[Paper Review] Uniform position alignment estimate of spherical flocking model with inter-particle bonding forces

Sun-Ho Choi, Dohyun Kwon+1 more|arXiv (Cornell University)|Jan 4, 2021

Distributed Control Multi-Agent Systems34 references4 citations

TL;DR

This paper establishes a sufficient condition for complete position flocking in a Cucker-Smale-type model on the unit sphere with inter-particle bonding forces, using energy dissipation and a transformed linear system to prove uniform exponential decay of position diameter. The condition depends only on communication rate and bonding strength, not the number of agents, ensuring asymptotic alignment to a single point with constant velocity.

ABSTRACT

We present a sufficient condition of the complete position flocking theorem for the Cucker-Smale type model on the unit sphere with an inter-particle bonding force. For this second order dynamical system derived in [Choi, S.-H., Kwon, D. and Seo, H.: Cucker-Smale type flocking models on a sphere. arXiv preprint arXiv:2010.10693, 2020] by using the rotation operator in three dimensional sphere, we obtain an exponential decay estimate for the diameter of agents' positions as well as time-asymptotic flocking for a class of initial data. The sufficient condition for the initial data depends only on the communication rate and inter-particle bonding parameter but not the number of agents. The lack of momentum conservation and the curved space domain make it difficult to apply the standard methodology used in the original Cucker-Smale model. To overcome this and obtain a uniform position alignment estimate, we use an energy dissipation property of this system and transform the Cucker-Smale type flocking model into an inhomogeneous system of differential equations of which solution contains the position and velocity diameters. The coefficients of the transformed system are controlled by the communication rate and a uniform upper bound of velocities obtained by the energy dissipation.

Motivation & Objective

• To establish a sufficient condition for complete position flocking in a second-order Cucker-Smale model on the unit sphere with inter-particle bonding forces.
• To overcome challenges from lack of momentum conservation and curved geometry in spherical space.
• To derive a uniform exponential decay estimate for the diameter of agents' positions independent of the number of agents.
• To ensure time-asymptotic flocking under initial data conditions depending only on communication rate and bonding parameter.
• To develop a framework applicable to multi-agent systems on manifolds where standard Cucker-Smale methods fail.

Proposed method

• Utilize energy dissipation property to derive a uniform upper bound on velocities, crucial for handling lack of momentum conservation.
• Transform the original nonlinear system into an inhomogeneous system of differential equations for position and velocity diameters.
• Apply Gronwall’s lemma to the transformed system to establish exponential decay of position diameter.
• Employ a rotation operator based on the Lohe model to preserve motion on the sphere and ensure geodesic consistency.
• Define a Lyapunov functional to track alignment and stability, leveraging the communication rate and bonding parameter.
• Introduce a modified inter-particle bonding force using the projection operator to maintain agents on the sphere and prevent antipodal clustering.

Experimental results

Research questions

• RQ1 Can complete position flocking be achieved in a Cucker-Smale-type model on the sphere with inter-particle bonding forces, despite the absence of momentum conservation and curved geometry?
• RQ2 What sufficient condition on initial data guarantees uniform exponential decay of the position diameter, independent of the number of agents?
• RQ3 How can energy dissipation be leveraged to control velocity bounds and enable uniform alignment estimates in non-Euclidean space?
• RQ4 To what extent does the bonding force parameter σ and communication rate ψ influence the convergence rate and stability of the flocking system?
• RQ5 Can a linearized system approach be used to derive asymptotic position alignment in a nonlinear, geometrically constrained multi-agent system?

Key findings

• The position diameter decays exponentially at rate μ/2, with the decay rate determined by the communication rate and bonding strength.
• A sufficient condition for complete position flocking depends only on the communication rate ψ and bonding parameter σ, not on the number of agents N.
• The velocity diameter is uniformly bounded via energy dissipation, enabling control of the system’s dynamics despite lack of momentum conservation.
• Numerical simulations confirm exponential decay of the maximum spatial diameter, matching theoretical predictions under admissible initial conditions.
• When initial data violate the admissible condition, exponential decay fails, as shown in simulations with increased σ=5.
• The system exhibits asymptotic convergence to a single point with constant velocity, confirming complete position flocking on the sphere.