Intelligent Information Management, 2009, 1, 139-144
doi:10.4236/iim.2009.13020 Published Online December 2009 (http://www.scirp.org/journal/iim)
Linear Problems of Optimal Control of Fuzzy Maps
Andrej V. PLOTNIKOV1, Tatyana A. KOMLEVA2
1Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture,
Odessa, Ukraine
2Department of Mathematics, Odessa State Academy of Civil Engineering and Architecture,
Odessa, Ukraine
Abstract: In the present paper, we show the some properties of the fuzzy R-solution of the control linear
fuzzy differential inclusions and research the optimal time problems for it.
Keywords: fuzzy differential inclusions, control problems
1. Introduction
The first study of differential equations with multivalued
right-hand sides was performed by A. Marchaud [1,2]
and S.C. Zaremba [3]. In early sixties, T. Wazewski [4,5],
A.F. Filippov [6] obtained fundamental results on exis-
tence and properties of the differential equations with
multivalued right-hand sides (differential inclusions).
One of the most important results of these articles was an
establishment of the relation between differential inclu-
sions and optimal control problems, that promoted to
develop the differential inclusion theory [7].
Considering of the differential inclusions required to
study properties of multivalued functions, i.e. an elabora-
tion the whole tool of mathematical analysis for multi-
valued functions [8–10].
Simultaneously there were appeared papers [11–14]
which used Hukuhara derivative [9,10] of multivalued
function for investigation of differential equations with
multivalued right-hand sides and solutions.
In works [15,16] annotate of an R-solution for differ-
ential inclusion is introduced as an absolutely continuous
multivalued function. Various problems for the R- solu-
tion theory were regarded in [17–22].
The basic idea for a development of an equation for
R-solutions (integral funnels ) is contained in [23].
Here the equation was considered as a particular case
of an approximated equation in a metric space. Ap-
proximated equations make possible to exclude an dif-
ferentiation operation and there by to avoid linearity re-
quirement for a solution space and to consider differen-
tial equations in linear metric spaces, equations with
multivalued solutions and dynamical systems in nonlin-
ear metric spaces by unified positions [14,20,23,24].
Therefore, an approximated equation is a quasidiffer-
ential equation for determination of the dynamical sys-
tem in a metric space.
The theory of mutational equations in metric spaces,
which deals with multivalued trajectories (pipers) and
trajectories in nonlinear spaces has been developed in
[25].
As well as in [23] it is taken an approach that does not
use a derivative in explicit form for description in
nonlinear metric spaces, while in [25] analogous results
are obtained by construction "differential calculus" in
nonlinear metric spaces.
Moreover in [23] quasidifferential equations were
considered for locally compact spaces, while in [20] for
complete metric space .
In the last years there has been forming new approach
to control problems of dynamic systems, which founda-
tion on analysis of trajectory bundle but not separate tra-
jectories. The section of this bundle in any instant is
some set and it is necessary to describe the evolution of
this set. Obtaining and research dynamic equations of
sets there is important problem in this case. The metric
space of sets with the Hausdorff metric is natural space
for description dynamic of sets. In theory of multivalued
maps definitions on derivative as for single-valued maps
is impossible because space of sets is nonlinear. This
bound possibility description dynamic sets by differential
equations. Therefore, the control differential equations
with set of initial conditions [26–28], the control differ-
ential inclusions [29–40], the control differential equa-
tions with Hukuhara derivative [14] and the control qua-
sidifferential equations [14,40,41] use for it.
In recent years, the fuzzy set theory introduced by
Zadeh [42] has emerged as an interesting and fascinating
branch of pure and applied sciences. The applications of
fuzzy set theory can be found in many branches of re-
gional, physical, mathematical, differential equations,
and engineering sciences. Recently there have been new
advances in the theory of fuzzy differential equations
[43–55] and inclusions [50,56–59] as well as in the the-
ory of control fuzzy differential equations [60–62] and

A. V. PLOTNIKOV ET AL.
140
inclusions [63–65].
In this article we consider the some properties of the
fuzzy R-solution of the control linear fuzzy differential
inclusions and research the optimal time problems for it.
2. The Control Differential inclusions
2.1. The Fundamental Definitions and Designations
Let






nn RconvRcomp be a set of all nonempty (con-
vex) compact subsets from the space ,
n
R
  ABSBASBAh rr
r


)(,)(min, 0
be Hausdorff distance between sets A and B, Sr(A) is
r
-neighborhood of set A.
Let En be the set of all such that u sat-
isfies the following conditions:
]1,0[: 
n
Ru
1) u is normal, that is, there exists an such
that ;
n
Rx 
0
1)(0xu
2) u is fuzzy convex, that is,
 )(),(min)1( yuxuyxu  


3) for any and
n
Ryx ,10 


;
4) u is upper semicontinuous,
5) is compact.   0)(:
0 xuRxclu n
If , then u is called a fuzzy number, and En is
said to be a fuzzy number space. For 0 < α
≤ 1, denote
n
Eu 
  

 )(: xuRxu n.
Then from 1)-4), it follows that the α-level set
for all 0 ≤ α ≤ 1.
 
n
Rconvu 



Theorem 1 (Negoita and Ralescu [66]). If , then
n
Eu 
1) for all  
n
Rconvu 

]1,0[

;
2) for
 

uu 10 


;
3) If

]1,0[
k

is a decreasing sequence converg-
ing to 0

then




1

k
k
uu

Conversely, if


10:  

A
n
R
1
is a family of convex
compact subsets of satisfying 1)-3), then
for  Au 


0

and

0
10
0AAu 




.
If is a function, then using Zadeh’s
extension principle we can extend
nnn RRRg:
g
~
to
by the equation
nnn EEE 
 )(),(minsup))(,(
~
),(
yvxuzvug
yxgz 
.
It is well known that





vugvug ,),(
~

for all and continuous function g.
Further, we have
10,,  
n
Evu







vuvu, ,  

ukku 
where Rk

.
Define by the relation ),0[:  nn EED
 




vuhvuD ,sup),(
10
,
where h is the Hausdorff metric defined in comp(Rn).
Then D is a metric in En.
Further we know that [67]:
1)


DEn, is a complete metric space,
2)



 vuDwvwuD ,,



for all ,
n
Ewvu ,,
3)



 vuDvuD ,,   for alland
n
Evu ,R


.
It can be proved that

 ),(),(, zvDwuDzwvuD 



for .
n
Ezwvu ,,,
Definition 1. A mapping is strongly
measurable if for all
n
ETF ],0[:
]1,0[


the set-valued map


n
RconvTF ],0[:

defined byis Leb-
esgue measurable.


()( FtF 

)t
Definition 2. A mapping is said to be
integrably bounded if there is an integrable function
such that
n
ETF ],0[:
)(th) )(tx()( thtx  for every . )(
0tF
Definition 3. The integral of a fuzzy mapping


n
ETF,0: is defined levelwise by
.The set of all such that
is a measurable selection for for all









T
dttF
0
)(

F


T
dttF
0
)(

n
RTf ],0[:

T
dttf
0
)(

 1,0


.
Definition 4. A strongly measurable and integrably
bounded mapping


n
ETF,0: is said to be inte-
grable over

 T,0 if .

T
n
EdttF
0
)(
Note that if


n
ET ,0:F is strongly measurable
and integrably bounded, then F is integrable. Further if

A. V. PLOTNIKOV ET AL. 141

n
ETF ,0: is continuous, then it is integrable.
Theorem 2. [43]. Let


n
ETGF ,0:, be integrable
and ,  Tc ,0R

. Then
a)

TT
c
c
dttFdttF
00
;)()(dttF )(

T
GtF
0
)(

T
dttF
0
)( 

GFD ,
tFD



)(
x

x

b) ;
 
TT
dttGdttFdtt
00
)()()(
c) ;


T
dttF
0
)( 
d) is integrable;

e) .  dttGtFDdttGdt
TTT  




000
)(),()(,
Consider the following control linear fuzzy differential
inclusions
,)(),,()( 00 xtxwtGxtA  (1)
and the following nonlinear fuzzy differential inclusions
,)(),,,( 00 xtxwxtFx
, (2)
where means dt
dx ; is the time; is
the state; is the control; A(t) is (n×n)- dimen-
sional matrix-valued function; ,
are the set-valued functions.

Rtn Rx 
nmE
m

Rw
mn RRRF 

:
x
RRG 

:
n
E
Let
)(: m RconvRW 
 (3)
be the measurable multivalued map.
Definition 5. Set LW of all single-valued branches of
the multivalued map W(·) is the set of the possible con-
trols.
Obviously, the control fuzzy differential inclusion (2)
turns into the ordinary fuzzy differential inclusion
 ,)(,, 00 xtxxt


 (4)
if the control  LWw 
~
is fixed and




xt,.
 )(
~
,, twxtF
 LWw
The fuzzy differential inclusions (3) has the fuzzy
R-solution, if right-hand side of the fuzzy differential
inclusion (3) satisfies some conditions [59].
Let X(t) denotes the fuzzy R-solution of the differen-
tial inclusion (3), then X(t,w) denotes the fuzzy
R-solution of the control differential inclusion (2) for the
fixed .
Definition 6. The set



 LWwwTXTY 

:,)(
be called the attainable set of the fuzzy system (2).
2.2. The some properties of the R-solution
In this section, we consider the some properties of the
R-solution of the control fuzzy differential inclusion (1).
Let the following condition is true.
Condition A:
A1. A(·) is measurable on


Tt ,
0;
A2. The norm


tA of the matrix is inte-
grable on

tA


Tt ,
0;
A3. The multivalued map



m
RconvTtW,: 0 is
measurable on

 Tt ,
0;
A4. The fuzzy map satisfies the
conditions
nmERRG 

:
1) measurable in t;
2) continuous in w;
A5. There exist

 
TtLv ,
02



and



TtLl ,
02


such that



 tlwtGtvtW  ,,
almost everywhere on

 Tt ,
0.
A6. The set






LWwtwtGtQ

:)(,

Tt ,
0)(tQ 
is compact
and convex for almost every

,i.e. .
)( n Econv
Theorem 3. Let the condition A is true.
Then for every

 LWw


there exists the fuzzy R- solu-
tion

 wX,

such that
1). the fuzzy map

 wX,

has form
 


t
t
dsswsGstxtwtX
0
))(,()(, 1
0,
where

 Ttt ,
0

;

 t

is Cauchy matrix of the differ-
ential equation xtA )(x

;
2). for every ;
n
EwtX ),(

Ttt ,
0

3). the fuzzy map

 wX,

is the absolutely continuous
fuzzy map on

 Tt ,
0.
Proof. The proof is easy consequence of the
[32,37,40,59] and theorem 1.
Theorem 4. Let the condition A is true.
Then the attainable set Y(T) is compact and convex.
Proof. The proof is easy consequence of the [32,37,
40,59] and theorem 1.
We obtained the basic properties of the fuzzy R- solu-
tion of systems (1). Now, we consider the some control

A. V. PLOTNIKOV ET AL.
142
Clearly, that there cases of the transversal condition of
the maximum principle correspond to the three ses of
pair
ca
fuzzy problems.
3. The Optimal Time Problems
Consider the following optimal control problem: it is
necessary to find the minimal time T and the control
such that the fuzzy R-solution of system (1)
satisfies one of the conditions:
 LWw
*



k
SwTX 
*
,, (5)


k
SwTX 
*
,, (6)


k
SwTX 
*
,, (7)
where is the terminal set.
n
kES 
Clearly, these time optimal problems are different
from the ordinary time optimal problem by that here
control object has the volume.
Definition 6. We shall say that the pair





** ,, wXw 
T satisfies the maximum principle on

, if there ex-
ists the vector-functio  

t,
0
n

, which is the solution of the
system

 )0(,)( 1 STtAT 

and the following conditions are true
1) the maximum condition
   )(,),(max)(,)(,(1
)(
1
*twtGCttwtGС
tWw









almost everywhere on ;

Tt ,
0
2) the transversal condition:
a) in the case (5):
   )(,)(,),( 1
1
*TSCTwTXC k
 





;
b) in the case (6): for all

1,0


    TSCTwTXC k 


,)(,),(*






and there exists
 1,0

such that
     TSCTwTXC k 


,,),( *





;
c) in the case (7): for all  1,0


   )(,)(,),( *TSCTwTXC k 









and there exists
 1,0

such that
   )(,)(,),( *TSCTwTXC k 








.
the time optimal problems.
Theorem 5. (necessary optimal condition). Let the
condition A are true and the




*
,wT is optimality.
Then the pair






** ,,wXw  satisfies the maximum
principle on

 Tt ,.
0
Proof. The proonsequence of the [32,37,
40].
f is easy co
where
Example. Consider the following control linear fuzzy
differential inclusions
,0)0(,
01
10








xFwxx




T
xxx21, is the state;  Wwww T 2 ,
1


0S1

is the control; 2 E
F

is the fuzzy set, where




194,41
2
2



194,0
9
2
2
1
2
2
2
ff
f2
Consider the following optimal control prob: it is
necessary to find the minimal time T and the control
2
11 fff
f  .
lem

 LW
* such that the fuzzy R-solution of system
satisfies of the conditions:
w


SwTX
*
,
k
where is the terminal set su, that
2
ESk ch
















 xQxxx ,,122
2
2
2
1
 

Qx
xQxxx
xQxx
x
0
1,121
11,21
11
2
2
2
2
1
2
2
1
































121
121
1212
:
2
1
2
2
1
1
2
2
1



x
xx
x
R
x
x
Q.

2T
Obviously, the optimal pair and





 tttw sin,cos)(
* satisfy of the co of the
th
1)
nditions
eorem 5:







 )(, t  for a.e.  ,
*WCttw
 

2,0t;
2)




 

TTwTXC  




,)(,, 1
1
*SC k
 ,

for a.e. 

2,0t, where








T
ttt sin,cos  



  ,0,2,cos, 1
*TTwT
   sin TT TTX 






11,2:, 2121
1 xxxxS T
k
 .

A. V. PLOTNIKOV ET AL. 143
REFERENCES
[1] A. Marchaud, “Sur Les Champs De Demicones Et Equ
es Du Premier Order,” (in French) Bull.
, No. 62, pp. 1–38, 1934.
egral funnel
ns of attainable set dynamics. I.
, pp. 367–377, 1990.
olstonogov, “On an equation of an integral funnel
equations in a metric
uous dependence of
, Blagoevgrad, 1995.
grad,
(in Rus-
,” (in Russian) Vyssh. Shkola, Moscow,
pan-United
e time-optimality
Linear control systems with multivalued
et of a
[34] A. V. Plotnikov, “A problem on the control of pencils of
a-
an
tions Differentiell
Soc. Math. France
[2] A. Marchaund, “Sur les Champs Des Deme-Droites Et
Les Equations Differentilles Du Premier Ordre,” (in
French) Bull. Soc. Math., France, No. 63, pp. 1–38, 1934.
[3] S. C. Zaremba, “Sur une extension de la notion d'equation
differentielle,” Comptes Rendus Acad. Sc., Paris, No. 199,
pp. 1278–1280, 1934.
[4] T. Wazewski, “Systemes De Commande Et Equations Au
Contingent,” (in French) Bull. Acad. Polon. Sci., Ser. Sci.
Math. Astronom. Phys., No. 9, pp. 151–155, 1961.
[5] T. Wazewski, “Sur Une Condition Equivalente E L'equa-
tion Au Contingent,” (in French) Bull. Acad. Polon. Sci.,
Ser. Sci. Math. Astronom. Phys., No. 9, pp. 865–867,
1961.
[6] A. F. Filippov, “Classical solutions of differential equa-
tions with multi-valued right-hand side,” SIAM J. Control,
No. 5, pp. 609–621, 1967.
[7] J. P. Aubin and A. Cellina, “Differential inclusions,”
“Set-valued maps and viability theory,” Springer-Verlag,
Berlin, Heidelberg, New York, Tokyo, 1984.
[8] J. P. Aubin and H. Frankovska, “Set-valued analysis,”
Birkhauser, Systems and Control: Fundations and Appli-
cations, 1990.
[9] M. Hukuhara, “Sur L'application Semi-Continue Dont La
Valeur Est Un Compact Convexe,” (in French) Funkcial.
Ekvac., No. 10, pp. 43–66, 1967.
[10] M. Hukuhara, “Integration Des Applications Mesurables
Dont La Valeur Est Un Compact Convexe,” (in French)
Funkcial. Ekvac., No. 10, pp. 205–223, 1967.
[11] F. S. De Blasi, and F. Iervolino, “Equazioni Differentiali
Con Soluzioni A Valore Compatto Convesso,” Boll. Un-
ione Mat.Ital., Vol. 2, No. 4–5, pp. 491–501, 1969.
[12] M. Kisielewicz, “Method of averaging for differential
equations with compact convex valued solutions,” Rend.
Mat., Vol. 9, No. 3, pp. 397–408, 1976.
[13] T. A. Komleva and A. V. Plotnikov, “Differential inclu-
sions with the Hukuhara derivative,” Nonlinear Oscil. (N.
Y.), Vol. 10, No. 2, pp. 229–245, 2007.
[14] V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, “Dif-
ferential equations with multivalued right-hand sides,”
Asymptotics Methods. (in Russian) Odessa, AstroPrint,
pp. 354, 1999.
[15] A. I. Panasyuk and V. I. Panasyuk, “Asymptotic optimi-
zation of nonlinear control systems,” (in Russian) Izdat.
Beloruss. Gos. Univ., Minsk, pp. 207, 1977.
[16] A. I. Panasjuk, and V. I. Panasjuk, “An equation gener-
ated by a differential inclusion,” (in Russian) Mat.
Zametki, Vol. 27, No. 3, pp. 429–437, 1980.
[17] A. I. Panasyuk, “Dynamics of sets defined by differential
inclusions,” (in Russian) Sibirsk. Mat. Zh. Vol. 27, No. 5,
pp. 155–165, 206, 1986.
[18] A. I. Panasyuk, “On the equation of an int
d its applications,” Differential Equations, Vol. 24, No.
11, pp. 1263–1271, 1988.
[19] A. I. Panasyuk, “Equatio
Integral funnel equations,” J. Optim. Theory Appl., Vol.
64, No. 2, pp. 349–366, 1990. “Equations of attainable set
dynamics. II. Partial differential equations,” J. Optim.
Theory Appl., Vol. 64, No. 2
[20] A. I. Panasyuk, “Quasidifferential equations in a com-
plete metric space under caratheodory-type conditions,”
Differential Equations, Vol. 31, No. 6, pp. 901–910, 1995.
Differential Equations, Vol. 31, No. 8, pp. 1308–1317,
1995.
[21] A. I. Panasyuk and V. I. Panasyuk, “Asymptotic turnpike
optimization of control systems,” (in Russian) “Nauka i
Tekhnika,” Minsk, pp. 296, 1986.
[22] A. A. T
of a differential inclusion,” (in Russian) Mat. Zametki,
Vol. 32, No. 6, pp. 841–852, 1982.
[23] A. I. Panasyuk, “Quasidifferential
space,” (in Russian) Differentsial'nye Uravneniya, Vol. 21,
No. 8, pp. 1344–1353, 1985.
[24] V. Plotnikov and P. Kitanov, “Contin
solutions of quasidifferential equations with impulses,”
Discrete Mathematics and Applications, Blagoevgrad, Vol.
5, pp. 238–245, 1994, Res. Math., “Neofit Rilski”
South-West Univ. Publ. House
[25] J. P. Aubin, “Mutational equations in metric spaces,” Set-
Valued Analysis, Vol. 1, No. 1, pp. 3–46, 1993.
[26] D. A. Ovsyannikov, “Mathematical methods for the con-
trol of beams,” (in Russian) Leningrad. Univ., Lenin
pp. 227, 1980.
[27] V. I. Zubov, “Dynamics of controlled systems,”
sian) Vyssh. Shkola, Moscow, pp. 286, 1982.
[28] V. I. Zubov, “Stability of motion. Lyapunov methods and
their application
pp. 232, 1984.
[29] N. Kikuchi, “On contingent equations,” Ja
States Seminar on Ordinary Differential and Functional
Equations, Kyoto, pp. 169–181, 1971, Lecture Notes in
Math., Springer, Berlin, Vol. 243, 1971.
[30] S. Otakulov, “A minimax control problem for differential
inclusions,” Soviet Math. Dokl., Vol. 36, No. 2, pp.
382–387, 1988.
[31] S. Otakulov, “On the approximation of th
problem for controlled differential inclusions,” Cybernet.
Systems Anal., Vol. 30, No. 3, pp. 458–462, 1994.
[32] A. V. Plotnikov, “
trajectories,” (in Russian) Kibernetika (Kiev), No. 4, pp.
130–131, 1987.
[33] A. V. Plotnikov, “Compactness of the attainability s
nonlinear differential inclusion that contains a control,”
(in Russian) Kibernetika (Kiev), No. 6, pp. 116–118,
1990.

A. V. PLOTNIKOV ET AL.
144
ons,” Cybernet. Systems Anal., Vol. 29, No. 4, pp.
s of control of trajectory bundles,” Cy-
ectories,” Cybern. Syst. Anal., Vol. 38
uncertainty,” J. Appl. Math
lications of
tial equations
ol. 98
Vol. 64, No. 5, 2006. pp. 895–900.
University, Vol. 27, No. 1, pp.
“Dif-
m, G. T. Bhaskar, and D. J. Vasundhara,
. N. Mohapatra, “Theory of
equations,” Int.
,
lear and fuzzy multivalued right-hand sides,” (in
ems, Vol. 24, No. 3, pp. 319–330, 1987.
25,
. 55–67, 1990.
o. 3, pp.
Vol. 54, No. 1, pp. 8–13, 1990.
rtain.
fuzzy control problems,” Elec-
Nonlinear
Devel., Vol.
eter,” Nonlinear Oscil. (N. Y.)
ity in the problems of
,” Ukrainian Math. J., Vol.
Press Book. John
trajectories,” Siberian Math. J., Vol. 33, No. 2, pp.
351–354, 1992.
[35] A. V. Plotnikov, “Two control problems under uncertainty
conditi
567–573, 1993.
[36] A. V. Plotnikov, “Necessary optimality conditions for a
nonlinear problem
bern. Syst. Anal., Vol. 36, No. 5, pp. 729–733, 2000.
[37] A. V. Plotnikov, “Linear problems of optimal control of
multiple-valued traj,
Ru
No. 5, pp. 772–782, 2002.
[38] A. V. Plotnikov and T. A. Komleva, “Some properties of
trajectory bunches of controlled bilinear inclusion,” Ukr. [55] D. V
orobiev and S. Seikkala, “Towards the theory of fuzzy
differential equations,” Fuzzy Sets and Systems, Vol. 1
Math. J., Vol. 56, No. 4, pp. 586–600, 2004.
[39] A. V. Plotnikov and L. I. Plotnikova, “Two problems of
encounter under conditions of.
No
Mech., Vol. 55, No. 5, pp. 618–625, 1991.
[40] V. A. Plotnikov and A. V. Plotnikov, “Multivalued differ-
ential equations and optimal control,” App
Mathematics in Engineering and Economics (Sozopol,
2000), Heron Press, Sofia, pp. 60–67, 2001.
[41] A. V. Plotnikov, “Controlled quasidifferen
567–
and some of their properties,” Differ. Equations, Vol. 34,
No. 10, pp. 1332–1336, 1998.
[42] L. A. Zadeh, “Fuzzy sets,” Information and Control, No. 8,
pp. 338–353, 1965.
[43] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and
Systems, Vol. 24 , No. 3, pp. 301–317, 1987.
[44] O. Kaleva, “The Cauchy problem for fuzzy differential
equations,” Fuzzy Sets and Systems, Vol. 35, No. 3, pp.
389–396, 1990.
[45] O. Kaleva, “The peano theorem for fuzzy differential
equations revisited,” Fuzzy Sets and Systems, V, No. Anal., Vol. 67, No. 5, pp. 1309–1315, 2007.
[62] N. D. Phu and T. T. Tung, “Existence of solutions of fuzzy
control differential equations,” J. Sci. Tech.
1, pp. 147–148, 1998.
[46] O. Kaleva, “A note on fuzzy differential equations,”
Nonlinear Anal.,
[47] T. A. Komleva, L. I. Plotnikova, and A. V. Plotnikov, “Av-
eraging of the fuzzy differential equations,” Work of the
Odessa Polytehnikal
185–190, 2007.
[48] T. A. Komleva, A. V. Plotnikov and N. V. Skripnik,
ferential equations with set-valued solutions,” Ukrainian
Mathematical Journal. ( Springer New York) Vol. 60, No.
10, pp. 1540–1556, 2008.
[49] V. Lakshmikantha
cont
“Theory of set differential equations in metric spaces,”
Cambridge Scientific Publishers, Cambridge, pp. 204,
2006.
[50] V
. Lakshmikantham and R
59, N
fuzzy differential equations and inclusions,” Series in
Mathematical Analysis and Applications, Taylor & Fran-
cis, Ltd., London, Vol. 6, pp. 178, 2003.
[51] J. Y. Park and H. K. Han, “Existence and uniqueness
theorem for a solution of fuzzy differential
J. Math. Math. Sci., Vol. 22, No. 2, pp. 271–279, 1999.
[52] J. Y. Park and H. K. Han, “Fuzzy differential equations,”
Fuzzy Sets and Systems, Vol. 110, No. 1, pp. 69–77
2000.
[53] A. V. Plotnikov and N. V. Skripnik, “Differential equations
with c
ssian) Asymptotics Methods, AstroPrint, Odessa, pp.
192, 2009.
[54] S. Seikkala, “On the fuzzy initial value problem,” Fuzzy
Sets and Syst
. 2, pp. 231–237, 2002.
[56] J. P. Aubin, “Fuzzy differential inclusions,” Probl. Control
Inf. Theory, Vol. 19, No. 1, pp
[57] V. A. Baidosov, “Differential inclusions with fuzzy right-
hand side,” Soviet Mathematics, Vol. 40, N
569, 1990.
[58] V. A. Baidosov, “Fuzzy differential inclusions,” J. of Appl.
Math. and Mechan.,
[59] E. Hullermeier, “An approach to modelling and simulation
of uncertain dynamical systems,” Internat. J. Unce
Fuzziness Knowledge-Based Systems, Vol. 5, No. 2, pp.
117–137, 1997.
[60] N. D. Phu and T. T. Tung, “Some properties of
sheaf-solutions of sheaf
tron. J. Differential Equations, No. 108, pp. 8, 2006.
(Electronic) URL: http://ejde.math.txstate.edu.
[61] N. D. Phu and T. T. Tung, “Some results on
sheaf-solutions of sheaf set control problems,”
10, No. 5, pp. 5–12, 2007.
[63] T. I. V. Molchanyuk and A. V. Plotnikov, “Linear control
systems with a fuzzy param
Vol. 9, No. 1, pp. 59–64, 2006.
[64] I. V. Molchanyuk and A. V. Plotnikov, “Necessary and
sufficient conditions of optimal
rol with fuzzy parameters,” Ukrainian Math. J., Vol.
61, No. 3, pp. 457–463, 2009.
[65] V. S. Vasil'kovskaya and A. V. Plotnikov, “Integro differ-
ential systems with fuzzy noise
o. 10, pp. 1482–1492, 2007.
[66] C. V. Negoito and D. A. Ralescu, “Applications of fuzzy
sets to systems analysis,” A Halsted
Wiley & Sons, New York-Toronto, Ont., pp. 191, 1975.
[67] M. L. Puri and D. A. Ralescu, “Fuzzy random variables,”
J. Math. Anal. Appl., No. 114, pp. 409–422, 1986.