英语翻译Remark:From attractiodrepulsion function g(.) in Eq.(2)o

英语翻译
Remark:From attractiodrepulsion function g(.) in Eq.(2)
one can see that one term in g(.) always gives attraction
and the other repulsion and the resultant force is their sum.
This leads to similar terms in the derivative of the Lyapunov
function in Eq.(4).If an individual is away from all the
other individuals,the second term in the Lyapunov function
is negligibly small compared to the first term and it moves
toward the center.If it is close to the other individuals (i.e.,
in their repulsion range),then the second term becomes significant.
Note that Lemma 2 does not imply that xi will converge
to -x for all i.Intuitively,once a member gets to the vicinity
of another member,then the repulsive force will be in
effect and the conditions of Lemma 2 will not be satisfied
anymore.However,it is important because it gives us an
idea of the tendency of the individuals to move toward the
center of the swarm.Therefore,it is normal to expect that
the members will (potentially) aggregate and form a cluster
around -x.To prove this we need to analyze the motion of
the members which are not necessarily free agents and that
is done in the next result.
Theorem 1 Consider the swarm described by the model in
Eq.( I ) with an attractiodrepulsion function g( .) as given
in Eq.(2).As time progresses all the members of the swarm
will converge to a hyperball
where E = -b&excp (-;).
Moreovel; the convergence will occur infinite time bounded
bY
Proof:Choose any swarm member i.Let Vi = ieiTei be
the corresponding Lyapunov function.From the proof of
Lemma 2 we know that
(6)
Therefore,if

备注：从attractiodrepulsion函数g （ .）在情商.（ 2 ）
人们可以看到,一学期在克（ ）总是给人的吸引力
和其他斥力以及由此产生的力量是他们的总和.
这导致了类似的条件,在衍生金融工具的李雅普诺夫
功能的均衡器.（ 4 ） .如果个人是远离所有
其他个人,第二任期内,在李雅普诺夫函数
是negligibly小相比,第一届和它的举动
对中心.如果是接近的其他个人（即,
在他们的斥力范围） ,那么第二个任期成为显着.
请注意,引理2 ,并不意味着席将衔接
到X的所有一.凭直觉,一旦会员愈附近
另一位议员的话,斥力将在
效应和条件引理2不会感到满意
了.不过,这是很重要的,因为它给了我们一个
的思想倾向,个人的走向
中心的群.因此,这是正常的期望
成员将（可能）总结,形成一个集群
靠近- X的.为了证明这一点,我们需要分析的议案
该成员不一定是免费的代理商,并
这样做是在未来的结果.
定理1考虑群所描述的模型在
情商.（一）与一attractiodrepulsion函数g （ .）为给
在情商.（ 2 ） .随着时间的进展,所有成员组成的群
将收敛到一超
其中e =乙＆ excp (-;).
moreovel ;收敛将出现无限的时间范围内
通过
证明：选择任何一窝蜂会员一让六= ieitei被
相应的Lyapunov函数.从证明
引理2我们知道,
（ 6 ）
因此,如果