数理解析研究所講究録
900 巻 1995 年 100-118

100

A Collision Avoidance Control Problem forMoving Bodies in the Plane
Jun-hong Ha, Jito Vanualailai and Shin-ichi Nakagiri

1Introduction
There appeared several works related with the applicability of direct method of

Liapunov to the find-path problems lies in the qualitive theory of differential games

and the avoidance control strategies. We refer to, e.g., Vincent andSkowronski and$\mathrm{V}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}[5],$game aspects, and$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[7]$$\mathrm{L}\mathrm{e}\mathrm{i}\mathrm{t}\mathrm{m}\mathrm{a}\mathrm{n}[3],$, and Skowronski and$\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$and$\mathrm{J}\mathrm{H}\mathrm{N}[9]$$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{l}6$$\mathrm{S}\mathrm{k}_{0}\mathrm{W}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{k}\mathrm{i}[4]$,] for differential

for avoidance strategy aspects.

In these works the generation of a suitable Liapunov function is the key in the viewof the Liapunov stability theory. Recently,$\mathrm{L}.\mathrm{T}$.

$\mathrm{G}\mathrm{r}\mathrm{u}\mathrm{j}\mathrm{i}\mathrm{c}[1,2]$

has established that an

asymptotically stable nonlinear system permits the construction of a Liapunov function
to guarantee the asymptotic stability. That is, there is no Liapunov function whichmakes a given system be asymptotically stable if the given system is not asymptoticallystable. But, it is very difficult or impossible to determine a suitable Liapunov function
for the given complexed nonlinear system, because we have not to integrate the dini-

derivative,$\mathrm{s}\mathrm{e}\mathrm{e}[1,2]$. Therefore, in many cases we need to construct a Liapunov function,

which implies that a system may be stable, and often we can obtain asymptotic stability

under some restricted conditions.
Approaching the findpath problem to a collision avoidance strategy of robot arms,$\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{L}8_{\mathrm{J}}^{1_{;}}\Gamma$

adopts the Liapunov theory$\mathrm{h}\mathrm{o}\mathrm{m}$the control and differential game literature

for capture within targets, and for avoidance of antitargets. It may be the first good$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}_{\mathrm{o}\mathrm{S}\mathrm{a}1}\sim$$\tau_{\lrcorner}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}$in [8] to solve the collision avoidance control problem in the basis of Liapunov. The essential feature of his approach is to construct Liapunov functions forthe approaching targets and collision avoiding of antitargets and to determine control

101$\mathrm{v}\mathrm{a}\mathrm{r}\dot{\mathrm{j}}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}$according to the time derivatives of Liapunov functions. However, in thetion of feedback control variables, he used assumption called “right-of way”,$\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}_{\sim}^{\vee}\cdot\min \mathrm{a}$which is reasonable in numerical simulations but not meaningful in mathematical sense,and unfortunately the generalized Liapunov functions do notdition of Liapunov stability theory;$\mathrm{a}\mathrm{S}\mathrm{S}\mathrm{U}\mathrm{n}\mathrm{l}\mathrm{P}^{\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}}$$\mathrm{s}\mathrm{e}\mathrm{e}[8]$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Phi$. In our previous work

the sufficient con-$\mathrm{J}\mathrm{H}\mathrm{N}[9]$, we can remove

such as “right-of way”, and we introduce the elliptic Liapunov function to

obtain good paths of orbit of moving objects. But, in [9] we have failed to treat control-oarameters which may make the path change freely, and futher the Liapunov function
does not satisfy the sufficient condition of the Liapunov stability.
In this paper, we introduce a new Liapunov function which satisfies the Liapunovstability sufficient conditions, and by using the Liapunov function we may easily change

the paths freely via the control parameters. Finally, we note that almost all are “regular cases” in that we are getting in nice, smooth paths for the collision avoidance innumericaliimulations. These are ilustrated by several examples, and the comparisionsof our numerical results with the cases of [8] and [9] are given.

2

Control plan for$\mathrm{m}$numbers of moving objects

Let us consider a system, containing $m$ numbers of moving objects and $m$ numbersof fixed targets in a plane workspace, for the trajectories of the moving objects being

controlled to obtain collision avoidance and to reach the targets. The collsion avoidanceis to control the movement of the -th moving object to reach the center$\dot{i}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\underline{\tau}\cdot \mathrm{o}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}$of the -th target, while ensuring the -th entire moving object to avoid the j-th target$\dot{i}$$\dot{i}$and the j-th entire moving object which is regarded as an antitarget with respect tothe$i_{i^{- \mathrm{t}\mathrm{h}}}$one, where$\dot{i}\neq j,$$1\leq\dot{i},j\leq m$. We will use the Liapunov technique as known

as a powerful mathematical method to accomplish the plan for solving the collision
avodience control problem. Therefore, to utilize the Liapunov technique, it is necessary

to introduce the Liapunov function which can be applied to the given system, and wegive it below.

102

The Liapunov techique

2.1

Let

be the set of positive real numbers. We will denote by$\mathbb{R}^{+}$

object and by$T_{j}$$A_{i}$$\dot{i}$the j-th target respectively, where $1\leq\dot{i},j\leq m$ . Let us regard thecenters of moving objects
objectthe -th moving$A_{i}$$A_{i}$

as the points$(x_{i}, y_{i})$moves continuously depending on

continuous function for$t\in \mathbb{R}^{+}$

on the plane. When each moving$t\in \mathbb{R}^{+}$, we can consider. In the paper, as studied in$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$suppose that the dynamics of $m$ point objects $(x_{i}, y_{i}),i=1,2,$$\cdots,$$m$$(x_{i}, y_{i})$

and

as a

$\mathrm{J}-\mathrm{H}- \mathrm{N}[9]$, we, are described by

the system of the controlled ordinary differential equations,$\dot{x}_{i}=z_{i}$$\dot{z}_{i}=u_{i}$

(2.1)

$\{$

$\dot{y}_{i}=w_{i}$$\dot{w}_{i}=v_{i}$Here in (2.1),
and$(^{\eta}.\cdot i, v_{i})$,$\dot{i}=1,2,$ $\cdots,m$.

denotes the time derivatives of the -th point object$\dot{i}$$(z_{i}, w_{i})=(\dot{x}_{i},\dot{y}_{i})$denotes the -th control variables pair. We remark that the special case$\dot{i}$where $m=2$ is considered inehe controlls$(u_{i}, v_{i}),$$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$$1\leq\dot{i}\leq m$and$\mathrm{J}-\mathrm{H}- \mathrm{N}[9]$. By the Liapunov technique,will be determined as feedback controls which are

obtained by the result of differentiating the Liapunov function associated with theof the i-th target with centersystem equation (2.1). Let us define the target set$T_{i}$$TS_{i}$$(p_{i}c_{1},pic_{2})$with centerand radius$(x_{j}, y_{j})$$rp_{i}$and the moving object set$AS_{j}$of the j-th moving object

and length rapj of the j-th moving object$TS_{i}=\{(X,y):(x-p_{i}c_{1})^{2}+(y-p_{i2}c)^{2}\leq rp_{i}^{2}\},$$AS_{j}=\{(x,y) :(x-x_{i^{)(y}}2+-yj)2\leq rap_{i}\}2,$$A_{j}$$A_{j}$

as follows:$1\leq\dot{i}\leq m$$1\leq j\leq m$,.

In order to determine the controls which give the trajectories to avoid collision, we
need to define the Liapunov functions such as approaching to the targets and avoiding$,he antitargets. Therefore, let us define such functions on the plane as follows. Let us

define the following (sub)-Lapunov functions:the Liapunov function to make the -th moving object$\dot{i}$$V_{i}$

target$A_{i}$

approach to the i-th

$T_{i\mathrm{i}}$$V_{i}= \frac{1}{2}\{(_{X}i-p_{i1}c)^{2}+(y_{i}-pic2)^{2}+zi+w_{i}^{2}2\},$$1\leq i\leq m$

,

103

$W_{ij}$$T_{j},$

the Liapunov function to make the -th moving object$\dot{i}$$\prime jJ\neq j$

avoid the j-th target;$W_{ij}= \frac{1}{2}\{(x_{ip_{j}c_{1})^{2}(-}-+y_{i}p_{j2}c)^{22}-rp_{j}\},$$1\leq i,j\leq m$. the Liapunov function to avoid the -th moving object$\dot{i}$$\overline{V}_{j}n$object$A_{i}$$A_{i}$,and the j-th moving

each other;$A_{j},\dot{i}\neq j$$V_{ij}= \frac{1}{2}\{(xi-xj)^{2}+(y_{i}-y_{j})2-\max\{rap^{2}i’ rap^{2}j\}\},$ $1\leq\dot{i},j\leq m$$G_{i}$,

the function which denotes the distance between centers of the i-th moving object

and the -th target;$\dot{i}$$G_{i}= \frac{1}{2}\{(xi-p_{i}c_{1})^{2}+(y_{i}-pic_{2})^{2}\},$

Using the above Liapunov functions
apunov function$\mathcal{L}$on$V_{i},$$W_{ij},$$V_{ij}$and$D(\mathcal{L})=\{(\mathrm{x}, \mathrm{z})\in \mathbb{R}^{2m}\cross \mathbb{R}^{2m}$$G_{i}$$1\leq i\leq m$.

, we can now define the total Li-: $V_{ij}(x_{i,y_{i,j}}x,y_{j})>0,$$W_{ij}(X_{i}, y_{i})>$for the system (2.1) as follows,$0,1\leq i,$ $j\leq 7n\}$”,$\mathcal{L}((\mathrm{x}, \mathrm{Z}1)=\sum_{=i1}^{n}V_{i}(X\mathrm{z}i, yi, z_{i},w_{i})+\sum_{i=}m1j\sum^{m}\frac{\alpha_{ij}Gi(x_{i},y_{i})}{W_{ij}(x_{i},y_{i})}+=1\frac{1}{2}\sum_{i=1j}\sum\frac{\beta_{ij}c_{i}(xiyi)cj(Xjy_{j})}{V_{ij}(xi,y_{i,j}x,y_{j})}mm=1$

where$.\dot{?},$$\mathrm{x}=lx_{1},$$y_{1}\backslash ’\cdots,$ $x_{m},$ $ym$$\alpha_{i}?.=\beta_{ii}=0,$$\alpha_{ij},$$\beta_{ij}>0$)

and$\in \mathbb{R}^{2m},$$\mathrm{z}=(z_{1}, w_{1,m}\ldots, Z, w_{m})\in \mathbb{R}^{2m}$$\beta_{ij}=\beta_{ji}$direct, calculations of the time derivativeand for all

for $1\leq\dot{i},j\leq m$ . Then it is verified by the$\dot{\mathcal{L}}_{(2.1)}((\mathrm{x}, \mathrm{z}))$along the equation (2.1) is givenby$\dot{\mathcal{L}}_{(2.1)}((_{\mathrm{X}}, \mathrm{z}))=-\sum_{i=1}^{m}(\gamma iz^{2}i+\mu iw_{i}^{2})$provided that the feedback control variables$u_{k}$$v_{k}$$\mathrm{v},\mathrm{v}\mathrm{h}\mathrm{e}\underline{\gamma}\cdot \mathrm{C}k=1,2,$$\gamma_{i}>0$.

$\mu_{i}>0$$(u_{k}, v_{k})$are given by$=$$-(x_{k}-p_{k1}c)(1+ \sum_{i=1}^{m}\frac{\alpha_{ki}}{W_{ki}}+\sum_{1i=}^{m}\frac{\beta_{ki}G_{i}}{V_{ki}}\mathrm{I}$$+$$\sum_{i=1}^{m}\frac{\alpha_{ki}G_{k}}{W_{ki}^{2}}(x_{k}-p_{i1}c)-\sum_{i=1}^{m}\frac{\beta_{ik}G_{i}G_{k}}{G_{ik}^{2}}(X_{i}-xk)-\gamma k^{Z}k$$=$$-(y_{k}-p_{k}c_{2})(1+ \sum_{=i1}^{m}\frac{\alpha_{ki}}{W_{ki}}+\sum_{i=1}^{m}\frac{\beta_{ki}G_{i}}{V_{ki}})$$+$$\sum_{i=1}^{m}\frac{\alpha_{ki}G_{k}}{W_{ki}^{2}}(y_{k}-p_{i}c_{2})-\sum_{=i1}m\frac{\beta_{ik}G_{i}c_{k}}{V_{ik}^{2}}(yi-y_{k})-\mu_{k}w_{k}$$\cdots,$,$m$. From now on we will call$\alpha_{ij},\beta_{ij}$

,

(2.2)

the control parameters andthe convergence parameters. The role of the numerators$G_{i}$

and$G_{i}G_{j}$104appeared in second and third trems ofand$V_{ij}$when$A_{i}$

approach to

can easily kr-ow that$(\mathrm{x}, \mathrm{z})\in D(\mathcal{L})$$\mathcal{L}(\mathrm{P}_{\text{ノ}^{})}=0$$T_{i}$or$\mathcal{L}((\mathrm{x}, \mathrm{z}))>0$$\mathcal{L}$is to wipe out the unnecessary effect ofapproach to$A_{j}$

and for$T_{j}$$W_{ij}$, where $1\leq\dot{i},j\leq m$ . Then we$\mathrm{z}\neq 0,\dot{\mathcal{L}}_{(2.1)}((\mathrm{x}, \mathrm{z}))\leq 0$

for the solution

associated with (2.1) and (2.2). Also, the Liapunov function$\mathcal{L}$satisfies

which becomes a sufficient condition for the stability, and simultaneously,

which guarantees that$\mathcal{L}((\mathrm{x}, \mathrm{z}))arrow \mathrm{O}$as $tarrow\infty$ implies$(\mathrm{x}, \mathrm{z})arrow \mathrm{P}$, i.e., each moving

object goes to each target, where $\mathrm{P}\equiv((p_{1^{C}}1,p1c2,p_{2}c1,p_{2}c_{2}), 0)$ is an equilibrium point

for the dynamics equation (2.1) with (2.2). But in$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$and JHN[9], for $m=2$

they required some restricted conditions that the control parameters $\alpha_{ij},\dot{i},j=1,2$ andsmau in order that$\beta_{ij}.\dot{\iota}.j=1,2$

are sufiiciently

thefunctions introduced by$\mathrm{L}_{\wedge};\mathrm{a}\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{o}\mathrm{v}$$\mathrm{V}((\mathrm{x}, \mathrm{z}))arrow 0$

as$tarrow\infty$, where V is$\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$$\mathrm{V}_{\mathrm{S}\mathrm{t}\mathrm{o}}=\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}(V_{1}+\frac{\alpha_{12}}{V_{12}}+\frac{\beta_{12}}{W_{12}},$ $V2+ \frac{\alpha_{21}}{V_{21}}+\frac{\beta_{21}}{W_{21}})$,and by JHN [9]Liapnov Function.

$\mathrm{V}_{\mathrm{J}\mathrm{H}\mathrm{N}^{=}}V_{1}+V_{2}+\frac{\alpha_{12}}{V_{12}}+\frac{\alpha_{21}}{V_{21}}+\frac{\beta_{12}}{W_{12}}+\frac{\beta_{21}}{W_{21}}+\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}$,andAs the result, since they have to demand the control parameters
sufficiently small, it is difficult or impossible to control the trajectories for the system$\alpha_{12},$$\alpha_{21},$$\beta 12$(2.1) with the controlls which they determined under the Liapunov functions,$\beta_{21}$$V_{\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{n}}\mathrm{i}\mathrm{e}\mathrm{r}$. That is to say, they failed to give their’s control parameters intrinsic meansowing to some constraints for all control parameters to be small. Beside, we canand$\vee V_{\mathrm{J}\mathrm{i}\mathrm{t}\mathrm{o}}$not expect the avoidance of collsion between moving objects or moving objects andtargets in the case where the control parameters are very small, because the effect of$V12,$$\mathrm{v}^{\mathit{7}}21,$

$W12$

and

$W_{21}$

disappeares for such the cases. For the new Liapunov function,

it is casily verified that$\mathrm{o}^{1}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{S}$. $A1,$$A_{2}$

moving object

and the$A_{1}$(resp.

$\beta_{12}$plays the role of adjusting the distance between moving$\alpha_{12}(\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{p}.

$A_{2}$\alpha 21)$

plays the part of modulating the distance between) and target$T_{2}$(resp.

$T_{1}$). Therefore, we have the advantage

point of turnnning a trajectory for the system (2.1) with (2.3) into the best trajectoryby artficial. We will survey such the points$\mathrm{h}\mathrm{o}\mathrm{m}$some examples.

105

2.2
For

the

Analysis of trajectories for $m=2$$\mathrm{t}1_{1}\mathrm{e}$case of $m=2$ , where it becomes an original problem introduced by$\underline{\mathrm{f}}_{\mathrm{O}}\mathrm{r}\mathrm{m}\mathrm{s}$$\mathrm{s}_{\mathrm{t}}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}[8]$,of new Liapunov function and controlls are given as follows:

Forms of Liapunov function and controlls for $m=2$$\mathcal{L}=V_{1}+V_{2}+\frac{\alpha_{12}G_{1}}{W_{12}}+\frac{\alpha_{21}G_{2}}{W_{21}}+\beta_{1}2^{\frac{G_{1}G_{2}}{V_{12}}}$.
,$u_{1}=-A(_{X_{1}}-p_{1}c1)+ \frac{\alpha_{12}G_{1}}{W^{2}}(x1-p_{2}c_{1})+\frac{\beta_{12}}{V_{12}^{2}}(x_{1}-X_{2})c1G2-\gamma 1z_{1}$(2.3)$u_{2}=-Bv_{1}=-A(y1-p(x_{2}-p_{2}C11C2))+(X_{2}+ \frac{\alpha_{12}H_{1}}{\frac{\alpha_{21}6_{2}W_{1}^{2}}{W^{2}}}(y1-p_{2^{C}2}-p1C1)+(2-x_{1})G_{1}G2-)+\frac{\beta_{12}}{\frac{V^{2}d_{12}^{2}}{V_{12}^{2}}}(y_{X^{-y2}}1)c1c_{2}-\mu_{1}\gamma_{2^{Z}}w_{1}2’$,,$v_{2}=-B(y2-p_{2}c_{2})+ \frac{\alpha_{21}\delta_{2}}{W_{21}^{2}}(y_{2}-p_{1}c_{2})+\frac{\beta_{12}}{V_{12}^{2}}(y2-y_{1})c1c2-\mu_{2}w2$

where$A=1+ \frac{\alpha_{12}}{W12}+\frac{\beta_{12}G_{2}}{V_{12}}$and$B=1+ \frac{\alpha_{21}}{W21}+L_{V^{\frac{2G_{1}}{12}}}1$. Since the asymptotic stabilityof the system (2.1) with (2.2) was not expected in general, there exists a possibility

such as$E=\{\mathrm{x}\in \mathbb{R}^{4} :\dot{\mathcal{L}}_{(2.1)}((\mathrm{x}, \mathrm{z}))=0, \mathrm{x}\neq(p_{1}c_{1,p_{1}}c2,p2C1,p2^{C}2)\}$

When the soiution $x(t)\equiv(X_{1}(t), y_{1(t}),$$X_{2(t}),$ $y_{2}(t))$

satisfies$\mathrm{x}(t)\in E$

is not empty.

for all $t\geq 0$ , weshaii call such one the singular solution. It is difficult to find the conditions for$E=\emptyset$

because of the complexity of controlls in (2.3), but we may search for the cases where
the singulrar solutions exist under some initial conditions. In particular, one may guessthat the trajectories caused by the symetricity of initial conditions belong to the set
E.

$\underline{\mathrm{T}}\mathrm{r}\perp \mathrm{d}\mathrm{e}\mathrm{e}\mathrm{d}$, firstly, let$\mathrm{x}(t)$satisfying$x_{1}(t)-p1c1=-(X_{2}(t)-p2c1),$ $y_{1}(t)=y_{2}(t),$ $p_{2}c_{1}=p_{2}C_{2},$be$\mathrm{t}_{\mathrm{A}}^{1,}1\mathrm{e}$(2.4)

solution of (2.1) with (2.3) under initial conditions satisfying $z_{1}(0)+z_{2}(0)=0$

and $w_{1}(0)=w_{2}(0)$ . Then either$\wedge(2,$$\forall t\geq 0$$,u_{1}=\mu_{2}$$\gamma_{1}=\gamma_{2}$and $rp_{1}=rp_{2}$ . Thus, either

that there is the time$t_{1}$

or$\alpha_{12}\neq\alpha_{21}$when trajectories

Secondly, for given $m,$$t_{f}$$n\in \mathbb{R}$$\alpha_{12}=\alpha_{21},$ $\gamma_{1}=$or $rp_{1}\neq rp_{2}$ implies the fact$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{i}^{r}\mathrm{i}\mathrm{n}\mathrm{g}$hold the equation (2.4), moreover the trajectories atthe equation (2.4), where

implies that$\mu_{1}=\mu_{2}$$t_{f}$

above initial conditions don’t
when $x(t_{f})\in E$ can’t satisfy

denotes the final time when all trajectories are stopped.
, let $p_{i}c_{2}=mp_{i}c_{1}+n,\dot{i}=1,2$ and let initial conditions

satisfy $z_{1}(0)+z_{2}(0)=0$ and $w_{i}(0)=mz_{i}(0)$ . Then we can easily see that for each

106

, the solutions $y_{i}(t)=mx_{i}(t)+n,\dot{i}=1,2$ satisfies (2.1) with (2.3) if. Therefore, the case where the -th target or trajectory is between j-thandand trajectory indicates $x(t)\in E$ . Similar to the first case, one can know when$\gamma_{1}=\mu_{1}$$t\geq 0$$i$$\gamma_{2}=\mu_{2}$$\tau.\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}$$\gamma_{i}\neq\mu_{i},\dot{i}--1$

or$\dot{i}=2$, for the trajectories to escape

the line $y_{i}(t)=mx_{i}(t)+n$$\mathrm{h}\mathrm{o}\mathrm{m}$and never to return to a parallel line with $y_{i}(t)=mx_{i}(t)+n$ , because of considering
the ffist case after rotating it proper.

EXAMPLE 2.1 We start to compare with three results through this example. This
example shows that an absolute value of controls is very small than two results, andthe same$\mathrm{t}\mathrm{i}\mathrm{n}\grave{\perp}\mathrm{e}$, reaching time to targets is to be shorten largely.

i) initial condition$x_{1}$$z_{1}$$y_{1}$$w_{1}$$x_{2}$$z_{2}$$y_{2}$$w_{2}$

$-20$

1

5

5

20

$-1$

2

2

of target and size of moving object, target and$\mathrm{R}\mathrm{K}4$(Runge-Kutta4th)$\underline{\mathrm{i}\mathrm{i}_{)\mathrm{P}^{\underline{\mathrm{o}\mathrm{s}}}}^{\backslash }}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$iii)$\mathrm{c}\mathrm{c}\mathrm{n}\mathrm{t}_{1}\mathrm{o}1$$p_{1}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$p_{2}c_{2}$$rp_{1}$$rp_{2}$$rap_{1}$$rap_{2}$

$RK4$

12

$0$

$-12$

$0$

6

6

6

6

0.01

and convergence parameter$\beta_{12}$$\alpha_{12}$$\alpha_{21}$$\gamma_{1}$$\mu_{1}$$\gamma_{2}$$\mu_{2}$

1

1

1

1

1

1

1

case 1;$\mathrm{i}\mathrm{v})\max\perp \mathrm{m}^{1}\mathrm{d}\mathrm{m}$and minimum value of controls$\max$$\min$$\underline{\mathrm{v}_{\mathit{1}}^{\backslash _{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{C}}}}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}$$u_{1}$$v_{1}$$u_{2}$$v_{2}$

Stonier

21.99

38.62

1287.61513.86

JHN

17.93

17.19

443.08148.12

New

25.71

7.60

10.16

3.60

Stonier$-69.14$$-54.99$$-86.48$$-51.99$

JHN$-46.02$$-74.95$$-37.27$-15.98New$-4.90$$-28.41$$-16.84$-10.75time to targets

107

Stonier

JHN

New$A_{1}arrow T_{1}$48.31

69.13 26.40$A_{2}arrow T_{2}$26.07

38.10 11.59

Trajectories for three results in example 2.1 are illustrated in picture 2.1.

EXAMPLE 2.2 In this example, we consider the case where initial condition andcenter of target are placed on the graph $\{(x, y) : y=mx+n, m, n\in \mathbb{R}\}$ . The case
1 where targets and initial points are put on two parallel lines is that the trajectories$\mathrm{d}\mathrm{c}\mathrm{L}’ \mathrm{t}$go to the targets, but we can make the trajectories move to the targets bychanging the value of$\mathrm{t}^{-}\mathrm{L})\mathrm{e}$$\alpha_{12}$different to$\alpha_{21}$. The case 2 where all datum are put online $y=2x+6$ can become asyptotically stable by virtue of varying the values of

convergent parameters, for example, $\mu_{1}=4$ .
i) initial condition$x_{1}$$z_{1}$$y_{1}$$w_{1}$$x_{2}$$z_{2}$$y_{2}$$w_{2}$

Case 1

$-20$

1

10

1

20

$-1$

10

1

Case 2

$-10$

1

$-14$

2

6

$-1$

18

$-2$

ii) position of target and size of moving object, target and RK4iii)$p_{1}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$p_{2}c_{2}$$rp_{1}$$rp_{2}$$rap_{1}$$rap_{2}$$RK4$Case 1

10

5

$-10$

5

5

5

5

5

0.05

Case 2

2

10

$-5$

$-4$

3

4

4

4

0.05

$\mathrm{c}\mathrm{c}\mathrm{n}\mathrm{t}_{1}\cdot 01$

and convergence parameter$\beta_{12}$$\alpha_{12}$$\alpha_{21}$$\gamma_{1}$$\mu_{1}$$\gamma_{2}$$\mu_{2}$case 1

1

1(2)

1

3

3

3

3

case 2

1

2

3

3

3(4)

3

3

108

$\mathrm{i}\mathrm{v})\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{m}$and minimum value of controls$\max$$\min$$\mathrm{v}^{\backslash }\mathit{1}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}$$u_{1}$$v_{1}$$u_{2}$$v_{2}$

case 1

28.32

17.95

22.88

28.23

case 2

9.61

17.22

7.63

15.49

case 1$-14.83$$-9.29$$-29.57$-11.02case 2$-5.78$$-13.79$$-7.95$-15.91time to targets
case 1 case 2$A_{1}arrow T_{1}$

23.6

29.6

$A_{2}arrow T_{2}$

20.6

24.0

Trajectories for the case 1 and 2 in example 2.2 are illustrated in picture 2.2 and picture

2.3, respectively.

Analysis of the trajectories for $m\geq 3$

2.,3In order to verify that even for $m\geq 3$ the new Liapunov function has no obstacle$\mathrm{n}\mathrm{n}\mathrm{d}oi\mathrm{g}arrow \mathrm{n}$for$\mathrm{t}\mathrm{l}\underline{\rceil}\mathrm{e}$a path for the collision avoidance control problem, we will give some examples
cases of $m=3,$ $m=4$ and $m=5$ . It may occure the case, similar to the case$m=2$ , that the solution of the system (2.1) with (2.2) belongs to an invariant set or

becomes asymptotically stable according to varying the values of parameters and initialconditions. Here we can get an information about the positions where the trajectories
stop on the way, which is occurred when the trajectories fall into a dead alley. Therefore,

it is necessary to block up the entrance of a dead alley for the trajectories not to enter

into a dead alley, which can be obtained by taking the control parametertarget where a dead alley arises sufficiently large.

2.3.1

An example for $m=3$

Form of Liapunov function and controllers for $m=3$$\mathcal{L}$$=$$V_{1}+V_{2}+V3+G_{1}( \frac{\beta_{12}G_{2}}{V_{12}}+\frac{\alpha_{12}}{W_{12}}+\frac{\alpha_{13}}{W_{13}})$$\alpha_{ij}$around a

109

$+$

$u_{1}$$v_{1}$$u_{2}$$v_{2}$$u_{3}$$u_{3}$$G_{2}( \frac{\beta_{23}G_{3}}{V_{23}}+\frac{\alpha_{21}}{W_{21}}+\frac{\alpha_{23}}{W_{23}})+G_{3}(\frac{\beta_{13}G_{1}}{V_{13}}+\frac{\alpha_{31}}{W_{31}}+\frac{\alpha_{32}}{W_{32}})$$=$$-(x_{1}-p1^{C}1)(1+ \frac{\alpha_{12}}{W_{12}}+\frac{\alpha_{13}}{W_{13}}+\frac{G_{2}\beta_{12}}{V_{12}}+\frac{G_{3}\beta_{13}}{V_{13}})-\gamma_{1}z_{1}$$+$$G_{1} \lfloor\frac{\beta_{12}G_{2}}{V_{12}^{2}}(x_{1}-arrow x2)-\frac{\beta_{13}G_{3}}{V_{13}^{2}}(x_{3}-X1)+\frac{\alpha_{12}}{W_{12}^{2}}(X_{1}-p_{2^{C_{1})+}}\frac{\alpha_{13}}{W_{13}^{2}}(X_{1}-p_{3^{C}}1)]$$=$$-( \prime v_{1}-p_{1^{C}2})(1+\frac{\alpha_{12}}{W_{12}}+\frac{\alpha_{13}}{W_{13}}+\frac{G_{2}\beta_{12}}{V_{12}}+\frac{G_{3}\beta_{13}}{V_{13}})-\mu_{1}w_{1}$$-|\tau$$G_{1}[ \frac{\mathcal{B}_{12}G_{2}’}{V_{12}^{2}}(y_{1}-y_{2})-\frac{\beta_{13}G_{3}}{V_{13}^{2}}(y3-y1)+\frac{\alpha_{12}}{W_{12}^{2}}(y1-p2c2)+\frac{\alpha_{13}}{W_{13}^{2}}(y_{1}-p_{3}c_{3})]$$=$$-(a_{2}-p_{2}c1)(1+ \frac{\alpha_{21}}{W_{21}}+\frac{\alpha_{23}}{W_{23}}+\frac{G_{1}\beta_{12}}{V_{12}}+\frac{G_{3}\beta_{23}}{V_{23}})-\gamma_{2}z_{2}$$+$$\llcorner \mathit{0}_{\tau}\underline,\lfloor\lceil_{\frac{\beta_{23}G_{3}}{V_{23}^{2}}(X}2-x3)-\frac{\beta_{12}G_{1}}{V_{12}^{2}}(x1-x2)+\frac{\beta_{21}}{W_{21}^{2}}(x_{2}-p_{1^{C}}1)+\frac{\beta_{23}}{W_{23}^{2}}(X_{2}-p3C_{1})]$$=$$-(y_{2}-p_{2^{C_{2}}})(1+ \frac{\alpha_{21}}{W_{21}}+\frac{\alpha_{23}}{W_{23}}+\frac{G_{1}\beta_{12}}{V_{12}}+\frac{G_{3}\beta_{23}}{V_{23}})-\mu_{2}w_{2}$$+$$G_{2}[ \frac{\prime\theta_{23}G_{3}}{V_{23}^{2}}(y2-y_{3})-\frac{\beta_{12}G_{1}}{V_{12}^{2}}(y1-y2)+\frac{\beta_{21}}{W_{21}^{2}}(y2-p1c_{2})+\frac{\beta_{23}}{W_{23}^{2}}(y2-p3^{C}2)]$$=$$-(x_{3}-p3^{C}1)(1+ \frac{\alpha_{31}}{W_{31}}+\frac{\alpha_{32}}{W_{32}}+\frac{G_{1}\beta_{13}}{V_{13}}+\frac{G_{2}\beta_{23}}{V_{23}})-\gamma_{3^{Z_{3}}}$$+$$G_{3}[ \frac{\beta_{13}G_{1}}{V_{13}^{2}}(_{X_{3}}-X_{1})-\frac{\beta_{23}c_{2}}{V_{23}^{2}}(x2-X3)+\frac{\alpha_{31}}{W_{31}^{2}}(_{X_{3p)+\frac{\alpha_{32}}{W_{32}^{2}}(c}}-1c1x3^{-}p_{21})]$$=$$-(y_{3}-p_{3^{C_{2})}}-\mu_{3}w_{3}$$+$$C \pi 3[\frac{\beta_{13}G_{1}}{V_{13}^{2}}(y_{3}-y_{1})-\frac{\beta_{23}G_{2}}{V_{23}^{2}}(y_{2}-y_{3})+\frac{\alpha_{31}}{W_{31}^{2}}(y_{3}-p_{1^{C_{2})}}+\frac{\alpha_{32}}{W_{32}^{2}}(y_{3}-P2^{C}2)]$

EXAMPLE 2.3 In this example, we consider the case where targets and initial points
are ccncentrated around the origin, which are considered as a difficult situation tocontrol the trajectory. In the case 1, the trajectories do not go to the targets inthe desired time, and asymptotically stable under the case 2 where we change thecontrol parameters
directicn,$A_{2}$

tosimultaneously.

$T_{1}$$\alpha_{12},$$\alpha_{23}$and$A_{3}$andto$T_{2}$$\alpha_{31}$, which play a role of making$A_{i}$

travel

$T_{3}$

in theand removing of the entrance into three dead alleys

110

condition$\underline{\mathrm{i})}\underline{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$$w_{1}$$x_{2}$$z_{2}$$y_{2}$$w_{2}$$x_{3}$$w_{3}$$y_{1}$$y_{3}$$z_{1}$$z_{3}$$x_{1}$

$-7$

1

6

1

7

1

6

1

$0$

1

$-5$

1

of target and size of moving object, target and RK4ii) $\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}_{\underline{\mathrm{O}}}\mathrm{n}$$p_{1}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$p_{2}c_{2}$$p_{3}c_{1}$$p_{3}c_{2}$

3.5

$0$

$-3.5$

$0$

$0$

6

, and $RK4=0.05$$rp_{i}=3.5,$ $rap_{i}=3,\dot{i}=1,2,3$iii) control and convergence parameter$\alpha_{31}$$\alpha_{32}$$\gamma_{1}$$\mu_{1}$$\mu_{3}$$\alpha_{21}$$\alpha_{23}$$\gamma_{3}$$\alpha_{12}$$\alpha_{13}$$\mu_{2}$$\mathcal{B}_{13}$$\beta_{23}$$\gamma_{2}$$\beta_{12}$case 1

5

5

5

1

1

1

1

1

1

5

5

5

5

5

5

case 2

5

5

5

15

1

1

15

15

1

5

5

5

5

5

5

tirne to targets

reaching$\underline{/}-$$\underline{\mathrm{i}}\mathrm{v}^{)}$$A_{1}arrow T_{1}$$A_{2}arrow T_{2}$$A_{3}arrow T_{3}$

20.0

20.0

20.5

case 2$\mathrm{T}\mathrm{r}\mathrm{a}_{d}\mathrm{i}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}$for the case 1 and 2 in example 2.3 are illustrated in picture 2.4.

An example for $m=4$

2.3.2

EXAMPLE 2.4 This example may not occure in a realistic system, but it is a very incase. Since the moving objects are closed up, they may get out of a workspace,collide each other or a moving object may collide with a target.
otherwise they$\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\vee’ \mathrm{t}\cup \mathrm{l}\mathrm{i}\mathrm{I}$$\mathrm{m}\hat{\circ}\mathrm{y}$$\mathrm{T}^{\tau}\mathrm{n}\llcorner \mathrm{s}$, it is necessary to adjust the strength between the moving objects to weak, whichmeans making the control parameters$\beta_{ij}$be small enough.

i) initial condition$y_{2}$$w_{2}$$x_{3}$$w_{3}$$x_{4}$$y_{4}$$w_{4}$$x_{2}$$z_{2}$$z_{4}$$w_{1}$$y_{3}$$z_{1}$$y_{1}$$z_{3}$$x_{1}$

$-16$

1

$-1$

$-13$

1

$0$

1

$-10$

1

$0$

$-1$

$-7$

1

$0$

$0$

1

of target and size of moving object, target and RK4$\mathrm{i}^{\mathrm{i}:}/\mathrm{p}\mathrm{o}\mathrm{i}\underline{\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}"$$p_{1}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$p_{2}c_{2}$$p_{3}c_{1}$$p_{3}c_{2}$$p_{4}c_{1}$$p_{4}c_{2}$

$0$

$0$

7

$0$

12

$0$

15

$0$

111

$rp_{1}=4$,$rp_{2}=3rp_{3}=2rp_{4}=1$$rap_{i}=2,\dot{i}=1,2,3,4$.

control and convergence parameter$\mathrm{i}\underline{\mathrm{i}}\mathrm{i})$1. $\gamma_{i}=\mu_{i}=5,1\leq i\leq 4,$ $\beta_{ij}=0.05,1\leq\dot{i}<j\leq 4$ and $\alpha_{ij}=1,1\leq i,j\leq 4,\dot{i}\neq$$j,$$RK4=0.005$ .

2. $\gamma_{i}=\mu_{i}=5,1\leq\dot{i}\leq 4,$ $\beta_{ij}=0.01,1\leq i<j\leq 4$ and $\alpha_{ij}=1,1\leq i,j\leq 4,$$j_{\iota}.RK4=0.01$$\prime \mathrm{r}_{\mathrm{r}}\mathrm{a}_{\vee}\mathrm{i}\mathrm{e}\mathrm{c}\mathrm{t}_{0}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{S}$

2.3.3

.

for the case 1 and 2 in example 2.4 are illustrated in picture$\mathrm{p}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{u}^{\sim}\underline{l}\mathrm{e}\sim 2.5(\mathrm{b})$$i\neq$$2.5(\mathrm{a})$and.
examples for $m=5$$\mathrm{S}\mathrm{o}\mathrm{r}_{-}:1\mathrm{e}$We present two interesting examples where the shape of located targets has four deadalleys and where all moving objects are concentrated on the very small workspace.

EXAMPLE 2.5 When we regard the$\mathrm{o}\mathrm{t}\mathrm{h}\cap \mathrm{r}$as a big pillar placed on the plane and thetargets as some small bodies which are attached on the$T_{1}$, there exist four deadwhich swallow all moving objects $A_{i},\dot{i}=2,3,4,5$ .

$\mathrm{a}^{\underline{]}}1\mathrm{e}\mathrm{y}\mathrm{s}$$\mathrm{i}_{j^{t}}$$T_{1}$

initial condition$\underline{\mathrm{z}\mathrm{i}\mathrm{i}\grave{)}}$$x_{1}$$z_{1}$$y_{1}$$w_{1}$$x_{2}$$z_{2}$$y_{2}$$w_{2}$$x_{3}$$z_{3}$$y_{3}$$w_{3}$

$0$

$0$

$0$

$0$

$-15$

1

$0$

$-1$

$0$

1

$-15$

1

$x_{4}$$z_{4}$$y_{4}$$w_{4}$$x_{5}$$z_{5}$$y_{5}$$w_{5}$

$15$

$-1$

$0$

1

$0$

$-1$

15

$-1$

positioa of target and size of moving object, target and RK4$p_{1}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$P2^{C}2$$P3^{C}1$$P\mathrm{s}c_{2}$$p_{4}c_{1}$$p_{4}c_{2}$$p_{5^{C}1}$

$0$

$0$

8

$0$

$0$

8

$-8$

$0$

$0$

$rp_{1}=5,$$rp_{i}=3.2\leq\dot{i}\leq 5$$rap_{i}=2,1\leq\dot{i}\leq 5$,

and $RK4=0.05$ .

$P_{5}c_{2}$

$-8$

112

iii) control

parameter$\mathrm{a}\mathrm{n}\mathrm{d}\underline{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}$$\gamma_{i}=\mu_{i}=3,1\leq i\leq 5,$$\beta_{ij}=1,1\leq\dot{i}<j\leq 4$,$\alpha_{21}=\alpha_{3}1=\alpha_{4}1=\alpha 51^{-}-20,$ $\alpha 25=\alpha 32=\alpha 43=\alpha_{54}=0.5$

and other than then$\alpha_{ij}=1$.

Trajectories for $m=5$ in example 2.5 are illustrated in picture 2.6.

EXAMPLE 2.6 Since all moving objects are closed up in the very small workspace,
we have to make the control parameters $\beta_{ii+1},\dot{i}=1,2,3,4$ be small to prevent moving
object and moving object or moving object and target

then it is necessary to arrange the control parameters$\mathrm{h}\mathrm{o}\mathrm{m}$$\alpha_{ij}$colliding each other, andto obtain the smooth of the

trajectories. The results are below.
$\underline{\mathrm{i}})$

initial condition$z_{1}$$x_{1}$

0.0 1.0$y_{1}$$w_{1}$$x_{2}$$-5.0$

1.0 4.75$x_{4}$$z_{4}$$-2.95$$y_{4}$

1.0 4.05$w_{2}$$x_{3}$$z_{3}$$z_{2}$$y_{2}$$-1.0$$-1.55$$w_{4}$$x_{5}$$z_{5}$$y_{5}$$w_{5}$$-1.0$$-4.75$1.0$-1.55$1.0$y_{3}$$w_{3}$

1.0 2.95 1.0 3.0 1.0$\underline{\mathrm{i}}\mathrm{i}_{\text{ノ^{}1}}^{\backslash }\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}T\mathrm{i}\mathrm{o}\underline{\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{i}\mathrm{Z}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{C}\mathrm{t},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{R}\mathrm{K}4}$$p_{4}\overline{|}c_{1}$$p_{1}c_{2}$$p_{2}c_{1}$$P2^{C}2$$p_{3}c_{1}$$p_{3}c_{2}$$p_{4}c_{1}$$p_{4}c_{2}$$p_{5}c_{1}$

$0$

10

$-9.5$

3.1$-5.9$$-8.1$

5.9$-8.1$

9.5$rp_{i}=4,1\leq i\leq 5$$rap_{i}=2,1\leq\dot{i}\leq 5$$\mathrm{i}\sim \mathrm{i}\mathrm{i})$$P_{5}c_{2}$3.1,

and $RK4=0.05$ .

control and convergence parameter

1. $\gamma_{i}=\mu_{i}=5,1\leq\dot{i}\leq 5,$ $\beta_{ij}=0.05,1\leq i<j\leq 5$ and $\alpha_{ij}=1,1\leq i,j\leq 4,\dot{i}\neq$$J$.

2. $\gamma_{i}=\mu_{i},=5,1\leq i\leq 5,$ $\beta_{ij}=0.05,1\leq i<j\leq 5$ ,$\alpha_{13}=\alpha_{2}4=\alpha_{35}=\alpha_{41}=\alpha_{52}=10$$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{j}\mathrm{e}\mathrm{C}\mathrm{t}_{0}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}$and other than them$\alpha_{ij}=1$.

lor the case 1 and 2 in example 2.6 are illustrated in picture 2.7.

113
3Conclusion

The most important feature of this paper is that the Liapunov function for the system(2.1) is setted up skilfully, so that the cotrolls and convergence parameters play their$\mathrm{p}\iota$

oper roles such as altering the trajectory of the system (2.1) into the desired one. It

is obvious that the system (2.1) with (2.2) is stable. However, under the new Liapunov
function, the asymptotic stability for the system (2.1) with (2.2) is not verified in. In$\mathrm{g}\mathrm{e}\mathrm{n}^{\mathrm{x}}\backslash \cdot \mathrm{r}\mathrm{a}1$$\mathrm{f}\mathrm{a}\mathrm{c}\tau’$, for $m=2$ there were many examples of the trajectories being stopped

on the way, but we could avoid it by means of manipulating every condition to break out

a symmetrical condition. We have hardly a stopping situation halfway, because the new
Liapumov function have many parameters which are not necessary symmetry. When$m\geq 3$, we were confronted with many situations that the trajectories belong to theinvariant set, but most situations were solved by adjusting the control parameters. If

one want to Lrlake the state which is not asymptotically stable be asymptotically stable,
we he.ve to compose another Liapunov function with relation to a neural system, but
it is a problezn in the future.

114

PICTURES

Picture 2.1. Trojectories for three results.

Picture 2.2. Trajectories for the case 1.

Picture 2.3. Trajectories for the case 2.

115

Picture 2.4. Trajectories for the case 1 and 2.

Picture 2.5. (a)$\mathrm{T}\mathrm{r}\mathrm{o}|\mathrm{e}\mathrm{C}\iota\acute{\mathrm{O}}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}$for the case 1.

116

Picture 2.5. (b) Trajectories for the case 2.

Picture 2.6. Trajectories for$\mathrm{m}=5$

.

117

Picture 2.7. Trajectories for the case 1 and 2.

118

References[1] L.T. Grujic, Exact determination of a Lyapunov function and the asymptotic stadomain, INT. J. S YSTEMS SIC., VOL. 23, NO. 11, 1871-1888, 1992.
$\mathrm{b}_{-}:1\mathrm{i}\mathrm{t}\mathrm{y}$$!^{2]}\mathrm{r}$L.T. Gru.ic, Complete exact solution to the Lyapunov stability problem: Timesystems with differentiable motions, Nonlin ar An alysis, Theory,varying$e$$\mathrm{r}\perp \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$Methods&Applica tion , Vol. 22, No. 8, 971-981, 1994.
$s$

G. Leitrnann, Guaranteed avoidance strategies, J. of Optimiza on Theory and Ap$t\mathrm{i}$$|\mathrm{r}_{3]}k$plica tions, Vol. 32, No. 4, 569-577, 1980.
[4] T. L. Vincent and$\mathrm{J}.\mathrm{M}$. Skowronski, Controllability with capture, J. of Optimiza tion

Theory and Appli cations, Vol. 29, No. 1, 77-86, 1979.

[5]$\mathrm{J}.\mathrm{M}$. Skowronski and T. L. Vincent, Playability with and without capture, J. of

Optirn$‘\lfloor \mathrm{r}_{6]}\mathrm{J}.\mathrm{M}$$iz\mathrm{a}$tion Theory and Applications, Vol. 36, No. 1, 111-128, 1982.

. Skowronski and$\mathrm{R}.\mathrm{J}$. Stonier, The barrier in a pursuit-evasion game with two, Comput. Math. Applic., Vol. 13, No. 1-3, 37-45, 1987.

$\mathrm{t}\mathrm{a}^{\gamma_{-}}\cdot \mathrm{g}\mathrm{e}\mathrm{t}\mathrm{s}$$\mathrm{r}7]\iota \mathrm{R}.\mathrm{J}$Th$r_{8^{\backslash }}\lfloor\rfloor$.

$\mathrm{S}\mathrm{t}\mathrm{c}\mathrm{r}_{1}\mathrm{i}\mathrm{e}\mathrm{r}$$\mathrm{e}o\mathrm{r}\gamma$, On qualitative differential games with two targets, J. of Optimization

an Application , Vol. 41, No. 4, 587-599, 1983.
$d$

$s$

R. J. Stonier, Use of Liapunov techniques for collision-avoidance of robort arms,

Control and Dynamic Systems, Vol. 35, 185-214, 1990.
[9] J.Vanualailai, Junhong Ha and Shin-ichi Nakagiri, Liapunov function-based control
functions for a collision avoidance control of two mass-points moving in a plane, toappear.