Fractional opinion formation models with leadership Ricardo Almeida
Agnieszka B. Malinowska
Tatiana Odzijewicz
Center for Research and Faculty of Computer Science Department of Mathematics Development in Mathematics Bialystok University and Mathematical Economics and Applications (CIDMA) of Technology, Poland Warsaw School Department of Mathematics Email:
[email protected] of Economics, Poland University of Aveiro, Portugal Email:
[email protected] Email:
[email protected]
Abstract—The paper studies two types of fractional opinion formation models with leader: the Hegselmann–Krause and the Cucker–Smale. We aim to design an optimal control strategy for the systems to reach a consensus. A numerical scheme, based on Grunwald–Letnikov ¨ approximation of the Caputo fractional derivative, is proposed for solving the fractional optimal control problem. The effectiveness of the proposed control strategy is illustrated by examples.
I. I NTRODUCTION One of the most fundamental phenomenons in multi–agent systems is the mutual agreement, or consensus [2], [5]. Because of that mathematical models describing interrelationships between individual agents- one can think of robots or other automatic units- point out these issues [9], [12]. Among the best known models are ones introduced by Hegselmann and Krause as well as by Cucker and Smale. The approach of Hegselmann and Krause investigates events when agents do not interact with neighbors if their opinions are too far apart [7], [8], [11], while the concept of Cucker and Smale studies situations, when all agents influence each other regardless of the configuration [4], [6]. Opinions of agents are expected to reach consensus. However, we can observe phenomena of opinion’s polarization or chaos. In such situations, one of the techniques is to introduce the virtual leader to the system and to design such control strategies so that its followers tend to consensus in the most efficient way (see e.g., [15], [17]). In this work, we are interested in optimal control strategies for opinion formation models with leader, both of Hegselmann–Krause and Cucker–Smale type, where the dynamics is described by fractional derivatives (that is, derivatives of real or complex order) [3], [10], [13], [14]. Such approach takes
into account the influence of the past on the system, by introducing memory parameter which is the order of the fractional derivative [10], [16]. The paper is organized as follows. Section II is a preparatory section where we introduce the notions of fractional derivatives. In Section III we define fractional opinion formation models. Section IV discusses the optimal control problem governed by the opinion formation model with the leader, and a numerical method of its solution. Finally, in Section V, two illustrative examples are given. II. P RELIMINARIES For f : [a, b] → R and α ∈ R+ the left Riemann–Liouville fractional integral of order α is defined by α Ia+ [f ](t)
1 := Γ(α)
Zt
(t − τ )α−1 f (τ )dτ, t > a,
a
and the right Riemann–Liouville fractional integral of order α by α Ib− [f ](t)
1 := Γ(α)
Zb
(τ − t)α−1 f (τ )dτ, t < b.
t
Let α ∈ (0, 1). We define the left and right Riemann–Liouville fractional derivatives by d α α Da+ [f ](t) := ◦I [f ](t), t > a, dt a+ and α Db− [f ](t) := −
d α ◦ Ib− [f ](t), t < b, dt
respectively. Note that, the Riemann–Liouville fractional derivatives of a constant are not zero and for regular functions they are singular. Therefore,
in order to avoid these anomalies, the Caputo fractional derivatives were introduced. Definition 1. The left and right Caputo fractional derivatives are defined as c
Now, using Definition 1 we obtain the following decomposition sum for the Caputo fractional derivatives c
α α Da+ [f ](t) := Da+ [f (τ ) − f (a)](t), t > a,
−
and c
α Da+ [f ](tk ) ≈
α α Db− [f ](t) := Db− [f (τ ) − f (b)](t), t < b,
f (a) ˜ α [f ](tk ), (tk − a)−α =: c D a+ Γ(1 − α) k = 1, . . . , M, (3)
respectively. For absolutely continuous functions f , we have d c α α Da+ [f ](t) = Ia+ ◦ [f ](t), t > a dt d c α α Db− [f ](t) = − Ib− ◦ [f ](t), t < b. dt Next, we define the Gr¨unwald–Letnikov fractional derivatives. Definition 2. The left and right Gr¨unwald– Letnikov fractional derivatives of function f , of order α, are given by GL
α Da+ [f ](t)
∞ 1 X α := lim α (wr )f (t − kh) h→0+ h r=0
and GL
α Db− [f ](t) := lim
h→0+
∞ 1 X α (w )f (t + kh), hα r=0 r
respectively. Here (wkα ) := (−1)k
α k
.
It is well known (see e.g., [14]) that the truncated Gr¨unwald–Letnikov fractional derivatives are firstorder approximations of the Riemann–Liouville fractional derivatives. Suppose that f is a continuously differentiable function and let T[a,b] = {tk }k=0,...,M = {a + kh}k=0,...,M be the usual regular partition of the interval [a, b] with M ≥ 2 and h = b−a M . Then, we have α Da+ [f ](tk ) ≈
k 1 X α (w )f (tk−r ) hα r=0 r
˜ α [f ](tk ), k = 1, . . . , M, (1) =: D a+
α Db− [f ](tk ) ≈
M−k 1 X α (w )f (tk+r ) hα r=0 r
α ˜ b− =: D [f ](tk ), k = 0, . . . , M − 1. (2)
c
α Db− [f ](tk ) ≈
−
k 1 X α (w )f (tk−r ) hα r=0 r
M−k 1 X α (w )f (tk+r ) hα r=0 r
f (b) ˜ α [f ](tk ), (b − tk )−α =: c D b− Γ(1 − α) k = 0, . . . , M − 1. (4)
III. F RACTIONAL
OPINION FORMATION MODEL
WITH LEADER
Consider a group of N interacting agents with one leader with opinions in Rd . Opinions of agents and the leader are denoted by (x1 , . . . , xN ) ∈ RN d and x0 ∈ Rd , respectively. Let α ∈ (0, 1), x : [0, T ] → R(N +1)d and u : [0, T ] → Rd . The fractional opinion formation model with the leader has the following form: c α D0+ [x0 ](t) = u(t), N c Dα [x ](t) = P a (x (t) − x (t)) ij j i 0+ i (5) j6=0,i +ci (x0 (t) − xi (t)), xi (0) = xi0 ∈ Rd , i = 0, 1, . . . , N.
The state of the system is denoted by x = (x0 , x1 , . . . , xN ) ∈ R(N +1)d . Functions aij = aij (kxi (t) − xj (t)kld ) quantify the way the agents 2 influence each other, while functions ci = ci (kxi (t) − x0 (t)kld ) represent the rates of rela2 tionship between the leader and the other agents, where k·kld denotes the Euclidean norm in Rd . 2 In this work, we focus on finding control strategy for model (5) driving all agents’ opinions to consensus i.e., we want to find such x∗ ∈ R(N +1)d , with x∗0 = x∗1 = · · · = x∗N , that limt→∞ x(t) = x∗ . Observe that in system (5) the control function is applied only to the leader. Namely, the main idea is to control the system throughout the leader since this type of the control can be easier implemented. This approach is close to the real-world phenomena (e.g., the relations between a sheepdog and sheep), and the real social behaviour (e.g., the influence of mass media on opinions of the society members, politics).
IV. O PTIMAL CONTROL
A. Fractional Cucker–Smale type model System (5) with coefficients given by aij = aij (kxi (t) − xj (t)kld ) 2
=
R 2
1 + kxi (t) − xj (t)kld 2
θ ,
and ci = ci (kxi (t) − x0 (t)kld ) 2
=
THROUGH LEADERSHIP
We want to enforce consensus in system (5) using the optimal control. To do so we propose the following cost functional
H 2
1 + kxi (t) − x0 (t)kld 2
γ
where θ, γ, H, R > 0 are fixed reals, we shall call a Fractional Cucker–Smale (FCS) type model. This system is motivated by the Cucker–Smale model in which each agent influences all of the others through the adjacency matrix, no matter what the configuration of the agents is [4]. Clearly, in system (5), the leader is not influenced by followers, and the weights aij , ci are functions of the distance between agents and rates of influence, respectively.
B. Fractional Hegselmann–Krause type model System (5) with aij = a(kxi (t) − x0 (t)kld ), a : 2 [0, ∞) → [0, 1], 0 ≤ r ≤ δ, 1, a(r) = a(r, δ, ε) := ϕ(r), δ < r < δ + ε, 0, δ + ε ≤ r,
where δ, ε > 0 are fixed reals, and ci = ci (kxi (t) − x0 (t)kld ) = φ(r), where φ : [0, ∞) → 2 (0, 1] is a smooth non-increasing function such that φ(0) = 1 and lim φ(r) = 0, we shall call a r→∞ Fractional Hegselmann–Krause (FHK) type model. In this case the first term in dynamics (5) originates from the Hegselmann–Krause (HK) model [7], [17] in which the interactions among agents are not necessarily all to all, and take place in the bounded domain of confidence. Each follower is influenced only by those agents that it has confidence in, δ is a bounded confidence and ε > 0 is a parameter defining the region, where a function a decays to zero. The second term in dynamics (5) models the action of the leader.
ZT "
N 1 X 2 J(x, u) = kxi (t) − xj (t)kld 2 2N 2 i,j=1 0 # N 1X ν 2 2 + kx0 (t) − xi (t)kld + ku(t)kld dt, 2 2 2 i=1 2 (6)
where ν > 0 is a weight constant. The first and the second terms in (6) are the energy of the system. The third term represents the cost of the control. Therefore, our optimal control problem is: min J(x, u) subject to system (5), where x : [0, T ] → R(N +1)d , u : [0, T ] → Rd . In order to solve the proposed fractional optimal control problem we apply a direct method proposed in [1] and call it a ”first discretize then optimize method”. Namely, objective functional (6) and system (5) are discretized by a simple Euler method with fixed step size and using the truncated Gr¨unwald– Letnikov fractional derivative, when approximating the Caputo fractional derivative. Let T[0,T ] . Then tZk+1"
N 1 X kxi (t) − xj (t)k2ld 2 2N 2 i,j=1 k=0 t k # N 1X ν 2 2 + kx0 (t) − xi (t)kld + ku(t)kld dt 2 2 2 i=1 2 " M−1 N X 1 X ≈ h kxi (tk ) − xj (tk )k2ld 2 2N 2 i,j=1 k=0 # N ν 1X 2 2 + kx0 (tk ) − xi (tk )kld + ku(tk )kld dt 2 2 2 i=1 2
J (x, u) =
M−1 X
and we can discretize (5) as follows c α ˜ [x0 ](tk+1 ) = u(tk ), D 0+ N c D ˜ α [xi ](tk+1 ) = P aij (xj (tk ) − xi (tk )) 0+ j6=0,i +ci (x0 (tk ) − xi (tk )), k = 0, . . . , M − 1, i = 1, . . . , N.
Moreover, we have
xi (0) = xi0 ∈ Rd , i = 0, 1, . . . , N. The described discretization method transforms optimal control problem (5)–(6) into a finite dimensional optimization problem and therefore allows
us to use finite dimensional optimization techniques to find approximate solutions to (5)–(6) in both the Hegselmann–Krause and the Cucker– Smale case. V. N UMERICAL
EXAMPLES
In this section, we give some numerical examples to demonstrate the effectiveness of the proposed control scheme. Systems are considered in the one dimensional case (d = 1) and M = 20. First, we analyze systems without the leader and without control, that is N c Dα [x ](t) = P a (x (t) − x (t)) ij j i 0+ i (7) j6=0,i x (0) = x ∈ R, i =, 1, . . . , N. i i0
Fig. 2. FCS type model without leader and control, for α = 0.8.
Next, we apply a control to the system through the leader and solve the appropriate optimal control problem. Let us consider the FCS type model with ν = 2, H = R = 1, γ = θ = 3/4, and N = 9. For α ∈ {0.3, 0.8, 1} Figures 1, 2 and 3 show approximate trajectory solutions to system (7) without the leader and without control, while Figures 4, 5 and 6 show approximate trajectory solutions to appropriate optimal control problem with the leader. Note that, in the latter case, all agents follow the leader.
Fig. 3. FCS type model without leader and control, for α = 1.
trajectory solutions to system (7) without the leader and without control, while in Figures 10, 11 and 12 we show approximate trajectory solutions to appropriate optimal control problem with the leader. In the first case, we observe the convergence to clusters in which all agents share the same opinion, while in the second case the consensus is achieved.
Fig. 1. FCS type model without leader and control, for α = 0.3.
VI. C ONCLUSIONS
In this paper, we studied two types of opinion Now, let us analyze the FHK type model with formation models: the Hegselmann–Krause and the N = 4 and Cucker–Smale. We designed a feedback control, 1 1 1 1 ϕ(r) = + tanh 2 + , where the control is implemented on the leader, 2 2 r − 0.5 r2 − (0.5 + 0.2) to globally achieve the consensus. A discretization scheme, based on the Gr¨unwald–Letnikov approx2 φ(r) = exp −r , imation of the Caputo fractional derivative, was that is δ = 0.5 and ε = 0.2. For α ∈ {0.5, 0.8, 1} proposed in order to find the numerical solution in Figures 7, 8 and 9 we report the approximate of the fractional optimal control problem.
Fig. 4. FCS type model with leader and control, for α = 0.3.
Fig. 6. FCS type model with leader and control, for α = 1.
Fig. 5. FCS type model with leader and control, for α = 0.8.
Fig. 7. FHK type model without leader and control, for α = 0.5.
ACKNOWLEDGMENT R. Almeida was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundac¸a˜ o para a Ciˆencia e a Tecnologia), within project UID/MAT/04106/2013; A. B. Malinowska and T. Odzijewicz were supported by Polish founds of National Science Center, granted on the basis of decision DEC2014/15/B/ST7/05270. R EFERENCES [1] R. Almeida, D. F. M. Torres, ’A discrete method to solve fractional optimal control problems’, Nonlinear Dynamics, vol. 80, issue 4, pp. 1811–1816, 2015. [2] G. R. Campos, L. Brinion–Arranz, A. Seuret, S-I. Niculescu, ’On the Consensus of Heterogeneous MultiAgent Systems: a Decoupling Approach’, IFAC Proceedings Volumes, vol. 45, no. 26, pp. 246–251, 2012.
[3] A. Dzieli´nski and P. M. Czyronis, ’Fixed final time and free final state optimal control problem for fractional dynamic systems - linear quadratic discrete-time case’, Bull. Pol. Acad. Sci., Tech. Sci., vol. 61, no. 3, pp. 681–690, 2013. [4] F. Cucker, S. Smale, ’On the mathematics of emergence’, Japan. J. Math., vol. 2, no. 1, pp. 197–227, 2007. [5] M. H. DeGroot, ’Reaching a Consensus’, J. Amer. Statist. Assoc., vol. 69, no. 365, pp. 118–121, 1974. [6] E. Girejko, L. Machado, A. B. Malinowska, N. Martins, ’On consensus in the Cucker-Smale type model on isolated times scales’, Discrete Contin. Dyn. Syst. Ser. S, vol. 11, no. 1, pp. 77–89, 2018. [7] R. Hegselmann, U. Krause, ’Opinion dynamics and bounded confidence, models, analysis and simulation’, J. Artif. Soc. Soc. Simul., vol. 5, no. 2, pp. 1–33, 2002. [8] R. Hegselmann, U. Krause, ’Opinion dynamics under the influence of radical groups, charismatic leaders and other constant signals. A simple unifying model’, Netw. Heterog. Media, vol. 10, no. 3, pp. 477–509, 2015. [9] A. Jadbabaie, J. Lin, A. S. Morse, ’Coordination of groups of mobile autonomous agents using nearest neighbor rules’, IEEE Trans. Automat. Control, vol. 48, no. 6, pp. 988–1001, 2003. [10] A. B. Malinowska, T. Odzijewicz, ’Optimal control of discrete-time fractional multi-agent systems’, J. Comput. Appl., https://doi.org/10.1016/j.cam.2017.10.014, in press.
Fig. 8. FHK type model without leader and control, for α = 0.8.
Fig. 10. FHK type model with leader and control, for α = 0.5.
Fig. 9. FHK type model without leader and control, for α = 1.
[11] D. Mozyrska, M. Wyrwas, ’Fractional discrete-time of Hegselmann-Krause’s type consensus model with numerical simulations’, Neurocomputingvol. 216, pp. 381–392, 2016. [12] R. Olfati-Saber, ’Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory’, IEEE Trans. Automat. Control, vol. 51, np. 3, pp. 401–420, 2006. [13] P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing, Singapore: World, Scientific Publishing Co. Pte. Ltd., 2016. [14] I. Podlubny, Fractional Differential Equations, San Diego: Mathematics in Science and Engineering, 198, Academic Press, 1999. [15] G. Ren, Y. Yu, S. Zhang, ’Leader-following consensus of fractional nonlinear multiagent systems’, Math. Probl. Eng., vol. 2015, Article ID 919757, 8pp., 2015. [16] L. Vazquez, ’A fruitful interplay: from nonlocality to fractional calculus’, Nonlinear Waves: Classical and Quantum Aspects. NATO Sci. Ser. II Math. Phys. Chem. Dordrecht: Springer, 2005, p. 129–133. [17] S. Wongkaew, M. Caponigro, A. Borz´ı, ’On the control through leadership of the Hegselmann-Krause opinion formation model’, Math. Meth. Appl. Sci. vol. 25, no. 3, pp. 565–585, 2015.
Fig. 11. FHK type model with leader and control, for α = 0.8.
Fig. 12. FHK type model with leader and control, for α = 1.
