{\lambda }^{\beta }}A\right)& =\frac{1}{\lambda }{u}_0+\frac{1}{{\lambda }^2}{u}_1+\frac{1}{{\lambda }^{\beta }}\nu (\lambda ),\hfill \\ \hfill \mu (\lambda )& ={\lambda }^{\beta -1}{({\lambda }^{\beta }-A)}^{-1}{u}_0+{\lambda }^{\beta -2}{({\lambda }^{\beta }-A)}^{-1}{u}_1+{({\lambda }^{\beta }-A)}^{-1}\nu (\lambda ).\hfill \end{array}$$

(4)

As a result, $t\ge 0$,

$$\mu (\lambda )={\lambda }^{\frac{\beta }{2}-1}{\int }_0^{\infty }{e}^{-{\lambda }^{\frac{\beta }{2}t}}C\left(t\right)u_{0}ds+{\lambda }^{-1}{\lambda }^{\frac{\beta }{2}-1}{\int }_0^{\infty }{e}^{-{\lambda }^{\frac{\beta }{2}t}}C\left(t\right)u_{1}ds+{\int }_0^{\infty }{e}^{-{\lambda }^{\frac{\beta }{2}t}}S\left(t\right)\nu (\lambda )d{C}_{t}ds.$$

(5)

Let

$${\psi }_q(\zeta )=\frac{q}{{\zeta }^{q+1}}{M}_q\left({\zeta }^{-q}\right),\zeta \in (0,\infty ).$$

Its Laplace transform is as follows:

$${\int }_0^{\infty }{e}^{-\lambda \zeta }{\psi }_q(\zeta )d\zeta =e^{-{\lambda }^q}$$

(6)

for $q\in \left(\frac{1}{2},1\right).$

To begin, we will use (6),

$$\begin{array}{cc}\hfill {\lambda }^{q-1}{\int }_0^{\infty }{e}^{-{\lambda }^{q}t}C\left(t\right)u_{0}dt& ={\int }_0^{\infty }q{(\lambda t)}^{q-1}{e}^{-{(\lambda t)}^q}C\left(t^q\right)u_{0}dt\hfill \\ \hfill & ={\int }_0^{\infty }-\frac{1}{\lambda }\frac{d}{dt}\left({\int }_0^{\infty }{e}^{-\lambda t\zeta }{\psi }_q(\psi )d\psi \right)C\left(t^q\right)u_{0}dt\hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }\zeta {\psi }_q(\zeta )e^{-\lambda t\zeta }C\left(t^q\right)u_{0}d\zeta dt\hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }{\psi }_q(\zeta )e^{-\lambda t}C\left(\frac{t^q}{{\zeta }^q}\right)u_{0}dtd\zeta \hfill \\ \hfill & ={\int }_0^{\infty }{e}^{-\lambda t}\left[{\int }_0^{\infty }{\psi }_q(\zeta )C\left(\frac{t^q}{{\zeta }^q}\right)u_{0}d\zeta \right]dt\hfill \\ \hfill & =\mathcal{L}\left[{\int }_0^{\infty }{M}_q(\zeta )C(t^q\zeta )u_{0}d\zeta \right](\lambda )\hfill \\ \hfill & =\mathcal{L}\left[C_q\left(t\right)u_0\right](\lambda ).\hfill \end{array}$$

(7)

Furthermore, by applying the Laplace convolution theorem, we obtain $\mathcal{L}\left[g_1\left(t\right)\right](\lambda )={\lambda }^{-1}$.

$$\begin{array}{cc}\hfill {\lambda }^{-1}{\lambda }^{q-1}{\int }_0^{\infty }{e}^{-{\lambda }^{q}t}C\left(t\right)u_{1}dt& =\mathcal{L}\left[C_q\left(t\right)u_1\right](\lambda ){\lambda }^{-1}{\lambda }^{q-1}{\int }_0^{\infty }{e}^{-{\lambda }^{q}t}C\left(t\right)u_{1}dt\hfill \\ \hfill & =\mathcal{L}[g_1*C_q\left(t\right)u_1](\lambda ).\hfill \end{array}$$

(8)

Similarly, we observe

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{e}^{-{\lambda }^{q}t}S\left(t\right)\nu (\lambda )dt& ={\int }_0^{\infty }q{t}^{q-1}{e}^{-{(\lambda t)}^q}S\left(t^q\right)\nu (\lambda )dt\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }q{t}^{q-1}{\psi }_q(\zeta )e^{-\lambda t\zeta }S\left(t^q\right)\nu (\lambda )d\zeta dt\hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }q\frac{t^{q-1}}{{\zeta }^q}{\psi }_q(\zeta )e^{-\lambda t}S\left(\frac{t^q}{{\zeta }^q}\right)\nu (\lambda )dtd\zeta \hfill \\ \hfill & ={\int }_0^{\infty }{e}^{-\lambda t}\left[{\int }_0^{\infty }q\frac{t^{q-1}}{{\zeta }^q}{\psi }_q(\zeta )e^{-\lambda t}\left(\frac{t^q}{{\zeta }^q}\right)\nu (\lambda )d\zeta \right]dt\hfill \\ \hfill & =\mathcal{L}\left[{\int }_0^{\infty }q{t}^{q-1}{M}_q(\zeta )S(t^q\zeta )d\zeta \right](\lambda ).\mathcal{L}\left[f\left(t\right)\right](\lambda )\hfill \\ \hfill & =\mathcal{L}\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f\left(s\right)d{C}_s+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)Bx\left(s\right)Cx\left(s\right)ds\right](\lambda ).\hfill \end{array}$$

(9)

Using the Laplace transform’s uniqueness theorem and combining (7)–(9), we have the following

$$\begin{array}{cc}\hfill u_t& ={\int }_0^{\infty }{\int }_0^{\infty }\zeta {\psi }_q(\zeta )e^{-\lambda t\zeta }C\left(t^q\right)u_{0}d\zeta dt+\mathcal{L}\left[C_q\left(t\right)u_1\right](\lambda )\hfill \\ \hfill & +{\lambda }^{-1}{\lambda }^{q-1}{\int }_0^{\infty }{e}^{-{\lambda }^{q}t}C\left(t\right)u_{1}dt+{\int }_0^{\infty }{\int }_0^{\infty }q{t}^{q-1}{\psi }_q(\zeta )e^{-\lambda t\zeta }S\left(t^q\right)\nu (\lambda )d\zeta dt.\hfill \\ \hfill u_t& =C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\left[f(s,u_s)\right]d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)Bx\left(s\right)Cx\left(s\right)ds.\hfill \end{array}$$

Assume that the following statements are true:

(H₁)

For $u_t,v_t\in C([0,T]\times (\Theta ,\mathcal{P},C_r),U),t\in [0,T]$. There exists a positive number m, such that

$$d_L({\left[f(t,u_t)\right]}^{\beta },{\left[f(t,v_t)\right]}^{\beta })\le m{d}_L({\left[u_t\right]}^{\beta },{\left[v_t\right]}^{\beta })$$

and

$$f(0,{\mathcal{X}}_{\left\{0\right\}}\left(0\right))\equiv 0.$$

(H₂)

$2cmKT\le 2$.

We know that (2) has solution $u_t$ because of Lemma 7. Thus, in Theorem 1 we show that the solution to (2) is unique. □

Theorem 1.

If $u_0\in E_N$, if $\left(H_1\right)$ and $\left(H_2\right)$ hold, (2) has a unique solution $u_t\in C\left([0,T]\right)\times (\Theta ,\mathcal{P},C_r),U)$.

Proof. 

For all ${\theta }_t\in C([0,T]\times (\Theta ,\mathcal{P},C_r),U),t\in [0,T]$, define

$$\begin{array}{cc}\hfill \varphi {\theta }_t& =C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\left[f(s,{\theta }_s)\right]d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)Bx\left(s\right)Cx\left(s\right)ds.\hfill \end{array}$$

As a result, one can illustrate that $\varphi \theta :[0,T]\times (\Theta ,\mathcal{P},C_r)\to C([0,T]\times (\Theta ,\mathcal{P},C_r),U)$ is continuous,

$$\varphi :C([0,T]\times (\Theta ,\mathcal{P},C_r),U)\to C([0,T]\times (\Theta ,\mathcal{P},C_r),U).$$

A fixed point of $\varphi$ is also an obvious solution for Equation (2). By Lemma 4 and hypothesis $\left(H_1\right)$, ${\theta }_t,{\mu }_t\in C([0,T]\times (\Theta ,\mathcal{P},C_r),U)$.

$$\begin{array}{cc}\hfill d_L({[\varphi {\theta }_t]}^{\beta },{[\varphi {\mu }_t]}^{\beta })& =d_L({\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\left[f(s,{\theta }_s)\right]d{C}_s+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)Bx\left(s\right)Cx\left(s\right)\right]}^{\beta },\hfill \\ \hfill & {\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\left[f(s,{\mu }_s)\right]d{C}_s+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)Bx\left(s\right)Cx\left(s\right)\right]}^{\beta })\hfill \\ \hfill & \le 2cmK{\int }_0^t{d}_L({\left[{\theta }_s\right]}^{\beta },{\left[{\mu }_s\right]}^{\beta })ds.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill D_L(\varphi {\theta }_t,\varphi {\mu }_t)& =\underset{\beta \in (1,2)}{sup}{d}_L({[\varphi {\theta }_t]}^{\beta },{[\varphi {\mu }_t]}^{\beta })\hfill \\ \hfill & \le 2cmK{\int }_0^t\underset{\beta \in (1,2)}{sup}{d}_L({\left[{\theta }_t\right]}^{\beta },{\left[{\mu }_t\right]}^{\beta })ds\hfill \\ \hfill & =2cmK{\int }_0^t{D}_L({\theta }_s,{\mu }_s)ds.\hfill \end{array}$$

As a consequence, by Lemma 1, for a.s. $\theta \in \Theta$,

$$\begin{array}{cc}\hfill E\left(H_1(\varphi \theta ,\varphi \mu )\right)& =E\left(\underset{t\in (0,T]}{sup}{D}_L(\varphi {\theta }_t,\varphi {\mu }_t)\right)\hfill \\ \hfill & \le E\left(2cmK\underset{t\in (0,T]}{sup}{\int }_0^t{D}_L({\theta }_t,{\mu }_t)\right)\hfill \\ \hfill & \le 2cmKTE\left(H_1(\theta ,\mu )\right).\hfill \end{array}$$

By hypothesis $\left(H_2\right)$, a contraction mapping is $\varphi$. This has a unique fixed point $x_t\in C([0,T]\times (\Theta ,\mathcal{P},C_r),U)$ by the Banach fixed point theorem in Equation (2). □

4. Exact Controllability for Fuzzy Fractional Evolution Equations

The exact controllability of Caputo fuzzy differential Equation (3) is examined in this section. For each x in $V(\subset E_N)$, we consider a solution for (3).

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t\hfill & =C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{x}_{s}C{x}_{s}ds\hfill \\ u\left(0\right)\hfill & =u_0,\hfill \\ u^{\prime }\left(0\right)\hfill & =u_1,\hfill \end{array}\right.\end{array}$$

(10)

where $S\left(t\right)$ is continuous with $S\left(0\right)=I$ and $S^{\prime }\left(0\right)=I,\left|S\left(t\right)\right|\le c,c>0,t\in [0,T]$. For Caputo fuzzy differential equations, we define the concept of controllability.

Definition 14.

Equation (3) is said to be controllable on $[0,T]$ if there is a control $u_t\in V$ for every $u_0\in E_N$ such that the solution u of (3) satisfies $u_t=u^{-1}\in U,$ a.s. ζ that is ${\left[u_t\right]}^{\beta }={\left[u^1\right]}^{\beta }$.

Define fuzzy mapping $\tilde{G}:\tilde{P}\left(R\right)\to U$

$${\tilde{G}}^{\beta }\left(y\right)=\left\{\begin{array}{cc}{\int }_0^T{(t-s)}^{q-1}{P}_q^{\beta }(t-s)B{y}_{s}C{y}_{s}ds,\hfill & y\subset {\overline{\Gamma }}_x,\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

where ${\overline{\Gamma }}_x$ is the closure of support x and $\tilde{P}\left(R\right)$ is a nonempty fuzzy subset of $\mathbb{R}$.

Then there is a ${\tilde{G}}_i^{\beta }(i=m,n)$,

$$\begin{array}{cc}\hfill {\tilde{G}}_m^{\beta }\left(y_m\right)& ={\int }_0^T{(t-s)}^{q-1}{P}_m^{\beta }(t-s)B{\left(y_s\right)}_{m}C{\left(y_s\right)}_{m}ds,{\left(y_s\right)}_m\in [{\left(y_s\right)}_m^{\beta },{\left(y_s\right)}^1],\hfill \\ \hfill {\tilde{G}}_n^{\beta }\left(y_n\right)& ={\int }_0^T{(t-s)}^{q-1}{P}_n^{\beta }(t-s)B{\left(y_s\right)}_{n}C{\left(y_s\right)}_{n}ds,{\left(y_s\right)}_n\in [{\left(y_s\right)}^1,{\left(y_s\right)}_n^{\beta }].\hfill \end{array}$$

We assume that ${\tilde{G}}_m^{\beta },{\tilde{G}}_n^{\beta }$ are bijective mappings. A $\beta$-level set of $x_s$ can be represented as follows:

$$\begin{array}{cc}\hfill {\left[x_s\right]}^{\beta }& =[{\left(x_s\right)}_m^{\beta },{\left(x_s\right)}_n^{\beta }]\hfill \\ \hfill & =\left[{\left({\tilde{G}}_m^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_m^{\beta }-C_q\left(t\right){\left(u_0\right)}_m^{\beta }-K_q\left(t\right)\left(u_1\right)-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f_m^{\beta }(s,u_s)d{C}_s\right.\right.\hfill \\ \hfill & \left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B_m^{\beta }\left(x_s\right)C_m^{\beta }\left(x_s\right)ds\right\},{\left({\tilde{G}}_n^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_n^{\beta }-C_q\left(t\right){\left(u_0\right)}_n^{\beta }-K_q\left(t\right)\left(u_1\right)\right.\hfill \\ \hfill & \left.\left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f_n^{\beta }(s,u_s)d{C}_s-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B_n^{\beta }\left(x_s\right)C_n^{\beta }\left(x_s\right)ds\right\}\right].\hfill \end{array}$$

The $\beta$-level of $x_t$ is obtained by substituting this expression into (10).

$$\begin{array}{cc}\hfill {\left[u_t\right]}^{\beta }& =\left[C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_s\right.\hfill \\ \hfill & {\left.+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{x}_{s}C{x}_{s}ds\right]}^{\beta }\hfill \\ \hfill & =\left[C_m^{\beta }\left(t\right){\left(u_0\right)}_m^{\beta }+K_m^{\beta }\left(t\right){\left(u_1\right)}_m^{\beta }+{\int }_0^t{(t-s)}^{q-1}{P}_m^{\beta }(t-s)f(s,{\left(u_s\right)}_m^{\beta })d{C}_s\right.\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_m^{\beta }(t-s)B{\left({\tilde{G}}_m^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_m^{\beta }-C_q\left(t\right){\left(u_0\right)}_m^{\beta }-K_q\left(t\right){\left(u_1\right)}_m^{\beta }\right.\hfill \\ \hfill & \left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f_m^{\beta }(s,u_s)d{C}_s-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B_m^{\beta }\left(x_s\right)C_m^{\beta }\left(x_s\right)ds\right\}ds,\hfill \\ \hfill & C_n^{\beta }\left(t\right){\left(u_0\right)}_n^{\beta }+K_n^{\beta }\left(t\right){\left(u_1\right)}_n^{\beta }+{\int }_0^t{(t-s)}^{q-1}{P}_n^{\beta }(t-s)f(s,{\left(u_s\right)}_n^{\beta })d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_n^{\beta }(t-s)B{\left({\tilde{G}}_n^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_n^{\beta }-C_q\left(t\right){\left(u_0\right)}_n^{\beta }-K_q\left(t\right){\left(u_1\right)}_n^{\beta }\right.\hfill \\ \hfill & \left.\left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f_n^{\beta }(s,u_s)d{C}_s-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B_n^{\beta }\left(x_s\right)C_n^{\beta }\left(x_s\right)ds\right\}ds\right]\hfill \\ \hfill & =\left[C_m^{\beta }\left(t\right){\left(u_0\right)}_m^{\beta }+K_m^{\beta }\left(t\right){\left(u_1\right)}_m^{\beta }+{\int }_0^t{(t-s)}^{q-1}{P}_m^{\beta }(t-s)f(s,{\left(u_s\right)}_m^{\beta })d{C}_s\right.\hfill \\ \hfill & +{\tilde{G}}_m^{\beta }{\left({\tilde{G}}_m^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_m^{\beta }-C_q\left(t\right){\left(u_0\right)}_m^{\beta }-K_q\left(t\right){\left(u_1\right)}_m^{\beta }-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f_m^{\beta }(s,u_s)d{C}_s\right.\hfill \\ \hfill & \left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B_m^{\beta }\left(x_s\right)C_m^{\beta }\left(x_s\right)ds\right\}ds,C_n^{\beta }t){\left(u_0\right)}_n^{\beta }+K_n^{\beta }\left(t\right){\left(u_1\right)}_n^{\beta }\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_n^{\beta }(t-s)f(s,{\left(u_s\right)}_n^{\beta })d{C}_s+{\tilde{G}}_n^{\beta }{\left({\tilde{G}}_n^{\beta }\right)}^{-1}\left\{{\left(u^1\right)}_n^{\beta }-C_q\left(t\right){\left(u_0\right)}_n^{\beta }-K_q\left(t\right){\left(u_1\right)}_n^{\beta }\right.\hfill \\ \hfill & =[{\left(u^1\right)}_m^{\beta },{\left(u^1\right)}_n^{\beta }]\hfill \\ \hfill & ={\left[u^1\right]}^{\alpha }.\hfill \end{array}$$

Hence this control $x_t$ satisfies $u_t=u^1,$ a.s. $\zeta$.

We now set

$$\begin{array}{cc}\hfill \psi u_t& =C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{\tilde{G}}^{-1}\left\{u^1-C_q\left(t\right)u_0-K_q\left(t\right)u_1\right.\hfill \\ \hfill & -{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_t\hfill \\ \hfill & \left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B\left(x_s\right)C\left(x_s\right)ds\right\}ds.\hfill \end{array}$$

Fuzzy mapping ${\tilde{G}}^{-1}$ satisfies the above statement.

Theorem 2.

If Lemma 4 and the hypotheses $\left(H_1\right)and\left(H_2\right)$ are satisfied, then Equation (3) is controllable on $[0,T]$.

Proof. 

We can easily verify that $\psi$ is continuous from $C\left(\right[0,T]\times (\Theta ,\mathcal{P},U)$ to $C\left(\right[0,T]$. For any given $\zeta$ with $C_r\{\zeta \}>0,x_t,y_t\in C([0,T]\times (\Theta ,\mathcal{P},C_r),U)$, we have by Lemma 4 and hypotheses $\left(H_1\right)$ and $\left(H_2\right)$ that

$$\begin{array}{cc}\hfill d_L\left({[\psi u_t]}^{\beta },{[\psi v_t]}^{\beta }\right])& =d_L(\left[C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_s+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\right.\hfill \\ \hfill & B{\tilde{G}}^{-1}\left\{u^1-C_q\left(t\right)u_0-K_q\left(t\right)u_1-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_t\right.\hfill \\ \hfill & \left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B\left(x_s\right)C\left(x_s\right)ds\right\}ds,C_q\left(t\right)v_0+K_q\left(t\right)v_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\hfill \\ \hfill & f(s,v_s)d{C}_s+{\int }_0^t{\left(t\right)}^{q-1}{P}_q(t-s)B{\tilde{G}}^{-1}\left\{v^1-C_q\left(t\right)v_0-K_q\left(t\right)v_1\right.\hfill \\ \hfill & \left.\left.-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,v_s)d{C}_t-{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B\left(x_s\right)C\left(x_s\right)ds\right\}ds\right]\hfill \\ \hfill & \le d_L\left({\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_s)d{C}_s\right]}^{\beta },{\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,v_s)d{C}_s\right]}^{\beta }\right)\hfill \\ \hfill & +d_L\left({\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{\tilde{G}}^{-1}\times {\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_t)d{C}_t\left(s\right)ds\right]}^{\beta },\right.\hfill \\ \hfill & \left.{\left[{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{\tilde{G}}^{-1}\times {\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,v_{t}d{C}_t\left(s\right)ds\right]}^{\beta }\right)\hfill \\ \hfill & \le cmK{\int }_0^t{d}_L\left({\left[u_s\right]}^{\beta },{\left[v_s\right]}^{\beta }\right)ds+d_L\left({\left[\tilde{G}{\tilde{G}}^{-1}{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,u_t)d{C}_t\left(s\right)\right]}^{\beta },\right.\hfill \\ \hfill & \left.{\left[\tilde{G}{\tilde{G}}^{-1}{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)f(s,v_t)d{C}_t\left(s\right)\right]}^{\beta }\right)\hfill \\ \hfill & \le cmK{\int }_0^t{d}_L\left({\left[u_s\right]}^{\beta },{\left[v_s\right]}^{\beta }\right)ds+cmK{\int }_0^t{d}_L\left({\left[f(s,u_s)\right]}^{\beta },{\left[f(s,u_s)\right]}^{\beta }\right)ds\hfill \\ \hfill & \le 2cmK{\int }_0^t{d}_L\left({\left[u_s\right]}^{\beta },{\left[v_s\right]}^{\beta }\right)ds.\hfill \end{array}$$

Therefore, by Lemma 1,

$$\begin{array}{cc}\hfill E\left(H_1(\psi u,\psi v)\right)& =E\left(\underset{t\in [0,T]}{sup}{D}_L(\psi u_t,\psi v_t)\right)\hfill \\ \hfill & =E\left(\underset{t\in [0,T]}{sup}\underset{1<\beta \le 2}{sup}{D}_L\left(|\psi u_t{|}^{\beta },{|\psi v_t|}^{\beta }\right)ds\right)\hfill \\ \hfill & \le E\left(\underset{t\in [0,T]}{sup}\underset{1<\beta \le 2}{sup}2cmK{\int }_0^t{D}_L({\left[u_s\right]}^{\beta },{\left[v_s\right]}^{\beta })ds\right)\hfill \\ \hfill & \le E\left(\underset{t\in [0,T]}{sup}2cmK{\int }_0^t{D}_L(u_s,v_s)ds\right)\hfill \\ \hfill & \le 2cmKTF\left(H_1(u,v)\right).\hfill \end{array}$$

Thus, $\left(2cmKT\right)<2$ is a sufficiently small T. As a consequence, $\psi$ represents a contraction mapping. Banach fixed point theorem is now used to prove that Equation (10) has a unique fixed point. As a consequence, (3) can be controlled on $[0,T]$. □

Example 1.

In credibility space, we consider the following Caputo fuzzy fractional differential equations

$$\left\{\begin{array}{c}{D}^{\beta }u(t,\zeta )=Au(t,\zeta )dt+f(t,x(t,\zeta ))d{c}_t+Bx\left(t\right)Cx\left(t\right)dt,\hfill \\ u\left(0\right)=u_0,\hfill \\ u^{\prime }\left(0\right)=u_1\in E_N,\hfill \end{array}\right.$$

(11)

where the state takes values from two bounded spaces $U(\subset E_N)$ and $V(\subset E_N)$. The set of all upper semi-continuously convex fuzzy numbers on R is $E_N$ and the credibility space is $(\Theta ,\mathcal{P},C_r)$.

The state function $u:[0,T]\times (\Theta ,\mathcal{P},C_r)\to U$ is a fuzzy coefficient. $f:[0,T]\times U\to U$ is a fuzzy process. $x:[0,T]\times (\Theta ,\mathcal{P},C_r)\to V$ is a regular fuzzy function, $x:[0,T]\times (\Theta ,\mathcal{P},C_r)\to V$ is a control function, and B is a V to U linear bounded operator. $u_0\in E_N$ is an initial value and $C_t$ is a standard Liu process.

Suppose $f(t,u_t)=\tilde{2}t{u}_t,S^{-1}\left(t\right)=e^{-\tilde{2}t}$, defining $w_t=S^{-1}\left(t\right)u_t.$ Then the equations of balance become

$$\left\{\begin{array}{cc}{u}_t\hfill & =C_q\left(t\right)u_0+K_q\left(t\right)u_1+{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)\tilde{2}t{u}_{t}d{C}_s\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{q-1}{P}_q(t-s)B{x}_{s}C{x}_{s}ds,\hfill \\ u\left(0\right)\hfill & =u_0,\hfill \\ u^{\prime }\left(0\right)\hfill & =u_1\in E_N.\hfill \end{array}\right.$$

(12)

Therefore, Lemma 7 is satisfied.

Since ${\left[2\right]}^{\beta }=[\beta +1,3-\beta ]$ is the β-level set of fuzzy number $\tilde{2}$ for all $\beta \in (1,2)$, the β-level set of $f(t,u_t)$ is

$${\left[f(t,u_t)\right]}^{\beta }=t[(\beta +1){\left(u_t\right)}_m^{\beta },(3-\beta ){\left(u_t\right)}_m^{\beta }].$$

Further, we have

$$\begin{array}{cc}\hfill d_L\left({\left[f(t,u_t)\right]}^{\beta },{\left[f(t,v_t)\right]}^{\beta }\right)& =d_L\left(t[(\beta +1){\left(u_t\right)}_m^{\beta },(\beta +1){\left(u_t\right)}_n^{\beta }],t[(\beta +1){\left(v_t\right)}_m^{\beta },(\beta +1){\left(v_t\right)}_n^{\beta }]\right)\hfill \\ \hfill & =tmax\left\{(\beta +1)|{\left(u_t\right)}_m^{\beta }-{\left(v_t\right)}_m^{\beta }|,(3-\beta )|{\left(u_t\right)}_n^{\beta }-{\left(v_t\right)}_n^{\beta }|\right\}\hfill \\ \hfill & \le 3Tmax\left\{|{\left(u_t\right)}_m^{\beta }-{\left(v_t\right)}_m^{\beta }|,|{\left(u_t\right)}_n^{\beta }-{\left(v_t\right)}_n^{\beta }|\right\}\hfill \\ \hfill & =m{d}_L({\left[u_t\right]}^{\beta },{\left[v_t\right]}^{\beta }),\hfill \end{array}$$

where $m=3T$ satisfies an inequality in the $\left(H_1\right),\left(H_2\right)$ hypotheses. After that, all of the conditions defined in Theorem 1 are satisfied.

Let $\tilde{1}$ be an initial value for $u_0$. $u^1=\tilde{2}$ is the target set. The β-level set of fuzzy number $\tilde{1}$ is $\left[\tilde{1}\right]=[\beta -2,2-\beta ],\beta \in (1,2)$. The β-level set of $x_s$ of (10) is introduced.

$$\begin{array}{cc}\hfill \left[x_s\right]& =[{\left(x_s\right)}_m^{\beta },{\left(x_s\right)}_n^{\beta }]\hfill \\ \hfill & =\left[{\left({\tilde{G}}_m^{\beta }\right)}^{-1}\left\{(\beta +2)-S_m^{\beta }(\beta -2)-{\int }_0^T{S}_m^{\beta }(T-s)s(\beta +1){\left(u_s\right)}_m^{\beta }d{C}_s\right\}\right.,\hfill \\ \hfill & \left.{\left({\tilde{G}}_n^{\beta }\right)}^{-1}\left\{(3-\beta )-S_n^{\beta }(3-\beta )-{\int }_0^T{S}_n^{\beta }(T-s)s(3-\beta ){\left(u_s\right)}_n^{\beta }d{C}_s\right\}\right].\hfill \end{array}$$

The β-level of $u_t$ is then obtained by substituting this expression into (12).

$$\begin{array}{cc}\hfill {\left[u_t\right]}^{\beta }& =\left[S_m^{\beta }\left(T\right)(\beta -2)+{\int }_0^T{S}_m^{\beta }(T-s)s(\beta +2){\left(u_s\right)}_m^{\beta }d{C}_s+{\int }_0^T{S}_m^{\beta }(T-s)B{\left({\tilde{G}}_m^{\beta }\right)}^{-1}\right.\hfill \\ \hfill & \left\{(\beta +2)-S_m^{\beta }\left(T\right)(\beta -2)-{\int }_0^T{S}_m^{\beta }(T-s)s(\beta +2){\left(u_s\right)}_m^{\beta }d{C}_s\right\}ds,\hfill \\ \hfill & S_n^{\beta }\left(T\right)(2-\beta )+{\int }_0^T{S}_n^{\beta }(T-s)s(3-\beta ){\left(u_s\right)}_n^{\beta }d{C}_s+{\int }_0^T{S}_n^{\beta }(T-s)B{\left({\tilde{G}}_n^{\beta }\right)}^{-1}\hfill \\ \hfill & \left.\left\{(3-\beta )-S_r^{\beta }\left(T\right)(2-\beta )-{\int }_0^T{S}_n^{\beta }(T-s)s(3-\beta ){\left(u_s\right)}_n^{\beta }d{C}_s\right\}ds\right]\hfill \\ \hfill & =\left[\right(\beta +2),(3-\beta \left)\right]\hfill \\ \hfill & ={\left[\tilde{2}\right]}^{\beta }.\hfill \end{array}$$

After that, all conditions described in Theorem 2 are satisfied. As a result, (13) can be controllable on $[0,T]$.

Example 2.

Assume the following fuzzy fractional evolution equation in credibility space

$${}_0^C{D}_t^{1.5}u(t,\zeta )=Au(t,\zeta )dt+f(t^3+2t^2+4t)d{c}_t+5Bx\left(t\right)Cx\left(t\right)dt,$$

(13)

with initial conditions $u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1\in E_N,\beta \in 1.5$, where the state takes values from two bounded spaces $U(\subset E_N)$ and $V(\subset E_N)$. The set of all upper semi-continuously convex fuzzy numbers on R is $E_N$ and the credibility space is $(\Theta ,\mathcal{P},C_r)$.

The state function $u:[0,T]\times (\Theta ,\mathcal{P},C_r)\to U$ is a fuzzy coefficient. $f:[0,T]\times U\to U$ is a fuzzy process. $x:[0,T]\times (\Theta ,\mathcal{P},C_r)\to V$ is a regular fuzzy function, $x:[0,T]\times (\Theta ,\mathcal{P},C_r)\to V$ is a control function, and B is a V to U linear bounded operator. $u_0\in E_N$ is an initial value and $C_t$ is a standard Liu process.

Since ${\left[4\right]}^{\beta }=[\beta +3,5-\beta ]$ is the β-level set of fuzzy number $\tilde{2}$ for all $\beta \in (1,2)$, the β-level set of $f(t,u_t)$ is

$${\left[f(t,u_t)\right]}^{\beta }=t[(1.5+3){\left(u_t\right)}_m^{1.5},(5-1.5){\left(u_t\right)}_m^{1.5}].$$

Further, we have

$$\begin{array}{cc}\hfill d_L\left({\left[f(t,u_t)\right]}^{1.5},{\left[f(t,v_t)\right]}^{1.5}\right)& =d_L\left(t[(1.5+3){\left(u_t\right)}_m^{1.5},(1.5+3){\left(u_t\right)}_n^{1.5}],t[(1.5+1){\left(v_t\right)}_m^{1.5},(1.5+1){\left(v_t\right)}_n^{1.5}]\right)\hfill \\ \hfill & =tmax\left\{(1.5+3)|{\left(u_t\right)}_m^{1.5}-{\left(v_t\right)}_m^{1.5}|,(5-1.5)|{\left(u_t\right)}_n^{1.5}-{\left(v_t\right)}_n^{1.5}|\right\}\hfill \\ \hfill & \le 5Tmax\left\{|{\left(u_t\right)}_m^{1.5}-{\left(v_t\right)}_m^{1.5}|,|{\left(u_t\right)}_n^{1.5}-{\left(v_t\right)}_n^{1.5}|\right\}\hfill \\ \hfill & =m{d}_L({\left[u_t\right]}^{1.5},{\left[v_t\right]}^{1.5}),\hfill \end{array}$$

where $m=5T$ satisfies an inequality in the $\left(H_1\right),\left(H_2\right)$ hypotheses.

5. Conclusions

If exact controllability is encouraged for fuzzy fractional evolution equations, it can serve as a benchmark for treating controllability for equations in credibility space, such as fuzzy semi-linear integro-differential equations and fuzzy delay integro-differential equations. As a result, this study’s theoretical result can be used to create stochastic extensions in credibility space. Moreover, future work may include expanding the ideas set out in this work, introducing observability, and generalizing other works. This is a fruitful field with wide research projects, which can lead to countless applications and theories. We plan to allocate notable attention to this direction.