\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 108, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/108\hfil Some properties of sheaf-solutions]
{Some properties of sheaf-solutions of sheaf fuzzy control problems}

\author[N. D. Phu, T. T. Tung\hfil EJDE-2006/108\hfilneg]
{Nguyen Dinh Phu, Tran Thanh Tung }  % in alphabetical order

\address{Nguyen Dinh Phu \newline
Department of Mathematics and Computer Sciences, 
University of Natural Sciences, 
Vietnam National University, 227 Nguyen Van Cu street,
District 5th, Ho Chi Minh City, Vietnam}
\email{ndphu@mathdep.hcmuns.edu.vn}

\address{Tran Thanh Tung \newline
Faculty of Pedagogy of Tay Nguyen University, 567 Le Duan Road,
Buon Ma Thuot City, Daklak Province, Vietnam}
\email{thanhtung\_bmt@yahoo.com}

\date{}
\thanks{Submitted April 11, 2006. Published September 11, 2006.}
\subjclass[2000]{34H05, 93C15, 93C42}
\keywords{Fuzzy theory; Differential equations;
Fuzzy differential equation; \hfill\break\indent Control theory}

\begin{abstract}
 In this paper we present some results on the dependence of
 sheaf-solutions of sheaf  fuzzy control problems on variation of
 fuzzy controls.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{definition}{Definition}[section]

\section{Introduction}
The problems of differential equations, differential inclusions and 
control differential equations have been studying in both theory and 
application. See \cite{CLR, PHU, PHU1, PHUH, XU}.
The concept sheaf solution of classical control problems was
introduced in \cite{OV}. Instead of studying each solution, one
studies sheaf-solution, that means, a set of solutions. In
\cite{PHUT1, PHUT2, PHUT3}, we presented some results of
sheaf-solutions in fuzzy control problems whose variables are
crisp and controls are fuzzy. In this paper, we study the fuzzy
control problems whose variables and controls are fuzzy sets and
the dependence of sheaf-solutions of sheaf  fuzzy control problems
on variation of fuzzy controls. The paper is organized as follows:
Section 2 reviews some concepts of fuzzy sets,  Hausdorff distance
and fuzzy diferential equations. The sheaf fuzzy control problem
and some new results of the dependence of sheaf-solutions on
variation of fuzzy  controls are presented in Section 3.


\section{ Preliminaries and notation}

We recall some notation and concepts which were presented in
detail in \cite{LAK1}-\cite {LAK4}, \cite{LAK6}.
 Let $K_c(\mathbb{R}^n)$ denote the collection of all nonempty, compact,
convex subsets of $\mathbb{R}^n$. Let $A,B$ be two nonempty
bounded subsets of ${\mathbb{R}^{n}}$. The Hausdorff distance
between $A$ and $B$ is defined as
\begin{equation}\label{eq21}
D[A,B]=\max\{\sup_{a \in A} \inf_{b \in B}\| a-b\|,\; \sup_{b \in B}
\inf_{a \in A}\| a-b\|\}.
\end{equation}
Note that $K_c(\mathbb{R}^n)$ with the metric $D$ is a complete
metric space. See \cite{TOL}. It is known that if the space
$K_c(\mathbb{R}^n)$ is equipped with the natural algebraic
operations of addition and nonnegative scalar multiplication, then
$K_c(\mathbb{R}^n)$ becomes a semilinear metric space which can be
embedded as a complete cone into a corresponding Banach space.

 Set
$\mathbb{E}^n =\{u:\mathbb{R}^n \to[0,1]: u \text{ satisfies
(i)--(iv) stated below }\}$
\begin{itemize}
\item[(i)] $u$ is normal, that is, there exists an $x_0\in \mathbb{R}^n$
such that $u(x_0)=1$;

 \item[(ii)] $u$ is fuzzy convex, that is,
for  $0\le {\lambda}\le 1$
$$
u(\lambda x_{1}+(1-\lambda)x_2)\ge \min \{u(x_1), u(x_2)\};
$$
\item[(iii)] $u$ is upper semicontinuous;

\item[(iv)] ${[u]}^0= \mathop{\rm cl}\{x \in \mathbb{R}^n: u(x)>0\}$ is compact.
The element $u\in \mathbb{E}^{n}$ is called a fuzzy number or fuzzy set.

\end{itemize}
For $0<\alpha\le 1$, the set $[u]^{\alpha}=\{x\in\mathbb{R}^n: u(x)\ge\alpha \}$ is called the $\alpha$-level set. From (i)-(iv), it follows that the $\alpha$-level sets are in $K_c(\mathbb{R}^n)$, for $0\le \alpha\le 1$. \\
Let us denote
$$
D_0[u, v]=\sup \{ D\big[ [u]^{\alpha}, [v]^{\alpha}\big]: 0 \le
\alpha\le 1 \}
$$
the distance between $u $ and $v$ in $\mathbb{E}^n$, where
 $D\big[ [u]^{\alpha}, [v]^{\alpha}\big]$ is Hausdorff distance between
two sets  $[u]^{\alpha}, [v]^{\alpha}$ of $K_c(\mathbb{R}^n)$.
Then $(\mathbb{E}^{n}, D_0)$ is a complete space. Some properties
of metric $D_0$ are as follows.
\begin{equation}\label{eq22}
\begin{gathered}
D_0[u+w, v+w]=D_0[u, v],\\
D_0[\lambda u, \lambda v ]=\mid \lambda \mid D_0[u, v],\\
D_0[u, v]\le D_0[u, w]+D_0[w, v],
\end{gathered}
\end{equation}
 for all $u, v, w \in\mathbb{E}^n$ and $\lambda\in \mathbb{R}$.
Let us denote $\theta^n \in \mathbb{E}^n$ the zero element of
$\mathbb{E}^n$ as follows
\begin{equation*}
\theta^n(z)=
\begin{cases}
1 & \text{if }z=\hat{0},\\
0 & \text{if }z\ne \hat{0},\\
\end{cases}
\end{equation*}
where $\hat{0}$ is the zero element of $\mathbb{R}^n$.

Suppose that $Q, G \subset \mathbb{E}^l$ for some positive integer
$l$. Here is some useful notation:
\begin{equation}\label{eq23}
\begin{gathered}
d^*[Q, G ]=\sup\{D_0[q,g]: q\in Q, g\in G\},\\
\mathop{\rm diam}[Q]=\sup\{D_0[q, q']: q, q'\in Q \},\\
d[Q]=d^*[Q, \theta^l],
\end{gathered}
\end{equation}
where $\theta^l$ is zero element of $\mathbb{E}^l$ which is
regarded as a one point set. Let $u, v\in \mathbb{E}^n$. The set
$z\in \mathbb{E}^n$ satisfying $u=v+z$ is known as the geometric
difference of the sets $u$ and $v$ and is denoted by the symbol
$u-v$. The mapping
$F:\mathbb{R_+}\supset I=[ t_0, T] \to \mathbb{E}^n$ is said to
have a Hukuhara derivative $D_{H}F(\tau)$
at a point $\tau\in I$, if
$$
\lim _{h\to 0^+}\frac{F(\tau+h)-F(\tau)}{h}\quad\text{and}\quad
\lim _{h\to 0^+}\frac{F(\tau)-F(\tau-h)}{h}
$$
 exist and are equal to $D_{H}F(\tau)$. Here limits are taken in the
 metric space $(E^n, D_0)$.
If $F:I \to \mathbb{E}^n$ is continuous, then it is integrable and
\begin{equation}
\int_{t_0}^{t_2}F(s)ds=\int_{t_0}^{t_1}F(s)ds+ \int_{t_1}^{t_2}F(s)ds.
\end{equation}
If $F, G:I \to \mathbb{E}^n$ are integrable,
$\lambda \in \mathbb{R}$, then some properties below hold
\begin{gather}
\int_{t_0}^t (F(s)+G(s))ds = \int_{t_0}^t F(s)ds+\int_{t_0}^t G(s)ds;\\
\label{eq26}
\int_{t_0}^{t} \lambda F(s)ds=\lambda \int_{t_0}^{t} F(s)ds, \quad
\lambda\in \mathbb{R}, t_0\le t\le T;\\
\label{eq27}
D_0\Big[ \int_{t_0}^{t}  F(s)ds, \int_{t_0}^{t}  G(s)ds \Big]
 \le \int_{t_0}^{t}D_0\big[ F(s), G(s)\big] ds.
\end{gather}
Let $F:I\to  \mathbb{E}^n$ be continuous. Then integral $\int_{t_0}^{t}  F(s)ds$ is differentiable and $D_{H}G(t)=F(t)$.\\
Recently, the study of fuzzy differential equation (FDE)
\begin{equation*}
D_{H}x=f(t, x(t)), \quad x(t_0)=x_0\in \mathbb{E}^n,
\end{equation*}
where  $f \in C[\mathbb{R}_+\times \mathbb{E}^n, \mathbb{E}^n ]$,
and $t\in \mathbb{R}_+, x(t) \in \mathbb{E}^n$
has gained much attention.
Some important results on the existence and comparison of solutions
of FDE have been obtained by Prof. V. Lakshmikantham and other authors
in \cite{CD, MM}, \cite{LAK3}- \cite{LAK5} and \cite{LAK6}.


\section{Main results}

We consider the initial valued problem (IVP) for a fuzzy control differential
equation (FCDE) as follows
\begin{equation}\label{eq31}
D_{H}x=f(t,x(t),u(t)), \quad x(t_0)=x_0\in \mathbb{E}^n,
\end{equation}
where  $f \in C[\mathbb{R}_+\times \mathbb{E}^n\times \mathbb{E}^p,
\mathbb{E}^n ]$, $t\in \mathbb{R}_+$, state $ x(t) \in \mathbb{E}^n$ and
fuzzy control $u(t) \in \mathbb{E}^p$.

The function $u: I\to \mathbb{E}^p $ is integrable, is called  an admissible
control. Let $U\subset \mathbb{E}^p$ be the set of all admissible
fuzzy controls. The mapping $x\in C^{1}[I, \mathbb{E}^n]$ is said
to be a solution of \eqref{eq31} on $I$ if it satisfies
(\ref{eq31}) on $I$. Since $x(t)$ is continuously differentiable,
we have
\begin{equation*}
x(t)=x_0+\int_{t_0}^t D_{H}x(s)ds, \quad t\in I.
\end{equation*}
Thus, we associate with the initial value problem (IVP) (\ref{eq31})
the function
\begin{equation}\label{eq32}
x(t)=x_0+\int_{t_0}^t f(s, x(s),u(s))ds, \quad t\in I
\end{equation}
where the integral is the Hukuhara integral. Observe that $x(t)$ is a
solution of (\ref{eq31}) if and only if it satisfies (\ref{eq32}) on $I$.

In \cite{PHUT3}, the authors proved the following theorem
(which is an adaptation of Theorems 4.1-4.2 of the set differential
equations in \cite{LAK3} to FCDE) on the existence of solutions of FCDE.

\begin{theorem}[\cite{PHUT3}] \label{theo31}
 Assume that $f \in C[\mathbb{R}_+\times \mathbb{E}^n\times \mathbb{E}^p,
 \mathbb{E}^n]$ and
\begin{itemize}
\item[(i)] $D_0\big[ {f(t,x(t),u(t)),\,\theta^n } \big]
\le g(t,\,D_0[ x(t), \theta^n ])$ for
 $(t,x,u) \in I\times \mathbb{E}^n\times U$,
where $g \in C[ {I\times \mathbb{R}_ + ,\mathbb{R}_ + }]$,
$g(t,w)$ is nondecreasing in $(t,w)$.

\item[(ii)] The maximal solution $r(t,w _{0})$ of the scalar differential
equation
$$
w'=g(t,w), \quad w(0)=w_0 \ge 0 \quad\text{exists  on } I.
$$
\end{itemize}
Then there exists a solution $x(t)=x(t, t_0, x_0, u(t))$ to
\eqref{eq31} which satisfies
$$
D_0[ {x(t), x_0 }] \le r(t,w_0 ) - w_0 ,\quad t \in I,
$$
where $w_0 = D_0[ x_0 ,\theta^n], u(t)\in U$.
\end{theorem}

In \cite{OV}, the author presented the concept of sheaf-solution of a
classical control differential equation.  Instead of studying each solution,
one studies sheaf-solution, that means, a set of solutions.
In this paper, we use this concept for FCDE.

\begin{definition} \rm
The sheaf-solution (or sheaf-trajectory) of \eqref{eq31} which gives at
the time $t$ a set
\begin{equation*}
 H_{t,u}=\{x(t)=x(t,x_0,u(t))-\text{solution of (\ref{eq31})} : x_0 \in H_0 \},
\end{equation*}
where
$ u(t) \in U \subset \mathbb{E}^p$, $t\in I$, $H_0\subset \mathbb{E}^n$.
\end{definition}

$H_{t,u}$  is called a $cross-area$ at $ (t,u)$ (or $(t,u)$-cut) of the
above sheaf-solution. 
System (\ref{eq31}) with its sheaf-solutions are called a
{\it sheaf fuzzy control problem}.
For two admissible controls $u(t)$ and $\bar{u}(t) $, we have two
sheaf-solutions whose cross-areas are
\begin{gather}\label{eq33}
 H_{t,u}=\big\{x(t)=x(t,x_0,u(t))-\text{solution of (\ref{eq31})} : x_0 \in H_0 \big\},
\\ \label{eq34}
\bar{H}_{t,\bar {u}}=\big\{\bar{x}(t)=x(t,\bar{x}_0,\bar {u}(t))
-\text{solution of (\ref{eq31})} : \bar{x}_0 \in \bar{H}_0 \big\}
\end{gather}
where $t \in I$; $u(t), \bar {u}(t)\in U$; $H_0 ,\bar{H}_0\subset
\mathbb{E}^n$.
In this paper, instead of comparison two sheaf-solutions, we compare
their cross-areas.

An immediate consequence of Theorem \ref{theo31} for estimate of one
sheaf-solution is the following result.

\begin{corollary}\label{coro31}
Under assumptions of Theorem \ref{theo31}, one has
$$  
d^*[H_{t,u}, H_0] \le r(t,w_0 ) - \alpha_0
$$
where $\alpha_0=\inf\{D_0[ x_0 ,\theta^n ]: x_0 \in
H_0\}$ for all $u(t)\in U$.
\end{corollary}

Now, we consider the following assumption on $f$:
The vector function $f:\mathbb{R}^+ \times \mathbb{E}^n \times
\mathbb{E}^p \to \mathbb{E}^n $ satisfies the condition
\begin{equation}\label{eq35}
D_0\big[f(t, \bar{x}(t), \bar{u}(t)),f(t, x(t), u(t))\big] \le c(t)
\big[ D_0\big[ \bar{x}(t), x(t)\big] +D_0[\bar{u}(t), u(t)]\big]
\end{equation}
for all $t\in I; u(t), \bar{u}(t) \in U; x(t),\bar{x}(t) \in
\mathbb{E}^n$, where $c(t)$ is a positive and integrable on $I$.

Let $C=\int_0^T c(t)dt$. Because $c(t)$ is integrable on $I$,
it is bounded almost everywhere (a.e) by some positive constant  $K$,
 that is,  $c(t)\le K$ a.e. $ t\in I$.
If we consider the RHS of assumption \eqref{eq35} as a scalar
function $g$, then $g$ depends on states and controls and the
structure of $g$ is simple. The scalar function $g$ is used to
estimate the variation of function $f$ whose value is in
$\mathbb{E}^n$.
 In the following theorem, we prove that the solutions of (\ref{eq31})
depend continuously on variation of controls and initials.

\begin{theorem}\label{theo32}
Suppose that $f$ is continuous and satisfies \eqref{eq35} and
$\bar{x}(t), x(t)$ are two solutions of (\ref{eq31}) originating
at different initials $\bar{x}_0, x_0$ with controls $\bar{u}(t), u(t)$,
respectively. Then
for every $\epsilon >0 $ there exists  a $\delta( \epsilon) >0$
such that
$$
D_0\big[\bar{x}(t), x(t) \big] \le \epsilon $$
if
\begin{equation}\label{eq36}
D_0\big[\bar{x}_0, x_0 \big]\le \delta( \epsilon)
\quad \text{and}\quad %\label{eq37}
D_0[\bar{u}(t), u(t)]\le \delta( \epsilon)
\end{equation}
where $t\in I$; $\bar{u}(t), u(t) \in U$.
\end{theorem}

\begin{proof}
 The solutions of (\ref{eq31}) for controls $u(t)$ and $\bar{u}(t)$
  originating at the points $x_{0}$ and $\bar{x}_{0}$, are equivalent
 to the following integral forms
\begin{gather*}
x(t)=x_{0}+ \int_{t_0}^{t}  f(s, x(s), u(s))ds,\\
\bar{x}(t)=\bar{x}_{0}+ \int_{t_0}^{t} f(s, \bar {x}(s), \bar {u}(s))ds.
\end{gather*}
Estimating $D_0[\bar{x}(t), x(t)]$, using (\ref{eq22}) and (\ref{eq27}),
we get
$$
D_0[\bar{x}(t), x(t)]  \le D_0[\bar{x}_0, x_0]
+\int_{t_0}^t D_0\big[f(s, \bar{x}(s), \bar{u}(s)), f(s,x(s), u(s)) \big]ds.
$$
By assumption \eqref{eq35}, we have
\begin{equation}\label{eq38}
\begin{aligned}
D_0[\bar{x}(t), x(t)] &\le D_0[\bar{x}_0, x_0]+\int_{t_0}^t c(s)\big[D_0\big[\bar{x}(s), x(s)\big]+D_0\big[\bar{u}(s), u(s)\big]\big]ds\\
&\le  D_0[\bar{x}_0, x_0]+\int_{t_0}^t c(s) D_0\big[\bar{x}(s), x(s)\big] ds + K \int_{t_0}^t D_0\big[\bar{u}(s), u(s)\big] ds,
\end{aligned}
\end{equation}
then using $D_0\big[\bar{u}(t), u(t)\big]\le \delta(\epsilon)$ and
$D_0[\bar{x}_0, x_0]\le \delta(\epsilon)$, we obtain
$$
D_0[\bar{x}(t), x(t)] \le [ 1+K(T-t_0)] \delta (\epsilon)
+ \int_{t_0}^t c(s)D_0[\bar{x}(t), x(t)] ds .
$$
By Gronwall's inequality, we have the  estimate
$$
D_0[\bar{x}(t), x(t)]\le [ 1+K(T-t_0)]\delta( \epsilon) \exp (C).$$
For a given $\epsilon>0$, if we choose
$$
0<\delta( \epsilon )\le \frac{\epsilon}{[ 1+K(T-t_0)] \exp (C)}
$$
 then
$D_0[\bar{x}(t), x(t)] \le \epsilon$.
 The proof is complete.
\end{proof}

An immediate consequence of Theorem \ref{theo32} is the following result.

\begin{corollary}\label{coro32}
Suppose that $f$ is continuous and satisfies \eqref{eq35} and
$\bar{H}_{t, \bar{u}}, H_{t, u}$ are two cross-areas of sheaf-solutions of (\ref{eq31})
corresponding to $\bar{H}_0, H_0$ and controls $\bar{u}(t), u(t)\in U$. Then
for every $\epsilon >0 $ there exists  a $\delta( \epsilon) >0$
such that
$$
d^*\big[\bar{H}_{t, \bar{u}}, H_{t, u} \big] \le \epsilon
$$
 if
\begin{equation*} % \label{eq36}
d^*\big[\bar{H}_0, H_0 \big]\le \delta( \epsilon)
\quad\text{and}\quad %\label{eq37}
D_0[\bar{u}(t), u(t)]\le \delta( \epsilon)
\end{equation*}
where $t\in I$; $\bar{u}(t), u(t) \in U$.
\end{corollary}

In Corollary \ref{coro32}, we remark that the sheaf-solutions of sheaf
fuzzy control problem concerning fuzzy control differential equation
(\ref{eq31}) depend continuously on variation of fuzzy controls and initials.

We have some theorems below on comparison two sheaf-solutions in case of
$\bar{H}_0, H_0, U$ are bounded subsets.

\begin{theorem}\label{theo33}
Suppose that $f$ is continuous and satisfies \eqref{eq35}  and $H_0, U$ are
bounded subsets. Then
 $$
d^*\big[\bar{H}_{t, \bar{u}}, H_{t, u} \big] \le \big[\mathop{\rm diam}[ H_0]+K.
\mathop{\rm diam} [U](T-t_0)\big]\exp{(C)}
$$
where  $\bar{H}_{t, \bar{u}}, H_{t, u}$ are any cross-areas of sheaf-solutions of
\eqref{eq31} with $\bar{H}_0=H_0$, for all
$t\in I$; $\bar{u}(t), u(t) \in U$.
\end{theorem}

\begin{proof}
Starting as in the proof of Theorem \ref{theo32}, we arrive at (\ref{eq38})
and use (\ref{eq23}), then
\begin{equation*}
\begin{aligned}
D_0[\bar{x}(t), x(t)]
&\le D_0[\bar{x}_0, x_0]+\int_{t_0}^t c(s)\big[D_0\big[\bar{x}(s), x(s)\big]
 +D_0\big[\bar{u}(s), u(s)\big]\big]ds\\
&\le  D_0[\bar{x}_0, x_0]+\int_{t_0}^t c(s) D_0\big[\bar{x}(s), x(s)\big] ds
 + K \int_{t_0}^t D_0\big[\bar{u}(s), u(s)\big] ds\\
&\le  \mathop{\rm diam}[ H_0]+\int_{t_0}^t c(s) D_0\big[\bar{x}(s), x(s)\big] ds
+ K.\mathop{\rm diam}[U](T-t_0).
\end{aligned}
\end{equation*}
Using Gronwall's inequality, then the proof is complete.
\end{proof}

The following is result more general  than Theorem \ref{theo33}.

\begin{theorem}\label{theo34}
Suppose that $f$ is continuous and satisfies \eqref{eq35}  and
$\bar{H}_0, H_0, U$ are bounded subsets. Then
 $$
d^*\big[\bar{H}_{t, \bar{u}}, H_{t, u} \big] \le
\big[d^*[\bar{H}_0, H_0]+K.\mathop{\rm diam} [U](T-t_0)\big]\exp{(C)}
$$
where $\bar{H}_{t, \bar{u}}, H_{t, u}$ are any cross-areas of sheaf-solutions of
\eqref{eq31}, for all $t\in I; \bar{u}(t), u(t) \in U$.
\end{theorem}

The proof is similar the proof of Theorem \ref{theo33}.\\
The next comparison result provides an estimate under assumption
that the structure of the scalar function $g$ is more general than
the one in \eqref{eq35}, but $g$ does not  depend on fuzzy controls.

\begin{theorem}\label{theo35}
Assume that
$f \in C\big[I\times \mathbb{E}^n\times \mathbb{E}^p,\mathbb{E}^n\big]$ and
\begin{equation}\label{eq39}
D_0\big[f( t, \bar{x}(t), \bar{u}(t)), f(t, x(t), u(t) )\big]  \le g(t, D_0[\bar{x}(t), x(t)]),
\end{equation}
 for $(t, \bar{x}(t),\bar{u}(t)),\,(t, x(t), u(t)) \in
I\times  \mathbb{E}^n\times U$,
 where $g\in C\big[\mathbb{R }_+ \times  \mathbb{R }_{+}, \mathbb{R}_+\big]$
and $g(t, w)$ is nondecreasing in $w$ for each $t\in I$. Suppose further
that  the maximal solution $r(t)=r(t, t_0, w_0) $ of  scalar differential
 equation
$$w'=g(t, w),\quad  w(t_0)=w_0\ge 0$$
exists for  $t\in I$.
Then, if  $\bar{x}(t)=\bar{x}(t, t_0, \bar{x}_0,
\bar{u}(t))$ and $x(t)=x(t, t_0, x_0, u(t)) $ are any solutions of
(\ref {eq31}) such that $\bar{x}(t_0)=\bar{x}_0, x(t_0)= x_0;
\,\bar{x}_0, x_0\in \mathbb{E}^n$ existing for $t\in I$, one has
\begin{equation}\label{eq310}
D_0[\bar{x}(t), x(t)] \le r(t, t_0, w_0),  t\in I,
\end{equation}
provided $D[\bar{x}_0, x_0] \le w_0$, for all $t\in I$;
$\bar{u}(t), u(t) \in U$.
\end{theorem}

\begin{proof}
The solutions of (\ref{eq31}) for controls $u(t)$ and $\bar{u}(t)$
originating at $x_{0}, \bar{x}_0$ are equivalent to the
integral forms
\begin{gather*}
x(t)=x_{0}+ \int_{t_0}^{t}  f(s, x(s), u(s))ds,\\
\bar{x}(t)=\bar{x}_{0}+ \int_{t_0}^{t}  f(s, \bar {x}(s), \bar {u}(s))ds.
\end{gather*}
Set $m(t)=D_0\big[ \bar{x}(t), x(t)\big]$, so that
$m(t_0)=D_0\big[ \bar{x}_0, x_0\big]\le w_0$. Using the properties
(\ref{eq22}), (\ref{eq27}) of the metric $D_0$, one has
\begin{align*}
m(t)
& = D_0\big[ \bar{x}_0 + \int_{t_0}^t f\big(s, \bar{x}(s), \bar{u}(s)\big)ds,
 x_0 + \int_{t_0}^t f\big(s, x(s), u(s)\big)ds \big]\\
&  \le D_0\big[ \bar{x}_0 + \int_{t_0}^t f\big(s, \bar{x}(s),
  \bar{u}(s)\big)ds, \bar{x}_0 + \int_{t_0}^t fb(s, x(s), u(s)\big)ds \big]\\
& \quad+D_0\big[ \bar{x}_0 + \int_{t_0}^t f\big(s, x(s), u(s)\big)ds, x_0
 + \int_{t_0}^t f\big(s, x(s), u(s)\big)ds \big]\\
& = D_0\big[ \int_{t_0}^t f\big(s, \bar{x}(s), \bar{u}(s)\big)ds,
 \int_{t_0}^t f\big(s, x(s), u(s)\big)ds \big]+D_0[\bar{x}_0, x_0]\\
& \le  m(t_0)+\int_{t_0}^t D_0\big[ f\big( s,  \bar {x}(s), \bar {u}(s)\big),
f\big(s, x(s), u(s)\big)\big] ds.
\end{align*}
Then, using (\ref{eq39}), we obtain
\begin{equation}\label{eq311}
\begin{aligned}
m(t) &\le  m(t_0) + \int_{t_0}^t g\big( s, D_0
[  \bar{x}(s),  x(s)]\big)ds \\
& = m(t_0)+\int_{t_0}^tg\big( s, m(s) \big)ds, \quad t\in I.
\end{aligned}
\end{equation}
Applying  \cite[Theorem 1.9.2]{LAK5}, we conclude that
$m(t) \le r(t, t_0, w_0 ), t\in I$,
which completes the proof.
\end{proof}

In the above theorem we can dispense with the monotone character
of $g(t,w)$ if we employ the theory of differential inequality
instead of integral inequalities. This result and some more results
on this direction will be presented in next works.

\begin{corollary}\label{coro33}
Under the assumptions of Theorem \ref{theo35}, one has
$$
d^*\big[\bar{H}_{t, \bar{u}}, H_{t, u} \big] \le r(t, t_0, w_0), \quad t\in I,
$$
provided $d^*[\bar{H}_0, H_0] \le w_0$, for all
$\bar{u}(t), u(t) \in U$.
\end{corollary}
It is easy to prove this corollary.

\subsection*{Acknowledgments}
The authors gratefully acknowledge the  careful reading
and many valuable remarks given by the anonymous referee.
They improved this paper. The authors wish to express their thanks to Prof. V.Lakshmikantham and the other authors for their results of fuzzy differential equations in \cite{LAK1, LAK4, LAK6} which are useful with this paper.


\begin{thebibliography}{00}

\bibitem{CLR} F. H. Clarke , Y. S. Ledyaev, R. J. Stern, P. R. Wolenski;
{\it Nonsmooth Analysis and Control Theory}, Spingger, New York, 1998.

\bibitem{CD} Branco P. J. Costa, J. A. Dente;
{\it Fuzzy systems modeling in practice}, Fuzzy sets and Systems 121 (2001)
73-93.

\bibitem{MM} M. Friedman, M., Ma, A. Kandel;
{\it Numerical solutions of fuzzy differential and integral equations},
Fuzzy sets and Systems 106 (1999) 35-48.

\bibitem{LAK1} T. Gnana Bhaskar, V. Lakshmikantham, Devi J. Vasundhara;
 {\it Revisiting fuzzy differential equations}, Nonlinear Analysis 58
(2004) 351-358.

\bibitem{LAK2} V. Lakshmikantham, S. Leela;
{\it Fuzzy differential systems and the new concept of stability},
Nonlinea Dynamics and Systems Theory, 1(2)(2001) 111-119.

\bibitem{LAK3} V. Lakshmikantham;
{\it Set differential equations versus fuzzy differential equations},
Applied Mathematics and Computation, 164 (2005) 277-294.

\bibitem{LAK4} V. Lakshmikantham, Bhaskar T. Gnana, Devi J. Vasundhara;
 {\it Theory of set differential equations in metric spaces},
Cambridge Scientific Publisher, 2006.

\bibitem{LAK5} V. Lakshmikantham, S. Leela;
{\it Differential and Intergral Inequalities}, Vol. I and II, Academic Press,
New York, 1969.

\bibitem{LAK6} V.  Lakshmikantham, R. Mohapatra;
{\it Theory of Fuzzy Differential Eqations and Inclusions},
Taylor and Francis, London, 2003.

\bibitem{OV} D. A. Ovsanikov; {\it Mathematical methods for sheaf-control},
Publisher of Leningrad University, Leningrad, 1980.

\bibitem{PHU} N. D. Phu; {\it Differential equations},
VNU-Publishing House, Ho Chi Minh City, 2002.

\bibitem{PHU1} N.D. Phu; {\it General views in theory of systems},
VNU-Publishing House, Ho Chi Minh City, 2003.

\bibitem{PHUH} N. D. Phu, N. T. Huong;
{\it  Multivalued differential equations}, VNU-Publishing House,
Ho Chi Minh City, 2005.

\bibitem{PHUT1} N. D. Phu, T. T. Tung;
{\it  Sheaf-optimal control problem in fuzzy type}, J.Science and
Technology Development, 8(12) (2005) 5-11.

\bibitem{PHUT2} N. D.Phu, T. T. Tung;
{\it  The comparison of sheaf-solutions in fuzzy control problems},
J. Science and Technology Development, 9(2)(2006) 5-10.

\bibitem{PHUT3} N. D. Phu, T. T. Tung;
{\it  Existence  of solutions of fuzzy control differential equations},
J. Science and Technology Development (accepted).

\bibitem{TOL} A. Tolstonogov;
{\it Differential inclusions in a Banach Space}, Kluwer Academic Publishers,
Dordrecht, 2000.

\bibitem{XU} X. M. Xue; {\it Existence of solutions for semilinear nonlocal
Cauchy problems in Banach spaces},
Electronic Journal of Differential Equations (http://ejde.math.txstate.edu),
Vol. 2005 (2005), No 64, 1-7.

\end{thebibliography}

\end{document}