Pareto optimality for nonlinear infinite dimensional control systems

PARETO

OPTIMALITY FOR NONLINEAR

INFINITE DIMENSIONAL

CONTROL SYSTEMS

by

EVGENIOS P. AVGERINOS

UniversityofThessaloniki Schoolof

Technology,

Thessaloniki54006,

GREECE

and

NIKOLAOS S. PAPAGEORGIOU UniversityofCalifornia 1015

Department

ofMathematics

Davis, California95616,

U.S.A.

(Received February 2, 1989 and in revised form April 3, 1989)

ABSTRACT.

In

this note weestablish theexistenceofParetooptimalsolutionsfor nonlinear, infinitedimensionalcontrolsystemswithstatedependentcontrolconstraintsand anintegral criteriontakingvaluesinaseparable,reflexive Banach lattice.

An

exampleisalsopresentedin

detail. Ourresult extendsearlieronesobtainedbyCesariand

Suryanarayana.

KEY

WORDS

AND

PHRASES. Paretoefficient points, Banach lattice,mildsolution, evolution operator, compactness,

Fatou’s

lemma,orientorfield,nonlinearparabolicequation.

1.

INTRODUCTION

In

recentyearsthere has been anincreasinginterestinoptimizationproblemswith

multiple objectivesconflictingwithoneanother. The subjecthasitsoriginsinmathematical

economicsandinparticularinwelfare theoryandfromthereitpassedintoother subjectslike game theory,operationsresearch,optimization andoptimalcontrol.

Suchproblems,inthe contextofoptimal controltheory, were recentlyconsideredby

Cesari and

Suryanarayana

inaseriesof interestingpapers

[5], [6],

[7].

Weshould alsomention

the earlier workof Olech

[9],

who,

motivatedfrom the fundamental work ofCesari

[4],

studied similarproblemsin

IR

n.

Theaimofthisnoteistoextend the finite dimensionalexistenceresultfor

Pareto

solutionsof Cesari-Suryanarayana

[5],

to infinite dimensionalcontrolsystems. Exploiting

[11],

weareable toprovetheclosedness of acertain orientorfield andthroughthat establish

theexistenceof

Pareto

optimalsolutions. 2.

PRELIMINARIES

Let

(,E)

be ameasurable space and

X

aseparable Banach space.

Throughout

this notewe willbe using thefollowingnotations:

and

Pf(c)(X)

{h c__

X: nonempty, closed,

(convex)}

P(w)k(c)(X)

{A

c_.

X: nonempty,

(w-)compact,

(convex)}.

A

multifunction

(set

valued

function)

F: -,

Pf(X)

_issaid to bemeasurableif andonly

ifforall y

X,

w-

d(y,

F(w))

inf{lly-xll:

x

F(w)}

ismeasurable. Ifthereexists a a-finite

measure

(.),

withrespect to which

E

iscomplete, then the above definitionofmeasurability

isequivalenttosaying that GrF

{(w,x)

e X:

xeF(w)}

ZB(X),

with

B(X)

beingthe

Borel a-fieldof

X (graph measurability).

For furtherdetails onmeasurablemultifunctions werefertothesurvey paper of

Wagner

[16].

By

S

we will denote the set ofselectorsof

F(.),

thatbelongintheLebesgue-Bochner space

LI(x)

i.e.

S

{f

LI(x)

f(w)

F(w)

#-a.e.}.

Thissetmaybe empty. Howeverif

F(.

isintegrablybounded

(i.e.

F(.)

is

measurable and w

IF(w)l

sup{llxll"

x

F(w)}

e

L),

then

S

#

q). Using

S

we can defineaset valuedintegralfor

F(.)

bysetting

F(w)d(w)=

{I

f(w)d(w)"

f

S}.

Let

Y,Z

beHausdorff topological spacesand let G:Y

2Z\{t)}.

We saythat

G(.)

is uppersemicontinuousbyinclusion

(u.s.c.i.),

ifforall

Yn

y in

Y

G(Yn)

{z

Z: z lim

znk’ znk

e

G(Ynk

),

n

<

n₂

<

...} c__

G(y).

Finally,inthe nextsectionwe will beusingsomenotionsand resultsfromthetheoryof ordered vectorspaces. Forthenecessarybackgroundwe refertothebooks ofPeressini

[14]

and

Schaefer

[15].

3.

EXISTENCE

THEOREM

Let Y

be alocallyconvex vectorspacewitha partial order inducedbyanonempty,

closed,convex and pointed cone

Y+.

For y,

y’

e

Y’

wewrite y

_<

y’

ifandonlyif

y’-y

Y+.

Let

A

c_

Y.

A

vector x₀

K

issaid to beParetoefficient for

A,

if

(x0-Y+)

N

),

where

Y+

Y+\(0}.

SotheParetoefficient

(or

Pareto optimal)pointsof

A,

are those pointsof whichareminimalfor the partial orderinducedby

Y/.

The setof Pareto

efficientpointsof

A

will bedenoted by

Eft(A).

(a)

All ordered vectorspaceswhichhave compact order intervals

(resp.

weakly compact,ifthe order is

normal).

(b)

Allsemi-reflexive ordered vectorspaces,withnormalorder.

(c)

Allorderedvectorspaces,with

Y/

completeandhavingaboundedbase

B

s.t.

0

I

t

(in

particularthen,if

Y/

islocally

compact).

(d)

All

(countably)

ordercomplete Banach lattices,unlesstheycontaina lattice isomorphic to

.

Notethatevery

(countably)

Daniellspaceis

(countably)

order complete. Thefollowingexistenceresultis well knownamong peopleworkinginPareto optimization

(see

for examplePenot

[13]).

PROPOSITION" If

(Y,Y/)

isaDaniell vectorspaceand

A

c_

Y

isnonemptyand

bounded below,

then

Eft(A)

#).

Nowlet T

[0,b],

abounded, closedinterval in

JR+,

X

aseparable Banach space

(the

state

space), Z

anotherseparableBanachspace

(the

control

space)

and

Y

aseparable, reflexive,ordercomplete,Banach lattice.

Wewill considerthefollowinginfinitedimensional,nonlinear

control

system:

[(t)=A(t)x(t)+

x(0)

x0,

u(t)

e

U(t,x(t))a.e.

f(t,x(t),u(t))](.)

By

a solutionofthissystem,we willunderstanda mild

(integral)

solution.

A

pair of functions

x(.)

e

C(T,X)

and

u(.)

LI(z),

thatsatisfythe dynamic constraints

(*),

aresaid

tobean "admissible pair".

In

particular

x(.)

isan "admissibletrajectory",while

u(.)

isan "admissible control".

We

willdenote the setofadmissible pairsby

A(x0).

Finally note that

system

(*)

hasfeedback typeconstraints,sincethe multifunction

U(.,-)

dependsalso on the state.

Tothiscontrolsystem,we associate aY-valued costcriterionof thefollowing form:

b

J(x,u)

|

L(t,x(t), u(t))dt

J₀

with

L" TXZ

Y.

Our goalistoproveatheorem saying thateveryvectorin

Eff(J(A(x0)))

isrealizedby an admissible pair.

Tothisend,weneed thefollowingsetof hypotheseson the dataoftheproblem.

H(A):

{A(t)"

e

T}

are linear, unboundedoperatorson

D(A(t)) c_

X,

thatgeneratea

stronglycontinuousevolutionoperator

S(t,s)

e

.2’(X),

0

_<

s

_<

_<

b, which iscompact for t--s

>

0.

H_:

f:

TXZ

X isa function s.t.

(1)

t

f(t,x,u)

ismeasurable,

(2) (x,u)

-

f(t,x,u)

issequentiallycontinuousfrom

XZ

Z

w denote the Banach spaces

X, Z

with theirrespective weak

topologies),

(3)

IIf(t,x,u)ll <

a(t)

+

b(t)

(llxll

+

Ilull)

a.e. with

a(.),

b(.)

e

L+.

H(L):

L:

TxXxZ

Y

isameasurablefunction.

H(Q):

For every

(t,x)

e

TxX,

the set

Q(t,x)

{(v,r/)

eXY: v

f(t,x,u),

u

U(t,x),

L(t,x,u)

<

r/}

isconvexand x

Q(t,x)

isu.s.c.i, from X into

X

w.

H(U)"

U: TxX

efc(Z)

isa multifunction s.t.

(1)

(t,x)

U(t,x)

ismeasurable,

(2)

x-,

U(t,x)

_isu.s.c.i, from

X

into

Z

_w,

(3) U(t,x)

c_

W a.e., where W

Pwkc(Z).

Ha:

A(x0)

#q)

(i.e.

thereexistadmissible"state-control" pairs).

Hb:

J(A(x0)

isorder boundedin X.

Hypothesis H_a isa controllability type hypothesis,whilehypothesis H_b issatisfiedif for example

IL(t,x,u)l <

a’(t)

+

b’(t)

(llxll

+

Ilull)

a.e. with

a’(.),

b’(.)

LI(y+).

Recall

lYl

=Y++Y--THEOREM1: Ifhypotheses

H(A),

H(f), H(L), H(Q), H(U),

H

_a and

H

_b hold,

then

Eff(J(A(x0)))

#

) andevery elementin this set can be realized

byan admissible"state-control" pair.

PROOF: Recalling thatareflexive Banach latticeisa Daniellspace and using

hypothesis

H

_b and the proposition, we deduce that

Eff(J(h(x0)))

#

).

Let

ee

nff(J(A(x0))).

Thenbydefinition ee

J(:A(x0)

). So

thereexist

Yk

e

J(A(x0)

),

k>l s.t.

Yk

_s

e. Wehave

Yk

J(Xk’Uk)’

with

(Xk,Uk) A(x0)

forall k>l.

Ourclaimisthat

{Xk}k>l

isrelativelycompactin

C(T,X).

Tothisend,we will first determinean apriori boundforthe trajectories of

(*).

Solet

x(.

e

C(T,X)

besuch a trajectory. Wehave

x(t)

S(t,O)x

₀

+

S(t,s)f(s,x(s)), u(s))ds,

T,

u(’)e

SI(.,x(.))

0

-<

IIS(t’O)xoII

+

I

IIS(t,s)I1.11 f(s,x(s),

x(t)]

ds

0

:=v

IIx(t)ll _<

M

IIXoI

+

M|

[a(s)

+

b(s)(llx(s)ll

+

Ilu(s)ll)]ds

(since

IIS(t,s)ll _<

M)

J₀

t

_<M(llXoI

I+llall

1+

IWI.IIDII1)+M

b(s)

JJx(t)JJ

ds

0

(IWl

:sup{lul’ueW})

InvokingGronwall’s inequality, wegetthat

JJx(t)Jl

_<

M

for all T and all

trajectories

x(.)

of

(*).

Next

let t,

t’

e

T,

_<

t’.

For any

(x,u)

eN

{(x

_k,

Uk)}k>l

c_.

A(xo)

t’

rt

]]S(t’,0)

x₀

+

J

|

S(t’,

s)

f(s,x(s),

u(s))

ds-

S(t,0)

x₀-|

S(t,s)

f(s,x(s), u(s))

0 J₀

t’

_< IlS(t’,0)x

₀

-S(t,0)x0]

+

|

]ls(t’,s)ll.

IIf(s,x(s),

u(s))]l

ds

J

/|

IIS(t’,s)- S(t,s)l[. [[f(s,x(s), u(s))l[

ds

J₀

Letusestimatethethreesummandsintheright handside of the above inequality.

Because

ofthestrongcontinuityoftheevolutionoperator,given

>

0 thereexists

1

>

0

s.t. for

]t’-t

<

1’

wehave

IIS(t’,0)

x

0

S(t,0)

x011

<

t’

t’

r

J

IIS(t’,s)ll IIf(s,x(s), u(s))l[

ds

<_

M

J

[

[a(s)

+

b(s)(M

₁

Also

1

b(.)e

L+,

wecanfind

2

>

0s.t. for

It’-t]

<

2’

wehave

t’

M

J

[a(s)

+

b(s)(M

₁

+

IWl)lds

<

(1)

+

Wl)]ds.

Since

a(.),

(e)

Finally for e

>

0 wehave

I

IlS(t’,s)- s(t,s)ll. IIf(s,x(s),

u(s))ll

ds

0

_<

[

t-el

I[S(t’,s)- S(t,s)]l.

Ils(s,x(s), u(s))]]

ds

+

2M

(a(s)

+b(s)(M

+

IWl))ds

0 t-e

1

Let

>

0 besuch that

I

2M(a(s)

+

b(s) (M

₁

+

IWl))ds

<

e/6.

Also from

t-e

proposition2.1 of

[11],

weknow thatbecause ofthe compactness hypothesis on

S(t,s)

for t-s

>

0

(see

hypothesis

H(A)),

wehavethat t

S(t,s)

iscontinuous intheoperator norm, uniformly forall s s.t. t--s isbounded awayfrom zero. Thus wecanfind

3

>

0s.t. for

It’-tl

<

a

wehave

IIS(t’,s)

S(t,s)ll <

e/6(llall

+

Ilbll

(M

+

IWl))

for all s

T

with

t--s

_>

1. Soweget -el

IIS(t’,s)-

S(t,s)ll.

IIf(s,x(s), u(s))ll

ds

<

e/

0 t

IlS(t’,s)- S(t,s)ll. IIf(s,x(s),

u(s))ll

ds

<

e/3

(3)

0

From

(1),

(2)

and

(3)

above,we havethatfor

t’-tl

<

b min

{1’

2’ 3

}’

we have

-x(t)ll

<

e for all x

{Xk}k>

xk(t

S(t,0)

x₀/

J

S(t,s) V(s)

ds 0

where

V(s)

{xe

X"

[[x[[

_<

a(s)

+

b(s)

(M

+

[W[)}.

So

V(s)

isalmosteverywhere bounded, closedandsince

S(t,s)

iscompact for t-s

>

0,

S(t,s) V(s)

is almosteverywhere

compact andclearly measurablein s.

So

by RadstrSm’s embeddingtheorem

(see

forexample

[11]),

weget that

I

S(t,s) V(s)

ds

Pkc(X),

==

{xk(t)}k>

Pkc(X).

So invokingthe 0

Arzela-Ascolitheorem,wededuce that

{Xk(. )}k>l

isrelatively compactin

C(T,X).

Observe that

{Uk}k>

c__

S

and thelatterisw-compact in

LI(z)

(see

proposition

3.1 of

[10]),

andbythe Eberlein-Smuliantheoremissequentially w-compact. So bypassingto

asubsequenceifnecessary,wemayassumethat x

k

-

x in

C(T,X)

and

u_k u in

LI(z).

Set

vk(t

f(t,

xk(t

),

Uk(t))

and

r/k(t

L(t,xk(t), uk(t)).

Notethat

I

b L

I

b

r/(s)

ds

J(Xk, Uk)

e. Applyingtheorem2.1 of Balder

[2],

weget

?e

(Y)

s.t.

?(t)

0 k 0

dt e and

(t)

convw-

lY-

{k(t)}k>_l

a.e.. Alsonotethat

Vke

SI(.)

where

K(t)

conv

f(t,xk(t), uk(t)).

But

becauseofthecompactnessof

{xk(t)}k>_l

in X andsince w

{uk(t)}k>_l

_

W and

f(t,.,.)is

continuouson

XxZw

into

Xw,

wededucethat

K(t)e

Pwkc(X)

a.e. andclearlyismeasurable. Soproposition 3.1 of

[10]

tells us that

SI(.)

is

sequentially w-compactin

LI(x).

Hencewemayassume that v_k v in

LI(x)

and

furthermore fromtheorem 3.1of

[12],

wehave

v(t)

conv

w-ff

{vk(t)}k>_l

a.e.. Thus we have

(v(t),

y(t))

econvw-l]

(vk(t), /k(t))

a.e.

(v, r/)

S1

convw-I]-

Q

(.

,x_k

(.))

But byhypothesis

H(Q),

weknow that x

Q(t,x)

isu.s.c.i, from

X

into

X xY

_{w w’}

Hence w-Ii-

q(t,

xk(t))

c__

q(t,x(t))

a.e.

==

(v,r/)

e

S((.,x(.

))"

Let

R(t)

{ue U(t,x(t)): v(t)

f(t,x(t), u), L(t,x(t),

u)

_<

r/(t)}.

Since

(v,)

S((.

,x(. ))’

we seethatfor all e

T

0,

A(T\T0)

0

(A

being the

Lebesgue

measure on

),

wehave

R(t)

). Set

hl(t,u

v(t)-

f(t,x(t), u)

and

h2(t,u

(t)- L(t,x(t), u).

Because

of hypotheses

H(f) (1),

(2)

and

H(L),

wededuce that both

hl(.,.

and

h2(.,.

are measurable. On

(T\T0)

Z

set

hl(t,u

0 in

X

and

h2(t,u

0 in Y. Alsolet p:

TZ

-

TXZ

be definedby

p(t,u)

(t,x(t), u).

Clearlythisismeasurable. So because of

hypothesis

H(U),

p-l(GrU)

e

B(T)

e(z).

But

observe that

p-l(GrU)

{(t,u):

u

hl(t,u

0,

h2(t,u

0}

B(T)B(Z).

Apply

Aumann’s

selectiontheorem toget u:

T

Z

measurable s.t.

u(t)

R(t)

forall

T

==

u(t)

Q(t,x(t))

a.e.

v(t)

f(t,x(t),

u(t))

a.e. and

L(t,x(t),

u(t)) < /(t)

a.e.. Recall that

xk(t

S(t,0)x

₀

+

J

0

S(t,s)

Vk(S

ds and x

k-

x in

C(T,X),

while

J[

S(t,s)

Vk(S

ds

S(t,s)v(s)

ds. Soin

0 0

thelimitas k-

x(t)

S(t,0)

x

0

+

|J₀

S(t,s)

f(s,x(s),

u(s))

ds,

u(t)

V(t,x(t))

a.e.i.e.

(x,u)

is

b rb

admissible

"state

control" pair and

J(x,u)

|

L(t,x(t),

u(t))

dt

_<

|

r/(t)

dt

_<

e an

0 J₀

J(x,u)

e.

Q.E.D.

REMARKS:

(1)

Our resultsextendsthefinite dimensionalwork of

Cesari-Suryanarayana

[3]

(theorem 4.1),

as wellastheinfinitedimensional ones bythesame authors

[6]

(theorem

1)

and

[7]

(theorem

8.1),

where

Z

wasreflexive,int

Z+

#

q) and the overallhypotheseson the data were more restrictive.

(2)

If

f(t,x,.)

islinear

(i.e.

f(t,x)u),

u

L(t,x,u)

is

Y+-convex

(i.e.

L(t,x, ,u

q-(1-A)u2)

_<

A

L(t,x,ul)

+

(l-A)

L(t,x,u2)

A

[0,1],

Ul,

u

2e

Z)

and

(x,u)

L(t,x,u)

is scalarly sequentially 1.s.c. on

XxZw

(i.e.

for every y e

Y+

(x,u)

(y

L(t,x,u))

is sequentially1.s.c. on

XxZ

w into

IR),

thenwe claimthat

H(Q)

issatisfied.

It

iseasytosee

that because of theabove hypotheses

Q(t,x)

is closed, convex forall

(t,x)

TX. Also if

x

n x in X and

(v,r/)

w-Ii

Q(t,

Xn)

thenbydefinitionwecan find

(Vk,

r/k

Q(t,

xnk

s.t.

(Vk,

r/k

w*w

(v,r/).

We

have

v

k

f(t,xk,Uk),

uk

U(t,xk)

c_.

W and

L(t,

Xk,

Uk)

_<

r/k

k_>l.

By

passing to asubsequenceifnecessary,wemayassume that u_k u. Then u

w-Ii

U(t,xk)

c__

V(t,x), f(t,

x_k,

Uk)

f(t,x,u)

==

v

f(t,x,u)

and for all y

eY+

(y

L(t,x,u))

_<

lira

(y

L(t,xk, Uk)

_< (y

r/)

L(t,x,u)

_<

r/==

(v,r/)

e

Q(t,x)

Q(t,.

is u.s.c.i..

4.

AN EXAMPLE

Let

T

[0,b]

and V aboundeddomain in

n

withsmoothboundary 0V F. On

240 E.P. AVGERINOS AND N.S. YAYAGEORGIOU

L(t,v,x(t,v), u(t,v))

dvdt inf

(x’u)

Io

Iv

/s.t.

O

t’v)

--Ax(t,v)

f(t,v,x(t,v)) u(t,v)

|

x(t,

v 0 on

Wr

x₀

/

x(0,v)

(v)

on

{0}

xV

_

Ilu(t,.)ll

₂

s

r _.3

L

(V)

We

willneedthefollowing hypotheses onthe dataof

(**).

H_’:

f:

TxVx

-

isamaps.t.

(1)

(t,v)

f(t,v,x)

ismeasurable

(2)

x

f(t,v,x)

iscontinuous,

(3)

If(t,v,x)l <_

a(t,v)

a.e., with

a(t,.)

L(R)(V),

-

Ila(t,’)ll

L(R)(V)

LI(T).

H_.("

L:

TxV.RxIR

m

isa function s.t.

(1) (t,v)

L(t,v,x,u)

ismeasurable

(2)

foreach k>_l

(x,u)

-

Lk(t,v,x,u

is1.s.c. and convexin

u,

where

L

m

L=(

k)k=l,

belongingin

(3)

k(t,v)

5

Lk(t,v,x,u

5

lk(t,v)

+

2k(t,v)(Ix[

2

+

[u[2),

where

1k(.,.

LI(TxV),

2k(t,.)e

L(R)(V)

and

1l2k(t,.)llL(R)(V)

L+.

H0:

x0(.)e

L2(V).

Let X

L2(V),

Z

L2(V)

and

Y

R

m.

Define

:

T

X

X

to be the Nemitsky operatorcorrespondingto

f(t,v,x)

i.e.

(t,x)

(.)

f(t,.

,x(.)).

Then

[(t,x)

u](v)

f(t,v,x(v)) u(v).

First let uscheckthat indeed

(t,x)

u(.)e

X

L2(V).

We

have

I(t,x)

u(v)l

2dv

f

]f(t,v,x(v))l

2

lu(v)l

2dv

V V

V

a(t’v)2 ]u(v)]2

dv

_<

Ila(t,.)ll2L(R)(V).

Ilull2L2(V

IIf(t,x)ll

<

(R).

L2(V)

Nextlet h

L2(V).-

We

have

(t,x)u)

h(v)f(t,v,x(v))

u(v)

dv

(h,

J_V

Fubini’stheoremwe know that t |

h(v) f(t,v,x(v))

u(v)

(h,

f(t,x) u)

is From

J

V

Nextlet

A

A with

D(A)

H(V)

n

H2(V).

It

iswell known

(see

for example

Barbu

[3])

that

A

generatesa semigroupofcontractions

S(t)"

L2(V)

L2(V)

whichare

compactfor

>

0. Sowehave satisfiedhy

pothesis H(A

).

Let L: WxXxY

-

m

be definedby

L(t,x,u)--

L(t,v,x(v), u(v))

dv. Then

V

(Lk)k=lA

m _and

for each k

(1,

m}

k(t,x,u)

V

Lk(t’v’x(v)’

u(v))

dt. Let

L:

TVx

-

beCaratheodory integrandss.t.

Ck(t,v)

_<

L(t,v,x,u)

_<

m and

L

mk

T

L

_k as m-o foreach k

e(1,

m}.

This ispossiblesince

Lk(.,.

,.)

isanormal integrand

(see

for

I

L(t,v,x(v),

u(v))

dv. FromFubini’s example lemma2 inBalder

[1]).

Set

(t,x,u)

V

theorem t

(t,x,u)

is

measurable,

while

(x,u)

(t,x,u)

isclearlycontinuous. So

(t,x,u)

L

Am

_k

(t,x,u)

ismeasurable andsinceby the monotone

convergence

theorem

L

^m

k

k’

weconclude

Lk(.

,.

,.)

ismeasurable.

So

wehave satisfied hypothesis

H(L).

In

fact

L(t,x,.)

is

---convex

(see

hypothesis

H(L)’(2))

andfrom theorem5 ofBalder

[1], (see

also

Ekeland-Temam

[8]),

wegetthat

L(t,.,.)

isscalarly1.s.c. on

L2(V) L2(V)

w.

So

H(Q)is

satisfied

(see

remark

(2)

in section

3).

_<

r}.

Then we have satisfied

H(U).

Also

Let

U(t,x)

W

(u

e

L2(V):

IlUlIL2(V

assumethat:

’- Thereexists

(x(.,.), u(.,.))

admissible pairfor

(**)

s.t.

J(x,u)

<

oo.

H

_a

This assumptionimplies that H

a issatisfied. Finally note that because of

H(L)’(3),

H

_b issatisfied too. Rewrite

(**)

inthefollowingabstractform:

jb

L(t,x(t),

u(t))

dt- inf

f

s.t.

i(t)

Ax(t)

+

(t,x(t))u(t)

x(0)

x

0

u(t)

W a.e.

This isaspecial case of the optimal controlproblemstudiedinsection 3. Sowecan

invoketheorem and

get

thefollowingexistenceresult.

THEOREM

2: If hypotheses

n(f)’, H(L)’,

H

₀ and

H

hold

the.___an

Eff(A(x0)

#

q) andthe

Pareto

efficient pointsof

(**)

are realized

byadmissible"state-control" pairsof

(**).

1)

E. Balder:

REFERENCES

"Lower

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2)

3)

4)

)

6)

7)

S)

9)

10)

11)

12)

13)

14)

E.Balder:

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lemmain infinitedimensions" J. Math. Anal. App_l.

13_6

(1988),

pp.450-465.

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Noordhoff InternationalPublishing, Leyden,The

Netherlands (1976).

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Math.

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Math. Soc. 136

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J.

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multivalued evolution equations and differential inclusionsinBanach

spaces"

Comm. Math.Univ.S.P. 36

(1987),

pp.21-39.

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"Convergence

theoremsforBanachspace valued integrable

multifunctions" Intern.J._

Math

and Math. Sci. 10

(1987),

pp.433-442.

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Penot:

"L’optimisation a la

Pareto: Deux

outroischosesquejesais d’elle"FacultedesSciences

Exactes, Departement

des Mathematiques,Univ. de

Pau,

reprint

(1978).

A.

Peressini: "Ordered Topological Vector

Spaces,"

Harper

and

Row,

New

York

(1967).

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Springer,Berlin

(1971).