英语翻译Note that since the distance from all the other members

英语翻译
Note that since the distance from all the other members to a
free agent is greater than 6,there will not be any repulsion
force and the total force on this member will be a combined
effect of all the attraction imposed by all the other members.
We will show that this force is pointing toward the center -x
of the swarm,and therefore,the member is moving toward
it.Before stating this result more rigorously,we define the
error variable as e' = x' - -x,for each individual i = 1,...,M.
Lemma 2 Assume that a member i of the swarm described
by the model in Eq.( I ) with an attractiodrepulsionfunction
g(.) as given in Eq.(2) is a free agent at time t and that its
distance to the center -x of the swarm is greater then 6,i.e.,
Ilei(t)ll = Ilx'(t) - -xll > S.
Then,at time t its motion is in a direction of decrease of
Ilei(t) 1) (i.e.,toward the center 3).
Proof:From the definition of the center -x of the swarm
we have ,=x Mj-x .Subtracting Mifrom both sides we
obtain M
(x' - x') = M(x' - -x) = Mk (3)
j= 1
Then,the motion of member i can be represented as
(x' -XI),
M
= -aMei+b I:exp
j= I,j#i
where on the first line we used the definition of function g(.)
in Eq.(2) and added a(x' - x') = 0,and substituted the value
of
Note that since Ix = 0,we have .b = 2.Choosing the Lyapunov
function candidate for member i as V; = ieiTei and
taking its derivative along the trajectory of the member we
obtain
(x' - x j ) from Eq.(3) on the second.
(4)
Note that bexp (- q)> 0 for all x' and xj.Therefore,
fl is bounded by
(5)
Since member i is a free agent,we have llx' -xi11 >
6,Vj # i and note that for that range the function
exp (- v) 112 - xjll is a decreasing function of the
distance with the maximum occuring at llx' -xJ 11 = 6.Using
these facts,we have
r',_< -aMllei112+b(M- 1)6exp ( -- T) Ile'll
..
= -allei112 - ( M - 1) [ a -e biS~ex~p (-f)] Ile'll.
For the second term to be negative semidefinite we need
Note,however,that exp (- $) = 1,which implies that
we need Ile'li 26,which,on the other hand,holds by our
hypothesis.Therefore,we have
fi 5 -allei112 = - 2 a j ,
which proves the assertion.

注意,由于距离的所有其他成员一免费代理大于6 ,将不会有任何斥力部队和总兵力在此的会员将是一个结合的影响,所有的吸引力所施加的所有其他成员.我们将表明,这股力量是指向对中心- X的该群,因此,会员是走向它.之前,说...