AN APPROACH TO CONSTRUCTING MATHEMATICAL MODELS

OF MOVING MECHANICAL SYSTEMS UNDER UNCERTAINTY

A. A. Martynyuk and V. I. Slyn’ko UDC 517.36

The paper outlines an approach to constructing a mathematical model of moving mechanical systems

under uncertainty. Generalized fuzzy differential equations are used to prove that the model is

mathematically correct. The motion of a control oscillator under uncertainty is considered as an example

Keywords: mechanical system under uncertainty, fuzzy differential equations, oscillator under uncertainty

Introduction. In constructing mathematical models of a moving mechanical system with a finite number of degrees of

freedom [6–8], it is generally assumed that all the most important factors affecting the dynamics of the system are known with

certainty and taken into account. Actually, no phenomenon can be represented in pure form, since however accurately the forces

acting on a material system are determined, there are always unaccounted, generally unknown perturbations. These

perturbations can be classified as follows: (i) perturbations of the first kind (the initial conditions for the evolution of the system

are not known with certainty, but some fuzzy set u En

02

containing these conditions is known) and (ii) perturbations of the

second kind (some fuzzy and uncertain forces).

Note that classical mechanics generally uses probability-theoretic approaches to describe uncertain perturbations. In the

process, the evolution of some distribution functions is considered. In quantum mechanics, the evolution of a system is described

by some function (I-function) that characterizes the probability distribution of states of the system. Thus, statistical or

probabilistic approaches are used to study the evolution of mechanical systems under uncertainty.

Some recent achievements in the theory of differential equations with multivalued solutions [1, 2, 5, 9] allow us to

propose an approach in a sense alternative to the probabilistic one. This approach analyzes the evolution of sets (fuzzy in the

general case) of attainability of a mechanical system.

1. Problem Formulation. Let some holonomic mechanical system with a finite number of degrees of freedom move

under the action of uncertain factors. Assume that the state of the system is characterized by vectors of generalized coordinates

q q qnT

� ( , , )1 � and generalized velocities � ( � , , � )q q qnT

� 1 � and by kinetic and potential energies,

T a q q q b q q q

i j

n

ij i j

i j

n

ij i j� �

� �

# #1

2

1

21 1, ,

( )� � , ( )P .

The equation of motion of the mechanical system can be written as Lagrange’s equations of the second kind:

d

dt

L

q

L

qQ i n

i i

i

�

�

�

�

��

�

��

�� �

�

, , , ,1 2 � ,

where L T� �P is the Lagrangian function of the mechanical system; Qi are given generalized forces.

International Applied Mechanics, Vol. 43, No. 10, 2007

1063-7095/07/4310-1157 ©2007 Springer Science+Business Media, Inc. 1157

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kyiv. Translated from Prikladnaya

Mekhanika, Vol. 43, No. 10, pp. 108–119, October 2007. Original article submitted October 30, 2006.

Assume that the vectors of initial position (q Rn

) and velocity (�q Rn

) are not known with certainty, and only some

fuzzy set u En

02

such that ( , � )q q uT T T

0 is known, where En2

is the space of fuzzy sets defined in [5]. The position of the

system at any time t t6 0 is characterized by some fuzzy set u t En

( ) 2

.

Let the system be subject to not only known forces, but also some unknown forces characterized by the vector of

generalized forces Q Q QnT* * *

( , , )� 1 � , which is estimated as

| | | | ( ) ( ( ), , ( ))*

Q p F l u l us� % 1 � ,

where lk , k s�1 2, , ,� , are some continuous functionals in the space En2

; p( )% is some nonincreasing function of the parameter

% [0, 1]; and F R Rs

: � is a continuous function.

The objective of the present paper is to outline an approach to the construction of a mathematical model of a moving

holonomic mechanical system under the assumptions introduced above. The equation of motion of this system is some fuzzy

differential equation in the space of fuzzy sets En2

describing the evolution of the fuzzy set of attainability of the system.

Let us consider the case where there are no unknown forces, i.e., Q*K 0. Assuming that the equations of motion are

resolvable for higher derivatives, they can be written as an ordinary differential equation:

dx

dtF t x x t x� �( , ), ( )0 0 , (1.1)

where x q qT T T

� ( , � ) and F C R R Rn n

R�1 2 2

( , ).

Let us study different approaches to include the trajectories of the ordinary differential equation (1.1) into the fuzzy set

u t( ), which is the solution of some fuzzy differential equation, i.e., it is necessary to set up a fuzzy differential equation

du t

dtF t u t u t u

( )( , ( )), ( )� �0 0 , (1.2)

such that the inclusion x t t x u t( ; , ) [ ( )]0 0 %

follows from the condition x u0 0 [ ]%

, where [ ( )]u t%

is the %-level of the fuzzy set

u t( ), x t t x( ; , )0 0 being the solution of the ordinary differential equation (1.1).

For the function F t x( , ) and the fuzzy set u En

2

, we define a family of sets [ ( , )] { ( , ) [ ] }F t u F t x x u% %� co , where

co{ }� is the closure of the convex hull of the set {...}. It can be proved that the solutions of the ordinary differential equation

(1.1) are included in the set of solutions of the fuzzy equation (1.2) if the following assumptions as to the functions F t x( , )hold.

Denote by KC

n2the set of nonempty compact subsets of the space R

n2.

Assumption 1.1. The function F t x( , ) is monotonic and continuous in the following sense:

(i) for any sets A B KC

n,

2such that A BW the inclusion F t A F t B( , ) ( , )W holds for all t R � ;

(ii) for any sequence of sets { }A Kk k C

n

�

�

1

2converging to the set A in Hausdorff metric, the sequence of sets

{ ( , )}F t Ak k�

�

1converges to the set F t A( , ) in Hausdorff metric.

Let us consider another approach to include the trajectories of the ordinary differential equation (1.1) into the fuzzy set

u t( ), which is the solution of some fuzzy differential equation of the form (1.2). Denote byUt

t

0

the operator of evolution of the

system of ordinary differential equations (1.1) and make the following assumptions.

Assumption 1.2. The function F t x( , ) is such that the operatorUt

t

0

has the following properties:

(i) for any sets A B KC

n,

2such that A BW , the inclusionU A U B

t

t

t

t

0 0

W holds for all t t R6 �0 ;

(ii) for any sequence of sets { }A Kk k C

n

�

�

1

2converging to the set A in Hausdorff metric, the sequence of sets

{ }U At

tk k

01�

�converges to the setU A

t

t

0

in Hausdorff metric for all t t6 0 .

1158

For each t t6 0 we define families of sets [ ( )] [ ]u t U ut

t% %�co

00 , % [0, 1]. It can be shown that Assumption 1.2

guarantees that for all t t6 0 the sets [ ( )]u t%

determine levels of some fuzzy set which can naturally be denoted by coU ut

t

00 .

Assume that the map u t( )is differentiable in the sense of the definition from [5, formula 2.5.1] for all t t a [ , )0 , t a0 � � �. Then

u t( ) is the solution of some fuzzy differential equation

du t

dtF t u t u t u

( )( , ( )), ( )� �0 0 ,

where F t u t D U ut

t( , ( )) � co

00 , D is the derivative in the sense of the definition [5, formula 2.5.1].

The major drawback of the first approach is that the diameter of the set of solutions u t( )of Eq. (1.1) is a nondecreasing

function, which is often physically inconsistent. Indeed, suppose that Eq. (1.1) has an asymptotically stable fixed point. Then any

small neighborhood of this point contracts; therefore, its diameter cannot be a nondecreasing function of time. The second

approach, however, cannot be used too in this situation because the multivalued function u t U ut

t( ) �co

00 can be

nondifferentiable. Thus, there is a need to generalize the concepts of differentiability of fuzzy multivalued maps and fuzzy

differential equations so as to adequately describe the motion of a mechanical system provided that the uncertainty of motion is

due only to the uncertainty of the initial data, rather than of the acting forces. Behind the generalization is the idea of embedding

the space of fuzzy sets En2

into some Banach space. This embedding is based on the statement that each fuzzy set u is in

one-to-one correspondence with its support function h S Rn n

[ , ]0 12 1

R ��

[5, p. 8].

Let us consider a Banach space� consisting of functions h S Rn

:[ , ]0 12 1

R ��

(where Sn2 1�

is a unit sphere of the space

Rn2

) satisfying the following conditions:

(i) h C Sn

( ,. ) ( )% �2 1

for any % [ , ]0 1;

(ii) the functions | | ( ,. )| |( )

hC S

n% �1 are bounded on the set [0, 1].

Then, as shown in [5, p. 9], the space En2

is isomorphic and is isometrically embedded into the space �. This

isomorphism allows us to construct a theory of fuzzy differential equations, proceeding not from fuzzy sets, but rather from their

support functions.

Let us define the derivative of a multivalued fuzzy map that would generalize the definition from [5, p. 42]. Let

u T En

: �2

, where T is an open set of a number semiaxis, and h t( ) � is the support function of the fuzzy set u t( ), t R � .

Definition 1.1. The map h t( )is called differentiable at a point t t T� 0 if there exists an element &( )t0 � such that

| | ( ) ( ) ( ) | | ( )h t h t t o0 0 0� � � �� & � �� as � � 0.

Note that the fundamental difference of Definition 1.1 from those given in [2, 5] is that the derivative of the map h t( )

may not belong to the space of fuzzy sets En2

.

The above definition can be used to generalize the concept of a fuzzy differential equation.

Definition 1.2. The mapU T: R �� � is called positive with respect to En2

if the inclusionU t h En

( , ) 2

holds for any

function h En

2

.

Let us consider the following differential equation in the Banach space�:

dh t

dtF t h t h t h p E

n( )

( , ( )), ( ) ( , )� � 0 02

% , (1.3)

where h t( ) �, F T En

: R �2

�, T RW � .

We additionally assume that the operator F t( ,. )satisfies the existence conditions for the evolution operatorU t t h( ; , )0 0

of Eq. (1.3) on some interval t t t a �[ , )0 0 1 , 0 1� � � �a a , defined on some set ( , )t h T D0 R , where D is some open set of the

space En2

.

Definition 1.3. The differential equation (1.3) is called a fuzzy differential equation if and only if the operatorU t t( ; ,. )0

is positive with respect to En2

for fixed ( , )t t0 at which this operator is defined.

We revert to (1.1) and assume that F t x D t x G t x( , ) ( ) ( , )� � , where the function G t x( , ) is estimated as

1159

| | ( , )| | ( )| | | | , ( ) ,G t x g t x g t� ; ;�1

0 0,

, . (1.4)

Let us consider the linear approximation of the system of equations (1.1):

dx

dtD t x x t x� �( ) , ( )0 0 . (1.5)

Let us derive a fuzzy differential equation for the fuzzy set u t t t u t t u( ) ( , ) ( , )� �co� �0 0 0 0 , where �( , )t t0 is the

evolution operator (matrizant) of the linear system (1.5). Denote by h t( ) the support function of the set u t( ) and analyze the

evolution of the system at successive times t and t t� = , where =t ; 0. Denote by u t t( )� = and u t( ) fuzzy sets of positions of the

phase vector of the system at times t and t t� = , respectively. Let x t u t( ) [ ( )] %

, then

[ ( )] {( ) ( ) ( ) | ( ) [ ( )]u t t I D t x t o t x t u t� � � � = = =% %

}.

Denote by h t p h pu t( , , ) ( , )( )% %K the support function of %-level of the fuzzy set u t( ). Then

h t t p h p o tI D t u t

( , , ) ( , ) ( )( )[ ( )]

� � ��

= ==

% %% .

Let us consider h pI D t u t( )[ ( )]

( , )� =

% % defined by

h p x pI D t u t

x I D t u t( )[ ( )]

( )[ ( )]

( , ) sup ( , )�

�

�=

=

%

%

%

� � � �

sup (( ) , ) sup ( , ( )

[ ( )] [ ( )]x u t x u t

TI D t x p x I D t p

% %

= = ) ( , , ( ) )� �h t I D t pT

% = .

Let h t C Sn

( , ,. ) ( )% � �2 1

. Then

h t I D t p h t p h t p D p t o tT

pT T

( , , ( ) ) ( , , ) ( ( , , )) ( )% % %� � � 5 �= = = ,

whence follows that the function h t p( , , )% satisfies the differential equation

dh t p

dth t p D t pp

T T( , , )

( ( , , )) ( )%

%� 5 (1.6)

with the initial condition h t p h p( , , ) ( , )0 0% %� , where h p0 ( , )% is the support function of the set u0 .

Equation (1.6) describes the motion of a mechanical system in the absence of perturbations of the second kind.

Let us consider the general case where Q*� 0. The nonlinear component G t x( , ) in (1.4) is referred to as a fuzzy

uncertain force. Such forces can be estimated as follows:

| | | | ( ) ( ( ( )), , ( ( )) ( )| | ( ,. )| |*

Q p F l u t l u t g t h ts� ��

%,

11

� .

In view of the general assumptions on uncertain forces, the mathematical model of a moving mechanical system is the

differential equation

dh t p

dth t p D t p F l u t l u tp

T Ts

( , , )( ( , , )) ( ) ( ( ( ), , ( (

%%� 5 � 1 � )) ( , )h pE %

� ��

g t h t h p h t h pB( )| | ( ,. )| | ( ), ( ) ( , )1

0 0,

% ,

where h pE ( , )% is the support function of a fuzzy set E defined levelwise:

[ ] {( , � ) , , | | � | | ( )}E q q q q pT T T%

%� � �0 ,

B is a subset of the space Rn2

,

B q q q qT T T

� � �{( , � ) , , | | � | |0 1}.

1160

2. Correctness of Problem Formulation. In Sec. 1 it was proved that the motion of a mechanical system under

uncertainty can be described by a differential equation in the Banach space�:

dh t

dtS t h t U t h t h t h

( )( ) ( ) ( , ( )), ( )� � �0 0 , (2.1)

where h �,U R E En n

: � R �2 2

, S t h t p D t ppT T

( ) ( ( , , )) ( )� 5 % is an unbounded, generally linear operator.

Here we establish conditions for the mathematical correctness of the problem formulation for a moving mechanical

system under uncertainty, which reduces to the following questions:

(i) Is the differential equation (2.1) a fuzzy differential equation in the sense of Definition 1.3?

(ii) What are the conditions for the Cauchy problem (2.1) to have a unique solution?

(iii) Are the solutions of the differential equation (2.1) continuous with respect to time and initial data?

To answer these questions, we first formulate a theorem on approximation of the support functions of fuzzy sets by

smooth support functions.

Lemma 2.1. If h p En

( , )% 2

, then there exists a sequence h pm ( , )% E h C Sn

mn2 2 1

, ( ,. ) ( )% � �

, such that

| | ( , ) ( , )| |h p h pm % %� �� 0 as m��.

Proof. Let A be a convex compact set such that h pA ( )is a support function. Let us consider the sequence of functions

h p g h p dgm

G

m g u( , ) ( ) ( )/ [ ]% + %� � 1 ,

where G SO n� ( )2 is a group of linear orthogonal transformations of the space Rn2

that preserve its orientation, the integral on

the right-hand side is an invariant Neumann integral over the compact group G [3], gA is the result of action of an element g G

on the set A, +� ( )g is an averaging kernel,

+ �

�

�

�

�( ) , | | | | ,

, | | | | ,

|| ||g Ce g e

g e

g e� � �

� 6

B

CE

DE

� �

2

2 2

0

(2.2)

C g dg

G

+� ( ) �� 1,

e is the unity of the group G, | |. | | is some norm on the group G induced by some matrix norm upon exact matrix representation of

the group G.

Let us show that h p Emn

( , )% 2

and h C Smn

( ,. ) ( )% � �2 1

for all % [0, 1]. It is obvious that for all % [0, 1] the

function hm ( ,. )% is convex and positively homogeneous. Let 0 1 2� � �% % 1. Then[ ] [ ]u u% %

2 1W and g u g u[ ] [ ]% %

2 1W for all

g G , h pg u[ ]

( )% 2� h p

g u[ ]( )%1

; therefore, the inequality h p h pm m( , ) ( , )% %2 1� holds. Let % [0, 1] be a fixed number. It is

easy to show that there exists a continuous map & :S Gn2 1�

� such that &( )q p q� , & � �

C S Gn

( ; )2 1

. Using the properties of the

invariant Neumann integral [3], we find

h q h q p g h q p dgm m

G

m g u( , ) ( , ( ) ) ( ) ( ( ) )/ [ ]% % & + &%� � � 1

� �� �G

m q g u

G

m gq g h q p dg q g h+ & & + &

&%1 1/ ( ) [ ] / [

( ( ) ) ( ( ) ) ( ( ) )u

p dg]

( )% .

Since +1/ ( ) ( ; )m g C G R �

and & � �

C S Gn

( , )2 1

, the function h qm ( , )% is infinitely differentiable on the set Sn2 1�

.

To complete the proof, it should be considered that+ �1/ ( ) ( )m g g e� � as m��, where the limit is taken in weak sense, �is the

Dirac delta function. Then

1161

h p g h p dg g e h p dg hm

G

m g u

G

g u( , ) ( ) ( ) ( ) ( )/ [ ] [ ]% + �% %� � � �� �1 [ ]

( ) ( , )u

p h p% %�

in the metric of the space�. Lemma 2.1 is proved.

Corollary 2.1. For any function h p KC

n( )

2there exists a sequence of functions h p K C Sm C

n n( ) ( ) X

� �2 2 1such

that | | ( ) ( )| |( )

h p h pm C Sn� ��2 1 0as m��.

It can be seen that the domain of definition of the operator S t( ) is a set of functions

D S t h K h C SC

n n( ( )) { , ( ,. ) ( )}�

� �2 2 1% ,

is an everywhere dense set in the space KC

n2, i.e.,

D S t KC

n( ( )) �

2.

Thus, it is possible to look into the existence of the closure of the operator S t( ).

Lemma 2.2. The operator S t( ) defined by

S t h t p D t ppT T

( ) ( ( , , )) ( )� 5 %

permits closure.

Let us describe the closure of S t( ). We define linear operators 0t

t

0

, t t6 0 ,

D K h p h t t p h Kt

t

C

n

t

t T

C

n( ) , ( ) ( ( , ) ),0 0 �

0 0

20

2� � ,

where�( , )t t0 is the matrizant of the linear system of ordinary differential equations (1.5).

The set of operators 0t

t

0

has the following properties:

(i) 0t

tid

0

0 � , where id is an identical operator;

(ii) 0 0 0st

t

s

t

t

0 0

� for all 0 0� � �t s t;

(iii) for any h C Sn

� �

( )2 1

, the limit limt t

t

th h

t t� �

�

�0

0

0 0

0

exists and equals S t h( )0 ;

(iv) for any t t0 , , 0 0� �t t, there exists a constant & &� ( , )t t0 such that the inequality

| | | | | | | |( ) ( )

0 0t

t

t

t

C S C Sh h h hn n

0 02 1 2 12 1 2 1� � �� �&

holds for all h C Sin

�

( )2 1

, i �1 2, .

(v) | | | |( )

0t

t

C Sh h n

01 0� �� as t t� �0 0 for any t0 06 and h C S

n

�( )

2 1.

The closure S t( )0 of the operator S t( )0 is defined as follows: the domain of definition of the operator S t( )0 consists of

those and only those functions h C Sn

�

( )2 1

for which the limit limt t

t

th h

t t� �

�

�0

0

0 0

0

exists. It is assumed

S t h

h h

t tt t

t

t

( ) lim00 00

0�

�

�� �

0

.

It is obvious that S t S t( ) ( )W and S t( ) is the closure of the operator S t( ). The operators S t( )and S t( )can be continued

to the space En2

, these continuations denoted by the same symbols. The domains of definition of these operators are denoted by~

( ( ))D S t and~

( ( ))D S t ,

1162

~( ( )) { , ( , . ) ( ), [ , ]}D S t h E h C S

n n�

� �2 2 10 1% % ,

~( ( )) { , ( , . ) ( ), [ , ]}D S t h E h D S

n�

20 1% .

Thus, for the differential equation of a fuzzy model of a mechanical system, it is necessary to take a differential equation

in the Banach space�:

dh t

dtS t h t U t h t h t h E

n( )

( ) ( ) ( , ( )), ( )� � � 0 02

, (2.3)

where h t( ) �, t R � .

Let us define the solution of Eq. (2.3) and establish its existence conditions.

Definition 2.1. The function h T En

: �2

is called a strong solution of Eq. (2.3) on the set T with the initial condition

h t h D S t( )~

( ( ))0 0 0� if and only if h t D S t( )~

( ( )) 0 for t T and Eq. (2.3) holds.

Definition 2.2. The function h T: �� is called a weak solution of Eq. (2.3) on the set T with the initial condition

h t h En

( )0 02

� if and only if h t( ) for t T and satisfies the integral equation

h t h U s h s dst

t

t

t

st

( ) ( , ( ))� � �0 00

0

0 . (2.4)

Note that any strong solution is weak, and if a weak solution satisfies the condition h t D S t( )~

( ( )) for all t T , then it is

a strong solution of Eq. (2.3).

In the space�, we introduce a partial order relation ;;as follows: h h2 1;; for h p1 ( , )% , h p2 ( , )% � if and only if the

condition h h KC

n2 1

2( , . ) ( , . )% %� holds for any % [0, 1].

Note that the inequality h t( ) ;; 0 is true for any solution of a fuzzy differential equation in the sense of Definition 1.3

with the initial condition h En

02

.

Definition 2.3. The operator U T E En n

: R �2 2

is called monotonic if for all t T the inequalities

U t h U t h( , ) ( , )2 1 0;; ;; follow from the inequalities h h2 1 0;; ;; , h Ein

2

(i �1 2, ).

Note that the operators 0t

t

0

are monotonic for all t t6 60 0.

The set D R Ea rn

, W R�2

is defined as follows: D t ha r, {( , )|� 0� � � � �t t a h h r0 0, | | | |� }.

Theorem 2.1. Let the operator U t t a E En n

:[ , )0 02 2

� R � satisfy the Lipschitz property on the set Da r, , i.e., there

exist a constant L0 0; and a non-negative function, summable on any interval R� , such that

| | ( , ) ( , )| | ( )| | | | , , ,U t h U t h L t h h h h E Dn

a r2 1 2 1 1 22

� � � X� � ,

L s ds L t t t t R

t

t

( ) ( ), ,� � � �0 2 1 1 2

1

2

.

Then Eq. (2.3) is a fuzzy differential equation in the sense of Definition 1.3 and for any h0 there exists a unique weak

solution of Eq. (2.3) defined on some interval t t t a �[ , )0 0 1 , a a1 � .

If the operatorU is monotonic, then for two weak solutions h t2 ( ) and h t1 ( ) with initial conditions h01, h En

022

, the

inequality h t h t2 1( ) ( )�� follows from the inequality h h20 01�� for all t from the interval of existence of solutions h ti ( ), i �1 2, .

Proof. Denote by C Da r( ),1

the space of continuous maps [ , ) { , | | | | }t t a h h h r0 0 1 0� � � �� � , where

a a L h r L1 0 0 01 4 1� �min{ , / ( (| | | | )), /� }, and assume that the inequalities | | ( , ) | | / | | | |� �T

t t I r h0 02� � , | | | ( , )| | |�T

t t0 1 1� �

hold for all 0 0 1� � �t t a . Let us introduce an operator E C D C Da r a r: ( ) ( ), ,1 1

� by the formula

1163

Eh t h U s h s dst

tst

t

t

( ) ( , ( ))� � �0 00

0

0 .

We will solve the equation h t Eh t( ) ( )� by the method of successive approximations, assuming that h Eh1 0� ,

h Eh2 1� ,� , h Ehk k� �1 ,� . It is easy to prove that the inclusion h D Ek a rn

X1

2, follows from the inclusion

h D Ea rn

02

0 X, . Using the inequality

| | | | | | | |( ) ( ), ,Eh Eh q h hk k C D k k C Da r a r� �� � �1 1

1 1

,

where q La� 1, we find the estimate | | | | | | | |( ) ( ), ,Eh Eh

q

qh hk m C D

k

C Da r a r� �

��

1 111 0 , m k6 , which leads to the conclusion that

the sequence { }hk k�

�

1is fundamental and converging (due to the completeness of the space C Da r( ),

1), i.e., there exists

h t C Da r( ) ( ), 1

such that h t h tk

k( ) lim ( )�

��

. It is obvious that h Eh� and the inclusion h t En

( ) 2

holds for all t t t a �[ , )0 0 1

due to the closedness of the set En2

. Uniqueness is obvious.

Let us prove that the solutions are monotonic. It is easy to prove by induction that h t h tk k2 1 0( ) ( );; ;; , whence

follows the inequality h t h t2 1( ) ( );; due to the closedness of the set KC

n2. The theorem is proved.

Let us establish the simplest sufficient conditions for the existence of strong solutions of the fuzzy differential equation

(2.3).

Theorem 2.2. Let the operatorU t t a E En n

:[ , )0 02 2

� R � satisfy the following conditions:

(i) the inclusionU t d D S t D S t( , /~

( ( ))~

( ( ))W holds for all t t t a �[ , )0 0 ;

(ii) the Lipschitz property with respect to norm | | | | | | | | | | | | | | ( ) | |~( ( ))

~( ( ))

7 � �D S t D S t

h h S t h� � holds on the set~

{( , )~

( ( )),D t h R D S ta r � R� t t t a h h rD S t0 0 0� � � � �, | | | | ~

( ( ))} for h D S t

~( ( )), i.e., there exist a constant L0 0; and a

non-negative function, summable on any interval R� , such that

| | ( , ) ( , )| | ( )| | | | ,~( ( ))

~( ( ))

U t h U t h L t h hD S t D S t2 1 2 1� � � h h Da r1 2,

~

, ,

L s ds L t t

t

t

( ) ( )

1

2

0 2 1� � � for all t t R1 2, � .

Then for any h D S t0 0 ~

( ( )), Eq. (2.3) has a unique strong solution defined on some interval t t t a �[ , )0 0 1 , a a1 � .

The proof is similar to that of the previous theorem.

Theorem 2.3. Under the conditions of the previous theorem, the set of all strong solutions is everywhere dense in the set

of all weak solutions in the sense of the topology C t t T En

([ , ], )0 02

� , T ; 0.

Proof. Let h En

02

and h t( ) be a weak solution of Eq. (2.3) with the initial condition h t h( )0 0� . Since

~( ( ))D S t E

n0

2� , there exists a sequence h D S t

N

0 0 ~

( ( )), | | | |h hN

0 00� �� as N �� and, according to Theorem 2.2, there

exist strong solutions h tN

( ), h t hN N

( )0 0� . The integral equation (2.4) yields

| | ( ) ( )| | ( , )| | | | ( , ) ( ) | |h t h t t t h h t s L s hN N N

� � � �� �& &0 0 0 0 ( ) ( )| |s h s ds

t

t

�� �

0

.

If & &�

�

sup ( , )

[ , ]t t t T

t s

0 0

0 , then the Gronwall–Bellman inequality yields

| | ( ) ( )| | | | | |([ , ]; )

h t h t e h hN

C t t T E

L T Nn� � �

�0 0

20

0 0&&

� ,

which completes the proof.

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Let h t( )be a weak solution of Eq. (2.3) with the initial condition h t h En

( )0 02

� defined on some interval[ , )t t a0 0 � .

Let us define a set of operators~

( )0t

th h t

00 � . This set has the following properties:

(i)~ ~ ~0 0 0s

t

t

s t

00

� for all 0 0 0� � � � �t s t t a;

(ii) for any t0 0; and � ; 0, there exists � � �� ;( , )t0 0 such that the inequality | |~ ~

| |0 0t

t th h

02 0 1� � � follows from the

inequalities 0 0� � �t t �and | | | |h h2 1� �� �;

(iii) for any t0 0; and T ; 0, there exists a constant L ; 0such that the inequality

| |~ ~

| | | | | |0 0 � �t

t

t

th h L h h

0 02 1 2 1� � �

holds for all t t t T �[ , ]0 0 .

The first property follows from the uniqueness theorem for weak solutions. The second and third properties follow from

the Gronwall–Bellman inequality.

If the matrix D t( ) does not explicitly depend on t and time does not explicitly appear in the operator U t( ,. ), then the

fuzzy differential equation (2.3) is autonomous, and the semigroup~0

t

t

0

is additionally invariant to shifts, i.e.,~ ~0 0

t

t t t

0

0

0�

�,

0 0� �t t.

3. Example. Let us consider the motion of a material point of mass mwith one degree of freedom (oscillator) under the

action of potential and dissipative forces characterized by potential energy P and Rayleigh R function:

P � �cq R q2 2

2 2/ , � /. ,

where q is the generalized coordinate; �q is the generalized velocity.

Assume that the initial phase vector ( , � )q q is a member of some fuzzy set u E02

and that the state of the point is not

completely known during motion—only its maximum total energy E E t� ( ) is known. Additionally assume that the point is

subject to some control force F t( )whose maximum magnitude is &�

E , where & and � are positive constants. Then, according to

the above results, the support function h( )Y , Y 3 [ , ]0 2 , of the fuzzy set of attainability satisfies the equation

dh t

dtc m m h t

( , . )(( / )sin cos / sin ) ( , ) (cos� � � � �1

2 2Y Y . Y Y Y . Y Y/ sin cosm

��

��c m

h tc m h t/ sin )

( , )max{( / ) , ( / ) | | ( )| | |

2 22 2Y

Y

Y"&

� � �cos | , ( ) ( )Y Yh t h0 0� , (3.1)

where h t C( , . ) [ , ] 0 23 , which is a space of continuous 23-periodic functions.

Equation (3.1) is a fuzzy differential equation and describes the motion of a material point under the action of uncertain

forces.

4. Discussion of the Results. Fuzzy differential equations can be used to construct mathematical models that would

take adequate account of fuzzy and uncertain factors and also incomplete data on the conditions of motion of the system. In

constructing such models, the need arises to generalize the concept of a fuzzy differential equation. Equation (2.1) is an adequate

model of a moving mechanical system under the assumptions made above. The correctness conditions for problem (1)–(3)

justify the problem formulation and allow the model to be applied to study real mechanical and other systems.

REFERENCES

1. A. A. Martynyuk and V. I. Slyn’ko, “On global existence of solutions of fuzzy differential equations,” Diff. Uravn., 42,

No. 10, 1324–1336 (2006).

2. V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, Differential Equations with Multivalued Right-Hand Side:

Asymptotic Methods [in Russian], AstroPrint, Odessa (1999).

3. L. S. Pontryagin, Continuous Groups [in Russian], Nauka, Moscow (1973).

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4. N. G. Chetaev, “On stable trajectories of dynamics,” in: Studies on Analytic Mechanics [in Russian], Izd. AN SSSR,

Moscow (1962), pp. 250–268.

5. V. Lakshmikantham and Ram Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Florida Institute of

Technology, Melbourne (2003).

6. V. B. Larin and A. A. Tunik, “Dynamic output feedback compensation of external disturbances,” Int. Appl. Mech., 42,

No. 5, 606–616 (2006).

7. A. A. Martynyuk and N. V. Nikitina, “On oscillations of a frictional pendulum,” Int. Appl. Mech., 42, No. 2, 214–220

(2006).

8. A. A. Martynyuk and N. V. Nikitina, “Complex behavior of a trajectory in single- and double-frequency systems,” Int.

App. Mech., 41, No. 3, 315–323 (2005).

9. A. A. Martynyuk and V. I. Slyn’ko, “On the boundedness of motions of mechanical systems described by fuzzy ordinary

differential equations,” Int. Appl. Mech., 41, No. 12, 1407–1412 (2005).

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