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

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 of α -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.


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 ock of birds, bees, insects, and sh, 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 ocking birds [1], the migration of honeybees to a new nest site [2], leadership in sh shoals [3], and the e ective 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 di erential 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][14][15][16][17][18][19]. New generalizations and de nitions of the fractional derivative are also of interest by choosing di erent kernels [20]. In particular, it motivates a generalization of the opinion formation model to involve fractional calculus.
Consider the fractional opinion formation model with the initial condition x i (0) ξ i ∈ R for i 0, 1, . . . , N. e coe cient a ij re ects the weight for which agents inuence each other. e fractional derivative of order α is used to describe the memory e ects of the interaction.
It is natural to introduce a leader to the above system in order to drive a consensus among all agents x i . e leader is regarded as a virtual agent whose opinion x 0 is independent of all the remaining agents' opinions. e fractional opinion formation model associated with a leader is described by subject to the initial condition x i (0) � ξ i ∈ R for i � 0, 1, . . . , N. e coefficient c i describes the influence of the leader on another agent. In particular, c i > 0, when the leader has an impact on the i th 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. e 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.
ey also studied the opinion formation model with external inputs h i (t) described by with the initial data x i (0) � ξ i ∈ 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 leaderfollow opinion formation model with x i (0) � ξ i ∈ R for i � 0, 1, . . . , N as well as the model with time-dependent external inputs subject to x i (0) � ξ i ∈ R for i � 0, 1, . . . , N. Here, we include the time-dependence in the source term f i (t, x j (t)). 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 leader x 0 . e rest of this paper is organized as follows. In Section 2, we provide some definitions and preliminary results in integral and differential fractional calculus and exponential stability of systems of fractional differential equations. e 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.

Fractional Calculus Framework
In this section, we briefly outline some notions from fractional calculus that will be used throughout the paper.
e Riemann-Liouville fractional integral of order α > 0 is given by provided the right-side integral exists point wisely, on (0, ∞).
Lemma 1 (see [23]). Let α > 0 be a real number. e general solution of the fractional Caputo type differential equation is a function satisfying for some constants for some constant k, if α ∈ N.
Lemma 2 (see [23]). For a positive constant α > 0 and y ∈ C n [0, T], we have for some constants Consider the differential equation with the Caputo fractional derivative defined by for some initial data satisfies the continuity in t and locally Lipschitz continuity in y.
Definition 3 (see [24]). We say the y * ∈ R m is an equilibrium of system (12) 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 x(t) and y(t) of system (6) subject to the initial conditions x 0 and y 0 , respectively, we have for t ≥ 0 where ‖.‖ denotes the Euclidean norm.
It should be noted that the notion of α-exponential stability concerns the closeness of solutions x(t) and y(t) subject to different initial conditions x 0 and y 0 , respectively, while the definition of the Mittag-Leffler stability in [12] concerns the convergence of a solution x(t) to an equilibrium point.

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) e functions f i , i � 1, 2, . . . , N are Lipschitz continuous with respect to the second variable on R with Lipschitz constants L i > 0, that is, for all x, y ∈ R uniformly with respect to t, And another prerequisite for proving results.
(H2) e constants c i is positive for i � 1, 2, . . . , N and satisfies 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
hence, next, we construct a function W by we compute the Caputo fractional derivative of W(t) using the assumptions (H1) and (H2) to obtain International Journal of Differential Equations where k � min 1≤i≤N k i . Lemma 4 implies that it follows that system (4) is exponentially stable. Next, we assume that the leader's opinion is kept constant as ξ 0 for all time, that is, u(t) � 0 in (4). Let y * � (y 1 , y 2 , . . . , y N ) T ∈ R N be an equilibrium of (4). Hence, it satisfies the following system Proof. Let c i y i � p i for i � 1, 2, . . . , N. We get from (25) that consider the map T: by the Lipschitz condition (H1), for any p, q ∈ R N , we have We have from (H1) that N j�1 |a ij |L j /c i < 1 for i � 1, 2, . . . , N. Hence, us, the map T is a contraction. erefore, T has a unique fixed point, showing that system (4) has a unique equilibrium. e next result shows that all the agents' opinions are in consensus with the leader. Proof. e proof follows the same argument as in [22, eorem 2] where the system is autonomous. We outline the proof when the nonautonomous source term is considered here. Let (x 0 (t), x 1 (t), . . . , x N (t)) be a solution of (4). Since we assume that u(t) � 0, we have that x 0 (t) � ξ 0 is a constant function. Let y i � x i − ξ 0 and define g i (t, y i ): � f i (t, y i + ξ 0 ), for i � 1, . . . , N.
en, system (4) can be written as  International Journal of Differential Equations we see that for any y, y ∈ R, for i � 1, 2, . . . , N. It follows from eorem 2 that system (30) has a unique equilibrium y * � (y 1 , y 2 , . . . y N ). Furthermore, the exponential stability in eorem 1 implies that every solution y of (30) converges to the equilibrium, that is,

Stability and Consensus of Leader-Follower Opinion Formation Model with Timedependent External Inputs
In this section, we extend the study to consider a leaderfollower opinion formation model with time-dependent external inputs. Consider the leader-follower opinion formation model with time-dependent external inputs h i (t), i � 1, 2, . . . , N in (5).
We assume the following conditions for the external inputs.
. , x N (t)) T and y(t) � (y 1 (t), y 2 (t), . . . , y N (t)) T be the solutions of system (5) subject to the different initial conditions x i (0) � ξ i and y i (0) � ζ i , i � 1, 2, . . . , N. By setting w * (t) � y(t) − x(t) and using a similar argument as in the proof of eorem 1, there exists M > 0 such that showing the exponential stability of (5). Next, we construct a function W by (36) Using assumptions (H1) and (H2), we compute the Caputo fractional derivative of W(t) as where k � min 1≤i≤n k i . Consider the fractional-order system It is easily seen that (38) is exponentially stable, so that any solution converges to the unique equilibrium V * � w/k, that is Hence, for any arbitrary ε > 0, there exists T > 0 such that By Lemma 3, we have W(t) ≤ V(t) with W(0) � V(0). us, there exists T ≥ 0 such that for any solution x(t),

International Journal of Differential Equations
Hence, the leader-follower for opinion formation model (48) is exponentially stable. e states of x 0 , x 1 , x 2 , and x 3 of system (48) converges to a consensus state (2, 2, 2, 2) as shown in Figure 2 for α � 0.8.

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 of α− 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. e 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. e results of this paper could be applied to obtain sufficient conditions for the system that ensures consensus with the leader.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  International Journal of Differential Equations