# Leaderless Consensus of Semilinear Hyperbolic Multiagent Systems with Semipositive or Seminegative Definite Convection

Authors: Jiashu Dai; Chengdong Yang; Dong Xu; Shiping Wen; Muwei Jian; Dongliang Yang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper deals with a leaderless consensus of semilinear first-order hyperbolic partial differential equation-based multiagent systems (HPDEMASs). A consensus controller under an undirected graph is designed. Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. Two sufficient conditions for leaderless consensus of HPDEMAS are presented by giving the gain range in the case of the symmetric seminegative definite convection coefficient and the semipositive definite convection coefficient, respectively. Two examples are presented to show the effectiveness of the control methods.


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## Body

## 1. Introduction

As one well-known group of dynamical behavior, multiagent systems (MASs) received many researchers’ attention in the last few decades [1]. There are a number of applications for MASs in engineering fields, for instance, power engineering [2], artificial intelligence [3], energy optimization [4], security [5, 6], traffic decision [7], and precision agriculture [8].Consensus control of MASs is to derive agents to do a designated task synchronously[9, 10]. Many meaningful control methods have been presented, such as event-triggered control [11, 12], containment control [13], pinning control [14], impulsive control [15], sampled-data control [16], and adaptive control [17].To put it another way, the mentioned literature assumed dynamics of MASs depending only on time. In practice, dynamics of all processes depends on both time and space. As a consequence, it is necessary to study spatio-temporal MASs [18]. Qi et al. proposed boundary control for PDE-modeled MASs(PDEMASs) under 3-D space with a control delay [19] and formation control for PDEMASs on a cylindrical surface [20]. An iterative learning algorithm was proposed for the consensus of multiagent system PDEMASs [21]. Yang et al. proposed several control methods for the consensus of semilinear PDEMASs or partial integro-differential equation-based MASs without and with time delays [22–24]. Several iterative learning methods were studied for the consensus of PDEMASs[21, 25–27].Most of the above references are modeled by parabolic PDE-based MASs, while there are few works considering hyperbolic PDE-based MASs (HPDEMASs). The consensus of HPDEMASs is meaningful and significant, as a result of existence of a number of hyperbolic PDE systems in practice [28, 29], including gas dynamics [30], reactor [31], traffic flow [32], and hyperbolic Hopfield neural networks [33]. There are several important studies about consensus of HPDEMASs. For example, Fu et al. proposed the containment control method for the consensus of linear parabolic PDEMASs and second-order HPDEMASs [34]. Wang and Huang proposed the boundary control approach for the finite-time consensus of HPDEMASs by assuming the convection coefficient to be 1 [35]. Zhang et al. proposed boundary control for the leader-following consensus of MASs with input delays by assuming the convection coefficient to be a positive definite diagonal matrix[36]. However, there are still technical difficulties in the consensus of semilinear first-order HPDEMASs for the cases of the convection coefficient to be symmetric seminegative definite or semipositive definite, which are motives of this paper.This paper mainly studies leaderless consensus control of a semilinear HPDEMAS with two sorts of boundary conditions in one-dimensional space. The contribution of this paper contains (1) a class of HPDEMAS models is given, assuming two sorts of conditions, one symmetric seminegative definite convection coefficient and the other semipositive definite convection coefficient. (2) Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. (3) A consensus controller based on communication is studied to drive HPDEMAS to reach leaderless consensus. (4) Dealing with two sorts of convection coefficients, two sufficient conditions for the consensus of the leaderless HPDEMAS are, respectively, reached.Notations: LetI denotes the identity matrix with proper order, λmaxmin⋅ denotes the maximum (minimum) eigenvalue of ⋅, λ2⋅ denotes the smallest nonzero eigenvalue of ⋅, and the superscript T denotes the transpose.


## 2. Problem Formulation

This paper studies a class of semilinear HPDEMASs with time delays(1)∂ziζ,t∂t=Θ∂ziζ,t∂ζ+Aziζ,t+fziζ,t+uiζ,t,where ζ,t∈0,L×0,∞ are space and time, respectively. ziζ,t,uiζ,t∈ℝn are the state and control input, respectively. 0<L∈ℝ, i∈1,2,…,N, A,Θ∈ℝn×n, and f⋅ are a nonlinear function.The boundary condition is(2)zi0,t=0,or(3)ziL,t=0.The initial condition is(4)ziζ,t=zi0ζ,t.This paper aims to study one controlleruiζ,t driving HPDEMAS (1) to the leaderless consensus. Let consensus error eiζ,t≜ziζ,t−1/N∑j=1Nzjζ,t and the controller is designed as follows:(5)uiζ,t=c∑j=1NgijΓzjζ,t−ziζ,t,where c is a control gain to be determined and Γ is symmetric positive definite. Assume that the topological structure G=gijN×N is defined gii=0; gij=gji>0i≠j if the agent i connects to j, otherwise gij=0i≠j.Remark 1.
Compared with papers [35, 36] using the information of only one neighbor, this controller (5) considers the whole communication information among all neighbors and takes full advantage of that.Definition 1.
([35]) HPDEMAS (1) reaches a consensus, if(6)limt⟶∞ziζ,t−z¯ζ,t=0,i∈1,2,…,N,where z¯ζ,t≜1/N∑j=1Nzjζ,t.Lemma 1.
([37]) For the Laplacian matrix ℒ, symmetric positive definite P and y∈ℝNn such that 1NnTy=0, the following inequality is satisfied:(7)λ2ℒyTIN⊗Py≤yTℒ⊗Py.Assumption 1.
([23]) For any ζ1,ζ2∈ℝ, there exists 0<X∈ℝ satisfying(8)fζ1−fζ2≤Xζ1−ζ2.


## 3. Consensus of HPDEMASs with the Seminegative Definite Convection Coefficient

Assumption 2.
AssumeΘ is symmetric seminegative definite.
Note that Assumption2 is sort of classical, which is extensively employed in the practice, see, e.g. [38, 39].
The error system of HPDEMAS (1), (2), and (4) can be obtained as follows(9)∂ϵζ,t∂t=Θ∂eζ,t∂ζ+IN⊗Aeζ,t+Feζ,t−cℒ⊗Γeζ,t,e0,t=0,eζ,0=e0ζ,where ei0ζ≜zi0ζ−1/N∑j=1Nzj0ζ, eζ,t≜e1Tζ,t,e2Tζ,t,…,eNTζ,tT, Feiζ,t≜fziζ,t−1/N∑j=1Nfzjζ,t, Feζ,t≜FTe1ζ,t,FTe2ζ,t, …, FTeNζ,tT, ℒ=D−G, D=diagd1,d2,…,dN, di=∑j=1Ngij, and so ℒ is a Laplace matrix [37].Theorem 1.
Suppose that Assumptions1 and 2 hold. The leaderless HPDEMAS shown in equations (1), (2), and (4) reaches the consensus under the controller (5), if(10)c>λmaxIN⊗A+AT/2+χIλ2ℒλminΓ.Proof.
We choose the Lyapunov functional candidate as shown in the following equation:(11)Vt=0.5∫0LeTζ,teζ,tdζ.
Taking the time derivative ofVt, we obtain(12)V˙t=∫0LeTζ,t∂eζ,t∂tdζ=∫0LeTζ,tIN⊗Θ∂eζ,t∂ζd+∫0LeTζ,tIN⊗A−cℒ⊗Γeζ,tdζζ+∫0LeTζ,tFeζ,tdζ.
Sinceℒ is a Laplace matrix and Γ is a symmetric positive definite matrix, using Lemma 1, one has(13)−c∫0LeTζ,tℒ⊗Γeζ,tdζ⩽−cλ2ℒ∫0LeTζ,tIN⊗Γeζ,tdζ⩽−cλ2ℒλminΓ∫0LeTζ,teζ,tdζ,where 0=λ1ℒ<λ2ℒ≤ℒ≤λNℒ [40].
For symmetric seminegative definiteΘ, employing integrating by parts, one gets(14)∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ=eTζ,tIN⊗Θeζ,tζ=0ζ=L−∫0L∂eTζ,t∂ζIN⊗Θeζ,t=eTL,tIN⊗ΘeL,t−∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ≤−∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ,which implies(15)∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ≤0.
Since∑i=0N∫0LeiTζ,tfy¯ζ,t−1/N∑j=0Nfyjζ,tdx=0, under Assumption 1, we can get(16)∫0LeTζ,tFeζ,tdζ=∑i=0N∫0LeiTζ,tfziζ,t−1N∑j=1Nfzjζ,tdζ=∑i=0N∫0LeiTζ,tfziζ,t−fz¯ζ,tdζ≤χ∫0LeTζ,teζ,tdζ.
Substitution of (12), (14), (15) into (11), we obtain(17)V˙t⩽∫0LeTζ,tΨeζ,tdζ,where Ψ≜IN⊗A+AT/2+χI−cλ2ℒλminΓI.
It is obvious that (9) implies(18)Ψ<0.
Substitution of (17) into (16), we obtain V˙t⩽−λ‖e˜⋅,t‖⩽−λ‖e⋅,t‖ for all nonzero eζ,t, implying consensus of HPDEMAS (1).


## 4. Consensus of HPDEMASs with the Symmetric Semipositive Definite Convection Coefficient

Assumption 3.
AssumeΘ is symmetric semipositive definite.
Note that Assumption3 is sort of classical, which is extensively employed in practice, see, e.g. [35, 36].
The error system of the HPDEMAS (1), (3), and (4) can be obtained as follows:(19)∂ϵζ,t∂t=Θ∂eζ,t∂ζ+IN⊗Aeζ,t+Feζ,t−cℒ⊗Γeζ,t,eL,t=0,eζ,0=e0ζ.Theorem 2.
Suppose that Assumptions1 and 3 hold. The leaderless HPDEMAS shown in equations (1), (3), and (4) reaches the consensus under controller (5) if (9) holds.Proof.
We choose the same Lyapunov functional candidate as in (10). Taking the time derivative of Vt, we obtain (11). For the symmetric semipositive definite Θ>0, employing integrating by parts, one has(20)∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ=eTζ,tIN⊗Θeζ,tζ=0ζ=L−∫0L∂eTζ,t∂ζIN⊗Θeζ,t=−eT0,tIN⊗Θe0,t−∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ≤−∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ,which implies(21)∫0LeTζ,tIN⊗Θ∂eζ,t∂ζdζ≤0.
Substitution of (12), (15), (19) into (11), we obtain (16). It is obvious that (9) implies(22)Ψ<0.
The later part of the proof is similar to that of Theorem2, and so it is omitted.Remark 2.
Different from the control design for consensus of parabolic PDEMASs in [41, 42], this paper deals with the consensus of a class of HPDEMASs.Remark 3.
Consensus of HPDEMASs has been studied by assuming the convection coefficient to be 1 in [35] and to be a positive definite diagonal matrix in [36]. Different from these results, this paper assumes the convection coefficient to be symmetric seminegative and semipositive definite.


## 5. Numerical Simulation

Example 1.
This example considers one HPDEMAS (1) as follows:(23)∂ziζ,t∂t=−0.800−1.6∂ziζ,t∂ζ+52.6−1.23.9ziζ,t+tanhziζ,t+uiζ,t,zi0,t=0,ziζ,t=zi0ζ,t,with random initial conditions. We get the following parameters:(24)Θ=−0.800−1.6,A=52.6−1.23.9,L=1,f⋅=tanh⋅.
The controller (5) is used with the following parameters:(25)Γ=1001,gij=1,fori,j=1,2,3,4andi≠j.
From Figure1, it can be seen that HPDEMAS (1) cannot reach the consensus without control. With Theorem 1, solving (9) by Matlab, c=1.59 is obtained. Figure 2 shows that the HPDEMAS (1) reaches the consensus under controller (5) with c=1.59. Controller (5) with the feedback gain c=1.59 is shown in Figure 3.Figure 1
The open-loop profile ofzζ,t.Figure 2
The closed-loop profile ofzζ,t.Figure 3
The profile ofuζ,t.Example 2.
This example considers one HPDEMAS (1) as follows:(26)∂ziζ,t∂t=0.2000.5∂ziζ,t∂ζ+52.6−1.23.9ziζ,t+tanhziζ,t+uiζ,t,ziL,t=0,ziζ,t=zi0ζ,t,with random initial conditions.
We getΘ=0.2000.5, and the other parameters are the same as (22). The parameters Γ and gij of the controller (5) are the same as (23).
From Figure4, it can be seen that HPDEMAS (1) cannot reach the consensus without control. With Theorem 2, solving (9) by Matlab, c=1.59 is obtained. Figure 5 shows that the HPDEMAS (1) reaches the consensus under controller (5) with c=1.59. The controller (5) with the feedback gain c=1.59 is shown in Figure 6.Figure 4
The open-loop profile ofzζ,t.Figure 5
The closed-loop profile ofzζ,t.Figure 6
The profile ofuζ,t.


## 6. Conclusion

This paper has dealt with leaderless consensus control of a class of semilinear HPDEMASs. One consensus controller of HPDEMASs under the structure of undirected graphs, making use of communication among agents, was established. Firstly, for the case of the symmetric seminegative definite convection coefficient, the boundary condition of the right endpoint was given. For the case of the symmetric semipositive definite convection coefficient, the boundary condition of the left endpoint was given. Two sufficient conditions for the consensus of HPDEMASs were obtained. Two examples illustrated the effectiveness of developed theoretical results. In future work, containment control, event-triggered control, and many other factors will be studied.



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