Directional Bounded-Confidence Cucker-Smale Flocking

Abstract

This paper introduces a directional bounded-confidence extension of the Cucker--Smale (CS) flocking model that decomposes pairwise relative velocities into independent radial and tangential channels within the instantaneous line-of-sight frame. Gating these components via compactly supported kernels creates a time-varying, state-dependent, matrix-weighted interaction that strictly generalizes classical scalar CS alignment. We establish an exact energy-dissipation identity with conserved mean velocity and prove exponential velocity consensus under a sliding-window persistent excitation (PE) condition formulated directly on the disagreement subspace. Furthermore, we derive a CS bounded-diameter criterion separating metric decay from aggregate directional coercivity, along with a cluster convergence guarantee. Simulations validate the exact dissipation balance and demonstrate that directional gating successfully restores global alignment in scenarios where traditional isotropic tied-confidence laws remain polarized.