Consensus Control of Leaderless and Leader-Following Coupled PDE-ODEs Modeled Multi-Agent Systems

Abstract

This paper discusses consensus control of nonlinear coupled parabolic PDE-ODE-based multi-agent systems (PDE-ODEMASs). First, a consensus controller of leaderless PDE-ODEMASs is designed. Based on a Lyapunov-based approach, coupling strengths are obtained for leaderless PDE-ODEMASs to achieve leaderless consensus. Furthermore, a consensus controller in the leader-following PDE-ODEMAS is designed and the corresponding coupling strengths are obtained to ensure the leader-following consensus. Two examples show the effectiveness of the proposed methods.

1. Introduction

Consensus in multi-agent systems (MASs) is to achieve a common group objective when agents have different initial states [1,2,3,4]. It has received great attention in the past decade as a result from its wide applications in flocking of mobile robots [5], opinion dynamics in social networks [6], formation of unmanned vehicles [7,8,9], microgrid energy management [10], traffic flow [11], etc.

In a pioneering contribution, many important control methods were proposed for consensus of MASs, focusing on models based on ordinary differential equations [12,13,14,15,16,17,18,19,20]. Actually, there are many practical cases in nature and discipline fields with spatio-temporal characteristics, modeled by coupled partial differential equations (PDEs) [21,22,23,24,25]. Applied to overhead cranes [26], hormonal therapy [27], traffic flow [28], etc., another class of spatio-temporal models is based on coupled partial differential equations—ordinary differential equations (PDE-ODEs) [29,30,31]. Therefore, it is important to research consensus control of PDE-based coupled MASs (PDEMASs) or coupled PDE-ODE-based MASs (PDE-ODEMASs).

More recently, there have been many important results related to PDEMASs. Ref. [32] studied a distributed adaptive controller of uncertain leader-following parabolic PDEMASs; ref. [33] studied consensus control for parabolic and second-order hyperbolic PDEMASs; ref. [34] studied distributed P-type iterative learning for PDEMASs with time delay; refs. [35,36] studied iterative learning control for PDEMASs without and with time delay; ref. [37] studied boundary control of 3-D PDEMASs with arbitrarily large boundary input delay; refs. [38,39] studied consensus and input constraint consensus of nonlinear PDEMASs using boundary control. However, consensus control for PDE-ODEMASs has not been addressed yet, which is a new challenge.

Motivated by the above, this paper studies consensus control of nonlinear coupled parabolic PDE-ODEMASs with Neumann boundary conditions. First, dealing with the leaderless case, a consensus controller of leaderless PDE-ODEMASs is designed. The leaderless consensus error system is obtained and one Lyapunov functional candidate is given. Using Wirtinger’s inequality and matrix properties, coupling strengths are obtained for leaderless PDE-ODEMASs to achieve cluster consensus. Furthermore, dealing with the leader-following case, a consensus controller of leader-following PDE-ODEMASs is designed. The leader-following consensus error system is obtained and another Lyapunov functional candidate is given. The corresponding coupling strengths are obtained to ensure leader-following consensus.

The remainder of this paper is organized as follows. The problem formulation is given in Section 2. Section 3 presents a consensus control design of the leaderless PDE-ODEMAS and Section 4 gives that of the leader-following PDE-ODEMAS. An example to illustrate the effectiveness of the proposed method is presented in Section 5 and Section 6 offers some concluding remarks.

Notations:${\lambda }_{max}(·),{\lambda }_2(·)$ stand for the maximum eigenvalue and smallest nonzero eigenvalue of ·, respectively. ⊗ is a Kronecker product of matrices. The identity matrix of n order is denoted by $I_n$. $\left|\right|·\left|\right|$ denotes the Euclidean norm for vectors in ${\mathit{R}}^n$ or the induced 2-norm for matrices in ${\mathit{R}}^{m\times n}$.

2. Problem Formulation

Consider a nonlinear PDE-ODEMAS as

$$\begin{array}{cc}\hfill & {\dot{x}}_i\left(t\right)=f\left(x_i\left(t\right)\right)+{\int }_0^{1}w\left(y_i(\xi ,t)\right)d\xi +u_i\left(t\right),\hfill \\ \hfill & \frac{\partial y_i(\xi ,t)}{\partial t}=\alpha \frac{{\partial }^2{y}_i(\xi ,t)}{\partial {\xi }^2}+p\left(y_i(\xi ,t)\right)\hfill \\ \hfill & +q\left(x_i\left(t\right)\right)+U_i(\xi ,t),\hfill \end{array}$$

(1)

such that

$$\begin{array}{cc}\hfill & {\left.\frac{\partial y_i(\xi ,t)}{\partial \xi }\right|}_{\xi =0}=0,{\left.\frac{\partial y_i(\xi ,t)}{\partial \xi }\right|}_{\xi =1}=0,\hfill \\ \hfill & x_i\left(0\right)=x_i^0,y_i(\xi ,0)=y_i^0\left(\xi \right),\hfill \end{array}$$

(2)

where $(\xi ,t)\in [0,1]\times [0,\infty )$, respectively, mean the spatial variable and time variable; $x_i\left(t\right),y_i(\xi ,t)\in {\mathbb{R}}^n$ are the states; $u_i\left(t\right),U_i(\xi ,t)\in {\mathbb{R}}^n$ are the control inputs; $x_i^0,y_i^0\left(\xi \right)$ are bounded and $y_i^0\left(\xi \right)$ is continuous; $\alpha$ is a positive scalar; $i\in \{1,2,\cdots ,N\}$; and $f(·),w(·),p(·),q(·)\in {\mathbb{R}}^n$ are sufficiently smooth nonlinear functions.

Define consensus error $e_i\left(t\right)\stackrel{\Delta }{=}{x}_i\left(t\right)-\frac{1}{N}{\sum }_{j=1}^N{x}_j\left(t\right)$ and ${\epsilon }_i(\xi ,t)\stackrel{\Delta }{=}{y}_i(\xi ,t)-\frac{1}{N}{\sum }_{j=1}^N{y}_j(\xi ,t)$.

Definition 1.

For the leaderless PDE-ODEMAS (1), (2) with any initial conditions, if

$$\underset{t\to \infty }{lim}{e}_i\left(t\right)\to 0,\underset{t\to \infty }{lim}{\epsilon }_i(\xi ,t)\to 0,$$

(3)

for any $i\in \{1,2,\cdots ,N\}$, then the leaderless PDE-ODEMAS (1), (2) achieves consensus.

Lemma 1

([40]). Let κ be a differentiable function with $\kappa \left(0\right)=0$ and $\kappa \left(1\right)=0$, then

$${\int }_0^1{\kappa }^T\left(s\right)\kappa \left(s\right)ds\le {\pi }^{-2}{\int }_0^1{\dot{\kappa }}^T\left(s\right)\dot{\kappa }\left(s\right)ds.$$

(4)

Lemma 2

([41]). For an undirected connected graph with Laplacian matrix L, and $x\in {\mathbb{R}}^n$ such that $1_N^{T}x=0$, then

$${\lambda }_2\left(L\right)x^{T}x\le x^{T}Lx.$$

(5)

If Laplacian matrix $L\in {\mathbb{R}}^{N\times N}$ is symmetric, then $0={\lambda }_1(·)<{\lambda }_2(·)\le \cdots \le {\lambda }_N(·)$. The smallest nonzero eigenvalue of ${\lambda }_2(·)$ is known as the algebraic connectivity of graphs [41].

Assumption A1.

Assume $f(·),p(·),q(·),w(·)$ satisfy the Lipschitz condition, i.e., for any ${\nu }_1$ and ${\nu }_2\in {\mathbb{R}}^n$, there exist scalars ${\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4>0$ such that

$$\begin{array}{cc}\hfill & |f\left({\nu }_1\right)-f\left({\nu }_2\right)|\le {\gamma }_1\left|{\nu }_1-{\nu }_2\right|,\hfill \\ \hfill & |p\left({\nu }_1\right)-p\left({\nu }_2\right)|\le {\gamma }_2\left|{\nu }_1-{\nu }_2\right|,\hfill \\ \hfill & |q\left({\nu }_1\right)-q\left({\nu }_2\right)|\le {\gamma }_3\left|{\nu }_1-{\nu }_2\right|,\hfill \\ \hfill & |w\left({\nu }_1\right)-w\left({\nu }_2\right)|\le {\gamma }_4\left|{\nu }_1-{\nu }_2\right|.\hfill \end{array}$$

(6)

3. Consensus Control of the Leaderless PDE-ODEMAS

To achieve consensus of the leaderless PDE-ODEMAS (1), the consensus controller is designed as:

$$\begin{array}{cc}\hfill & u_i\left(t\right)=d\sum _{j=1}^N{a}_{ij}(x_j\left(t\right)-x_i\left(t\right)),\hfill \\ \hfill & U_i(\xi ,t)=k\sum _{j=1}^N{b}_{ij}(y_j(\xi ,t)-y_i(\xi ,t)),\hfill \end{array}$$

(7)

where d and k are the coupling strengths to be determined, $i\in \{1,2,\cdots ,N\}$. Assume that the topological structure $A={\left(a_{ij}\right)}_{N\times N}$ is defined as: $a_{ij}=a_{ji}>0(i\ne j)$ if the agent i connects to j, otherwise $a_{ij}=0(i\ne j)$; $a_{ii}=0$. The topological structure $B={\left(b_{ij}\right)}_{N\times N}$ is defined the same as A.

The consensus error system can be obtained from (1), (2), and (7) that

$$\begin{array}{cc}\hfill & {\dot{e}}_i\left(t\right)=f\left(x_i\left(t\right)\right)-\frac{1}{N}\sum _{j=1}^{N}f\left(x_j\left(t\right)\right)+{\int }_0^{1}w\left(y_i(\xi ,t)\right)d\xi -\frac{1}{N}\sum _{j=1}^N{\int }_0^{1}w\left(y_j(\xi ,t)\right)d\xi \hfill \\ \hfill & +d\sum _{j=1}^N{a}_{ij}(x_j\left(t\right)-x_i\left(t\right)),\hfill \\ \hfill & \frac{\partial {\epsilon }_i(\xi ,t)}{\partial t}=\alpha \frac{{\partial }^2{\epsilon }_i(\xi ,t)}{\partial {\xi }^2}+p\left(y_i(\xi ,t)\right)-\frac{1}{N}\sum _{j=1}^{N}p\left(y_j(\xi ,t)\right)+q\left(x_i\left(t\right)\right)\hfill \\ & -\frac{1}{N}\sum _{j=1}^{N}q\left(x_j\left(t\right)\right)+k\sum _{j=1}^N{b}_{ij}(y_j(\xi ,t)-y_i(\xi ,t)),\hfill \end{array}$$

(8)

such that

$$\begin{array}{cc}\hfill & {\left.\frac{\partial {\epsilon }_i(\xi ,t)}{\partial \xi }\right|}_{\xi =0}=0,{\left.\frac{\partial {\epsilon }_i(\xi ,t)}{\partial \xi }\right|}_{\xi =1}=0,\hfill \\ \hfill & e_i\left(0\right)=e_i^0\left(\xi \right),{\epsilon }_i(\xi ,0)={\epsilon }_i^0\left(\xi \right),\hfill \end{array}$$

(9)

where $e_i^0\stackrel{\Delta }{=}{x}_i^0-\frac{1}{N}{\sum }_{j=1}^N{x}_j^0$ and ${\epsilon }_i^0\left(\xi \right)\stackrel{\Delta }{=}{y}_i^0\left(\xi \right)-\frac{1}{N}{\sum }_{j=1}^N{y}_j^0\left(\xi \right)$.

Theorem 1.

Under Assumption 1, assume the graphs A and B are connected. Using the controller (7), the leaderless PDE-ODEMAS (1), (2) achieves consensus if

$$\begin{array}{cc}\hfill & d>\frac{{\gamma }_1+\frac{1}{2}{\gamma }_3^2+\frac{1}{2}}{{\lambda }_2\left(L_a\right)},\hfill \\ & k>max\{\frac{{\gamma }_2+\frac{1}{2}{\gamma }_4^2+\frac{1}{2}-\alpha {\pi }^2}{{\lambda }_2\left(L_b\right)},0\}.\hfill \end{array}$$

(10)

Proof. 

Consider the following Lyapunov function as

$$\begin{array}{cc}\hfill & V_1\left(t\right)=\frac{1}{2}\sum _{i=1}^N{e}_i^T\left(t\right)e_i\left(t\right)+\frac{1}{2}\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t){\epsilon }_i(\xi ,t)d\xi .\hfill \end{array}$$

(11)

We have

$$\begin{array}{cc}\hfill \dot{V_1}\left(t\right)=& \sum _{i=1}^N{e}_i^T\left(t\right){\dot{e}}_i\left(t\right)+\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)\frac{\partial {\epsilon }_i(\xi ,t)}{\partial t}d\xi \hfill \\ \hfill =& \sum _{i=1}^N{e}_i^T\left(t\right)[f\left(x_i\left(t\right)\right)-\frac{1}{N}\sum _{j=1}^{N}f\left(x_j\left(t\right)\right)]\hfill \\ \hfill & +\sum _{i=1}^N{e}_i^T\left(t\right)[{\int }_0^{1}w\left(y_i(\xi ,t)\right)d\xi -\frac{1}{N}\sum _{j=1}^N{\int }_0^{1}w\left(y_j(\xi ,t)\right)d\xi ]\hfill \\ \hfill & +\sum _{i=1}^N{e}_i^T\left(t\right)d\sum _{j=1}^N{a}_{ij}(e_j\left(t\right)-e_i\left(t\right))+\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)\Theta \frac{{\partial }^2{\epsilon }_i(\xi ,t)}{\partial {\xi }^2}d\xi \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)(p\left(y_i(\xi ,t)\right)-\frac{1}{N}\sum _{j=1}^{N}p\left(y_j(\xi ,t)\right))d\xi \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)(q\left(x_i\left(t\right)\right)-\frac{1}{N}\sum _{j=1}^{N}q\left(x_j\left(t\right)\right))d\xi \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)k\sum _{j=1}^N{b}_{ij}({\epsilon }_j(\xi ,t)-{\epsilon }_i(\xi ,t))d\xi .\hfill \end{array}$$

(12)

According to the matrix property,

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{e}_i^T\left(t\right)d\sum _{j=1}^N{a}_{ij}(e_j\left(t\right)-e_i\left(t\right))\hfill \\ \hfill =& -d{e}^T\left(t\right)(L_a\otimes I_n)e\left(t\right)\hfill \\ \hfill \le & -d{\lambda }_2\left(L_a\right)e^T\left(t\right)e\left(t\right),\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)k\sum _{j=1}^N{b}_{ij}({\epsilon }_i(\xi ,t)-{\epsilon }_j(\xi ,t))d\xi \hfill \\ \hfill =& -k{\int }_0^1{\epsilon }^T(\xi ,t)(L_b\otimes I_n)\epsilon (\xi ,t)d\xi \hfill \\ \hfill \le & -k{\lambda }_2\left(L_b\right){\int }_0^1{\epsilon }^T(\xi ,t)\epsilon (\xi ,t)d\xi ,\hfill \end{array}$$

(14)

where ${\lambda }_2(·)$ denotes the smallest nonzero eigenvalue of ·, $L_{a,ij}=-a_{ij}$ when $i\ne j$, $L_{a,ii}={\displaystyle \sum _{j=1}^N}{a}_{ij}$, $L_{b,ij}=-b_{ij}$ when $i\ne j$, $L_{b,ii}={\displaystyle \sum _{j=1}^N}{b}_{ij}$. Therefore, $L_a$, $L_b$ are Laplacian matrices. Using Lemma 1, for $\alpha >0$,

$$\begin{array}{cc}\hfill & {\int }_0^1\sum _{i=1}^N{\epsilon }_i^T(\xi ,t)\alpha {\epsilon }_{i,\xi \xi }(\xi ,t)d\xi \hfill \\ \hfill =& -\alpha {\int }_0^1{\epsilon }_{\xi }^T(\xi ,t){\epsilon }_{\xi }(\xi ,t)d\xi \hfill \\ \hfill \le & -\alpha {\pi }^2{\int }_0^1{\epsilon }^T(\xi ,t)\epsilon (\xi ,t)d\xi .\hfill \end{array}$$

(15)

Using Assumption 1, owing to ${\displaystyle \sum _{i=1}^N}{e}_i^T\left(t\right)(f\left(\frac{1}{N}{\sum }_{j=1}^N{x}_j\left(t\right)\right)-\frac{1}{N}{\sum }_{j=1}^{N}f\left(x_j\left(t\right)\right))=0$ and ${\displaystyle \sum _{i=1}^N}{\epsilon }_i^T(\xi ,t)(f\left(\frac{1}{N}{\sum }_{j=1}^{N}p\left(y_i(\xi ,t)\right)\right)-\frac{1}{N}{\sum }_{j=1}^{N}p\left(y_i(\xi ,t)\right)=0$, we have

$$\begin{array}{c}\hfill \sum _{i=1}^N{e}_i^T\left(t\right)(f\left(x_i\left(t\right)\right)-\frac{1}{N}\sum _{j=1}^{N}f\left(x_j\left(t\right)\right))\le {\gamma }_1\sum _{i=1}^N{e}_i^2\left(t\right),\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)(p\left(y_i(\xi ,t)\right)-\frac{1}{N}\sum _{j=1}^{N}p\left(y_j(\xi ,t)\right))d\xi \le {\gamma }_2{\int }_0^1{\epsilon }_i^2(\xi ,t)d\xi .\end{array}$$

(17)

In the same way,

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)(q\left(x_i\left(t\right)\right)-\frac{1}{N}\sum _{j=1}^{N}q\left(x_j\left(t\right)\right))d\xi \hfill \\ \hfill =& \sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t)(q\left(x_i\left(t\right)\right)-q\left(\frac{1}{N}\sum _{j=1}^N{x}_j\left(t\right)\right))d\xi \hfill \\ \hfill \le & \frac{1}{2}\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t){\epsilon }_i(\xi ,t)d\xi +\frac{1}{2}\sum _{i=1}^N{\int }_0^1{(q\left(x_i\left(t\right)\right)-q\left(\frac{1}{N}\sum _{j=1}^N{x}_j\left(t\right)\right))}^{2}d\xi \hfill \\ \hfill \le & \frac{1}{2}\sum _{i=1}^N{\int }_0^1{\epsilon }_i^T(\xi ,t){\epsilon }_i(\xi ,t)d\xi +\frac{1}{2}{\gamma }_3^2\sum _{i=1}^N{e}_i^2\left(t\right),\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{e}_i^T\left(t\right){\int }_0^1(w\left(y_i(\xi ,t)\right)-\frac{1}{N}\sum _{j=1}^{N}w\left(y_j(\xi ,t)\right))d\xi \hfill \\ \hfill =& \sum _{i=1}^N{e}_i^T\left(t\right){\int }_0^1(w\left(y_i(\xi ,t)\right)-w\left(\frac{1}{N}\sum _{j=1}^N{y}_j(\xi ,t)\right))d\xi \hfill \\ \hfill \le & \frac{1}{2}\sum _{i=1}^N[e_i^T\left(t\right)e_i\left(t\right)d\xi +{\int }_0^1{(w\left(y_i(\xi ,t)\right)-w\left(\frac{1}{N}\sum _{j=1}^N{y}_j(\xi ,t)\right))}^{2}d\xi ]\hfill \\ \hfill \le & \frac{1}{2}\sum _{i=1}^N{e}_i^T\left(t\right)e_i\left(t\right)+\frac{1}{2}\sum _{i=1}^N{\gamma }_4^2{\int }_0^1{\epsilon }_i^2(\xi ,t)d\xi .\hfill \end{array}$$

(19)

Substituting (13)–(19) into (12),

$$\begin{array}{cc}\hfill \dot{V_1}\left(t\right)\le & -{\rho }_1{e}^T\left(t\right)e\left(t\right)-{\rho }_2{\int }_0^1{\epsilon }^2(\xi ,t)d\xi \hfill \\ \hfill \le & -2\rho V\left(t\right),\hfill \end{array}$$

(20)

where $e\stackrel{\Delta }{=}{[e_1^T,e_2^T,\cdots ,e_N^T]}^T$, $\epsilon \stackrel{\Delta }{=}{[{\epsilon }_1^T,{\epsilon }_2^T,\cdots ,{\epsilon }_N^T]}^T$, ${\rho }_1\stackrel{\Delta }{=}-{\gamma }_1-\frac{1}{2}{\gamma }_3^2-\frac{1}{2}+d{\lambda }_2\left(L_a\right)$, ${\rho }_2\stackrel{\Delta }{=}-{\gamma }_2-\frac{1}{2}{\gamma }_4^2-\frac{1}{2}+\alpha {\pi }^2+k{\lambda }_2\left(L_b\right)$, and $\rho \stackrel{\Delta }{=}min\{{\rho }_1,{\rho }_2\}$.

Taking d and k as (10) yields,

$$\begin{array}{c}\hfill {\rho }_1>0,{\rho }_2>0.\end{array}$$

(21)

It follows from (20) and (21) that $V_1\left(t\right)\le V_1\left(0\right)exp\{-2\rho t\}$, which implies $e_i\left(t\right)\to 0$ and ${\epsilon }_i(\xi ,t)\to 0$ as $t\to \infty$ This completes the proof. □

4. Consensus Control of the Leader-Following PDE-ODEMAS

The leader agent is supposed to be

$$\begin{array}{cc}\hfill & {\dot{x}}_0\left(t\right)=f\left(x_0\left(t\right)\right)+{\int }_0^{1}w\left(y_0(\xi ,t)\right)d\xi ,\hfill \\ \hfill & \frac{\partial y_0(\xi ,t)}{\partial t}=\alpha \frac{{\partial }^2{y}_0(\xi ,t)}{\partial {\xi }^2}+p\left(y_0(\xi ,t)\right)+q\left(x_0\left(t\right)\right),\hfill \end{array}$$

(22)

such that

$$\begin{array}{cc}\hfill & {\left.\frac{\partial y_0(\xi ,t)}{\partial \xi }\right|}_{\xi =0}=0,{\left.\frac{\partial y_0(\xi ,t)}{\partial \xi }\right|}_{\xi =1}=0,\hfill \\ \hfill & x_0\left(0\right)=x_0^0,y_0(\xi ,0)=y_0^0\left(\xi \right),\hfill \end{array}$$

(23)

where $x_0^0,y_0^0\left(\xi \right)$ are bounded and $y_0^0\left(\xi \right)$ is continuous.

The leader-following consensus controller is designed as:

$$\begin{array}{cc}\hfill & u_i\left(t\right)=d[\sum _{j=1}^N{a}_{ij}(x_j\left(t\right)-x_i\left(t\right))+{\delta }_i(x_0\left(t\right)-x_i\left(t\right))],\hfill \\ \hfill & U_i(\xi ,t)=k[\sum _{j=1}^N{b}_{ij}(y_j(\xi ,t)-y_i(\xi ,t))+{\rho }_i(y_0(\xi ,t)-y_i(\xi ,t))],\hfill \end{array}$$

(24)

where ${\delta }_i>0$ if $x_i$ can obtain the information of $x_0$; otherwise, ${\delta }_i=0$; and ${\rho }_i>0$ if $y_i$ can obtain the information of $y_0$; otherwise, ${\rho }_i=0$.

Let ${\tilde{e}}_i\left(t\right)=x_i\left(t\right)-x_0\left(t\right)$ and ${\tilde{\epsilon }}_i(\xi ,t)=y_i(\xi ,t)-y_0(\xi ,t)$. The leader-following consensus error system is obtained as

$$\begin{array}{cc}\hfill & {\dot{\tilde{e}}}_i\left(t\right)=f\left(x_i\left(t\right)\right)-f\left(x_0\left(t\right)\right)+{\int }_0^{1}w\left(y_i(\xi ,t)\right)d\xi \hfill \\ \hfill & -{\int }_0^{1}w\left(y_0(\xi ,t)\right)d\xi -d\sum _{j=1}^N{g}_{ij}{\tilde{e}}_j\left(t\right),\hfill \\ \hfill & \frac{\partial {\tilde{\epsilon }}_i(\xi ,t)}{\partial t}=\alpha \frac{{\partial }^2{\tilde{\epsilon }}_i(\xi ,t)}{\partial {\xi }^2}+p\left(y_i(\xi ,t)\right)-p\left(y_0(\xi ,t)\right)\hfill \\ & +q\left(x_i\left(t\right)\right)-q\left(x_0\left(t\right)\right)\hfill \\ \hfill & -k\sum _{j=1}^N{h}_{ij}{\tilde{\epsilon }}_j(\xi ,t),\hfill \end{array}$$

(25)

such that

$$\begin{array}{cc}\hfill & {\left.\frac{\partial {\tilde{\epsilon }}_i(\xi ,t)}{\partial \xi }\right|}_{\xi =0}=0,{\left.\frac{\partial {\tilde{\epsilon }}_i(\xi ,t)}{\partial \xi }\right|}_{\xi =1}=0,\hfill \\ \hfill & {\tilde{e}}_i\left(0\right)={\tilde{e}}_i^0\left(\xi \right),{\tilde{\epsilon }}_i(\xi ,0)={\tilde{\epsilon }}_i^0\left(\xi \right),\hfill \end{array}$$

(26)

where $G=\left[g_{ij}\right]=L_A+diag\left\{{\delta }_i\right\},H=\left[h_{ij}\right]=L_B+diag\left\{{\rho }_i\right\}$, ${\tilde{e}}_i^0\stackrel{\Delta }{=}{x}_i^0-x_0^0$ and ${\tilde{\epsilon }}_i^0\left(\xi \right)\stackrel{\Delta }{=}{y}_i^0\left(\xi \right)-y_0^0\left(\xi \right)$.

Definition 2.

For the leader-following PDE-ODEMAS (22), (23) with any initial conditions, if

$$\underset{t\to \infty }{lim}{\tilde{e}}_i\left(t\right)\to 0,\underset{t\to \infty }{lim}\left|\right|{\tilde{\epsilon }}_i(\xi ,t)\left|\right|\to 0,$$

(27)

for any $i\in \{1,2,\cdots ,N\}$, then the leader-following PDE-ODEMAS (22), (23) achieves consensus.

Theorem 2.

Under Assumption 1, assume the graphs A and B are connected. Using the controller (26), the leader-following PDE-ODEMAS (1) achieves consensus if

$$\begin{array}{cc}\hfill & d>\frac{{\gamma }_1+\frac{1}{2}{\gamma }_3^2+\frac{1}{2}}{{\lambda }_{min}\left(G\right)},\hfill \\ & k>max\{\frac{{\gamma }_2+\frac{1}{2}{\gamma }_4^2+\frac{1}{2}-\alpha {\pi }^2}{{\lambda }_{min}\left(H\right)},0\}.\hfill \end{array}$$

(28)

Proof. 

Consider the Lyapunov functional candidate as

$$\begin{array}{cc}\hfill V_2\left(t\right)=\frac{1}{2}& \sum _{i=1}^N{\tilde{e}}_i^T\left(t\right){\tilde{e}}_i\left(t\right)+\frac{1}{2}\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t){\tilde{\epsilon }}_i(\xi ,t)d\xi .\hfill \end{array}$$

(29)

One has

$$\begin{array}{cc}\hfill \dot{V_2}\left(t\right)=& \sum _{i=1}^N{\tilde{e}}_i^T\left(t\right){\dot{\tilde{e}}}_i\left(t\right)+\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)\frac{\partial {\tilde{\epsilon }}_i(\xi ,t)}{\partial t}d\xi \hfill \\ \hfill =& \sum _{i=1}^N{\tilde{e}}_i^T\left(t\right)(f\left(x_i\left(t\right)\right)-f\left(x_0\left(t\right)\right))\hfill \\ \hfill & +\sum _{i=1}^N{\tilde{e}}_i^T\left(t\right)[{\int }_0^{1}w\left(y_i(\xi ,t)\right)d\xi -{\int }_0^{1}w\left(y_0(\xi ,t)\right)d\xi ]\hfill \\ \hfill & -\sum _{i=1}^N{\tilde{e}}_i^T\left(t\right)d\sum _{j=1}^N{g}_{ij}{\tilde{e}}_j\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)\alpha \frac{{\partial }^2{\tilde{\epsilon }}_i(\xi ,t)}{\partial {\xi }^2}d\xi \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)(p\left(y_i(\xi ,t)\right)-p\left(y_0(\xi ,t)\right))d\xi \hfill \\ \hfill & +\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)(q\left(x_i\left(t\right)\right)-q\left(x_0\left(t\right)\right))d\xi \hfill \\ \hfill & -\sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)k\sum _{j=1}^N{h}_{ij}{\tilde{\epsilon }}_j(\xi ,t)d\xi .\hfill \end{array}$$

(30)

Since G and H are symmetric positive definite matrices,

$$\begin{array}{cc}\hfill & -\sum _{i=1}^N{\tilde{e}}_i^T\left(t\right)d\sum _{j=1}^N{g}_{ij}({\tilde{e}}_j\left(t\right))\hfill \\ \hfill =& -d{\tilde{e}}^T\left(t\right)(G\otimes I_n)\tilde{e}\left(t\right)\hfill \\ \hfill \le & -d{\lambda }_{min}\left(G\right){\tilde{e}}^T\left(t\right)\tilde{e}\left(t\right),\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{\int }_0^1{\tilde{\epsilon }}_i^T(\xi ,t)k\sum _{j=1}^N{h}_{ij}{\tilde{\epsilon }}_j(\xi ,t)d\xi \hfill \\ \hfill =& -k{\int }_0^1{\tilde{\epsilon }}^T(\xi ,t)(H\otimes I_n)\tilde{\epsilon }(\xi ,t)d\xi \hfill \\ \hfill \le & -k{\lambda }_{min}\left(H\right){\int }_0^1{\tilde{\epsilon }}^T(\xi ,t)\tilde{\epsilon }(\xi ,t)d\xi ,\hfill \end{array}$$

(32)

where $\tilde{e}\stackrel{\Delta }{=}{[{\tilde{e}}_1^T,{\tilde{e}}_2^T,\cdots ,{\tilde{e}}_N^T]}^T$, $\tilde{\epsilon }\stackrel{\Delta }{=}{[{\tilde{\epsilon }}_1^T,{\tilde{\epsilon }}_2^T,\cdots ,{\tilde{\epsilon }}_N^T]}^T$, ${\lambda }_{min}(·)$ denotes the smallest nonzero eigenvalue and G, H are symmetric positive definite matrices.

Considering (13)–(19), and substituting (31)–(32) into (30),

$$\begin{array}{cc}\hfill \dot{V_2}\left(t\right)\le & ({\gamma }_1+\frac{1}{2}{\gamma }_3^2+\frac{1}{2}-d{\lambda }_{min}\left(G\right)){\tilde{e}}^T\left(t\right)\tilde{e}\left(t\right)+({\gamma }_2+\frac{1}{2}{\gamma }_4^2\hfill \\ \hfill & +\frac{1}{2}-\alpha {\pi }^2-k{\lambda }_{min}\left(H\right)){\int }_0^1{\tilde{\epsilon }}^T(\xi ,t)\tilde{\epsilon }(\xi ,t)d\xi .\hfill \end{array}$$

(33)

In a similar way to the analysis in Theorem 1, the proof can be completed. □

Remark 1.

Many papers have investigated stabilization control methods for PDE-ODE systems [29,30,31,42], while this paper investigates consensus control for PDE-ODE-based MASs, considering control based on coupling.

Remark 2.

Many significant results were obtained for consensus control modeled by PDEMASs [32,33,34,35,36,37,38,39]. Different from PDEMASs, this paper investigates consensus control methods for PDE-ODEMASs, as well as considering leaderless and leader-following models.

5. Numerical Simulation

Example 1.

Consider the leaderless PDE-ODEMAS (1) and (2) with coefficients as

$$\begin{array}{cc}\hfill & \alpha =0.8,f(·)=w(·)=p(·)=q(·)=tanh(·),\hfill \\ \hfill & a_{ij}=b_{ij}=1,\mathrm{and}i\ne j,\mathrm{for}i,j=1,2,3,4,\hfill \\ & n=2,\hfill \end{array}$$

(34)

and with random initial conditions.

It is obvious that $f(·),p(·),q(·)$, and $w(·)$ satisfy the Lipschitz condition with ${\gamma }_1={\gamma }_2={\gamma }_3={\gamma }_4=1$.

With Theorem 1, according to (10), $d>0.50$ and $k>0$ are obtained. Therefore, we take $d=0.51$ and $k=0.01$. It can be seen in Figure 1 and Figure 2 that the leaderless PDE-ODEMAS achieves consensus with control gains $d=0.51$ and $k=0.01$.

Figure 1. $e_i\left(t\right)$ withthe control gains $d=0.51$ and $k=0.01$ in Example 1.

Figure 2. ${\epsilon }_i(\xi ,t)$ with the control gains $d=0.51$ and $k=0.01$ in Example 1.

From another point of view, $d=0.49$ and $k=0$ do not satisfy (10). It can be seen in Figure 3 and Figure 4 that the leaderless PDE-ODEMAS cannot achieve consensus with control gains $d=0.49$ and $k=0$.

Figure 3. $e_i\left(t\right)$ with the control gains $d=0.49$ and $k=0$ in Example 1.

Figure 4. ${\epsilon }_i(\xi ,t)$ with the control gains $d=0.49$ and $k=0$ in Example 1.

Example 2.

Consider a nonlinear leader-following PDE-ODEMAS composed of 1 leader agent (22) and (23) and 4 following agents (1) and (2) with coefficients the same as Example 1. In the same way, ${\gamma }_1={\gamma }_2={\gamma }_3={\gamma }_4=1$ are obtained. Choose ${\delta }_i={\rho }_i=1$. With Theorem 2, according to (28), $d>2.0$ and $k>0$ are obtained. Therefore, we take $d=2.1$ and $k=0.1$. It can be seen in Figure 5 and Figure 6 that the leader-following PDE-ODEMAS achieves consensus.

Figure 5. ${\tilde{e}}_i\left(t\right)$ with the control gains $d=2.1$ and $k=0.1$ in Example 2.

Figure 6. ${\tilde{\epsilon }}_i(\xi ,t)$ with the control gains $d=2.1$ and $k=0.1$ in Example 2.

From another point of view, $d=1.9$ and $k=0$ do not satisfy (28). It can be seen in Figure 7 and Figure 8 that the leader-following PDE-ODEMAS cannot achieve consensus with control gains $d=0.49$ and $k=0$.

Figure 7. ${\tilde{e}}_i\left(t\right)$ with the control gains $d=1.9$ and $k=0$ in Example 2.

Figure 8. ${\tilde{\epsilon }}_i(\xi ,t)$ with control gains $d=1.9$ and $k=0$ in Example 2.

6. Conclusions

This paper has studied consensus control of the PDE-ODEMASs. First, a consensus controller of the leaderless PDE-ODEMASs was designed. We have shown that the cluster consensus behavior can be reached for the given coupling strengths for the leaderless PDE-ODEMASs. Then, a consensus controller in the leader-following PDE-ODEMASs was designed. Leader-following consensus behavior can be arrived at for the given coupling strengths for the leader-following PDE-ODEMASs. In numerical simulations, it shows the obtained gains according to the proposed methods can ensure consensus of both leaderless and leader-following PDE-ODEMASs. On the contrary, the control with gains a little bit less than those according to the proposed methods cannot achieve consensus. There are often a great number of agents in the real world and, in future, pinning consensus, only controlling a few agents of the PDE-ODEMASs, will be studied, as well as time delays.