数理解析研究所講究録
1048 巻 1998 年 26-32

26The Limiting Behavior of Fuzzy Statesin Dynamic Fuzzy Systems(Yuji YOSHIDA)

吉田祐治

北九州大学経済学部

1. Introduction and notations
The limiting behavior of fuzzy states in dynamic fuzzy systems has been studied by
Kurano et al. [4] and Yoshida et al. [7]. Under a contractive condition for the fuzzy
relation, [4] showed that the limiting fuzzy state is a unique solution of a fuzzy relational
equation. Also, [7] discussed the limit theorem in a monotone case. In this paper, weconsider the limit theorem when fuzzy relations satisfy the transitive property. We show
that, in this case, the limiting fuzzy state is a solution of the fuzzy relational equation.
But the equation does not necessarily have a unique solution similarly to the monotone
case, therefore we need to investigate the space of the solutions of the equation.
The existence and the uniqueness of the solutions of the fuzzy relational equation has
been studied by Kurano et al. [5] under some assumptions. In the transitive case, thispaper makes clear the structure of the space of the solutions of the fuzzy relational equation, and we give a simple characterization of the limiting fuzzy state by the fundamental
solutions for the numerical calculation of the limiting fuzzy state.
We use some notations in [5]. Let
be a compact metric space. Let $C(E)$ be thecollection of all non-empty closed subsets of , and let be the Hausdorff metric on$C(E)$ . Then it is well-known ([3]) that $(C(E), \rho)$ is a compact metric space. Let
be the set of all fuzzy sets : $Earrow[0,1]$ which are upper semi-continuous and satisfy$\sup_{x\in E}\tilde{s}(x)=1$ . For$\alpha\in[0,1]$ , is defined by, the -cut
$E$

$E$

$\rho$$\mathcal{F}(E)$$\tilde{s}$$\tilde{s}\in \mathcal{F}(E)$$\alpha$$\tilde{s}_{\alpha},$$\tilde{s}_{\alpha}:=\{x\in E|\tilde{s}(x)\geq\alpha\}(\alpha\neq 0)$

and$\tilde{s}_{0}:=\mathrm{c}1\{x\in E|\tilde{s}(x)>0\}$,where cl means the closure of a set. Let : $E\cross Earrow[0,1]$ be a fuzzy relation on
satisfying
for $x\in E$ . A fuzzy relation is called “transitive” (see Klir andYuan [2] if it satisfies$E$$\tilde{q}$$\tilde{q}(x, \cdot)\in \mathcal{F}(E)$$\tilde{q}$$)$$\tilde{q}(x, y)\geq\sup_{z\in E}\{\tilde{q}(X, Z)\wedge\tilde{q}(Z, y)\}$,$x,$ $y\inE$.

Throughout this paper, we assume is transitive. In Sections 1 and 2, we consider thespace of solutionsof the following fuzzy relational equation (see [5]) :$\tilde{q}$$\tilde{p}\in \mathcal{F}(E)$$\tilde{p}(y)=\sup_{Ex\epsilon}\{\tilde{p}(x)\wedge\tilde{q}(X, y)\}$

where

$a$

A $b:= \min\{a, b\}$ for real numbers

$a$

and

$b$

,$y\in E$,(1). By the solutions of (1) (see $[4, 7]$ for

fuzzy relations), in Section 3 we discuss the limiting behavior of
the sequence of fuzzy stateswith an initial fuzzy state
which
is defined by$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}/\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}$$\{\tilde{s}_{n}\}_{n=}^{\infty}0\subset \mathcal{F}(E)$$\tilde{s}_{0}:=\tilde{s}$,and$\tilde{s}_{n+1}(y):=\sup_{x\in E}\{\tilde{s}_{n}(x)\wedge\tilde{q}(x, y)\}$$\tilde{s}\in \mathcal{F}(E)$,$y\in E$

for $n=0,1,2,$$\cdots$.

(2)

27
Crisp sets $\tilde{q}_{\alpha}(x)(x\in E, \alpha\in[0,1])$ are defined byforfor $\alpha=0$ .

$\{y\in E|\tilde{q}(x, y)\geq\alpha\}$$\alpha\neq 0$$\tilde{q}_{\alpha}(X):=\{$$\mathrm{c}1\{y\in E|\tilde{q}(x, y)>0\}$: $E[]arrow C(E)$ is continuous for all $\alpha\in[0,1]$ . We
In this paper, we assume the map
for $D\in C(E)$ , and then we note that
: $C(E)arrow C(E)$ .
also define
is defined iteratively by
For $x\in E$ and $\alpha\in[0,1]$ , a sequence$\tilde{q}_{\alpha}(\cdot)$$\tilde{q}_{\alpha}(D):=\bigcup_{x\in D}\tilde{q}\alpha(X)$$\tilde{q}_{\alpha}$$\{\tilde{q}_{\alpha}^{n}(x)\}_{n}^{\infty}=0\subset C(E)$$\tilde{q}_{\alpha}^{0}(x):=\{x\}$and,$\tilde{q}_{\alpha}^{n+1}(x):=\tilde{q}_{\alpha}(\tilde{q}_{\alpha}(nX))$,$n=0,1,2,$

Then we have the following lemma for a sequence of fuzzy relations$\tilde{q}^{1}:=\tilde{q}$,and$\tilde{q}^{n+1}(x, y):=\sup_{z\in E}\{\tilde{q}^{n}(x, z)\wedge\tilde{q}(z, y)\}$,$x,$ $y\in$\cdots$.

$\{\tilde{q}^{n}\}_{n=1}^{\infty}$

defined by

for $n=1,2,$E$$\cdots$.

(3)

Lemma 1.1. The following (i) and (ii) hold:(i) For all $n=1,2,$$\cdots$,,$\tilde{q}^{n}(x, y)\geq\tilde{q}(n+1x,y)$and,$\tilde{q}_{\alpha}^{n+1}(X)\subset\tilde{q}_{\alpha}^{n}(x)$(ii)$\tilde{q}^{n}(x, x)=\tilde{q}(x, x)$$x,$ $y\in$x\in E,$for all $x\in E$ an $dn=1,2,$$\cdots$

E$,(4)$\alpha\in[0,1]$.

(5)

.

Let
$R_{\alpha}:=\{x\in E|x\in\cup^{\infty}n=1\tilde{q}_{\alpha}(nX)\}$

Each state of$R_{\alpha}$,$\alpha\in[0,1]$.

is called “ -recurrent” (see [8]). From Lenuna l.l(i), we have$\alpha$$R_{\alpha}=\{x\in E|\tilde{q}(X, X)\geq\alpha\}$,$\alpha\in(0,1]$.

(6)

Let $x\in E$ . The following crisp sets are used in [5] to analyze the solutions of (1):$F_{\alpha}(x):= \bigcup_{n=0}^{\infty}\tilde{q}^{n}\alpha(x)$and$\hat{F}_{\alpha}(x):=,\bigcap_{\alpha<\alpha}\mathrm{c}1\{F_{\alpha’}(X)\}(\alpha\neq 0)$,

and$\hat{F}_{0}(x):=\mathrm{c}1\{F_{0}(X)\}$.

In the transitive case, by Lemma l.l(i) they are reduced to the following (7):$\hat{F}_{\alpha}(x)=\{x\}\cup\tilde{q}_{\alpha}(x)$Especially we have$\hat{F}_{\alpha}(z)=\tilde{q}_{\alpha}(z)$,,$x\in E,$$z\in R_{1},$$\alpha\in[0,1]$$\alpha\in[0,1]$.

(7)

.

(8)

Therefore, we obtain the following lemma.

Lemma 1.2

(Kurano et al. [5, Lemma$1.2(\mathrm{i})$and Theorem 2.1$(\mathrm{i}\mathrm{i})]$).

28(i) For$\tilde{p}\in \mathcal{F}(E),\tilde{p}$satisfies (1) if and only if$\tilde{q}_{\alpha}(\tilde{p}_{\alpha})=\tilde{p}\alpha$’$\alpha\in[0,1]$.

(9)(ii) Let $z\in R_{1}$ . Defin a fuzzy state$e$$\tilde{p}^{z}(x):=\sup_{\alpha\in 10,1]}\{\alpha\wedge 1_{\hat{F}_{\Phi}(}z)(X)\}=\tilde{q}(_{Z,x})$Then,$x\in E$.

(10)satisfies (1).

$\tilde{p}^{z}\in \mathcal{F}(E)$

2. The space of the solutions
We putproperty:$\mathcal{P}:=${Lemma 2.1$\tilde{p}\in \mathcal{F}(E)|\tilde{p}$is a solution of (1)}. Then the space(Kurano et al. [5, Theorem
sa tisfy$\{\alpha^{k}\in[0,1]|k=1,2, \cdots, l\}$$2.2(\mathrm{i}\mathrm{i})]$$\tilde{p}\in \mathcal{P}$

has the following). Let $\tilde{p}^{k}\in P(k=1,2, \cdots, l)$ , an let
. Pu
$d$

$\sup_{k=1,2,\cdots,l}\alpha^{k}=1$$\tilde{p}(x):=k=1,2\max,\cdots,\{l\alpha^{k}\wedge\tilde{p}^{k}(x)\}$Then$\mathcal{P}$$t$,$x\in E$.

(11).

The purpose of this section is to prove an inverse of the statement of Lemma 2.1.
Namely, we represent general solutionsof (1) by the fundamental solutionsof(10). From now on, we assume. We identify the states ofwith respect to thefollowing equivalent(see [5] and (8)): For,$\tilde{p}\in P$$\tilde{p}^{z}$$R_{1}\neq\emptyset$$R_{1}$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\sim \mathrm{o}\mathrm{n}R_{1}$$z_{1},$

means that$z_{1}\sim z_{2}$Then we put$R_{1}^{\sim}:=R_{1}/\sim$Assumption A. Let$z_{1}\in\tilde{q}_{1}(z_{2})$and$\alpha\neq 0$

and $A\in C(E)$ . If$\tilde{q}_{\alpha}(A)=A$$\{\alpha^{z}\in[0,1]|z\in R_{1}^{\sim}\}$.

holds, then.

From now on, we suppose that Assumption A holds (Let$z_{2}\in\tilde{q}_{1}(z_{1})$.

$R_{\alpha}\cap A\subset\cup\tilde{q}\alpha(zz\in R_{1}\sim_{\mathrm{n}}A)$

Theorem 2.1.

$z_{2}\in R_{1}$$\mathrm{c}.\mathrm{f}$.

$[5$, Assumption A3]).

be ation of (1). Then, there exists a family of coefficientssatisfying
and$\tilde{p}$$sol\mathrm{u}$$\sup_{z\in R_{1}^{\sim\alpha^{z}}}=1$$\tilde{p}(x)=z\in R_{1}\sup_{\sim}\{\alpha\wedge z\tilde{p}^{z}(_{X})\}=$ $\sup_{\sim,z\epsilon R1}\{\alpha\wedge z\tilde{q}(Z,x)\}$,$x\in E$.

(12)

29
3. A Limit Theorem
We uses the convergency of fuzzy states in following sense.

Definition (see [7]). Let$\lim_{narrow\infty}\tilde{s}_{n}=\tilde{p}$where$\tilde{s}_{n,\alpha}$$\tilde{s}_{n},\tilde{p}\in \mathcal{F}(E)$means

are the -cuts of$\alpha$$\tilde{s}_{n}$. Then$\rho(\tilde{s}_{n,\alpha},\tilde{p}_{\alpha})arrow \mathrm{O}$and$\rho$$(narrow\infty)$

for all $\alpha\in[0,1]$ ,is the given Hausdorff metric.

. In this section, first we discuss the convergenceFix an initial fuzzy state
defined by (2), and we prove the limiting fuzzyof the sequence of fuzzy statesstate is a solution of the fuzzy relational equation (1). Next, we give a representation ofthe limiting fuzzy state for the numerical calculation, by using the characterization (12).
$\tilde{s}\in \mathcal{F}(E)$$\{\tilde{s}_{n}\}_{n=}^{\infty}0$

Lemma 3.1. Let $\alpha\in[0,1]$ . Then$\lim_{narrow\infty}\tilde{s}_{n},\alpha=\cap n\geq 1x\in\cup\tilde{S}\alpha\tilde{q}^{n}\alpha(_{X})=x\in\bigcup_{\overline{s}_{\alpha}n}\cap\tilde{q}\alpha(n)\geq 1x$

.

(13)

We use the following lemma to construct the limiting fuzzy state.

Lemma 3.2 ([4, 6]). Let a family of subsets $\{D_{\alpha}|\alpha\in[0,1]\}\subset C(E)$ satisfies theand $(b)$ :followin condition$s(\mathrm{a})$$g$for $0\leq\alpha’<\alpha\leq 1$ .

(a)$D_{\alpha}\subset D_{\alpha’}$(b)$\lim_{\alpha’\uparrow}\alpha D_{\alpha’}=D_{\alpha}$

Then

for $\alpha\in(0,1]$ .

$\alpha\in[0,1]$, where$1_{D}$$x\in E$ ,satisfies
anddenotes the characteristic fun ction of a set $D\in C(E)$ .

$\tilde{s}(x):=\sup_{\alpha\in[0,1}]\{\alpha\wedge 1_{D_{\alpha}}(x)\},$$\tilde{s}\in \mathcal{F}(E)$$\tilde{s}_{\alpha}=D_{\alpha}$

for all

From Lemma 3.1, we define$\tilde{p}(x):=\sup\{\alpha\wedge 1_{D_{\alpha}}(x)\}$,,$x\in E$

(14)$\alpha\in[0,1]$

where’$D_{\alpha}:= \lim_{narrow\infty}\tilde{s}n,\alpha=\bigcap_{n\geq 1x}\bigcup_{\alpha}\tilde{q}_{\alpha}(\epsilon\tilde{s}nX)=x\in\cup \mathrm{n}\tilde{q}_{\alpha}^{n}(x)\overline{S}\alpha n\geq 1$

Theorem 3.1.
(i)(ii)$\tilde{p}h$.

(15)

as the following property (i) and (ii):$\tilde{p}=\lim_{narrow\infty n}\tilde{S}\in \mathcal{F}(E)$$\tilde{p}$$\alpha\in[0,1]$.

is a solution of (1).

Finally, by using the limit theorem (Theorem 3.1) and the characterization of the, we give a simple representation
solutions of the fuzzy relational equationof the limiting fuzzy state for the numerlcal calculation (Section 4).
$\vee(.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}2.1)$$\tilde{p}$30Theorem 3.2. The coefficients in Theorem 2.1 are given by$\alpha^{z}=\tilde{q}(_{\tilde{S}})(Z)=\tilde{p}(Z)$

where

,

$z\in R_{1}^{\sim}$

,

(16)

. Namely,$\tilde{q}(\tilde{s})(z)=\sup_{x\in E}\{\tilde{\mathit{8}}(x)\wedge\tilde{q}(x, z)\}$,$\tilde{p}(x)=\sup_{z\in R^{\sim}1}\{\tilde{q}(\tilde{S})(_{Z})\wedge\tilde{q}(z,x)\}=\sup_{1}\{\tilde{p}(Z)\wedge\tilde{q}z\epsilon R\sim(Z, x)\}$$x\in E$.

(17)

4. A numerical example‘al example to illust.r ate our results in Sections 2
We consider a one-dimensionaland 3. Let $E=[-2,2]$ . We give the transitive fuzzy relation (see Fig.1) by$\mathrm{n}\mathrm{u}‘ \mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}_{\mathrm{C}}\backslash$$\tilde{q}$’$( \frac{j(y)+y-2x}{y-!(y)}0)\wedge 1$$\tilde{q}(x, y)=$111$\backslash 0$

if $xy>0,$ $y\neq-1,1$
if $y=0$
if $x\geq 1,$ $y=1$
if $x\leq-1,$ $y=-1$
otherwise,(18)

where $f(y):=y^{5}-2y^{3}+2y$ , and we put $a \mathrm{O}:=\max\{a, 0\}$ and $a \wedge 1:=\min\{a, 1\}$ for
real numbers . The -cut of fuzzy relation is written as follows (see Fig.2):$a$$\alpha$$\tilde{q}$$[0,\tilde{q}_{\alpha}(X)*]$$\tilde{q}_{\alpha}(x):=\{$$[\tilde{q}_{\alpha}(x)^{*}, 0]$

where$\max\tilde{q}_{\alpha}(x)$$\tilde{q}_{\alpha}(x)^{*}:=\{$$\min\tilde{q}_{\alpha}(x)$if $x\geq 0$
if $x<0$ ,if $x\geq 0$
if $x<0$ .

(19)

31

Figure 2. The -cut of the fuzzy relation .
$\tilde{q}$$\alpha$We can easily check$\tilde{q}$is transitive(5) since the trnasitivity is equivalent to$)$$(\mathrm{c}.\mathrm{f}.$$\tilde{q}_{\alpha}^{2}(x)\subset\tilde{q}_{\alpha}(x)$,$x\in E,$$\alpha\in[0,1]$.

(20): $Erightarrow C(E)$ is continuous for all $\alpha\in[0,1]$ . Then, we have
It is trivial that the map$R_{1}^{\sim}=R_{1}=\{-1,0,1\}$ and $R_{\alpha}=\{-1,0,1\}(\alpha\in[0,1])$ . If $A\in C(E)$ satisfies
for some $\alpha\in(0,1]$ , then $A=\{0\}$ or -1, or $[0,1]$ or [-1, 1]. Therefore, we can easily
check that satisfies Assumption A. From Theorem 3.2, we have the coefficients$\tilde{q}_{\alpha}(\cdot)$$\tilde{q}_{\alpha}(A)=A$$[$$0]$$\tilde{q}$$\alpha^{z}=\tilde{q}(\tilde{S})(Z)=x\in E\mathrm{s}\mathrm{u}\mathrm{p}\{_{\tilde{S}}(x)\wedge\tilde{q}(x, z)\}=(\sup_{x\geq 1}\tilde{s}(_{X})\sup_{x\leq}1-1\tilde{s}(_{X)}$$\mathrm{i}\mathrm{f}z\mathrm{i}\mathrm{f}_{\mathcal{Z}}=\mathrm{i}\mathrm{f}z=1=0-1$(21)

since we get $\tilde{q}(x, 0)=1,\tilde{q}(x, -1)=1_{[]()}-2,-1x$ and $\tilde{q}(x, 1)=1_{[1,2]}(x)$ from (18). From(18) and (10), we get the fundamental solutions of (1):1 $\{0\}(x)$$p^{z}(x)=\tilde{q}(z,X)=\{$$1_{1-1,0}](X)$$1_{1}0,1](x)$For example, we put an initial fuzzy state$\tilde{s}$if $z=0$
if $z=-1$
if $z=1$ .

(22)

by

$\tilde{s}_{0}(x)=\tilde{s}(x)=\max\{1-|x-1/2|, 0\}$,$x\in E$.

32$\alpha^{-1}=0$
Then, from (21), we have
and $\alpha^{1}=1/2$ . By Theorem 3.1, the sequenceof fuzzy states
converges to the the limiting fuzzy state . Therefore, from (21),(22) and Theorem 3.2, we obtain the the limiting fuzzy state$\alpha^{0}=1,$$\{\tilde{s}_{n}\}_{n}\infty=0$$\tilde{p}(x)=z=-1\max,\{\alpha^{z}\wedge 0,1p(zX)\}=1_{\{0\}(}X)\mathrm{v}\{1/2\wedge 1_{[0,1]}(x)\}=\{$$\tilde{p}$1if $x=0$
1/2 if $0<x\leq 1$$0$

otherwise.

References[1] R.E.Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management
Sci. Ser B. 17 (1970) 141-164.
[2] G.J.Klir and B.Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications(Prentice-Hall, London, 1995). Interscience Publishers, New York, 1958).

[3] K.Kuratowski, Topology I (Academic Press, New York, 1966).
[4] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, A limit theorem in some dynamic
fuzzy systems, Fuzzy Sets and Systems 51 (1992) 83-88.
[5] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, A fuzzy relational equation indynamic fuzzy systems, to appear in Fuzzy Sets and Systems.

[6] V.Nov\’ak, Fuzzy Sets and Their Applications (Adam Hilder, Bristol-Boston, 1989).

[7] Y.Yoshida, M.Yasuda, J.Nakagami and M.Kurano, A limit theorem in some dynamic
fuzzy systems with a monotone property, Fuzzy Sets and Systems 94 (1998) 109-119.

[8] Y.Yoshida, The recurrence of dynamic fuzzy systems, to appear in Fuzzy Sets andSyste m8.