数理解析研究所講究録
第 899 巻 1995 年 53-62

53

Dynamic Fuzzy Systems with Time Average Rewards$\mp\not\cong$$\mp$(Masami Kurano)(Masami Yasuda)-(
Nakagami)
Jun-ichiF
ffiff liffl ffiffi (Yuji Yoshida)

X. aefi i

ae .a

g$*$ .

$\mp$$*$

Jb jtlll.

$\Phi$$\star$$\cong$$\mathfrak{k}$

iE
E iE

$\not\equiv$$\mathfrak{F}\mathfrak{B}$$\yen$$|$$\lessgtr$$\hslash \mathfrak{p}\ovalbox{\tt\small REJECT}$

Abstract
In this paper, using a fuzzy relation we define a dynamic fuzzy system with a boundedof an n-dimensional Euclidean space. As
convex fuzzy reward on the positive orthanta measure of the system’s performance we introduce the time average fuzzy reward, which
is characterized by the limiting fuzzy state under the contractive properties of the fuzzy
relation. In one-dimensional case, the average fuzzy reward is expressed explicitly by thefunctional equations concerning the extreme points of its -cuts. Also, a numerical example
is given to illustrate the theoretical results.
$R_{+}^{n}$$\alpha$Keywords : dynamic fuzzy system, time average fuzzy reward, contractive properties, fuzzy relational equation.

1. Introduction and notations
In the previous paper [3, 8, 9], we defined a dynamic fuzzy system using a fuzzy relationand gave limit theorems for the transition of fuzzy states of the system under the contractive
and nonexpansive properties of the fuzzy relation. Here, the dynamic fuzzy system definedin [3, 8] will be extended to the one with a bounded fuzzy reward on the positive orthantof an n-dimensional Euclidean space and the time average fuzzy reward is introduced asa measure of the system’s performance and characterized by the limiting fuzzy state or byvarious fuzzy relational equations.
Let $X$ and be convex compact subsets of some Banach space or the positive orthants. We denote by $C(X)$ the collection of all closed convex compact subsets of $X$ . Let be
the Hausdorff metric on $C(X)$ .
Throughout this paper we define a fuzzy set on $X$ by its membership function$[0,1]$ and its-cut by$R_{+}^{n}$$Y$$R_{+}^{n}$$\rho$$\tilde{s}:X\mapsto$$\alpha$$\tilde{s}_{\alpha}$$:=\{x\in X|\tilde{s}(x)\geq\alpha\}(\alpha>0)$

and% $:=cl\{x\in X|\tilde{s}(x)>0\}$ ,where means the closure of a set. For the details, we we refer to Nov\’ak [6] and Zadeh [11].
A fuzzy set on $X$ is called convex if$cl$$\tilde{s}$$\tilde{s}(\lambda x+(1-\lambda)y)\geq\tilde{s}(x)\wedge\tilde{s}(y)$where $a \wedge b=\min\{a, b\}$ for real numbers
is called convex if$a$for any $x,$ $y\in X$ and $\lambda\in[0,1]$ ,and . Also, a fuzzy relation$b$$\tilde{p}$defined on $X\cross Y$$\tilde{p}(\lambda x_{1}+(1-\lambda)x_{2}, \lambda y_{1}+(1-\lambda)y_{2})\geq\tilde{p}(x_{1}, y_{1})\wedge\tilde{p}(x_{2}, y_{2})$$x_{2}\in X,$ $y1,$ $y_{2}\in Y$ , and.
for anyLetbe the set of all convex fuzzy sets on $X$ , which are upper semi-continuous.
implies
Clearly
for all $\alpha\in[0,1]$ .
multiplicative
operation of fuzzy sets are defined as follows (see [4])The addition and the$\lambda\inR_{+};=[0,\infty)$
,
any
and
: For
$\lambda\in[0,1]$$x_{1},$$\mathcal{F}(X)$$\tilde{s}\in \mathcal{F}(X)$$\tilde{s}$$\tilde{s}_{\alpha}\in C(X)$$\tilde{s},\tilde{v}\in \mathcal{F}(X)$$(\tilde{s}+\tilde{v})(x)$$:= \sup_{y,z\in Xy+z=x}\{\tilde{s}(y)\wedge\tilde{v}(z)\}$$x\in X$

(1.1)

54

and

if
if

$(\lambda\tilde{s})(x):=\{\begin{array}{l}\tilde{s}(xT)I_{\{0\}}(x)\end{array}$$\lambda>0$$\lambda=0$$x\in X$,

. is the classical indicator function of a subset
where
It is easily seen that, for $\alpha\in[0,1])$$I_{A}$ $($

$A$

$)$

and

$(\tilde{s}+\tilde{v})_{\alpha}=\tilde{s}_{\alpha}+\tilde{v}_{\alpha}$

,

(1.2)

of $X$ .

$(\lambda\tilde{s})_{\alpha}=\lambda\tilde{s}_{\alpha}$,$:=\{\lambda x|x\in A\}$ for any subsets $A,$
where $A+B$ $:=\{x+y|x\in A, y\in B\}$ and
The following results have appeared in Chen-wei Xu [1].
$\lambda A$$B$

of $X$ .

Lemma 1.1. ([1]).
(i) For any$\tilde{s}\tilde{v}$)$\in \mathcal{F}(X)$an$d\lambda\in[0, \infty)$,$\tilde{s}+\tilde{v}\in \mathcal{F}(X)$(ii) Let$\tilde{p}$

an

$d$

$\lambda\tilde{v}\in \mathcal{F}(X)$.

be an lower semi-continuous convex fuzzy relation on$y$for all$\sup_{x\epsilon x}\tilde{s}(x)\wedge\tilde{p}(x, \cdot)\in \mathcal{F}(Y)$$X\cross$$\tilde{s}\in \mathcal{F}(X)$

Y. Then.

Here, we give the notion of convergency for a sequence of fuzzy sets, which is used in

Section 2.

Definition 1.1 ([3, 5]). Letasif$\{\tilde{\eta}\}_{t=0}^{\infty}$

be a sequence fuzzy sets in$\mathcal{F}(S)$. Then we write$tarrow\infty$$\tilde{v}_{l}arrow\tilde{v}\in \mathcal{F}(S)$

(1.3)

,

$\lim_{\ell\vee\infty}\sup_{\alpha\in[0,1]}\rho(\tilde{v}_{t,\alpha},\tilde{v}_{\alpha})=0$

where$\tilde{ll}_{l,\alpha}$and$\tilde{v}_{\alpha}$are -cuts of$\alpha$and$\tilde{v}_{t}$Note that for a sequence of sets$\tilde{v}$respectively.

$\{A_{t}\}_{t=1}^{\infty}\subset C(X)$

and $A \in C(X),\lim_{tarrow\infty}A_{t}=A$ means that$\varlimsup_{tarrow\infty}A_{\ell}=\varliminf_{\ellarrow\infty}A_{\ell}=A$

,

where
$\varlimsup_{tarrow\infty}A_{\ell}:=\{z\in X|\varliminf_{\ellarrow\infty}d(z, A_{\ell})=0\}$,$\varliminf_{tarrow\infty}A_{t}:=\{z\in X|\varlimsup_{tarrow\infty}d(z, A_{\ell})=0\}$,$d(z, D)$ $:= \min_{z\in D}d(z, z’)$

and
that

$d$

$\tilde{v}_{l}$$D\in C(X)$

is a metric on $X$ . It is known [2} that $\lim_{tarrow\infty}\rho(A_{\ell}, A)=0$ iff $\lim_{tarrow\infty}A_{t}=A$ , so
uniformly forconverges to as
in the sense of (1.3) means that$\alpha\in[0,1]$

$($

$)$

$tarrow\infty$$\tilde{v}$$\lim\tilde{\eta}_{\alpha}=\tilde{v}_{\alpha}$.

Now, extending a discrete dynamic fuzzy system in [3], we consider the one with a fuzzy
reward, which is characterized with the elements
as follows :$(S,\tilde{q}_{1}\tilde{r},\tilde{s})$(i) The state space is a convex compact subset of some Banach space. In general, thesystem is fuzzy, so that the state of the system is called a fuzzy state denoted as an.
element of$S$$\mathcal{F}(S)$(ii) The law of the motion and the fuzzy reward for the system are denoted by the timeinvariant fuzzy relations : $S\cross S\mapsto[0,1]$ and : $S\cross R_{+}^{n}\mapsto[0,1]$ respectively. We
restrict the convex fuzzy number
has the finite support contained in the. We alsointerval $[0, M]^{n}\subset R_{+}^{n}$ , that is,
assume that $\tilde{q}:S\cross S\mapsto[0,1]$ and $\tilde{r}:S\cross[0, M]^{n}\mapsto[0,1]$ are convex and continuous.
$\tilde{r}$$\tilde{q}$$\tilde{n}\in \mathcal{F}(R_{+}^{n})$$\mathcal{F}([0, M]^{n});=\{\tilde{n}\in \mathcal{F}(R_{+}^{n})|\tilde{n}_{0}\subset[0, M]^{n}\}$55

If the system is in a fuzzy state, whereto a new fuzzy stateby$\tilde{s}\in \mathcal{F}(S)$, a fuzzy reward$Q:\mathcal{F}(S)\mapsto \mathcal{F}(S)$$Q(\tilde{s})$and$R( \tilde{s})(z):=\sup_{x\in S}\tilde{s}(x)\wedge\tilde{r}(x, z)$$R(\tilde{s})$$Q( \tilde{s})(y):=\sup_{x\in S}\tilde{s}(x)\wedge\tilde{q}(x,y)$$R$(iii) The initial fuzzy stateis arbitrary.

$\tilde{s}\in \mathcal{F}(S)$For the dynamic fuzzy system, where$[0, M]^{n},$$(S,\tilde{q})\tilde{r},\tilde{s})$$y\in S$.

(1.5)

are well-defined.

Note that by Lemma l.l(ii) the maps

$Q$

(1.4)

$z\in[0, M]^{n}$

and
and

is incurred and we moveare defined$R:\mathcal{F}(S)\mapsto \mathcal{F}([0, M]^{n})$, we can define a sequence of fuzzy rewards on$\{R(\tilde{s}_{t})\}_{t=0}^{\infty}$$\tilde{s}0:=\tilde{s}$and$\tilde{s}_{t+1}:=Q(\tilde{s}_{t})(t\geq 0)$

(1.6)

.

In Section 2, we define the time average fuzzy reward, which is characterized by thelimiting fuzzy state under the contractive properties of of the fuzzy relation .
In Section 3, the one-dimensional case is treated and by introducing relative value functions the average fuzzy reward is expressed by the functional equations concerning the extreme points of its -cuts.
Also, a numerical example is given to illustrate the theoretical results in this paper.
$\tilde{q}$$\alpha$2. The average fuzzy reward
In this paper we specify the time average reward as a measure of the system’s performanceand discuss its characterization under the contractive assumption given in [3].
by
We define the total T-time fuzzy reward$\tilde{R}_{T}(\tilde{s})$$\tilde{R}_{T}(\tilde{s})$$:= \sum_{t=0}^{T-1}R(\tilde{s}_{t})$$T\geq 1$

(2.1)

,

is given in (1.6).
where
are the corresponding maps
Associated with the fuzzy relation and fuzzy reward: $C(S)\mapsto C([0, M]^{n})(\alpha\in[0,1])$ defined as follows :: $C(S)\mapsto C(S)(\alpha\in[0,1])$ and
For $D\in C(S)$ ,$\{\tilde{s}_{t}\}_{t=0}^{\infty}$$\tilde{q}$$\tilde{\Gamma}_{)}$$R_{\alpha}$$Q_{\alpha}$$Q_{\alpha}(D):=\{\begin{array}{ll}\{y\in S|\tilde{q}(x, y)\geq\alpha for some x\in D\} \alpha>0cl\{y\in S|\tilde{q}(x, y)>0 for some x\in D\} \alpha=0,\end{array}$(2.2)$R_{\alpha}(D):=\{\begin{array}{ll}\{z\in[0, M]^{n}|\tilde{r}(x, z)\geq\alpha for some x\in D\} \alpha>0cl\{z\in[0, M]^{n}|\tilde{r}\langle x, z)>0 for some x\in D\} \alpha=0.\end{array}$

(2.3)

and

The iterates$Q_{\alpha}^{t}(t\geq 0)$

are defined by setting$Q_{\alpha}^{0}$$:=I$ (identity) and iteratively,$t\geq 1$$Q_{\alpha}^{t}:=Q_{\alpha}Q_{\alpha}^{t-1}$.

We have the following lemma, which is easily verified by the ideas in the proof of [3, Lemma
1 and the property (1.1).
$]$

Lemma 2.1.
(i)

(ii)
(iii)

for $T\geq 1$ .
for $t\geq 0$ , where$\tilde{R}_{T}(\tilde{s})\in \mathcal{F}([0,TM]^{n})$$\tilde{s}_{t,\alpha}=Q_{\alpha}^{t}(\tilde{s}_{\alpha})$$( \tilde{R}_{T}(s\gamma)_{\alpha}=\sum_{t=0}^{T-1}R_{\alpha}(\tilde{s}_{t,\alpha})$$\tilde{s}_{t,\alpha}=(\tilde{s}_{t})_{\alpha}$

for $T\geq 1$ .

.

56

From Lemma 2.1(ii),(iii),$(\tilde{R}(\tilde{s}))_{\alpha}$

is dependent only on$\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$

and

$T$

, so that we put

for $T\geq 0$ and $\alpha\in[0,1]$ .

$\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha}):=(\tilde{R}_{T}(\tilde{s}))_{\alpha}$

From this$\alpha,\tilde{s}_{\alpha}$

we try to estimate the increasing amount of fuzzy reward per unit time.

For $K>0$ and $\alpha\in[0,1]$ , we define$G_{K,\alpha}:=\{r\in R_{+}^{n}|thereexists\{z_{T}\}_{T=1}^{\infty}suchthatz_{T}\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})and||z_{T}-rT||\leq K$

for all$T\geq 1\}$ ‘

(24)

where
is the poeitive orthant of a n-dimensional Euclidean space and $||x||=( \sum_{i=1}^{n}x_{i}^{2})8$$x=(x_{I},x_{2},\cdots, x_{n})\in R_{+}^{n}$ . The following properties offorare formulated in a lemma.
$R_{+}^{n}$$G_{K,\alpha}$

Lemma 2.2. Let $K>0$ . Then :
(i)
(ii)
(iii)

$\{G_{K,\alpha}|\alpha\in[0,1]\}\subset C(R_{+}^{n})$$G_{K,\alpha}\subset G_{K,\alpha’}$for.

$0\leq\alpha’\leq\alpha\leq 1$$\lim_{\alpha’\uparrow\alpha}G_{K,\alpha’}=G_{K,\alpha}$for.

$\alpha\in(0,1]$, i.e.,$\lim_{\alpha’\uparrow\alpha}\delta(G_{K,\alpha’}, G_{K,\alpha})=0$.

Proof. (ii) From Lemma 2.1(i), we have$\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})\subset\tilde{R}_{T,\alpha’}(\tilde{s}_{\alpha’})$for $T\geq 0,0\leq\alpha’\leq\alpha\leq 1$ .

Therefore we obtain (ii) from the definition (2.4).
(i) Let $\alpha\in[0,1]$ . We prove thatis closed and convex. Let
be a sequencesuch that $r_{k}\in G_{K,\alpha}(k\geq 1)$ and $r_{k}arrow r(karrow\infty)$ . From 2.4 , there exists a sequencesuch thatand $||z_{T}^{k}-r_{k}T||\leq K$ for all $T\geq 1,$ $k\geq 1$ .
(2.5)
$G_{K,\alpha}$$\{r_{k}\}_{k=1}^{\infty}$$\{z_{T}^{k}\}$

$)$

$($

$z_{T}^{k}\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$Since

is compact, there exist a convergent subsequence of. From (2.5), we obtain$\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$$z_{T}\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$$||z_{T}-rT||\leq K$

Therefore we get
Next let$r_{1},$$G_{K,\alpha}$$||z_{T}^{1}-r_{1}T||\leq K,$

Let $T\geq 1$ . Since

for all $T\geq 1$ .

andis closed.
. From (2.4), there exists$r\in G_{K,\alpha}$$r_{2}\in G_{K,\alpha}$$\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$and its limit$\{z_{T}^{k}\}_{k=1}^{\infty}$$z_{T}^{2}\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$$z_{T}^{1},$$||z_{T}^{2}-r_{2}T||\leq K$

such thatfor all $T\geq 1$ .

(2.6)

is convex, we have$\lambda z_{T}^{1}+(1-\lambda)z_{T}^{2}\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$for all$\lambda\in[0,1]$

.

(2.7)

From (2.6), we have$||(\lambda z_{T}^{1}+(1-\lambda)z_{T}^{2})-(\lambda r_{1}+(1-\lambda)r_{2})T||$$\leq$$\lambda||z_{T}^{1}-r_{1}T||+(1-\lambda)||z_{T}^{2}-r_{2}T||$$\leq$$\lambda K+(1-\lambda)K=K$

From (2.7) and this inequality, we obtainis convex. Thus we get (i).
(iii) Let $\alpha\in(0,1]$ . It is sufficient to prove
From (ii), there exist sequencesandfor all$\lambda\in[0,1]$$\lambda r_{1}+(1-\lambda)r_{2}\in G_{K,\alpha}$.

for all $\lambda\in[0,1]$ . Therefore$G_{K,\alpha}$$\{\alpha_{k}\}_{k=1}^{\infty}$$\alpha_{k}\uparrow\alpha(karrow\infty)$,$\{r_{k}\}_{k=1}^{\infty}$$r_{k}\in G_{K,\alpha_{k}}(k=1,2, \cdots)$

From (2.4), there exists a sequence$z_{T}^{k}\in\tilde{R}_{T,\alpha_{k}}(\tilde{s}_{\alpha_{k}})$$\varlimsup_{\alpha’arrow\alpha}G_{K,\alpha’}\subset G_{K,\alpha}$$\{z_{T}^{k}\}_{k=1}^{\infty}$. Let$r\in\varlimsup_{\alpha’arrow\alpha}G_{K,\alpha’}$such that,and$r_{k}arrow r(karrow\infty)$

such thatand $||z_{T}^{k}-r_{k}T||\leq K$

for all $T\geq 1,$ $k=1,2,$$\cdots$

.

.

.

57

From the compactness, there exists a convergent subsequence ofThen we obtain$||z\tau-rT||\leq K$
for all $T\geq 1$ .

$\{z_{T}^{k}\}_{k=1}^{\infty}$

and its limit$z\tau$.

(2.8)

Further, from Lemma 2.1(i), we have$\lim_{\alpha’arrow\alpha}\tilde{R}_{T,\alpha’}(\tilde{s}_{\alpha’})=\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$Therefore we also obtain.

$z\tau\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$.

. Together with (2.8), we get$r\in G_{K,\alpha}$$\varlimsup_{\alpha^{J}arrow\alpha}G_{K,\alpha^{l}}\subset G_{K,\alpha}$. We obtainq. e. d.

From [3, Lemma 3], we can define a fuzzy number$r\in[0, M]^{n}$$\tilde{g}(\tilde{s})(r):=\sup_{\alpha\in[0,1]}\{\alpha\wedge I_{G_{K.\alpha}}(r)\}$for$\tilde{s}\in \mathcal{F}(S)$

.

(2.9)

Then,
and
for all $\alpha\in[0,1]$ .
an average fuzzy reward for the dynamic fuzzy systems, which depends on
We callthe initial fuzzy statewith suppression of $K$ . In the remainder of this section,
we will investigate the average fuzzy reward from the limiting behaviour of the fuzzy states.
The following lemma is useful in the sequel.
$\tilde{g}(\tilde{s})\in \mathcal{F}([0, M]^{n})$$(\tilde{g}(\tilde{s}))_{\alpha}=G_{K,\alpha}$$\tilde{g}(\tilde{s})$$\tilde{s}\in \mathcal{F}(S)$Lemma 2.3. Let
For any

and $D\in C(S)$ such th at $\lim_{tarrow\infty}D_{t}=D$ . Let $\alpha\in(0,1]$ .
, there exists
such that$\{D_{t}\}_{t=1}^{\infty}\subset C(S)$$\epsilon(\alpha>\epsilon>0)$$T\geq 1$$R_{\alpha-\epsilon}(D)\supset R_{\alpha}(D_{t})$Proof. Suppose that for somesuch that$t_{k}arrow\infty(karrow\infty)$,$\epsilon(\alpha>\epsilon>0)$

andfor all $t\geq T$.

, there exist sequencesand$\{t_{k}\}_{k=1}^{\infty}$$z_{k}\in R_{\alpha}(D_{t_{k}})\backslash R_{\alpha-\epsilon}(D)(k=1,2, \cdots)$$\{z_{k}\}_{k=1}^{\infty}$.

Then we have$\tilde{r}\langle x,$

and there exists a sequence

for all $x\in D,$ $k=1,2,$$z_{k})<\alpha-\epsilon$$\{x_{k}\}_{k=1}^{\infty}$$\tilde{r}\langle x_{k},$for $k=1,2,$$z_{k})\geq\alpha$From the compactness, we may assume that the sequences
andgent. We put the limits$\lim_{tarrow\infty}D_{t}=D$ . From $(2.10),(2.11)$ , we obtain$x^{*}= \lim_{karrow\infty}x_{k}$$\tilde{r}\langle x^{*},$$z^{*})\geq\alpha$,(2.10)

such thatand$x_{k}\in D_{t_{k}}$$\cdots$

and.

(2.11)

andare conver. Then we have $x^{*}\in D$ since$\{x_{k}\}_{k=1}^{\infty}$$z^{*}= \lim_{karrow\infty}z_{k}$$\tilde{r}\langle x,$$\cdots$$\{z_{k}\}_{k=1}^{\infty}$

for all $x\in D$ .

$z^{*})\leq\alpha-\epsilon$

q. e. d.

It is a contradiction. Thus we get this lemma.

, we need the following two asIn order to characterizing the average fuzzy reward
sumptions, the first one is a contractive property concerning the fuzzy relation whichguarantee the existence of the limiting fuzzy state and the second is a Lipschitz condition
related with the fuzzy reward .
$\tilde{g}(\tilde{s})$$\tilde{q}$$\tilde{r}$Assumption
There exists$A$(Contraction and ergodic property).
and $\beta(0<\beta<1)$ satisfying that$t_{0}\geq 1$$\rho(Q_{\alpha}^{t_{O}}(D_{1}), Q_{\alpha}^{t_{O}}(D_{2}))\leq\beta\rho(D_{1}, D_{2})$for all$D_{1},$$D_{2}\in C(S),$$\alpha\in[0,1]$.

(Lipschitz conditions).
Assumption
There exists a constant $C>0$ such that$B$$\delta(R_{\alpha}(D_{1}), R_{\alpha}(D_{2}))\leq C\rho(D_{1}, D_{2})$where$\delta$

for allis the Hausdorff metric on $C([0, M]^{n})$ .

Lemma 2.4 [3, Theorem 1]$D_{1},$$D_{2}\in C(S),$$\alpha\in[0,1]$,(2.12)

58(i) There exists a unique fuzzy state
state , sa tisfyiiig$wh$$\tilde{p}\in \mathcal{F}(S),$ich is independen $tly$ of the initial fuzzy$\tilde{s}$for all $y\in S$ .

$\tilde{p}(y)=\max_{x\in S}\{\tilde{p}(x)\wedge\tilde{q}(x, y)\}$(ii) For $\alpha\in[0,1]$ , the -cut$\alpha$(2.13)

is a uniq $ue$ set of $C(S)$ such $tAat$$\tilde{p}_{\alpha}$.

$Q_{\alpha}(\tilde{p}_{\alpha})=\tilde{p}_{\alpha}$(iii) Let $\alpha\in[0,1]$ . It holds th atfor all $D\in C(S),$ $t\geq 1$ ,$\rho(Q_{\alpha}^{t}(D),\tilde{p}_{\alpha})\leq\beta^{[\ell/t_{0}]}K_{\alpha}(D,\tilde{p}_{\alpha})$and, for a $reaI$ number

wherein teger equal to or less th an .

$K_{\alpha}(D, \tilde{p}_{\alpha});=\sum_{l=0}^{\ell_{O}-1}\rho(Q_{\alpha}^{l}(D),\tilde{p}_{\alpha})$

$c,$

$[c]$

is the largest$c$

Recently, Yoshida [10] has given the notion of -recurrent set for the fuzzy relationand shown that the -cut of the limiting fuzzy set in Lemma 2.4 is characterized as the
maximum -recurrent set.
$\alpha$$\alpha$$\tilde{p}$$\alpha$Now, we can state one of main results, which shows that
limiting fuzzy state .

$\tilde{g}(\tilde{s})$is represented using the$\tilde{p}$Theorem 2.1. Suppose that Assumption
th at

$s$

$A$

aiid

$B$

hold. For sufficient large all $K$ , it holds,(2.14)$\tilde{g}(\tilde{s})=R(\tilde{p})$where is the liniitin fuzzy state given in Leman 2.3. Further this is in dependent of thein itial fuzzy state .
$g$$\tilde{p}$$\tilde{s}$Proof. Let $\alpha\in[0,1]$ . First we show that.

(2.15)$(\tilde{g}(\tilde{s}))_{\alpha}=G_{K,\alpha}\subset R_{\alpha}(\tilde{p}_{\alpha})=(R(\tilde{p}))_{\alpha}$By (2.9) and Lemma 2.1(iii), the two equality in (2.15) hold obviously. We prove. Then. Suppose that there exists$G_{K,\alpha}\subset$$r\in G_{K,\alpha}\backslash R_{\alpha}(\tilde{p}_{\alpha})$$R_{\alpha}(\tilde{p}_{\alpha})$$\tilde{r}\langle x,$$r)<\alpha$

for all

From the continuity of and the compactness of$\tilde{r}$$\tilde{r}\langle x,$$\tilde{p}_{\alpha}$$x\in\tilde{p}_{\alpha}$, there exists

for all$r)\leq\epsilon<\alpha$.

$x\in\tilde{p}_{\alpha}$such that$\epsilon>0$.

Therefore$f\not\in R_{\frac{\alpha*e}{2}}(\tilde{p}_{\alpha})$Since$R_{\frac{a*e}{2}}(\tilde{p}_{\alpha})$.

(2.16)

is closed and convex, there exists a uniquefor all$0<\gamma:=||z_{0}-r||\leq||z-r||$$z_{0}\in R_{\frac{\alpha\neq e}{2}}(\tilde{p}_{\alpha})$$z\in R_{\frac{a\neq e}{2}}(\tilde{p}_{\alpha})$.

such that(2.17)

From Lemma 2.1(ii) and Lemma 2.4(iii), we have$\tilde{s}_{l,\alpha}arrow\tilde{p}_{\alpha}$$(tarrow\infty)$.

By Lemma 2.3, there exists $\tau*>0$ such that
$R_{\frac{\alpha\neq e}{2}}(\tilde{p}_{\alpha})\supset R_{\alpha}(\tilde{s}_{t,\alpha})$On the other hand, from$f\in G_{K,\alpha}$, there exists$r\tau\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$for all$\{r\tau\}_{T=0}^{\infty}$$t\geq T^{*}$.

(2.18)

such thatand $||r\tau-rT||\leq K$ for all $T\geq 1$ .

$($

2.19

$)^{-}$

59

From Lemma 2.1(iii), there exists a sequence$r\tau,t\in R_{\alpha}(\tilde{s}_{t,\alpha})(t=0,1,2, \cdots , T-1)$

Noting the supporting hyperplane of$\langle z_{0}-r,$such that$\{r_{T,t}\}$and$R_{\frac{\alpha+e}{2}}(\tilde{p}_{\alpha})$$r_{T}= \sum_{t=0}^{T-1}r_{T,t}$at$r_{T_{1}t}-r)\geq||z_{0}-r||^{2}=\gamma^{2}$$z_{0}$

for all $T\geq 1$ .

(2.20), from (2.17),(2.18),(2.20) we havefor all$t,$$T(T>t\geq T^{*})$ .

So
for all $T>T^{*}$ ,$\{z_{0}-r,\sum_{t=T}^{T-1}.(r_{T,t}-r)\}\geq(T-T^{*})\gamma^{2}$

where$\langle\cdot).\}$

is an inner product on:
for $a=(a_{1}, a_{2}, \cdots, a_{n}),$ $b=$
. By Cauchy-Schwartz inequality, we obtain$\langle a,$$R_{+}^{n}$$b \}=\sum_{i=1}^{n}a_{i}b_{i}$$(b_{1}, b_{2}, \cdots, b_{n})\in R_{+}^{n}$for all $T>T^{*}$ .

$\geq(T-T^{*})\gamma$$\sum_{t=T^{s}}^{T-1}(r_{Tt,)}-r)$(2.21)

Since$||r\tau-rT||\geq$$\sum_{t=0}^{T-1}(r_{T,t}-r)$$\sum_{t=0}^{T-1}(r_{T_{1}t}-r)$

$-$

and
$\sum_{\ell=0}^{T-1}(r_{T,t}-r)$$\leq 2nT^{*}M$

for all $T>T^{*}$ ,we have$||r_{T}-rT||=$$\sum_{t=0}^{T-1}(r_{T,t}-r)$$(Tarrow\infty)$$arrow\infty$.

This contradicts (2.19). Thus we obtain (2.15).
Next we provefor sufficient large all K.
$R_{\alpha}(\tilde{p}_{\alpha})\subset G_{K,\alpha}$

From Assumption$B$(2.22), we have$\delta(R_{\alpha}(\tilde{s}_{t,\alpha}), R_{\alpha}(\tilde{p}_{\alpha}))\leq C\rho(\tilde{s}_{t,\alpha},\tilde{p}_{\alpha})$for $t\geq 0$ .

(2.23)

From Lemmas 2.1(ii) and 2.4(iii), we also havefor $t\geq 0$ .

$\rho(\tilde{q}_{\alpha},\tilde{p}_{\alpha})\leq\beta^{[t/t_{0}]}K_{\alpha}(\tilde{s}_{\alpha},\tilde{p}_{\alpha})$(2.24)

From $(2.23),\{2.24)$ and the compactness of $E$ , there exists a constant $C^{*}>0$ such that$\delta(R_{\alpha}(\tilde{s}_{t,\alpha}), R_{\alpha}(\tilde{p}_{\alpha}))\leq C^{*}\beta^{t}$

Therefore, for any$f\in R_{\alpha}(\tilde{p}_{\alpha})$, there exists$r_{t}\in R_{\alpha}(\tilde{s}_{t,\alpha})$and$\{r_{t}\}_{t=0}^{\infty}$

for $t\geq 0$ .

such that$||r_{t}-r||\leq C^{*}\beta^{t}$for $t\geq 0$ .

(2.25)

Then$\Vert\sum_{t=0}^{T-1}r_{t}-rT\Vert=\Vert\sum_{t=0}^{T-1}(r_{t}-r)\Vert\leq\sum_{t=0}^{T-1}||r_{t}-r||\leq\sum_{t=0}^{T-1}C^{*}\beta^{t}\leq\frac{c*}{1-\beta}$

for all $T\geq 1$ .

Thus we getfor all $K\geq C^{*}/(1-\beta)$ . Therefore (2.22) holds for all $K\geq C^{*}/(1-\beta)$ .
Together with (2.15), we get (2.14) for sufficient large all $K$ . It is trivial that (2.14) is
independent of the initial fuzzy state from Lemma 2.4(i).
q. e. d.
$r\in G_{K,\alpha}$$\tilde{s}$60

From now on we take $K\geq C^{*}/(1-\beta)$ . The following corollary shows thatby the method of Cesaro averaging.
as the limit of$\tilde{g}(\tilde{s})$is given$\{R(\tilde{s}_{t})\}_{t=0}^{\infty}$

Corollary 2.1. Under the same condition as Theorem 2.1, it holds that$\lim_{Tarrow\infty}\frac{1}{T}\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})=(\tilde{g}(s\gamma)_{\alpha}$

Proof. Let $\alpha\in[0,1]$ . Let$f\in(\tilde{g}(\tilde{s}))_{\alpha}=G_{K,\alpha}$$r\tau\in\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$for $aIl\alpha\in[0,1]$ .

. Then there exists

and $||r_{T}-rT||\leq K$(2.26)

such that$\{r\tau\}_{T=0}^{\infty}$

for $T=0,1,2,$$\cdots$

for $T=1,2,$$\cdots$.

Then$\frac{\prime\tau}{T}\in\frac{1}{T}\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$and$\Vert\frac{r_{T}}{T}-r\Vert\leq\frac{K}{T}$.

Letting $Tarrow\infty$ , we obtain$r \in\varliminf_{Tarrow\infty}\frac{1}{T}\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})$.

Therefore$( \tilde{g}(\tilde{s}))_{\alpha}\subset\varliminf_{Tarrow\infty}\frac{1}{T}\tilde{R}_{T_{J}\alpha}(\tilde{s}_{\alpha})$Conversely we let
that$r\in\varlimsup\tauarrow\infty T^{\tilde{R}_{T,\alpha}(\tilde{s}_{\alpha})}1$.

. Then there exists a subsequence$r\tau_{j}\in\tilde{R}_{T_{j},\alpha}(\tilde{s}_{\alpha})(j=1,2, \cdots)$and$\lim\underline{r_{T_{j}}}=f$$\{r_{T_{j}}\}_{j=1}^{\infty}$

.

such

(2.27)$jarrow\infty T_{j}$Let $j=1,2,$$\cdots$. From Lemma 2.1(iii), there exists$\{r_{T_{j},t}\}_{t=1}^{\infty}$$r_{T_{j},t}\in R_{\alpha}(\tilde{s}_{t,\alpha})(t=0,1,2, \cdots, T_{j}-1)$

andsuch that$r\tau_{j}=\sum_{t=0}^{T_{j}-1}r_{T_{j},t}$In similar way to (2.25), from Lemma 2.4(iii) and Assumption$C^{*}>0$ andsuch that$B$.

, there exist a constant$\{z_{T_{j},t}\}_{j=1}^{\infty}$$z_{T_{j},t}\in R_{\alpha}(\tilde{p}_{\alpha})$Since$R_{\alpha}(\tilde{p}_{\alpha})$andis convex, putting$\frac{z_{\mathcal{T}_{j}}}{T_{j}}\in R_{\alpha}(\tilde{p}_{\alpha})$and$||z_{T_{j},t}-r_{T_{j},t}||\leq C^{*}\beta^{t}$$:= \sum_{t=0}^{T_{j}-1}z_{T_{j},t}$$z_{T_{j}}$for $t=1,2,$$\cdots$, $T_{j}-1$ .

, we have$\Vert\frac{z_{T_{j}}}{T_{j}}-\frac{r\tau_{j}}{T_{j}}\Vert\leq\frac{1}{T_{j}}\sum_{t=0}^{T_{j}-1}C^{*}\beta^{t}\leq\frac{c*}{T_{i}(1-\beta)}$for $j\geq 1$ .

(2.28)

From (2.27) and (2.28), we obtain$\Vert\frac{z_{T_{j}}}{T_{j}}-r\Vert\leq\Vert\frac{z_{T_{j}}}{T_{i}}-\frac{r_{T_{j}}}{T_{j}}\Vert+\Vert\frac{r_{T_{j}}}{T_{j}}-r\Vertarrow 0(jarrow\infty)$

Since$R_{\alpha}(\tilde{p}_{\alpha})$is closed, we obtain$r\in R_{\alpha}(\tilde{p}_{\alpha})=(\tilde{g}(\tilde{s}))_{\alpha}$(2.26).

.

from Theorem 2.1. Thus we get
q. e. d.

3. One-Dimensional Case
, and characterize an
In this section we consider the case of $n=1$ , i.e.
by the functional equations concerning with the extremal points
average fuzzy reward
of its -cuts. Throughout this section it is assumed that Assumptions and hold.
: $C(S)\mapsto$
Since $C([0, M])$ is the set of all closed intervals, we can write the map$C([0, M])$ by the following notation:$\tilde{r}\in \mathcal{F}(S\cross[0, M])$$\tilde{g}(\tilde{s})$$A$$\alpha$$B$$R_{\alpha}$$R_{\alpha}(D)$$:=[ \min R_{\alpha}(D), \max R_{\alpha}(D)]$for all $D\in C(S)$ .

(3.1)

61

Let
$\tilde{R}_{T,\alpha}(D)$$:= \sum_{t=0}^{T-1}R_{\alpha}(Q_{\alpha}^{t}(D))$

for $D\in C(S)$ .

Then, by Lemma 2.1(iii), it holds that
(3.2)$\min\tilde{R}_{T,\alpha}(D)=\sum_{t=0}^{T-1}\min R_{\alpha}(Q_{\alpha}^{t}(D))$

and$\max\tilde{R}_{T_{l}\alpha}(D)=\sum_{t=0}^{T-1}\max R_{\alpha}(Q_{\alpha}^{t}(D))$

,

where
$\tilde{R}_{T,\alpha}(D)=[\min\tilde{R}_{T_{2}\alpha}(D),\max\tilde{R}_{T,\alpha}(D)]$From Lemma 2.4(iii) and Assumption
we observe that
exponentially first as. Thus, by (3.2) and (3.2),
$B$

(3.3)

.

$R_{\alpha}(Q_{\alpha}^{t}(D))$

converges to$R_{\alpha}(\tilde{p}_{\alpha})$$tarrow\infty$$\underline{h}_{\alpha}(D)$(3.4)$:= \lim_{Tarrow\infty}(\min\tilde{R}_{\mathcal{T},\alpha}(D)-T\cross\min R_{\alpha}(\tilde{p}_{\alpha}))$and$\overline{h}_{\alpha}(D)$(3.5)$:= \lim_{Tarrow\infty}(\max\tilde{R}_{T,\alpha}(D)-T\cross\max R_{\alpha}(\tilde{p}_{\alpha}))$converge for all $D\in C(S)$ . The function(
resp.) is called a lower (upper) relativevalue function, whose basic ideas are appearing in the theory of Markov decision processes$\overline{h}_{\alpha}$$\underline{h}_{\alpha}$(c.f. [7]). By Theorem 2.1, we have$\tilde{g}(\tilde{p})_{\alpha}=[\min R_{\alpha}(\tilde{p}_{\alpha}), \max R_{\alpha}(\tilde{p}_{\alpha})]$

,

(3.6)

where the extremal points are characterized in the following theorem.
Theorem 3.1. Let $\alpha\in[0,1]$ . Then the follo wing (i) and (ii) hold.
(i) Let$\underline{h}_{\alpha}$and$\overline{h}_{\alpha}$

be defined by (3.4) and (3.5). Then, the follo wing $eq$ uation hold:$s$$\underline{h}_{\alpha}(D)+\min R_{\alpha}(\tilde{p}_{\alpha})=\min R_{\alpha}(D)+\underline{h}_{\alpha}(Q_{\alpha}(D))$(3.7)$\overline{h}_{\alpha}(D)+\max R_{\alpha}(\tilde{p}_{\alpha})=\max R_{\alpha}(D)+\overline{h}_{\alpha}(Q_{\alpha}(D))$

(3.8)

and
for all $D\in C(S)$ .
(ii) Conversely, if there exist bounded $fu$ nctions
satisfying th at$\underline{h}_{a}$an$d\overline{h}_{\alpha}$

on $C(S)$ an constants$d$$\underline{If}_{\alpha}$

an

$d$

$\overline{K}_{\alpha}$$\underline{h}_{\alpha}(D)+\underline{IC}_{\alpha}=\min R_{\alpha}(D)+\underline{h}_{\alpha}(Q_{\alpha}(D))$

an

(3.9)

$d$

$\overline{h}_{\alpha}(D)+\overline{K}_{\alpha}=\max R_{\alpha}(D)+\overline{h}_{\alpha}(Q_{\alpha}(D))$

for $aIlD\in C(S)$ , then$\tilde{g}(\tilde{s})_{\alpha}=[\underline{K}_{\alpha}, \overline{K}_{\alpha}]$.

Proof. (i) (3.4) implies$arrow h(D)$$= \lim_{Tarrow\infty}\sum_{t=0}^{T-1}(\min R_{\alpha}(Q_{\alpha}^{t}(D))-\min R_{\alpha}(\tilde{p}_{\alpha}))$$= \min R_{\alpha}(D)-\min R_{\alpha}(\tilde{p}_{\alpha})$$+ \sum_{t=1}^{\infty}(\min R_{\alpha}(Q_{\alpha}^{t-1}(Q_{\alpha}(D)))-\min R_{\alpha}(\tilde{p}_{\alpha}))$$= \min R_{\alpha}(D)-\min R_{\alpha}(\tilde{p}_{\alpha})+\underline{h}_{\alpha}(Q_{\alpha}(D))$

which leads to (3.7). Also, (3.8) can be shown analogously to (3.7).

,(3.10)

62(ii) Let$\underline{h}_{\alpha}(D)$and$\underline{K}_{\alpha}(D)$be as in (3.9). Then, it holds that for each $t(t\geq 0)$ ,$\underline{h}_{\alpha}(Q_{\alpha}^{t}(D))+\underline{K}_{\alpha}=\min R_{\alpha}(Q_{\alpha}^{t}(D))+\underline{h}_{\alpha}(Q_{\alpha}^{t+1}(D))$

By summing (3.11) for $t=0,1,$$\cdots,$.

(3.11)$T-1$ , we get.

$\underline{h}_{\alpha}(D)+T\cross\underline{If}_{\alpha}=\sum_{t=0}^{T-1}\min R_{\alpha}(Q_{\alpha}^{t}(D))+\underline{h}_{\alpha}(Q_{\alpha}^{T}(D))$

So$\underline{K}_{\alpha}=\lim_{Tarrow\infty}\frac{1}{T}\sum_{t=0}^{T-1}\min R_{\alpha}(Q_{\alpha}^{t}(D))$

for $D\in C(S)$ .

Thus, from Theorem 2.1 and Corollary 2.1,$\underline{K}_{\alpha}=\min R_{\alpha}(\tilde{p}_{\alpha})$We also obtainq. e. d.

$\overline{K}_{\alpha}=\max R_{\alpha}(\tilde{p}_{\alpha})$.

similarly. Therefore we get$\tilde{g}(\tilde{s})_{\alpha}=\underline{\lceil K}_{\alpha},\overline{K}_{\alpha}]$by (3.6).

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