mjÞ

2a 2a 2a 2a

where q0   ¼ max    q1     ; 2 max    q2     ; q3     ðk   Þ 。 If the minimum of Dðj/m jÞ can be obtained and is larger than zero, (22) will

be satisfied。

It  follows  from  j/m j ¼ jð a   1 — x ÞfðtÞj that  j/mj 6 kð a   1 — x Þk2kfðtÞk2   based  on  the  Ho€lder  inequality,  which  implies

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi

that j/mj 6 a2  þ ð1 — xÞ  kfðtÞk2。 From (21), one can see that the state of the system asymptotically converge to the origin,

which means that kfðtÞk2   6 kfð0Þk2  where fð0Þ is the initial state and can be given in advance。 Furthermore, /m  – 0 by Claim

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi

1。 Thus,  we  have 0 < j/mj 6 a2  þ ð1 — xÞ  kfð0Þk2。

In order to obtain the minimum of Dðj/mjÞ, which is a nonlinear function for j/m j, two cases are considered: one is the set

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffi

v1  ¼ fj/m j : 1 6 j/m j 6 a2  þ ð1 — xÞ  kfð0Þk2 g while  the  other  is  the  set  v2   ¼ fj/m j : 0 < j/mj < 1g。  It  is  worth  mentioning

that v1  ¼ ; if

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi

a2  þ ð1 — xÞ  kfð0Þk2  < 1。 On the one hand, if v1  – ;, it is a compact set, for any j/mj 2 v1; Dðj/mjÞ – 0。 Hence,

k2   ¼ minj/m j2v1 Dðj/m jÞ exists, and is larger than zero。 On the other hand, if v1  ¼ ; or j/mj  R v1, we have j/mj 2 v2, which im-

   2 4a 2a

plies that j/m j1þa   < 1。 Hence, j/m j1þa   < j/mj

Let

。 In  this case, Dðj/m jÞ > 1。

k ¼ ð

1 — xÞ2b2

q0

。 1。

·min    k2; 2

ð25Þ

Since V ðtÞ is  radially unbounded and V_ ðtÞ is  negative definite, global finite-time stability holds。      Q

Remark 6。 From Theorem 1, one can see that the proposed continuous-time FPSO algorithm possesses a good tracking per- formance。 Especially, when pi ðtÞ is time-varying, the proposed continuous-time FPSO algorithm can always track pi ðtÞ over a finite-time  interval,  which  enables  this  algorithm  to  efficiently  deal  with  the  problem  of  odor  source   localization。

Remark 7。 It should also be pointed out that Theorem 1 gives a convergence condition for the deterministic system, i。e。, a is a constant。 If a is a random number, Theorem 1 describes an expected convergence condition in the mean square。