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 of -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.
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 modelwith the initial condition for . The coefficient reflects the weight for which agents influence each other. The fractional derivative of order 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 agents . The leader is regarded as a virtual agent whose opinion is independent of all the remaining agents’ opinions. The fractional opinion formation model associated with a leader is described bysubject to the initial condition for . The coefficient describes the influence of the leader on another agent. In particular, , when the leader has an impact on the 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 described bywith the initial data for . 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 modelwith for as well as the model with time-dependent external inputssubject to for . Here, we include the time-dependence in the source term .
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 .
The 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. 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]). For , the left Caputo fractional derivative of order is given bywhere is a natural number such that and .
Definition 2 (see [23]). The Riemann–Liouville fractional integral of order is given byprovided the right-side integral exists point wisely, on .
Lemma 1 (see [23]). Let be a real number. The general solution of the fractional Caputo type differential equationis a function satisfyingfor some constants , where if ,andfor some constant , if .
Lemma 2 (see [23]). For a positive constant and , we havefor some constants , where Here, is the Caputo fractional derivative.
Consider the differential equation with the Caputo fractional derivative defined byfor some initial data , where , , and the nonhomogeneous term satisfies the continuity in and locally Lipschitz continuity in .
Definition 3 (see [24]). We say the is an equilibrium of system (12) whenever for all .
Definition 4 (see [24]). System (12) is -exponentially stable if there exist two positive constants and such that for any solutions and of system (6) subject to the initial conditions and , respectively, we havefor where denotes the Euclidean norm.
It should be noted that the notion of -exponential stability concerns the closeness of solutions and subject to different initial conditions and , respectively, while the definition of the Mittag-Leffler stability in [12] concerns the convergence of a solution to an equilibrium point.
Lemma 3 (see [17]). If with and , then .
Lemma 4 (see [17]). Let be a function such that , where and is an arbitrary constant. Then, we haveWe define the signum function as
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 functions are Lipschitz continuous with respect to the second variable on with Lipschitz constants , that is, for all uniformly with respect to , And another prerequisite for proving results. (H2) The constants is positive for 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 (4) is exponentially stable.
Proof. Let and be the solutions of system (4) subject to the different initial conditions and , .
Let for , then . By (4) we getIf is positive, thenif is negative, thenhence,next, we construct a function bywe compute the Caputo fractional derivative of using the assumptions (H1) and (H2) to obtainwhere . Lemma 4 implies thatit follows that system (4) is exponentially stable.
Next, we assume that the leader’s opinion is kept constant as for all time, that is, in (4). Let be an equilibrium of (4). Hence, it satisfies the following system
Theorem 2. Assume that (H1) and (H2) hold. Then, system (4) has a unique equilibrium.
Proof. Let for . We get from (25) thatconsider the map defined by , where andby the Lipschitz condition (H1), for any , we haveWe have from (H1) that for . Hence,Thus, the map is a contraction. Therefore, 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 that for and for all . Then, all agents’ opinions of (4) converge to the consensus opinion .
Proof. The proof follows the same argument as in [22, Theorem 2] where the system is autonomous. We outline the proof when the nonautonomous source term is considered here. Let be a solution of (4). Since we assume that , we have that is a constant function. Let and define , for . Then, system (4) can be written aswe see that for any ,for . It follows from Theorem 2 that system (30) has a unique equilibrium . Furthermore, the exponential stability in Theorem 1 implies that every solution of (30) converges to the equilibrium, that is, . Since for , we see that . Consequently, we have for every 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 inputs , in (5).
We assume the following conditions for the external inputs.
(H3) The functions are bounded that is, there exists such thatfor each .
Theorem 4. Assume that (H1), (H2), and (H3) hold. Then, system (5) is exponentially stable. Moreover, for any solution , there exists such thatwhere and is arbitrary small.
Proof. Let and be the solutions of system (5) subject to the different initial conditions and , . By setting and using a similar argument as in the proof of Theorem 1, there exists such thatshowing the exponential stability of (5). Next, we construct a function byUsing assumptions (H1) and (H2), we compute the Caputo fractional derivative of aswhere .
Consider the fractional-order systemIt is easily seen that (38) is exponentially stable, so that any solution converges to the unique equilibrium , that isHence, for any arbitrary , there exists such thatBy Lemma 3, we have with . Thus, there exists such that for any solution ,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 that for and for all . Then, all agents’ opinions of (5) converge to the consensus opinion .
Proof. Let be a solution of (5). Since we assume that , the leader’s opinion is a constant function . Let and define , for . Then, system (5) can be written asit can be seen that satisfies Lipschitz continuity condition asfor any and . Hence, by applying Theorem 4 for (42), that is, considering instead of and taking in (5), and using the assumption for , we see that the solution of (42) satisfiesfor some wheresince for , we have whenever is sufficiently large for all . Hence, if we consider (37) for some sufficiently large time , then the bound of in (H3) can be chosen as an arbitrary small . Consequently, we obtainAs was arbitrary, we have as . Consequently, we obtain the consensus opinion for every .
5. Numerical Examples
In this section, we give examples of the fractional opinion formation model of order 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 leadersubject to the initial condition .
In this system of opinion formation model (47) with leader , we set , , , , , , , , , , , , , , , , , , , , , , , and , respectively. We see that , and for 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 and of system (47) converges to a consensus state as shown in Figure 1 for .
Example 2. Consider the leader-follower opinion formation modelwhere subject to the initial condition .
In this system of opinion formation model (48) with leader , we set , , , , , , , , , , , , respectively. We see that , and for so that all assumptions of Theorem 5 are satisfied. Hence, the leader-follower for opinion formation model (48) is exponentially stable. The states of , and of system (48) converges to a consensus state as shown in Figure 2 for .
Example 3. Consider leader-follower for the opinion formation modelSubject to the initial condition . In this system of opinion formation model (49) with leader , we set , , , , , , respectively. We see that , and for so that all assumptions of Theorem 5 are satisfied. Hence, the leader-follower for opinion formation model (49) is exponentially stable. The states of , and of system (49) converges to a consensus state as shown in Figure 3 for.
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 of 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 -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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author was supported by the Kanchanaburi Rajabhat University. The second author was supported by the King Mongkut’s University of Technology Thonburi.