# Exponential Stability for an Opinion Formation Model with a Leader Associated with Fractional Differential Equations

Authors: Dussadee Somjaiwang; Parinya Sa Ngiamsunthorn
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

This paper studies the dynamics of an opinion formation model with a leader associated with a system of fractional differential equations. We applied the concept ofα-exponential stability and the uniqueness of equilibrium to show the consensus of the followers with the leader. A sufficient condition for the consensus is obtained for both fractional formation models with and without time-dependent external inputs. Moreover, numerical results are provided to illustrate the dynamical behavior.


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## Body

## 1. Introduction

Humans and some animals are social animals. Living together as a society requires mutual assistance and generosity, which will lead to happiness. Whether it is a small or a large society, social animals, for instance, a flock of birds, bees, insects, and fish, have a complex thought structure and they tend to have a leader to lead their lives to peacefulness. Some research focuses on the behavior of these social animals, including an all-leader agent-based model for turning and flocking birds [1], the migration of honeybees to a new nest site [2], leadership in fish shoals [3], and the effective leadership and decision-making in animal groups on the move [4]. In these systems, individuals are mutually dependent and communicate some information to their peers in a certain region around them. Many researchers have studied and developed a group of opinion formation models, which have been developed into many approaches. Examples of these approaches are opinion formation models on a gradient [5], opinion formation models based on game theory [6], Boltzmann and Fokker–Planck equations for opinion formation models associated with strong leaders [7], opinion formation models based on kinetic equations [8], agent-based models for opinion formation [9], consensus and clustering in opinion formation on networks [10], and Boltzmann-type control of consensus of the opinion with leaders [11].In recent years, fractional differential and integral operators have been utilized in various real-world problems and areas of study, such as physical and biological sciences [12], environmental science, signal and image processing, and engineering [13–19]. New generalizations and definitions of the fractional derivative are also of interest by choosing different kernels [20]. In particular, it motivates a generalization of the opinion formation model to involve fractional calculus.Consider the fractional opinion formation model(1)DC0+αxit=∑j=1Naijfjxjt−fixit,i=1,2,3,…,N,with the initial condition xi0=ξi∈ℝ for i=0,1,…,N. The coefficient aij reflects the weight for which agents influence each other. The fractional derivative of order α is used to describe the memory effects of the interaction.It is natural to introduce a leader to the above system in order to drive a consensus among all agentsxi. The leader is regarded as a virtual agent whose opinion x0 is independent of all the remaining agents’ opinions. The fractional opinion formation model associated with a leader is described by(2)DC0+αx0t=ut,DC0+αxt=∑j=1Naijfjxjt+cix0t−xit,subject to the initial condition xi0=ξi∈ℝ for i=0,1,…,N. The coefficient ci describes the influence of the leader on another agent. In particular, ci>0, when the leader has an impact on the ith agent’s opinion.During the last few years, the opinion formation model with leadership based on the fractional differential system has attracted tremendous attention and has been extensively studied by many researchers. For example, Almeida et al. [21] discussed an optimal control strategy for two types of fractional opinion formation models with leadership to reach a consensus. The result was given by a numerical scheme to approximate the Caputo fractional derivative based on the Grunwald–Letnikov approximation. Next, in 2019, Almeida et al. [22] used Mittag-Leffler stability to discuss sufficient conditions of (2) to guarantee that all agents have a consensus opinion approaching the leader’s opinion. They also studied the opinion formation model with external inputs hit described by(3)DC0+αx0t=ut,DC0+αxit=∑j=1Naijfjxjt+cix0t−xit+hit,with the initial data xi0=ξi∈ℝ for i=0,1,…,N. In the above system, optimal control strategies were designed for the leader to obtain a consensus opinion.Motivated by [21, 22], this paper aims to study the dynamics of a nonautonomous nonlinear fractional leader-follow opinion formation model(4)DC0+αx0t=ut,DC0+αxit=∑j=1Naijfjt,xjt+cix0t−xit,with xi0=ξi∈ℝ for i=0,1,…,N as well as the model with time-dependent external inputs(5)DC0+αx0t=ut,DC0+αxit=∑j=1Naijfjt,xjt+cix0t−xit+hit,,subject to xi0=ξi∈ℝ for i=0,1,…,N. Here, we include the time-dependence in the source term fit,xjt.Rather than focusing on the design of an optimal control, we mainly focus on the exponential stability of the solutions and establish some sufficient conditions for the consensus opinion of all agents with the leaderx0.The rest of this paper is organized as follows. In Section2, we provide some definitions and preliminary results in integral and differential fractional calculus and exponential stability of systems of fractional differential equations. The existence of a unique consensus equilibrium point of fractional opinion formation models associated with a leader will be investigated in Section 3 and Section 4 for the systems described by (4) and (5), respectively. In Section 5, we provide examples and numerical simulations to demonstrate our analytical results.


## 2. Fractional Calculus Framework

In this section, we briefly outline some notions from fractional calculus that will be used throughout the paper.Definition 1 (see [23]).
Forα>0, the left Caputo fractional derivative of order α is given by(6)Dαft=dαdtαft=1Γn−α∫att−τn−α−1Dnfτdτ,where n is a natural number such that α∈n−1,n and D=d/dτ.Definition 2 (see [23]).
The Riemann–Liouville fractional integral of orderα>0 is given by(7)Iαft=1Γα∫0tt−sα−1fsds,provided the right-side integral exists point wisely, on 0,∞.Lemma 1 (see [23]).
Letα>0 be a real number. The general solution of the fractional Caputo type differential equation(8)dαdtαyt=0,is a function satisfying(9)yt=k0+k1t+k2t2+⋯+kn−1tn−1,for some constants ki, i=0,1,…,n−1 where n=α+1 if α∉ℕ,and(10)yt=ktn−1,for some constant k, if α∈ℕ.Lemma 2 (see [23]).
For a positive constantα>0 and y∈Cn0,T, we have(11)Iαdαdtαyt=yt+k0+k1t+⋯+kn−1tn−1,for some constants ki, i=0,1,…,n−1 where n=α+1 Here, Dα=dα/dtα⋅ is the Caputo fractional derivative.Consider the differential equation with the Caputo fractional derivative defined by(12)Dαyt=ft,ytt∈0,∞,y0=y0,for some initial data y0∈ℝm, where y=y1,y2,…,ymT∈ℝm, 0<α<1, and the nonhomogeneous term f:0,∞×ℝm⟶ℝm satisfies the continuity in t and locally Lipschitz continuity in y.Definition 3 (see [24]).
We say they∗∈ℝm is an equilibrium of system (12) whenever ft,y∗=0 for all t∈0,∞.Definition 4 (see [24]).
System (12) is α-exponentially stable if there exist two positive constants M>0 and λ>0 such that for any solutions xt and yt of system (6) subject to the initial conditions x0 and y0, respectively, we have(13)xt−yt≤Mx0−y0exp−λtα,for t≥0 where . denotes the Euclidean norm.
It should be noted that the notion ofα-exponential stability concerns the closeness of solutions xt and yt subject to different initial conditions x0 and y0, respectively, while the definition of the Mittag-Leffler stability in [12] concerns the convergence of a solution xt to an equilibrium point.Lemma 3 (see [17]).
Ifdαx/dtα≤dαy/dtα with α∈0,1 and x0=y0, then xt≤yt.Lemma 4 (see [17]).
LetW∈C0,∞ be a function such that dαy/dtα≤δWt, where α∈0,1 and δ is an arbitrary constant. Then, we have(14)Wt≤W0expδΓα+1tα.
We define the signum function as(15)signx=1x∈0,∞,−1x∈−∞,0..


## 3. Stability and Consensus of Fractional Leader-Follower Opinion Formation Model

In this section, we present a sufficient condition for the consensus opinion in a leader-follower opinion formation model (4) based on the exponential stability concept and the uniqueness of the equilibrium. We first outline the following assumptions for the problem (4)(H1) The functionsfi,i=1,2,…,N are Lipschitz continuous with respect to the second variable on ℝ with Lipschitz constants Li>0, that is,(16)fit,x−fit,y≤Lix−y,for allx,y∈ℝ uniformly with respect to t,And another prerequisite for proving results.(H2) The constantsci is positive for i=1,2,…,N and satisfies(17)ki≔ci−∑j=1NaijLj>0.Under the Lipschitz condition of the source term (H1), we obtain the existence and uniqueness of the solution to the Cauchy problem (4) from the standard result [25].Motivated by [24], we prove the following result.Theorem 1.
Under the assumptions (H1) and (H2), system (4) is exponentially stable.Proof.
Letxt=x1t,x2t,…,xNtT and yt=y1t,y2t,…,yNtT be the solutions of system (4) subject to the different initial conditions xi0=ξi and yi0=ζi, i=1,2,…,N.
Letwi∗t=yit−xit for i=1,2,…,N, then wi∗0≠0. By (4) we get(18)Dαwi∗t=−ciwi∗t+∑j=1Naijfjt,yjt−fjt,xjt.
Ifwi∗t is positive, then(19)Dαwi∗t=1Γ1−α∫t0tt−s−αwi∗′sds=Dαwi∗t,
ifwi∗t is negative, then(20)Dαwi∗t=−1Γ1−α∫t0tt−s−αwi∗′sds=−Dαwi∗t,
hence,(21)Dαwi∗t=signwi∗tDαwi∗t,
next, we construct a functionW by(22)Wt=∑i=1Nwi∗t,
we compute the Caputo fractional derivative ofWt using the assumptions (H1) and (H2) to obtain(23)DαWt=∑i=1NDαwi∗t=∑i=1Nsignwi∗t−ciwi∗t+∑j=1Naijfjt,yjt−fjt,xjt≤∑i=1N−ci|wi∗t|+∑j=1Naijfjt,yjt−fjt,xjt=−∑i=1Nci−∑j=1NaijLjwi∗t≤−kWt,where k=min1≤i≤Nki. Lemma 4 implies that(24)Wt≤W0expkΓα+1tα,
it follows that system (4) is exponentially stable.
Next, we assume that the leader’s opinion is kept constant asξ0 for all time, that is, ut=0 in (4). Let y∗=y1,y2,…,yNT∈ℝN be an equilibrium of (4). Hence, it satisfies the following system(25)−ciyi+∑j=1Naijfjt,yj+ciξ0=0,i=1,2,…,N.Theorem 2.
Assume that (H1) and (H2) hold. Then, system (4) has a unique equilibrium.Proof.
Letciyi=pi for i=1,2,…,N. We get from (25) that(26)pi=∑j=1Naijfjt,pjcj+ciξ0,i=1,2,…,N,
consider the mapT:ℝN⟶ℝN defined by Tp=T1p,T2p,…,TNpT, where p=p1,p2,…,pnT∈ℝN and(27)Tip=∑j=1Naijfjt,pjcj+ciξ0,i=1,2,…,N,
by the Lipschitz condition (H1), for anyp,q∈ℝN, we have(28)|Tip−Tiq|=∑j=1Naijfjt,pjcj−fjt,qjcj≤∑j=1Naijfjt,pjcj−fjt,qjcj≤∑j=1NaijLjcjpj−qj.
We have from (H1) that∑j=1NaijLj/ci<1 for i=1,2,…,N. Hence,(29)Tp−Tq=∑j=1NTip−Tiq≤∑i=1N∑j=1NaijLjcjpj−qj=∑i=1N∑j=1NaijLjcjpj−qj<∑i=1Npi−qi=p−q.
Thus, the mapT is a contraction. Therefore, T has a unique fixed point, showing that system (4) has a unique equilibrium.
The next result shows that all the agents’ opinions are in consensus with the leader.Theorem 3.
Assume that (H1) and (H2) hold. Suppose thatfit,ξ0=0 for t∈0,∞ and for all i=1,2,…,N. Then, all agents’ opinions xit of (4) converge to the consensus opinion ξ0.Proof.
The proof follows the same argument as in [22, Theorem2] where the system is autonomous. We outline the proof when the nonautonomous source term is considered here. Let x0t,x1t,…,xNt be a solution of (4). Since we assume that ut=0, we have that x0t=ξ0 is a constant function. Let yi=xi−ξ0 and define git,yi:=fit,yi+ξ0, for i=1,…,N. Then, system (4) can be written as(30)Dαyit=∑j=1Naijgjt,yjt−ciyit,i=1,2,…,N,
we see that for anyy,y¯∈ℝ,(31)git,y−git,y¯≤fit,y+ξ0−fit,y¯+ξ0≤Liy−y¯,for i=1,2,…,N. It follows from Theorem 2 that system (30) has a unique equilibrium y∗=y1,y2,…yN. Furthermore, the exponential stability in Theorem 1 implies that every solution y of (30) converges to the equilibrium, that is, limt⟶∞yt−y∗=0. Since git,0=fit,ξ0=0 for i=1,2,…,N, we see that y∗=0,…,0T. Consequently, we have limt⟶∞xit=ξ0 for every i=1,…,N showing the consensus of opinion.


## 4. Stability and Consensus of Leader-Follower Opinion Formation Model with Time-dependent External Inputs

In this section, we extend the study to consider a leader-follower opinion formation model with time-dependent external inputs.Consider the leader-follower opinion formation model with time-dependent external inputshit, i=1,2,…,N in (5).We assume the following conditions for the external inputs.(H3) The functionshi,i=1,2,…,N are bounded that is, there exists Oi>0 such that(32)hit≤Oi,for each i=1,2,…,N.Theorem 4.
Assume that (H1), (H2), and (H3) hold. Then, system (5) is exponentially stable. Moreover, for any solution xt, there exists T≥0 such that(33)xt≤wk+ε,t≥T,where k=min1≤i≤Nki(34)w=∑i=1N∑j=1Naijsupt∈0,∞fjt,0+ciξ0+Oi,and ε>0 is arbitrary small.Proof.
Letxt=x1t,x2t,…,xNtT and yt=y1t,y2t,…,yNtT be the solutions of system (5) subject to the different initial conditions xi0=ξi and yi0=ζi, i=1,2,…,N. By setting w∗t=yt−xt and using a similar argument as in the proof of Theorem 1, there exists M>0 such that(35)limt⟶∞yt−xt≤Mζ0−ξ0exp−λtα=0,t≥0,showing the exponential stability of (5). Next, we construct a function W by(36)Wt=xt=∑i=1Nxit.
Using assumptions (H1) and (H2), we compute the Caputo fractional derivative ofWt as(37)DαWt=∑i=1NDαxit=∑i=1Nsignxit−cixit+∑j=1Naijfjt,xjt−fjt,0+∑i=1Nsignxit∑j=1Naijfjt,0+∑i=1Nsignxitciξ0+hit≤∑i=1N−ci|xit|+∑j=1NLjaij‖xjt|+∑j=1N|aij‖fjt,0|+ciξ0+Oi=−∑i=1Nci−∑j=1NaijLj|xit|+∑i=1N∑j=1N|aij‖fjt,0|+ciξ0+Oi≤−kWt+w,where k=min1≤i≤nki.
Consider the fractional-order system(38)DαVt=−kVt+w.
It is easily seen that (38) is exponentially stable, so that any solution converges to the unique equilibrium V∗=w/k, that is(39)limt⟶∞Vt−V∗=limt⟶∞Vt−wk=0.
Hence, for any arbitraryε>0, there exists T>0 such that(40)Vt≤wk+ε,whenevert>T.
By Lemma3, we have Wt≤Vt with W0=V0. Thus, there exists T≥0 such that for any solution xt,(41)xt≤wk+ε,t≥T.
We next state the consensus result for an opinion model with time-dependent external inputs.Theorem 5.
Assume that (H1), (H2), and (H3) hold. Suppose thatfit,ξ0=0 for t∈0,∞ and limt⟶∞hit=0 for all i=1,2,…,N. Then, all agents’ opinions xit of (5) converge to the consensus opinion ξ0.Proof.
Letx0t,x1t,…,xNt be a solution of (5). Since we assume that ut=0, the leader’s opinion is a constant function x0t=ξ0. Let yi=xi−ξ0 and define git,yi:=fit,yi+ξ0, for i=1,…,N. Then, system (5) can be written as(42)Dαyit=∑j=1Naijgjt,yjt−ciyit+hit,
it can be seen thatgi satisfies Lipschitz continuity condition as(43)git,y−git,y¯≤fit,y+ξ0−fit,y¯+ξ0≤Liy−y¯,for any y,y¯∈ℝ and i=1,2,…,N. Hence, by applying Theorem 4 for (42), that is, considering gi instead of fi and taking ξ0=0 in (5), and using the assumption git,0=fit,ξ0=0 for t∈0,∞, we see that the solution yt=y1t,y2t,…,yNt of (42) satisfies(44)yt≤wk+ε,t≥T,for some T>0 where(45)w=∑i=1N∑j=1Naijsupt∈0,∞gjt,0+ci0+Oi=∑i=1NOi,
sincelimt⟶∞hit=0 for i=1,2,…,N, we have hit≤ε whenever t is sufficiently large for all i=1,2,…,N. Hence, if we consider (37) for some sufficiently large time t≥T^, then the bound of hi in (H3) can be chosen as an arbitrary small ε. Consequently, we obtain(46)yt≤Nεk+ε,t≥T^.
Asε>0 was arbitrary, we have yit⟶0 as t⟶∞. Consequently, we obtain the consensus opinion limt⟶∞xit=ξ0 for every i=1,…,N.


## 5. Numerical Examples

In this section, we give examples of the fractional opinion formation model of order0<α<1 to illustrate the dynamics of the system. In particular, we demonstrate through numerical examples that consensus opinion can be achieved when the assumptions of our results are satisfied.Example 1.
Consider the following system of opinion formation model with leader(47)DC0+αx0t=0,DC0+αx1t=a110.5tanhx1t+a12x2x22t+1+a16x6x62t+1+9e−t−0.3x1t+0.3x0t,DC0+αx2t=a210.5tanhx1t+a22x2x22t+1+a240.03tanhx4t+0.09e−t−0.3x2t+0.3x0t,DC0+αx3t=a330.3tanhx3t+a340.3tanhx4t+0.1e−t−0.2x3t+0.2x0t,DC0+αx4t=a440.3tanhx4t+a46x6x62t+1+a450.3tanhx5t−0.4x4t+0.4x0t,DC0+αx5t=a52x2x22t+1+a530.3tanhx3t+a56x6x62t+1+0.02e−t−0.1x5t+0.1x0t,DC0+αx6t=a610.5tanhx1t+a62x2x22t+1+a630.3tanhx3t+a66x6x62t+1−0.4e−t−0.4x6t+0.4x0t,subject to the initial condition x00=0,x10=25,x20=10,x30=30,x40=1.5,x50=−10,x60=−4.5.
In this system of opinion formation model (47) with leader x0t=ξ0=0, we set a11=0.06, a12=−0.01, a16=−0.02, a21=0.03, a22=−0.02, a24=−0.02, a33=−0.01, a34=0.01, a44=0.03, a45=−0.02, a46=0.02, a52=0.05, a53=−0.09, a56=0.08, a61=0.03, a62=0.05, a63=0.09, a66=0.02, c1=0.3, c2=0.3, c3=0.2, c4=0.4, c5=0.1, and c6=0.4, respectively. We see that fit,ξ0=0, limt⟶∞hit=0 and Li=1 for i=1,2,…,6 so that all the assumptions of Theorem 5 are satisfied. Hence, the leader-follower for opinion formation model (47) is exponentially stable. The states of x0,x1,x2,x3,x4,x5 and x6 of system (47) converges to a consensus state 0,0,0,0,0,0,0 as shown in Figure 1 for α=0.5.Figure 1
The states of the system of opinion formation model (47) with α=0.5.Example 2.
Consider the leader-follower opinion formation model(48)DC0+αx0t=0,DC0+αx1t=−0.2x1t+a11ft,x1t+a12ft,x2t+a13ft,x3t+0.2x0t,DC0+αx2t=−0.4x2t+a21ft,x1t+a22ft,x2t+a23ft,x3t+1t2+0.4x0t,DC0+αx3t=−x1t+a31ft,x1t+a32ft,x2t+a33ft,x3t+0.005+x0t,where ft,x=x−2/x2+et subject to the initial condition x00=2,x10=1.5,x20=−5,x30=3.
In this system of opinion formation model (48) with leader x0t=ξ0=2, we set a11=−0.06, a12=0.01, a13=−0.04, a21=0.01, a22=0.01, a23=−0.02, a31=0.06, a32=−0.2, a33=0.01, c1=0.2, c2=0.4, c3=1, respectively. We see that ft,ξ0=0, limt⟶∞hit=0 and Li=1 for i=1,2,3 so that all assumptions of Theorem 5 are satisfied. Hence, the leader-follower for opinion formation model (48) is exponentially stable. The states of x0,x1,x2, and x3 of system (48) converges to a consensus state 2,2,2,2 as shown in Figure 2 for α=0.8.Figure 2
The states of the system of opinion formation model (48) with α=0.8.Example 3.
Consider leader-follower for the opinion formation model(49)DC0+αx0t=0,DC0+αx1t=a110.5x1+1−2x1−4+a120.5x2+1−2x2−4+sintt+0.2x0t−0.2x1tDC0+αx2t=a210.5x1+1−2x1−4+a220.5x2+1−2x2−4−2e−t+0.3x0t−0.3x2t.,Subject to the initial condition x00=5,x10=−1,x20=15. In this system of opinion formation model (49) with leader x0t=ξ0=5, we set a11=−0.06, a12=0.01, a21=−0.03, a22=−0.02, c1=0.2, c2=0.3, respectively. We see that fit,ξ0=0, limt⟶∞hit=0 and Li=1 for i=1,2 so that all assumptions of Theorem 5 are satisfied. Hence, the leader-follower for opinion formation model (49) is exponentially stable. The states of x0,x1, and x2 of system (49) converges to a consensus state 5,5,5 as shown in Figure 3 for. α=0.5Figure 3
The states of the system of opinion formation model (5.3) withα=0.5.


## 6. Discussion and Conclusion

We study the exponential stability of a leader-follower opinion formation model given by the system of nonlinear fractional differential equations. We mainly focus on the sufficient conditions for a consensus opinion of the system when time-dependent inputs are involved. In particular, under the Lipschitz continuity of the source terms, the boundedness condition on the interaction of agents, and the boundedness of external inputs, we apply the concept ofα− exponential stability to prove the consensus of the followers with the leader. Our main theoretical result is illustrated by a numerical model to verify that consensus opinion can be achieved. The result is complementary to the optimal control problem for the leader-follower opinion formation model where external control is added to drive all agents’ opinions to consensus, for example, in [21, 22]. A general framework for α-exponential stability can also be considered for fractional-order neural networks as in [24]. In many real-world phenomena, a group of agents can be described by a network where the information exchange between agents is represented by some weight. The results of this paper could be applied to obtain sufficient conditions for the system that ensures consensus with the leader.



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