Asymptotic Behavior of Some Linear Differential Systems

Consider the following chasing game in amongbparticles whose trajectories are governed by the following differential equations: Fori S={1, 2, …, b}, whereFi(t) is the position at timetof the particleiandqij(t), which measures the instantaneous rate that particleiis attracted toward particlej, is given byqij(t)=pijλ(t)U(i, j)forj≠i. The rate functionλ(t) is positive with limt→∞ λ(t)=0,U(i, j) [0, ∞] is the “cost” function indicating the level of difficulty fromitojand the nonnegative numberspijfori≠jreflect the neighborhood structure of the particles. Particles are attracted to one another but such attraction becomes weaker as time goes on. A sufficient condition onλ(t) is given for allFi(t) converging to a common finite limit.