Boundary Feedback Control of a Cascade System of Fractional PDE-ODEs

Abstract

The fractional PDE-ODE coupled system with Neumann boundary condition is considered in this paper. We design a boundary state feedback controller using the Backstepping method to stabilize the considered system. According to operator semigroup theory, we obtain a unique solution of the investigated system. Based on this, by using the fractional Lyapunov scheme, we prove that the system is Mittag-Leffler stable.

1. Introduction

Fractional calculus has attracted extensive attention across the broad of spectrum of areas such as mathematics [1], physics [2], engineering, and biology [3,4,5,6]. In recent years, different from integer derivatives, fractional derivatives can characterize complex dynamic behaviors with memory and heredity by introducing non-local integral kernels [7,8,9,10]. This characteristic enables fractional models to exhibit unique advantages when describing practical phenomena such as viscoelastic materials, anomalous diffusion processes, and signal transmission in biological systems. Particularly, the fractional diffusion equation can describe the anomalous diffusion phenomenon in porous media. Compared with the integer diffusion equation, it can more realistically reflect the transport behavior of particles in complex media. In biological system modeling, fractional differential equations are helpful for characterizing the growth and interaction of biological populations, taking into account the influence of individual biological memory and historical factors in the ecological environment on the current state. In engineering, fractional control theory provides new ideas for designing more efficient and robust control systems, showing potential advantages in the control strategy design and the power system stability analysis.

In recent years, the control research on hierarchical systems formed by coupling integer ordinary (ODEs) and partial differential equations (PDEs) has made significant progress. Many scholars have proposed various methods to study such systems. For instance, the related stability issues of ODE coupled string equation systems and ODE coupled heat equation systems [11,12]; sliding mode stabilization of ODE-PDE (diffusion equation) coupled systems [13]; boundary energy stability problems of PDE-ODE coupled systems [14]; the solution of the exponential stabilization problem for linear ODE coupled with 1Dl linear Korteweg-de Vries Equation [15]; the boundary stabilization problem of heat systems presented by integer delay ODE [16]; etc.

Research on fractional coupled systems mainly focuses on single fractional PDE or ODE coupling models. However, actual complex systems often involve multiple dynamic processes, and it is necessary to consider the coupling of fractional partial differential and fractional ordinary differential processes simultaneously. For example, in a thermal conduction and chemical reaction coupled system, the temperature distribution satisfies a PDE, while the chemical reaction rate follows an ODE, and the two processes interact with each other. When considering the non-local characteristics and memory effects of the system, constructing a fractional PDE-ODE coupled model becomes inevitable.

However, through detailed research, it has been found that the theoretical study of the ODE-PDE coupled system with fractional derivatives has just begun. Only a few scholars have conducted research on fractional ODE-PDE systems, such as in reference [17], where the state feedback control problem of fractional ODE-PDE was studied; in reference [18,19], where the authors studied the coupled system of linear fractional ODE-PDE under Dirichlet boundary conditions (the coupling only occurs between the ordinary differential component system and the boundary conditions) and the output feedback control problem by using the observer; and the authors of [18] pointed out that the coupled system of fractional ODE-PDE has practical physical significance, such as representing a ship moving in a continuous medium. To date, no relevant articles have been reported on the fractional ODE-PDE system under Neumann boundary conditions.

Based on the above discussions, we are interested in the PDE-ODE system with $\alpha$-th Caputo fractional derivative

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hspace{1em}{}_0{}^{C}D_t^{\alpha }X\left(t\right)=AX\left(t\right)+BZ(t,0),\hfill \\ \hspace{1em}{}_0{}^{C}D_t^{\alpha }Z(t,x)=Z_{xx}(t,x)+aZ(t,x)+CX\left(t\right),x\in (0,l),\hfill \\ \hspace{1em}Z(t,l)=Z_c\left(t\right),\hfill \\ \hspace{1em}{Z}_x(t,0)=-qZ(t,0),\hfill \end{array}\right.\end{array}$$

(1)

where $Z_c\left(t\right)$ is the input, the matrix $A\in R^{n\times n},B\in R^{n\times 1},C\in R^{1\times n}$, $a>0$ is a constant, $q\in R$ of the last equation of (1) is also a constant, the initial values are $X\left(0\right)=X_0,Z(0,x)=Z_0\left(x\right),\alpha \in (0,1),X\left(t\right)\in R^n,Z(x,t)\in R,t>0$ are the ODE state and the PDE state, respectively.

The system (1) can be used to describe the dynamic characteristics of thermoelectric cascaded systems [20,21], where the thermal energy system is modeled by a fractional partial differential equation, and the vibration caused by the thermal energy system is modeled by a fractional ordinary differential equation. As is well known, when the coefficient “a” of the PDE subsystem is sufficiently large, the PDE may be unstable [22]. This article aims to establish an appropriate state feedback controller (SFC) to stabilize system (1). The innovation and contribution of this article mainly lie in the following: The system (1) we considered has coupling not only in the fractional ordinary differential equations and at the boundary, but it also contains the states of the fractional ODE system in the fractional PDE subsystem, and its form is more complex and has a wider applicability. This makes the studied partial differential equation subsystem more complex and closer to reality than the coupling only occurs between the ordinary differential component system and the boundary conditions cases in the existing literature [19,20].

In addition, fractional derivatives are non-local operators and do not satisfy semigroup properties. There are many classical methods applicable to integer derivatives, such as the Hille–Yosida theory, spectral analysis, which are difficult to be applied to fractional derivatives. By defining an overall operator and utilizing the properties of its adjoint operator, we established the existence and uniqueness of the solution for system (1). Based on this, we obtained the Mittag-Leffler (ML) stability of the system via the Lyapunov function (LF), the Poincar inequality, and so on.

The paper is arranged as follows: Section 2 presents some basic knowledge. Section 3 introduces a state-feedback controller (SFC) for system (1) based on the Backstepping integral transformation, as well as the existence and uniqueness of the solution. Section 4 demonstrates system (1) is Mittag-Leffler stable through Lyapunov function.

2. Preliminaries

We introduce several important definitions and lemmas of fractional calculus in this section.

Let $\mathrm{\Gamma }$ be the Gamma function, then we have the following definitions:

Definition 1

([23]). Riemann-Liouville fractional integral:

$$\begin{array}{c}\hfill {}_{t_0}D_t^{-q}x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(q\right)}}{\int }_{t_0}^t{(t-s)}^{q-1}x\left(s\right)ds,(n-1<q\le n).\end{array}$$

(2)

Definition 2

([23]). Caputo fractional derivative:

$$\begin{array}{c}\hfill {}_{t_0}{}^{C}D_t^{q}x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-q)}}{\int }_{t_0}^t{\displaystyle \frac{x^{\left(n\right)}\left(s\right)}{{(t-s)}^{q+1-n}}}ds,(n-1<q\le n).\end{array}$$

(3)

Definition 3.

One-parameter Mittag-Leffler function:

$$\begin{array}{c}\hfill E_{\alpha }\left(t\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{t^k}{\mathrm{\Gamma }(k\alpha +1)}},(\alpha >0)\end{array}$$

(4)

Definition 4.

Two-parameter Mittag-Leffler function:

$$\begin{array}{c}\hfill E_{\alpha ,\beta }\left(t\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{t^k}{\mathrm{\Gamma }(k\alpha +\beta )}},(\alpha ,\beta >0)\end{array}$$

(5)

Lemma 1

([24]). Given a differentiable continuous function $u\left(t\right)\in \mathbb{R}$, we readily have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}({}_{t_0}{}^{C}D_t^{\alpha }{u}^2\left(t\right))\le u\left(t\right){}_{t_0}{}^{C}D_t^{\alpha }u\left(t\right),0<\alpha <1, \mathit{for}\mathit{any}\mathit{time}t\ge t_0\ge 0.\end{array}$$

(6)

Lemma 2

([16]). Let $V:[t_0,\infty ]\times {\mathbb{R}}^n\to \mathbb{R}$ be a continuous function, and it satisfies

$$\begin{array}{c}\hfill {}_{t_0}{}^{C}D_t^{\alpha }V(t,X)\le -\gamma V(t,X),\end{array}$$

(7)

where $\alpha \in (0,1],X\left(t\right)\in {\mathbb{R}}^n$ is a vector of differentiable function, and $\gamma >0$, then

$$\begin{array}{c}\hfill V(t,X\left(t\right))\le V(t_0,X\left(t_0\right))E_{\alpha ,1}(-\gamma {(t-t_0)}^{\alpha }),\end{array}$$

(8)

Definition 5

([25]). (Mittag-Leffler stability) For ${}_{t_0}{}^{C}D_t^{\alpha }u\left(t\right)=f(t,u)$, where $\alpha \in (0,1),\gamma \ge 0$, and $t_0$ is the initial value of time, its solution is Mittag-Leffler stable if

$$\begin{array}{c}\hfill \left|\right|u\left(t\right)\left|\right|\le \left(m\left[u\left(t_0\right)\right)\right]E_{\alpha }(-\lambda {(t-t_0)}^{\alpha }){)}^b\end{array}$$

(9)

where $m\left(u\right)$ satisfies the local Lipschitz condition for $u\in \mathbb{B}\subset {\mathbb{R}}^n$, $m\left(u\right)\ge 0$, and $m\left(0\right)=0$ , and $b\ge o$.

3. SFC Design

To stabilize the system (1), by using an invertible integral transform

$$X\left(t\right)=X\left(t\right),$$

(10)

$$V(t,x)=Z(t,x)-{\int }_0^{x}e(x,y)Z(t,y)dy-Q\left(x\right)X\left(t\right),$$

(11)

we convert it into a chosen target system, i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_0{}^{C}D_t^{\alpha }X\left(t\right)=UX\left(t\right)+BV(t,0),\hfill \\ {}_0{}^{C}D_t^{\alpha }V(t,x)=V_{xx}(t,x)-bV(t,x),x\in (0,l),\hfill \\ V_x(t,0)=0,\hfill \\ V(t,l)=0,\hfill \end{array}\right.\end{array}$$

(12)

where the matrix U in (12) is the closed-loop system matrix, defined as $U=A+BK$, the matrix K is the state feedback gain matrix, designed to stabilize the target ODE system, that is, $U=A+BK$ must be Hurwitz (all eigenvalues lie in the left half-plane), $U\in R^{n\times n}$, $A\in R^{n\times n}$, $B\in R^{n\times 1},K\in R^{1\times n}$, $b>0$ is a constant, $Q\left(x\right)\in R^{n\times 1}$(Q is twice differentiable) and $e(x,y)\in R$ are unknown functions needed to be established later. By selecting the gain matrix K and the constant $b>0$(b is a design parameter (scalar constant) chosen for stability of (12)), then the target system (12) is stable under certain conditions. At this point, since the integral transformation (10) and (11) is invertible and the target system is stable, the stabilization of system (1) can be established under the controller

$$\begin{array}{c}\hfill Z_c\left(t\right)={\int }_0^{l}e(l,y)Z(t,y)dy+Q\left(l\right)X\left(t\right).\end{array}$$

(13)

The key to designing the SFC (13) lies in determining the appropriate value of $e(x,y)$ and $Q\left(x\right)$ that meets specific conditions. Next we will elaborate in detail on the transformation process from system (1) to form (12) and at the same time provide the conditions that the functions $e(x,y)$ and $Q\left(x\right)$ must satisfy, thereby designing the controller.

According to (11), we calculated the terms $V_x(t,x)$ and $V_{xx}(t,x)$ by the following:

$$\begin{array}{c}\hfill V_x(t,x)=Z_x(t,x)-e(x,x)Z(t,x)-{\int }_0^x{e}_x(x,y)Z(t,y)dy-Q^{\prime }\left(x\right)X\left(t\right)\end{array}$$

(14)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{xx}(t,x)=& Z_{xx}(t,x)-e(x,x)Z_x(t,x)-({\displaystyle \frac{d\left(e\right(x,x\left)\right)}{dx}}+e_x(x,x))Z(t,x)\hfill \\ \hfill & -{\int }_0^x{e}_{xx}(x,y)Z(t,y)dy-Q^{″}\left(x\right)X\left(t\right).\hfill \end{array}\end{array}$$

(15)

Taking the $\alpha$-th Caputo derivative of (11) follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }V(t,x)=& {}_0{}^{C}D_t^{\alpha }Z(t,x)-{\int }_0^{x}e(x,y){}_0{}^{C}D_t^{\alpha }Z(t,y)dy-Q\left(x\right){}_0{}^{C}D_t^{\alpha }X\left(t\right)\hfill \\ \hfill =& Z_{xx}(t,x)+aZ(t,x)+cX\left(t\right)-{\int }_0^{x}e(x,y)Z_{yy}(t,y)dy-{\int }_0^{x}e(x,y)aZ(t,y)dy\hfill \\ \hfill & -({\int }_0^{x}e(x,y)dyc+Q\left(x\right)A-C)X\left(t\right)-Q\left(x\right)BZ(t,0)-{\int }_0^{x}e(x,y)Z_{yy}(t,y)dy,\hfill \end{array}\end{array}$$

(16)

the matrix C is the coefficient matrix of $X\left(t\right)$ in the second equation of system (1). For the last term of (16), according to the method of partial integration, we have that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -{\int }_0^{x}e(x,y)Z_{yy}(t,y)dy=& -e(x,x)Z_x(t,x)+e(x,0)Z_x(t,0)+e_y(x,x)Z(t,x)\hfill \\ \hfill & -e_y(x,0)Z(t,0)-{\int }_0^{x}Z(t,y)e_{yy}(x,y)dy.\hfill \end{array}\end{array}$$

(17)

According to $Z_x(t,0)=-qZ(t,0)$ and (17), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -{\int }_0^{x}e(x,y)Z_{yy}(t,y)dy=& -e(x,x)Z_x(t,x)-[qe(x,0)+e_y(x,0)]Z(t,0)\hfill \\ \hfill & +e_y(x,x)Z(t,x)-{\int }_0^{x}Z(t,y)e_{yy}(x,y)dy.\hfill \end{array}\end{array}$$

(18)

Thus, we can obtain by (16)–(18) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }V(t,x)=& Z_{xx}(t,x)+[a+e_y\left(xx\right)]Z(t,x)-{\int }_0^x[e_{yy}(x,y)+ae(x,y)]Z(t,y)dy\hfill \\ \hfill & -[Q\left(x\right)B+e_y(x,0)+qe(x,0)]Z(t,0)-e(x,x)Z_x(t,x)\hfill \\ \hfill & -({\int }_0^{x}e(x,y)dyc+Q\left(x\right)A-C)X\left(t\right).\hfill \end{array}\end{array}$$

(19)

Combining (14) and (15) with (19), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }V(t,x)-V_{xx}(t,x)+bV(t,x)=& [{\displaystyle \frac{d\left(e\right(x.x\left)\right)}{dx}}+a+e_y(x,x)+e_x\left(xx\right)]Z(t,x)\hfill \\ \hfill & -{\int }_0^x[e_{yy}(x,y)+ae(x,y)-e_{xx}(x,y)+be(x,y)]Z(t,y)dy\hfill \\ \hfill & -[Q\left(x\right)B+e_y(x,0)+qe(x,0)]Z(t,0)\hfill \\ \hfill & -[{\int }_0^{x}e(x,y)dyc+Q\left(x\right)(A+bI)-C-Q^{″}\left(x\right)]X\left(t\right).\hfill \end{array}\end{array}$$

(20)

By the Backstepping transform (11) and (14), we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V(t,0)=-Q\left(0\right)X\left(t\right)+Z(t,0)\end{array}.\end{array}$$

(21)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V_x(t,0)=Z_x(t,0)-Q^{\prime }\left(0\right)X\left(t\right)-e(0,0)Z(t,0)\end{array}.\end{array}$$

(22)

Let $x=l$ in (11)

$$\begin{array}{c}\hfill V(t,l)=Z(t,l)-{\int }_0^{l}e(l,y)Z(t,y)dy-Q\left(l\right)X\left(t\right)=0,\end{array}$$

(23)

that is, the SFC of the system can be constructed as

$$\begin{array}{c}\hfill Z_c\left(t\right)={\int }_0^{l}e(l,y)Z(t,y)dy+Q\left(l\right)X\left(t\right).\end{array}$$

According to the boundary condition $V_x(t,0)=0$ with $Z_x(t,0)=-qZ(t,0))$, one obtains

$$\begin{array}{c}\hfill \begin{array}{c}\hfill Q^{\prime }\left(0\right)=0ande(0,0)=-q\end{array}.\end{array}$$

(24)

The sufficient condition for systems (1) and (12) to be equivalent is that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Q}^{″}\left(x\right)-Q\left(x\right)(A+bI)-{\int }_0^{x}e(x,y)dyC+C=0,\hfill \\ Q\left(0\right)=K,\hfill \\ Q^{\prime }\left(0\right)=0,\hfill \end{array}\right.\end{array}$$

(25)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{e}_{xx}(x,y)-e_{yy}(x,y)=(a+b)e(x,y),\hfill \\ e(x,x)=-{\displaystyle \frac{a}{2}}x+q(q\ge 0),\hfill \\ e_y(x,0)=-Q\left(x\right)B-qe(x,0),\hfill \end{array}\right.\end{array}$$

(26)

where $C\in R^{1\times n}$, $K\in R^{1\times n}$, $Q\left(x\right)\in R^{1\times n}$, $A\in R^{n\times n}$, $B\in R^{n\times 1}$, I is the n-order identity matrix, $a, b, q$ are constants. According to the form of the controller (13), it remains to determine the solutions of the Equations (25) and (26). Furthermore, by the similar proof of ([26], Theorem 2.2, Section 4) and [27], one can easily obtain the following lemma.

Lemma 3.

The Equations (25) and (26) uniquely determine the solutions, i.e., the functions $e\in C^2\left(\overline{\mathcal{K}}\right)$ and $\mathrm{\Phi }(·)\in C^2[0,l]$, where $\mathcal{K}=\left\{\right(x,y):l>x>y>0\}$ and $\overline{\mathcal{K}}$ represents the closure of the set $\mathcal{K}$.

Using (10) and (11), we have the following inverse transform:

$$X\left(t\right)=X\left(t\right)$$

(27)

$$Z(t,x)=V(t,x)+{\int }_0^{x}j(x,y)V(t,y)dy+R\left(x\right)X\left(t\right)$$

(28)

Combining these terms with the system (1) and (12), we can also obtain the sufficient condition for that system (12) that can be converted into the system (1) by (27) and (28)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{j}_{yy}(x,y)-j_{xx}(x,y)=(a+b)j(x,y),\hfill \\ j_y(x,0)=-R\left(x\right)B,\hfill \\ j(x,x)=-{\displaystyle \frac{1}{2}}(a+b)+q.\hfill \end{array}\right.\end{array}$$

(29)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{R}^{″}\left(x\right)-R\left(x\right)(A+BK-aI)+C=0,\hfill \\ R\left(0\right)=K,\hfill \\ R^{\prime }\left(0\right)=-qK,\hfill \end{array}\right.\end{array}$$

(30)

where $R\left(x\right)\in R^{1\times n}$, $A\in R^{n\times n}$, $B\in R^{n\times 1}$, $K\in R^{1\times n}$, $C\in R^{1\times n}$, $a,q>0$ are constants, and I is an nth-order identity matrix. Similar to Equations (25) and (26), the solution of Equations (29) and (30) exist and are unique.

Lemma 4.

The Equations (29) and (30) uniquely determine the solutions, i.e., the functions $j\in C^2\left(\overline{\mathcal{J}}\right)$ and $R(·)\in C^2[0,l]$, where $\mathcal{J}=\left\{\right(x,y):l>x>y>0\}$ and $\overline{\mathcal{J}}$ represents the closure of the set $\mathcal{J}$.

Remark 1.

From Lemmas 3 and 4, it can be concluded that systems (1) and (12) are equivalent under the action of the state feedback controller (13).

Before we gave the existence and uniqueness of the solution, we introduce the Hilbert space $\mathcal{H}=R^2\times H^1(0,l)$ with norm ${\left|\right|(f,g)\left|\right|}_{\mathcal{H}}^2={\left|f\right|}^2+{\left|\right|g\left|\right|}_2^2+\left|\right|g_x{\left|\right|}_2^2$; here, $H^1(0,l)$ denotes the Sobolev space of functions on domain $(0,l)$ with square-integrable values and first derivatives, equipped with the norm ${\parallel g\parallel }_{H^1(0,1)}^2={\parallel g\parallel }_2^2+{\parallel g_x\parallel }_2^2$.

Theorem 1.

The system (1) with the SFC (13) has a unique solution $(X^T,Z)\in (0,\infty ;\mathcal{H})$ for any initial values $u(t,0)\in H^1(0,l)$ and $X\left(0\right)\in R^n$.

Proof. 

Define a linear operator $G:=D\left(G\right)\subset \mathcal{H}\to \mathcal{H}$ as follows:

$$G\left(\begin{array}{c}X\left(t\right)\\ V(t,·)\end{array}\right)=\left(\begin{array}{c}(A+Bk)X\left(t\right)+BV(t,0)\\ V_{xx}(t,x)-bV(t,x)\end{array}\right)$$

Given that System (1) and System (12) are equivalent, the proof of the existence and uniqueness of System (1) therefore suffices to prove (12) admits a unique solution. In fact,

$$D\left(G\right)=\{{(X^T\left(t\right),V(t,·))}^T\in R^n\times H^1(0,l)|V_x(l,0)=0,V(t,l)=0\}$$

Then, we rewrite system (12) in an abstract form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_0{}^{C}D_t^{\alpha }Y\left(t\right)=GY\left(t\right),\hfill \\ Y\left(0\right)=Y_0,\hfill \end{array}\right.\end{array}$$

where $Y\left(t\right)={(X^T\left(t\right),V(t,·))}^T\in D\left(G\right),Y_0=(X_0^T,V_0\left(x\right))$. Straightforwardly, we calculate the adjoint operators of $G^{*}$ of G by

$$G^{*}\left(\begin{array}{c}{X}^{*}\left(t\right)\\ V^{*}(t,·)\end{array}\right)=\left(\begin{array}{c}{(A+Bk)}^{T}X\left(t\right)\\ V_{xx}^{*}(t,x)-bV(t,x)\end{array}\right)$$

where $D\left(G^{*}\right)=\{{(X^{*}\left(t\right)}^T,V^{*}(t,·))\in R^n\times H^2(0,1)|V_x^{*}(t,0)=-B^T{X}^{*},V^{*}(t,l)=0\}$. According to Propositions 2.8.1 and 2.8.5 of [28], we can get that G gives a $C_0$-semigroup on $\mathcal{H}$. Then, system (12) has a unique solution $(X^T,Z)\in (0,\infty ;\mathcal{H})$ according to Lemma 1.6 of [27], which means that system (1) admits a unique solution $(X^T,V)\in (0,\infty ;\mathcal{H})$. □

4. Mittag-Leffler Stability

We will prove system (1) under the controller (13) we designed is Mittag-Leffler stable in this section.

Theorem 2.

Let system (1) under SFC (13) be a closed-loop system, if there exist $\gamma ,\beta ,b>0$ and a positive definite symmetric matrix P and a matrix $K\in R^{1\times n}$ such that U(defined as $U=A+BK$) is Hurwitz, then, system (1) is Mittag-Leffler stable, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left|\right|X\left(t\right),u(t,·)\left|\right|}^2={\left|X\left(t\right)\right|}^2+{\left|\right|u(t,·)\left)\right||}_{H^1(0,l)}^2.\end{array}\end{array}$$

Proof. 

An LF

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{V}\left(t\right)=X^T\left(t\right)PX\left(t\right)+{\displaystyle \frac{\beta }{2}}{\int }_0^l{V}^2(t,x)dx+{\displaystyle \frac{\gamma }{2}}{\int }_0^l{V}_x^2(t,x)dx\end{array}\end{array}$$

(31)

is used, where $P=P^T$ serves as the weighting matrix for the state vector $X\left(t\right)$, $\beta ,\gamma >0$ are constants. From Lemma 1, taking the $\alpha$-th Caputo derivative of (31) yields

$${}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)\le 2X^T\left(t\right)P{}_0{}^{C}D_t^{\alpha }X\left(t\right)+\beta {\int }_0^{c}V{(t,x)}_0^c{D}_t^{\alpha }V(t,x)dx+\gamma {\int }_0^c({}_0{}^{C}D_t^{\alpha }{V}_x^2(t,x))dx.$$

Based on the system (12) and partial integration, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)& \le 2X^T\left(t\right)P[UX\left(t\right)+BV(t,0)]-\beta {\int }_0^c{V}_x^2(t,x)dx-\beta b{\int }_0^c{V}^2(t,x)dx\hfill \\ \hfill & +\gamma {\int }_0^c{V}_x(t,x){}_0{}^{C}D_t^{\alpha }{V}_x(t,x)dx.\hfill \end{array}\end{array}$$

(32)

To compute the last term of (32),

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \gamma {\int }_0^c{V}_x(t,x){}_0{}^{C}D_t^{\alpha }{V}_x(t,x)dx=-\gamma {\int }_0^c{V}_{xx}^2(t,x)dx-\gamma b{\int }_0^c{V}_x^2(t,x)dx,\end{array}\end{array}$$

(33)

we use the fact $V_x(t,0)={}_0{}^{C}D_t^{\alpha }V(t,0)={}_0{}^{C}D_t^{\alpha }V(t,l)=0$. By (32) and (33), one can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)& \le X^T\left(t\right)[PU+U^{T}P]X\left(t\right)+2X^T\left(t\right)PBV(t,0)\hfill \\ \hfill & -(\beta +\gamma b)\left|\right|V_x{(t,·)\left|\right|}_2^2-{\beta b\left|\right|V(t,·)\left|\right|}_2^2-\gamma \left|\right|V_{xx}(t,·){\left|\right|}_2^2.\hfill \end{array}\end{array}$$

(34)

By Young’s inequality, i.e., $ab\le {\displaystyle \frac{ϵa^2}{z}}+{\displaystyle \frac{b^2}{2ϵ}}$ for any $a, b$, and $ϵ>0$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2X^T\left(t\right)PBV(t,0)& \le ϵX^{\top }X\left(t\right)+{\displaystyle \frac{1}{ϵ}}{\left[PBV(t,0)\right]}^{\top }\left[PBV(t,0)\right]\hfill \\ \hfill & =ϵX^{\top }X\left(t\right)+{\displaystyle \frac{1}{ϵ}}{\left|PB\right|}^2{V}^2(t,0).\hfill \end{array}\end{array}$$

Letting $ϵ={\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2}}$, we can get

$$2X^T\left(t\right)PBV(t,0)\le {\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2}}{X}^T\left(t\right)X\left(t\right)+2{\displaystyle \frac{{\left|PB\right|}^2{V}^2(t,0)}{{\lambda }_{min}\left(Q\right)}},$$

(35)

where $Q=PU+U^{T}P$ for some $Q=Q^T>0$, and ${\lambda }_{min}\left(Q\right)$ is the minimum eigenvalue of Q. We can get by (34) and (35) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)& \le -{\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2}}{\left|X\left(t\right)\right|}^2+2{\displaystyle \frac{{\left|PB\right|}^2{V}^2(t,0)}{{\lambda }_{min}\left(Q\right)}}\hfill \\ \hfill & -(\beta +\gamma b)\left|\right|V_x{(t,·)\left|\right|}_{c^2}^2-{\beta b\left|\right|V(t,·)\left|\right|}_2^2-\gamma \left|\right|V_{xx}(t,·){\left|\right|}_2^2.\hfill \end{array}\end{array}$$

(36)

By using the boundary condition of (12) and the Cauchy–Schwarz inequality, one has

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -\gamma \left|\right|V_{xx}(t,c){\left|\right|}_2^2\le -{\displaystyle \frac{2\gamma }{c^3}}{V}^2(t,0).\end{array}\end{array}$$

(37)

Thus, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)& \le -{\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2}}{\left|X\left(t\right)\right|}^2+(2{\displaystyle \frac{{\left|PB\right|}^2}{{\lambda }_{min}\left(Q\right)}}-{\displaystyle \frac{2\gamma }{c^3}})V^2(t,0)\hfill \\ \hfill & -(\beta +\gamma b)\left|\right|V_x(t,·){\left|\right|}_2^2-{\beta b\left|\right|V(t,·)\left|\right|}_2^2.\hfill \end{array}\end{array}$$

(38)

Taking $\lambda >{\displaystyle \frac{c^3{\left|PB\right|}^2}{{\lambda }_{min}\left(Q\right)}}$ yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }\mathcal{V}\left(t\right)& \le -{\lambda }_1{\left(\right|X\left(t\right)|}^2+{\left|\right|V(t,·)\left|\right|}_2^2+\left|\right|V_{xx}{(t,·)\left|\right|}_2^2)\hfill \\ \hfill & \le {\displaystyle \frac{-{\lambda }_1}{max\{{\lambda }_{max}\left(P\right),\frac{\beta }{2},\frac{\gamma }{2}\}}}{(|X\left(t\right)|}^2+{\left|\right|V(t,·)\left|\right|}_2^2+\left|\right|V_{xx}{(t,·)\left|\right|}_2^2)\hfill \\ \hfill & \le -\lambda \mathcal{V}\left(t\right).\hfill \end{array}\end{array}$$

where ${\lambda }_1=min\{{\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2}},\beta +\gamma b,\beta b\}$, $\lambda ={\displaystyle \frac{{\lambda }_1}{max\{{\lambda }_{max}\left(P\right),\frac{\beta }{2},\frac{\gamma }{2}\}}}$ It yields from Lemma 2 that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{V}\left(t\right)\le \mathcal{V}\left(0\right)E_{\alpha ,1}(-\lambda t^{\alpha })\end{array}\end{array}$$

Thus, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_{min}{\left(P\right)\left|X\left(t\right)\right|}^2+{\displaystyle \frac{\gamma }{2}}\left|\right|V_x(t,.){\left|\right|}_2^2& \le X^{T}PX+{\displaystyle \frac{\gamma }{2}}\left|\right|V_x{(t,.)\left|\right|}_2^2+{\displaystyle \frac{\beta }{2}}{\left|\right|V(t,.)\left|\right|}_2^2\hfill \\ \hfill & \le \mathcal{V}\left(0\right)E_{\alpha }(-\lambda t^{\alpha })\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left|X\left(t\right)\right|}^2+\left|\right|V_x(t,.){\left|\right|}_2^2\le {\displaystyle \frac{1}{min\{\lambda min\left(P\right),\frac{\gamma }{2}\}}}V\left(0\right)E_{\alpha }(-\lambda t^{\alpha })\end{array}\end{array}$$

(39)

Furthermore, it follows by the Poincare inequality and (39) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left|X\left(t\right)\right|}^2+\left|\right|V_x(t,.){\left|\right|}_2^2& \le {\displaystyle \frac{1}{min\{\lambda min\left(P\right),\frac{\gamma }{2}\}}}max\{\lambda max\left(P\right),{\displaystyle \frac{\gamma }{2}}+2\beta c^2\}{\left(\right|X\left(0\right)|}^2\hfill \\ \hfill & +\left|\right|V_x{(t,0)\left|\right|}_2^2)E_{\alpha }(-\lambda t^{\alpha })\hfill \\ \hfill & \le {\lambda }_2{\left(\right|X\left(0\right)|}^2+\left|\right|V_x{(t,0)\left|\right|}_2^2)E_{\alpha }(-\lambda t^{\alpha })\hfill \end{array}\end{array}$$

(40)

where ${\lambda }_2={\displaystyle \frac{max\{\lambda max\left(P\right),\frac{\gamma }{2}+2\beta l^2\}}{min\{\lambda min\left(P\right),\frac{\gamma }{2}\}}}$.

According to the ML definition, system (12) is ML stable, i.e., ${(X\left(t\right){|}^2+{\int }_0^c{V}_x^2(t,x)dx)}^{{\displaystyle \frac{1}{2}}}$. Given that integral transforms are both invertible and bounded, similar to the discussion for Lemma 4.1 of [29], there exist constants $C_1,C_2>0$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left|X\left(t\right)\right|}^2+\left|\right|Z_x(t,.){\left|\right|}_2^2& \le C_1{\left(\right|X\left(t\right)|}^2+\left|\right|V_x{(t,0)\left|\right|}_2^2)\hfill \\ \hfill {\left|X\left(t\right)\right|}^2+\left|\right|V_x(t,.){\left|\right|}_2^2& \le C_2{\left(\right|X\left(t\right)|}^2+\left|\right|Z_x{(t,0)\left|\right|}_2^2)\hfill \end{array}\end{array}$$

Thus, we can rewrite the inequality (40) as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{1}{C_1}}{|(X\left(t\right)|}^2+\left|\right|Z_x{(t,.)\left|\right|}_2^2)& \le \left(\right|X\left(t\right){|}^2+\left|\right|V_x(t,0){\left|\right|}_2^2)\hfill \\ \hfill & \le {\lambda }_2{\left(\right|X\left(0\right)|}^2+\left|\right|V_x{(t,0)\left|\right|}_2^2)E_{\alpha }(-\lambda t^{\alpha })\hfill \\ \hfill & \le {\lambda }_2{C}_2{\left(\right|X\left(0\right)|}^2+\left|\right|Z_x{(t,0)\left|\right|}_2^2)E_{\alpha }(-\lambda t^{\alpha }).\hfill \end{array}\end{array}$$

We find that

$${\left|X\left(t\right)\right|}^2+\left|\right|Z_x(t,.){\left|\right|}_2^2\le {\lambda }_2{C}_1{C}_2{\left(\right|X\left(0\right)|}^2+\left|\right|Z_x{(t,0)\left|\right|}_2^2)E_{\alpha }(-\lambda t^{\alpha }).$$

(41)

Thus, system (1) is ML stable based on (41). □

5. Conclusions

This study investigates a fractional PDE-ODE coupled system under Neumann boundary conditions. We designed a SFC for system (1) utilizing the Backstepping procedure and established the existence and uniqueness of the solution through operator theory. By selecting an appropriate Lyapunov functional and incorporating fractional inequalities, sufficient conditions for achieving an ML stable system (1) under the SFC are derived. Future research will focus on addressing the output feedback stabilization issues of the disturbed systems of fractional PDE-ODEs under Neumann boundary conditions.