y_ 2ðtÞ¼ —ð1 — xÞy2 ðtÞ— ay1 ðtÞ— ð1 — xÞp_ i ðtÞ— p€i ðtÞ

Consider the stagnant and deterministic case, that is, pi ðtÞ is stable and a is a constant。 We obtain

。 y_ 1ðtÞ¼ y2ðtÞ

y_ 2ðtÞ¼ —ð1 — xÞy2 ðtÞ— ay1 ðtÞ

ð12Þ

ð13Þ

The system (13) is asymptotically stable at the origin when t ! 1, if the parameters x and a satisfy x < 1 and a > 0, respectively。 The asymptotical convergence of the system (13) is illustrated in Fig。 2 where the logarithmic scale is used。

Remark 4。 It is obvious that the system (12) is not stable at the origin if piðtÞ is time-varying, which motivates us to consider the stable system (13)。 Moreover, one can also see that the system (12) does not possess a good tracking performance, which may result in the higher oscillation of convergence results。 Therefore, this issue also motivates us to introduce the control theory to modify the system (12) derived by the continuous-time PSO algorithm for a given decision on piðtÞ。

2

10

0

10

−2

10

−4

10

−6

10

0 10 20 30 40 50 60 70 80

Time

2

10

0

10

−2

10

−4

10

−6

10

−8

10

0 10 20 30 40 50 60 70 80

Time

Fig。 2。  The  convergence  curves of the  system states  in (13) (x ¼ 0:8;  a ¼ 6;  y1 ð0Þ ¼ 5, and y2 ð0Þ ¼ —9)。

3。3。The  continuous-time  model  of  finite-time  particle  swarm optimization

In this subsection, based on the continuous-time model of the PSO algorithm (6), we will first give a continuous-time FPSO algorithm。 Then, we will present some remarks about parameters introduced by the continuous-time FPSO algorithm。 In what follows, we propose a continuous-time FPSO algorithm described by (14), where a nonlinear damping item and a parameter c are introduced into the system (13) such that the system (13) can be stable in finite time and the magnitude  of

state oscillation is controlled。

。 y_ 1 ðtÞ¼ y2 ðtÞ

y_ 2 ðtÞ¼ —cðð1 — xÞy2 ðtÞþ ay1ðtÞÞ — bsigðð1 — xÞy2 ðtÞþ ay1 ðtÞÞ

ð14Þ

where 0 < a < 1; b > 0, and 0 < c 6 1。

It is worth noting that the system (14) becomes the linear system (13), which was studied by Fernández Martínez et al。 [15], if we set b ¼ 0 and c ¼ 1。 Hence, the system (13) can be regarded as a special case of the system (14)。 The finite-time convergence of the system states in (14) under the same parameter values x, a and initial states y1ð0Þ; y2ð0Þ is elaborated in Fig。 3。