Cybernetics and Systems Analysis, Vol. 46,

No.

2, 2010

ANALYSIS AND CONTROL OF SECOND-ORDER

DIFFERENTIAL-OPERATOR INCLUSIONS WITH

-COERCIVE DAMPING

N. V. Zadoyanchuk

a†

and P. O. Kasyanov

a‡

UDC 517.9

Second-order differential-operator inclusions with weakly coercive pseudomonotone mappings are

considered. Function-topological properties of a resolving operator are investigated. The results are

applied to mathematical models of the nonlinearized theory of viscoelasticity.

Keywords: second-order differential inclusion,

-coercive operator, optimal control, viscoelasticity.

INTRODUCTION

In investigating mathematical models of nonlinear processes and fields of the nonlinearized theory of viscoelasticity

and piezoelectrics and studying waves of different nature, the following scheme is often used: such a model is reduced to

some differential-operator inclusion or a multivariational inequality in an infinite-dimensional space [1–3]. Next, using some

method of approximation or other, the existence of a generalized solution of this problem is proved, constructive methods of

searching for approximate solutions are substantiated, and functional-topological properties of the resolving operator [2] are

studied. If the character of the mentioned process is evolutionary, its mathematical model is described with the help of a

second-order differential-operator inclusion [4–6]. In this case, relationships between key parameters of the original problem

provide definite properties for a multivalued (in the general case) mapping in the differential-operator scheme of

investigation. We note that, in the majority of works devoted to this line of investigation, rather stringent conditions

connected with uniform coercitivity, boundedness, and generalized pseudo-monotonicity are imposed on “damping” [4, 6].

As a rule, these conditions not only provide the existence of solutions for such problems but also guarantee the dissipation of

all solutions and sometimes the existence of a global compact attractor, which does not always naturally reflect the actual

behavior of the geophysical process or field being considered [7]. Therefore, the need arises for the investigation of

functional-topological properties of the resolving operator for a differential-operator inclusion that describe, in particular,

new wider classes of nonlinear processes and fields of the nonlinearized theory of viscoelasticity with an adequate essential

weakening of the above-mentioned properties of differential operators with corresponding applications to concrete

mathematical models.

In this work, problems of analysis and control of second-order differential-operator inclusions with weakly coercive

-pseudo-monotone mappings are considered. The dependence of solutions on functional parameters of a problem is

investigated, and an optimal control problem is considered. The results obtained are applied to mathematical models of the

nonlinearized theory of viscoelasticity.

PROBLEM STATEMENT

Let

and

be real reflexive separable Banach spaces with the corresponding norms

|| ||

and

|| ||

, and let

be a real Hilbert space whose scalar product is denoted by

(,)

××

and that is identified with its topologically conjugate space

. We assume that the embedding

is compact and dense and that the embedding

is continuous and

305

1060-0396/10/4602-0305

2010 Springer Science+Business Media, Inc.

Taras Shevchenko National University, Kiev, Ukraine,

†

[email protected];

‡

[email protected]. Translated from

Kibernetika i Sistemnyi Analiz, No. 2, pp. 152–160, March–April 2010. Original article submitted September 15, 2009.

dense. We obtain the following chain of continuous and dense embeddings [2,8]:

VZHZV

00 000

ÌÌ ÌÌ

, where

and

are corresponding spaces topologically conjugate with

and

with the corresponding norms

|| ||

and

|| || .

introduce the following denotations:

ST=[, ]t

-¥ < < <+¥t T

p ³ 2

>+=1

HLSH=

(; )

ZLSZ

= (; )

VLSV

= (; )

HLSH

(; )

ZLSZ

= (; )

VLSV

(; )=

WyVyV=Î

Î{}|

where

is the derivative of an element

yVÎ

in the sense of

DSV

(; )

[2, 8]. Note that the embeddings

VZHZ VÌÌ Ì Ì

are continuous and dense. Moreover, the embedding

WZÌ

is compact [9,10], and the

embedding

WCSHÌ (; )

is continuous [8, 10].

It should also be noted that canonical pairings

áñ ´ ®

××

VV R

and

áñ ´ ®

××

ZZ R

coincide on

with the scalar product in

. Then the pairing

áñ ´®

××

VV R

and, accordingly,

áñ ´®

××

ZZR

coincides on

HV´

with the scalar product in

, i.e., we have

áñ=áñ=áñ= =

fu fu fu fu fs us ds

,: , , (,) ((),())

fHÎ

uVÎ

Assume that

and

are separable locally convex linear topological spaces (LTSs),

UUÌ

and

KKÌ

are some

nonempty sets,

AV U: ´®

()

and

CZ K C Z:()´®

are multivalued mappings with nonempty convex weakly

compact values in the corresponding spaces

and

BV V: ®

is a linear operator,

fVÎ

, aHÎ

, and

bVÎ

are

arbitrary fixed elements.

We state the problem of investigating functional-topological properties of the resolving operator

Ku abf(,,,, )u

the following problem:

¢¢

++ '

yAyuByCy f

yby a

(,)()(,),

() , () .

(1)

Here,

Ku ab f yy CSV W y(,,,, ) (, ) (; ) |u =

Î´{

is a solution of problem (1), and the derivative

of an element

understood in the sense of the space of distributions

DSV

(; )

Note that since the embedding

WCSHÌ (; )

is continuous, the initial conditions in problem (1) make sense.

CLASSES OF MULTIVALUED MAPPINGS

Let

be some reflexive Banach space, let

be its topologically conjugate space, let

áñ ´ ®

××

YYR

be a

pairing, and let

AY Y: Ã

be a strict multivalued mapping, i.e., we have

Ay()¹Æ" ÎyY

. For it, we define the upper

[(),] sup ,

()

Ay d

dAy

=áñ

and lower

[(),] inf ,

()

Ay d

dAy

=áñ

support functions, where

yY, w Î

, and also the upper

|| ( ) || sup || ||

()

Ay d

dAy

and lower

|| ( )|| inf || ||

()

Ay d

dAy

norms. Let us consider mappings

co AY Y: Ã

and

coAY Y:,

that are connected with

and are specified by the relationships

()()(())co coAy Ay=

and

(()) (())co coAy Ay=

, respectively, where

co ( ( ))Ay

is the weak closure of the convex envelope of the set

Ay()

in the space

As is well known [2, 11], strict multivalued mappings

ABY Y,:Ã

have the following properties:

(1)

[(), ] [(), ] [(), ]Ay Ay Ayuu u u

12 1 2

+£ +

+++

[(), ] [(), ]Ay Ayuu u

12 1

+³ +

[(), ]Ay u

2 -

"ÎyY,,uu

;

(2)

[(),] [(), ]Ay Ayuu

=- -

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

() () ()

Ay By Ay By+= +

+- +- +-

uuu"ÎyY, u

;

(3)

[(),] [ (),]

() ()

Ay Ayuu

+- +-

= co "ÎyY, u

;

(4)

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

() ()

Ay Ay

+- +-

|| ( ) ( ) || || ( ) || || ( ) ||Ay By Ay By+£ +

+++

"ÎyY

306

In particular, the inclusion

dAyÎco ( )

takes place if and only if we have

[(),] ,Ay d

³á ñ " Îu Y

If we have

aYYR(,):

××

´®

, then, for each

yYÎ

, a functional

Yay' wwa (, )

is positively homogeneous, convex,

and lower semicontinuous if and only if there is a strict multivalued mapping

AY Y: Ã

such that we have

ay Ay(, ) [ (), ]ww=

"ÎyY, w

Let

be some normalized space that is embedded into

, let

be some separable LTS, and let

XXÌ

be some

nonempty set. Let us consider a parametrized multivalued mapping

AY X Y: ´

Hereafter,

¾®¾

will mean that

converges weakly to

Definition 1. A strict multivalued mapping

AY X Y: ´

is called

· l

-quasimonotone on

WX´

if, for any sequence

{}ya WX

nnn

Ì´

such that

¾®¾

, and

¾®¾

n ®+¥

, where

dAya

nnn

Îco ( , ) "³n 1

, the inequality

lim ,

dy y

®¥

á-ñ£

implies the

existence of a subsequence

{}yd a

nn nk

kk k

³1

from

{}yda

nnnn

³1

for which we have

lim ,

®¥

á-ñ³w

[( , ), ]Ay a y Y

00 0

-"Î

;

bounded if, for each

L > 0

and a set

DXÌ

bounded in the topology of the space

, there is some

l > 0

such that we

have

|| ( , )||Ayu l

£" Î´ £yu Y D y L

, :|| ||

;

demiclosed if, for an arbitrary sequence

{}yu

nnn

YX´

such that

, and

¾®¾

where

dAyu

nnn

Îco ( , ) "³n 1

, we have

dAyu

000

Îco ( , )

With each multivalued mapping

AY Y: Ã

, we associate a parametrized multivalued mapping

:AY X Y´

according to the following rule:

(, ) ()Ayu Ay=

yYÎ

uXÎ

Definition 2. A strict multivalued mapping

AY Y: Ã

is called

· l

-pseudo-monotone on

if the corresponding

:AY X Y´

-quasimonotone on

WX´

;

bounded if the corresponding mapping

:AY X Y´

is bounded.

MAIN RESULTS

Let us study functional-topological properties of the resolving operator of problem (1) that are connected with

closedness in definite topologies. The necessity of substantiation of such properties is connected with problems of control of

mathematical models of nonlinear geophysical processes and fields that are described with the help of problem (1) and with

the investigation of the dynamics of solutions of such problems.

THEOREM 1. Let

AV U C V:()´®

-quasimonotone on

WU´

and bounded, let

BV V: ®

be linear and

continuous, and let a multimapping

CZ K C Z:()´®

be bounded and demiclosed. In addition, let us consider a sequence

{}fabu V HVUK

mmmmmm

,,,,u

Ì´ ´´´

100

. We also assume that, for all

m ³ 1

, we have

(, )yy

Î Ku

(, ,u

ab f

mmm

,,)

and that the following convergences take place:

¾®¾yg

. (2)

Then there is some

yCSVÎ (; )

such that we have

ÎyW

and

=yg

(, )yy

ÎKu a b f(, , , ,)

00000

. Moreover, we

have

CSV m(; ),

®+¥

, (3)

¾®¾

Wm, ®+¥

, (4)

"Î

¾®¾

tSy t yt

() ()

, ®+¥

. (5)

307

Proof. Let the condition of the theorem be satisfied. For a fixed

bVÎ

, we consider a Lipshitz continuous operator

RZ Z

: ®

(accordingly,

VV®

) specified by the relationship

( )() ()Ryt b ysds y Z

=+ "Î

(accordingly,

"ÎyV

"ÎtS

Let us consider a multivalued operator

AV U C V:()´®

Ayu Ay u B R y CRy y V u u b U U K

(, ) (, ) () ( , ), , (,, )=+ + Î=Î=´o uu´V

(, ) (,,,, )yy Ku abf

Î u

, then

zy= '

is a solution of the problem

zAzu b f

(,,,) ,

() .

(6)

On the contrary, if

zWÎ

is a solution of problem (6), then we have

(, ) ( ,) (,,,, )yy Rzz Ku abf

=Îu

Let

{}fabu V HVUK

mmmmmm

,,,, ,u

Ì´ ´´´

100

(, )yy

Î Ku

(, ,u ab f m

mmm

,,)"³1

, and let

convergences (2) take place. We put

"³m 1

. Then we have

yRz

mbm

="³m 1

. It may be noted that inclusion

(6) implies that,

"³m 1

$ÎdAzu b

mmmmm

(,,,)u

such that we have

dfzAzu b

mmm mmmm

Î (, , , )u

. (7)

Convergences (2) imply the boundedness of sequences

{}z

{}u

{}b

, and

{}f

in the corresponding

topologies of the spaces

VU KV,

, and

Then the boundedness of

{}d

follows from the boundedness of

the mappings

AV U C V:()´®

BV V:,®

and

CZ K C Z:()´®

, the continuity of the embeddings

VZÌ

and

, and the following estimates:

|| || [|| || || || ] ,Rx Rx b b x x xx V

12 0

12 12 12 12

-£-+-"Îa ,,bb V

12 0

|| || [|| || || || ] ,Rx Rx b b x x xx Z

12 0

12 12 12 12

-£-+-"Îb ,,bb V

12 0

where

and

are constants that do not depend on

and

Thus, we obtain

$>"³ £

cmd c

01:||||

. (8)

The boundedness of

{}

mm1

follows from the boundedness of

{}f

and inequality (8) and, hence, we

have

$>"³

££

cmz zc

01: |||| ||||

. (9)

Since the embedding

WCSHÌ (; )

is continuous (see [2, 8]), from inequalities (9) we obtain

$>"³"Î £cmtSztc

: , || ( )||

. (10)

With the help of convergences (2) and estimates (8–(10) and with allowance for the continuity of the mapping

yya

DSV

(; )

, we obtain

zgWddfgV

tSz t gt H

¾®¾¾®¾=-

"Î ¾®¾

in in

() ()

as m ®¥.

(11)

From this and also convergences (2), we obtain, in particular, that

gWÎ

and

ga()t =

. (12)

Let us show that

satisfies the inclusion

+'gAgu f(, )

, where

uu b

0000

= (, ,)u

. Since

+=gdf

, it suffices

to prove that

dAguÎ (, )

308

We first make sure that we have

lim ,

dz g

®¥

á-ñ£0

. (13)

In fact, owing to relationship (7),

"³m 1

, we have

á-ñ=áñ-á

ñ-á ñdz g fz zz dg

mm mm mm m

,,,,

=á ñ-á ñ+ -fz dg z zT

mm m m

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

. (14)

Next, in the left and right sides of equality (14), we take the upper limit as

m ®¥

lim , lim , lim ,

dz g fz d g

®¥ ®¥ ®¥

á-ñ£áñ+á-ñ

+-£áñ-áñ

®¥

lim (|| ( )|| || ( )|| ) , ,

zzTfgdg

+-=á-ñ-áñ=

(|| ( ) || || ( ) || ) , ',ggTfdggg

The latter takes place by virtue of [8, Lemma I.5.3] and convergences (11). Inequality (13) is verified.

The definition of

implies that,

"³m 1

$Îz

mmm

Az u(, )

and

$Îxu

mbmm

CR z(,)

such that we have

"³ = + +md BRz

mm bm m

1 zxo ()

. (15)

From convergences (2) and (11), the boundedness of

CZ K C Z:()´®

, and the compactness of the embedding

WZÌ

, we have

Rz Rg

bm b

¾®¾

, ®+¥

. (16)

The demiclosedness of

CZ K C Z:()´®

implies that

xuÎCR g

(,)

. (17)

Since we have

z sds ts z cts

( ) | | || || | |

£- £ -

"Îts S,

"³m 1

, where

is a constant independent

m ³ 1

and

st S, Î

, by virtue of convergences (2), we have

Rz R

bm b

CSV m(; ),

®+¥

, (18)

in particular, we have

BR z BRg

bm b

oo®

®+¥,

. (19)

Thus, we obtain

á+-ñ®®+¥BR z z g m

bm mm

o () , ,x 0

, (20)

lim ( ) , ( ) ,

bm mm b

BR z z BR g g

®¥

á+-ñ=á+-ñooxw xw

³á - ñ+ - " Î

BR g g CRg g V

(), [( , ), ]wuww

. (21)

From relationships (13), (15), and (20) we have

lim ,

®¥

á-ñ£z 0

(22)

It also follows from (15)–(18) and convergences (11) that we have

zzx

dBRg¾®¾=--o

()

®+¥,

(23)

309

Thus, owing to convergences (11), inequality (22), convergence (23), and the

-quasimonotonicity of

WU´

,we

obtain that, up to a subsequence

{}{}zud zud

mmmk mmmm

kkk

,, ,,

³³

lim , [ ( , ), ]

zAgug

®+ ¥

á-ñ³ -zw w

"Îw V

. This, inequality (13), and convergence (20) imply, in particular, that

lim ,

®¥

á-ñ=z 0

and

á-ñ=zw, g

lim ,

®¥

á-ñzw³-

[(, ), ]Ag u g

w "Îw V

. Together with relationships (21), the latter relationships guarantee that

the inequality

á-ñ³dg, w

[(,,,), ]Agu b g

000

uw-

"Îw V

holds true. This means that

dAgu bÎ (, , , )

000

. Putting

yRg

and

=yg

, we obtain

(,) (,,,,)yy Ku a b f

0000

. Note that convergence (3) directly follows from

convergence (19) and that convergences (4) and (5) follow from convergences (11).

The theorem is proved.

We additionally assume that there are real Hilbert spaces

and

such that the embeddings

VVV H

ÌÌ Ì

are continuous and dense. Then the embedding

is compact. We assume that

WyVyLSV

qss

=Î

{}|(;)

, where

is a space topologically conjugated with

and

is the derivative of an element

yVÎ

in the sense of

DSV

(; )

[8].

THEOREM 2. If, for some

uUÎ

u ÎK

aHÎ

, and

bVÎ

, a mapping

AuV CV(, ): ( )

-pseudo-

monotone on

, if we have

$>"Î ³-

cc c yVAyuy c y c

Ayu

123 1 2

0,, : [(,),] |||| ,

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

£+cy

(24)

and if a mapping

BL SV L SV:(; ) (; )

satisfies the property

"ÎuV()() ()Bu t B u t=

for almost all

tSÎ

where

BV V

is a linear bounded self-conjugate monotone operator, if a mapping

CZCZ(, ): ( )

demiclosed, and if we have

$>"Î

e 0: yZ sup || || ( ( ) )( || ||

(,)

dC y

ppq

dcT y

--*

£--+

gte

(25)

where

ggº£

××

const: || || || ||

, then, for an arbitrary

fVÎ

we have

Ku ab f(,,,, )u ¹Æ

Proof. Let us consider some

w ÎW

such that we have

wt()= a

and the mapping

AV C V:()®

Ay Ay u CR y

() ( ,) ( ( ),)=+ + +wwu

yVÎ

. Conditions (24) and (25) imply that (see [5]), for some

0, >

, we have

[(),] ||||Ay y y

³-aa

yVÎ

. Repeating the reasoning from [5, p. 207] and taking into account that the embedding

is compact, we obtain that the mapping

is bounded and

-pseudo-monotone on

. Next, following the proof of

Theorem 1 from [5], we obtain that the problem

¢¢

+'-

yAy Byf BR

yby

() ,

() , '()

(26)

has a solution

yCSVÎ (; )

such that

ÎyW

. Applying the replacement

zyR=+

to problem (26), we obtain the

required statement.

The theorem is proved.

We additionally assume that

(,(,))KX XX=

, where

is some real separable or reflexive Banach space,

KKÌ

is a

-weakly compact nonempty subset, and the mapping

AV C V:()®

does not depend on a parameter

uUÎ

. Then,

instead of

Ku abf(,,,, )u

, we will write

Kabf(,,, )u

. Let us fix arbitrary

fXaVÎÎ

and

bHÎ

.Weput

GyyCSVWKyyKabf

Î´´

Î{( , , ) ( ; ) ( , ) ( , , , )}uu

310

THEOREM 3. Let

LCSV W W W X X X: ( ; ) ( ; ( ; )) ( ; ( ; ))

´´

be a lower semicontinuous functional such that,

"ÎuCSV(; )

u ÎW ,

and

wXÎ

, we have

Lu w(, , )u ³+ju j

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

, where

RR:

are such that

s()®+¥

and

s ®+¥

i =12,

. We also assume that

AV C V:()®

BV V:,®

and

CZ K C Z:()´®

satisfy the

conditions of Theorems 1 and 2. Then the problem

Lyy

yy G

(, ,) inf,

(, ,)

(27)

has a solution.

The proof immediately follows from the statements of Theorems 1 and 2 and the generalized Weierstrass theorem [2].

APPENDIX

Let us consider a viscoelastic body that, in its undeformed state, fills a limited domain

WÌR

, d = 23,.

We assume

that the boundary

GW=¶

is regular [8] and that

is divided into the following three pairwise nonintersecting measurable

parts:

, and

so that we have

meas

()G>0

[4]. The body is clamped on

so that the displacement field

vanishes. We also assume that a given vector of the body force

is distributed in

and that the surface force

distributed over

. The body can come into contact with the base over the potential contact surface

. We put

QT=´W (, )0

for

0 <<+¥T

. We denote by

uQ R

: ®

the displacement field, by

s : QS

the stress tensor, by

ee() ( ())uu

ij i j j i

uuu() ( )

, the deformation tensor, where

ij d,,= 1

, and by

the space

dd´

of symmetric

matrices of order

. Following [12], we will consider a multivalued analogue of the Kelvin–Voigt viscoelastic defining

relationship

see(, ) (( )) (())uu u u

ÎÀ

+Á

, where

is a multivalued nonlinear mapping and

is a single-valued linear

defining mapping. Note that, in the classical linear viscoelasticity, the law defined above assumes the form

se e

i j ijkl kl ijkl kl

cugu

() ()=

, where

À={}c

ijkl

and

Á={}g

ijkl

ijkl d,,, ,=1

, are the tensors of viscosity and elasticity,

respectively.

We denote the normal and tangential components of a displacement

and

uun

=×

uuun

where

is the unit vector of the outer normal to

. Similarly, the normal and tangential components of the stress field on

specified through

nn=×()

and

sss

nn=-

, respectively. On the contact surface

, we consider boundary

conditions. The normal stress

and normal displacement

satisfy the nonmonotone normal condition of compliance

reaction of the form

-Î¶sz

NN N

jxtu(,, , )

T´ (, )0

. The law of friction between the friction force

and

tangential displacement

is of the form

-Î¶sx

TT T

jxtu(,, , )

T´ (, )0

. Here,

(,,, ):

×××

TR R´´®(, )0

and

jTRR

(,,, ): (, )

×××

´´®x G 0

are locally Lipshitzian with respect to the last variables of the

function, and

¶j

and

¶j

are Clark’s subdifferentials of the corresponding functionals

jxt

(,,, )

and

jxt

(,,, )

.In

particular cases, such boundary conditions include classical conditions on the domain boundary (see, for example, [4,

Chapter 2.3; 6]). We denote the initial displacement and initial speed by

and

. The classical formulation of the contact

problem is of the form: find

uQ R

: ®

and

s : QS

such that we have

¢¢

ÎÀ

+Á

=´

uf Q

uu Q

div in

se e

(( )) (( )) ,

(, )G ,

(, ),

(,, , ), (,, ,

szs

nf T

jxtu jxtu

NN N TT T

=´

-Î¶ -Î¶

0on G

x )(,),

() , () .

uuu u

(28)

For the variational statement of this problem, we put

HL R

= (; )W

HL S

d02

= (; )W

HuHuHH R

10 0

=Î Î ={}|() ( ; )e W

D01

0=Î ={on}uu| G

311

Using Green’s formula the definition of the Clark subdifferential [13, 14], under conditions of the corresponding

smoothness of initial data (see [4, 12] for more detail), one can obtain [4, 12] the variational statement of problem (28)

on the search for

uTV:[ , ]0 ®

and

s :[ , ]0

TH®

such that we have

{á

¢¢

ñ+ +

ut t

jxtu j

(), ( (), ( ))

((,, , ;)

useu

xtu d x

ft V

(,, ; ; )) ()

(),

³á ñ Îfor all and almost all tT

uuu u

[, ],

() , () ,

(29)

where

áñ= +ft f t f t

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

(; )

uu u

for all

u ÎV

and almost all

Let

VL TV=

0(, ; )

, let

WwVwV=Î

{}|

, and let

:(;): (;)

HR ZH R

WG=® Ì

(; )G

be a trace

operator,

d Î(/ ; )121

. Mappings

:,®

and

BV V

00 0

: ®

are defined as follows:

[(,),] sup{(,())| ,() (,,((Atu d dV d t u

ueu e

=ÎÎÀ

×× ×

)))}

tSÎ , uV, u Î

Au d V dt A t yt() | () (,())=Î Î

{

for almost all

tSÎ }

uVÎ

áñ=Á "ÎÎBu xt u u V t T

0, ( (,,()),()) , , [, ]ueeuu

Here,

[(,),]Atu u

is the upper support function of a set

Atu V(, )Ì

. A functional

JTL R KR

:[ , ] ( ; )0

´´®G

defined as follows:

Jt j xt x j xt x d x

NN TT

(,,) ( (,, (),) (,, (),)) ()uh u z u x=+

for

tTÎ[, ]0

u ÎLR

(; )G

, and

hzx=Î(, ) K

, and

CZ K C Z:()´®

is defined as follows:

Cu d Z dt Jt ut( , ) | ( ) ( ( , ( ), ))hggh=Î Î ¶

{

for almost all

tTÎ[, ]0}

where

is the operator conjugate to

. Thus, we have obtained the problem on the search for

uVÎ

and

ÎuW

such

that we have

¢¢

++ '

uAu BuCu f

uuu u

() (,) ,

() , () .

(30)

It may be proved (for more details, see [4, 12]) that each solution of problem (30) is a solution of problem (29). Thus,

imposing conditions on the parameters of problem (28) in such a way that the mappings

AB,,

and

satisfy the

conditions of Theorems 1–3 (for more details, see [1, 2, 3, 5, 12]), we obtain results on some properties of the

resolving operator of problem (30) and, in particular, of optimal control problem (27). Note that the investigation

scheme proposed in this work (in contrast to the existing approaches [1, 2, 3, 5, 12]) makes it possible, for example, to

relax the technical condition of the uniform “

”-coercivity by reducing it to the “

”-coercivity and that of the

generalized pseudo-monotonicity by reducing it to the

-pseudo-monotonicity, etc. It should be noted that concrete

classes of differential operators of pseudomonotone type occurring in problem (28) are considered in detail in [1–14]

(see these works and references to sources).

312

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