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Optimal control for N-Person stochastic inclusions. IIBui An Ton aa Department Of Mathematics , University Of British Columbia , Vancouver, B C, V6T 1Z2,Canada E-mail:Published online: 03 Apr 2007.

To cite this article: Bui An Ton (2000) Optimal control for N-Person stochastic inclusions. II , Stochastic Analysis andApplications, 18:6, 1031-1053, DOI: 10.1080/07362990008809710

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STOCHASTIC ANALYSIS AND APPLICATIONS, 18(6), 1031-1053 (2000)

OPTIMAL CONTROL FOR N-PERSON STOCHASTIC INCLUSIONS. I1

BUI AN TON

DEPARTMENT OF ~ ~ A T H E M A T I C S , UNIVERSITY OF BRITISH COLUMBIA , VANCOUVER ,B.C. CANADA V6T 122

E - m a d address: bui@math.ubc.ca

ABSTRACT. The existence of a solution of a large class of nonlinear stochastic inclusions is shown . The existence of an open loop problem for those stochastic inclusions , is established. Application to an optimal strategy problem arising in the fight against drugs is given.

The purpose of this paper is to establish the existence of an open loop optimal co~ltrol problem for a large class of nonlinear stochastic inclusions.

lu the study of stochastic equations . the convergence of the approximating so- lutions is usually obtained via C'auchy criterion since there is no compact injection mapping in the natural spaces associated with the problem and thus restrictions on the type of nonlinearity a te ilnposed . Using the notion of fractional time derivative, i~~troduced in 121 by Bensoussan in the study of stochastic Navier-Stokes equations, the Prokorov theorem on the tightness of measures together with the Storokhod the- orem, we have proved in [9] the existence of an open loop control problem for a large class of stochastic inclusions .

Let H be a Hilbert space, h' be a compact convex subset of H . Let E , L be upper semi-co~~tinuons set-valued mappings of H x U into the closed convex subsets of IJ w ~ t h U being a compact convex subset of the space of controls C(0 ,T ; O;) and where U3, j = 1, ., N are Hilbert spaces .

Consider the stochastic inclusion

dy(t; w) + A(y)dt E B ( y ; u,; x,v)dt + L(y; u,; a,v)dw(t; w) on ( 0 , T ) a . s .

(1.1) ~ ( 0 ; w) = E ; E E K ; A(y) E ~ I K ( Y )

where I K ( . ) is the indicator of the set K and r,v = (v l , ., vj-1, vj+l, ., V N ) .

1991 Malhema l zc s Snbjert Classzfiralzon Primary 49 J 22, 49 N 40 , Secondary 49 N 55 , 90 D 06 , 92 H 35

103 1

Copyright O 2000 by Marcel Dekker, Inc.

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1032 BUI AN TON

In 191 , we have shown that there exists a probability space (R ,A ,P ) , a f - standard Wiener process w and y such that

{(Q, A, P ) ; Y , w )

is a solution of 1.1 for a given {u, , x ,v) in U . In this paper, instead of the Prokorov and the Storokhod theorems, we shall use a

result of Von-Neumann on sections of closed graph set-valued mappings and consider the case when the probability space and the Wiener process w are given , not un- knowns of the problem. A similar approach has been used in the study of stochastic Navier-Stokes equations in Ton [ l o ]

In Section 4 , the set R S t , ( ( ; u,; x,v) of all solutions of 1.1 is shown to be non- empty .

Let { f , , gj}y=, be some given convex functionals and for y E R, to( ( ; u,; ?r,v), let

The existence of ii E U and of tj E ii,; r j i i ) such that

E { J , ( t ; y; O,; " , a ) ) 5 E{J , (E; y; v j ; r j i i ) } ; j = 1, ., N

for all v E 2.4 and all y E RSt , ( ( : v,; a,iL), is shown in Section 5 of the paper . The notations , some known results are given in Section 2 . The deterministic

case is considered in Sections 3 and 4 . The results of Sections 5 are applicable to the study of an optimal strategy for an international investment portfolio and to the study of American options as done earlier in Ton [9] . Instead of mathematical finance

we shall consider some new applications and study the problem of optimal resources allocation in the fight against drug addiction . The model studied in Section 6 seems ttrw allti ex(rnds in several directions the 2-player deter~ninistic model considered by 1l.l)awid and G.Feichtinger [GI .

2. NOTATIONS , ASSUMPTIONS, K N O L I N RESULTS

Let H and let {U,)~,, the space of controls , be real Hilbert spaces and let kr be a compact convrx subset of H with 0 E K.

Let y, be positive numbers going to 0 and let

2 = { Y : y E L 2 ( 0 , T ; H ) , y ( t ) E K a.e.; I Y 1z< m}

with

and where y = 0 outside of [O,T].

Lemma 2.1. For each positive constant c ,

{Y : 1 Y I E _ < c )

is a compact subset of L 2 ( 0 , T ; H ) Dow

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N-PERSON STOCHASTIC INCLUSIONS. I1 1033

The lemma is s roved in the same wav as in Aubin's theorem with the fractional time-derivative replacing the usual derivative . A proof can be found in [9] .

Let U be a compact convex subset of C ( 0 , T ; $=, I J j ) and let Q, L be upper semi-continuous (u.s.c.) set-valued mappings of H X U into the closed convex subsets of H . We assume that

Assumpt ion 2.1. Let Q, L be u.s.c. set-valued mappings of H x U into the closed convex subsets of H . Suppose that there exists a constant C such that

Let (R,A, P ) be a probability space and let P be a filtration on the space . We assume that the measure P is regular in the interior , i.e. for every E , there exists a compact li, such that P ( R - K c ) 5 E and the restriction of P to I(, is continuous.

Since h' is a compact convex subset of H , it is known that IK( .) is a 1.s.c. convex function on H and hence 81K is defined .

Definition 2.1. Let (R, A, P ) be a probability space with P being regular in the interior , let F be a filtration on the space and let w be a F - standard Wiener process on the space .

Let { E ; u,, n,v) E L 2 ( R , A , P ; K ) x U , then y is said to be a solution of 1.1 if 0 y is .P measurable and

y E L 2 ( R , A, P : Lm(O,T; H ) ) ; y ( t ) E K a.e. on [O,T] a.s.

and

with

+ J,' L ( y ; u j ; ~ ~ v ) d w ( s ; w )

a.s. for all t E [O,T] and for { A ( y ) , G, L ) E 8 1 K ( y ( t ) ) x Q x L.

The equation 2.2 implies that y is in C ( 0 , T ; H) a.s. We associate with 2.2 the cost functions

where y is a solution of 2.2

Assumpt ion 2.2. Let { f j , g j ) be continuous convex functions from H x L2(0 , T ; H ) x U into R: . W e assume that

1)

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1034 BUI AN TON

2) f , , g j ( x ; . ; .rrjv) are strictly convex and that 8g,(x; . ; ~ , v ) are strictly maximal monotone.

The following theorem of Von Neumann on sections of set-valued mappings will be used throughout the paper . Theorem 2.1. Let X, Y be two separable Banach spaces and let A he a set-valued mappzng of X znto the non-empty closed subsets of Y. Suppose that the graph of A zs closed Then A has a urtzversally Radon measurable sectzoii, z e. there exzsts a slngle-valued mappzng u of X znto Y wzth u ( x ) E A ( x ) for all x E X , whzch 1s

measurable for any Radon measure on the Borel field of X .

The proof of the theorem call be found in the Appendix of [dl. For deterministic differential inclusions with singular inputs , we shall need the

following assumption.

Assumption 2.3. Let w be a cor~tznuous functzon on [O, TI and lrt N = [Tlh] , t k = I11 jor some small posztzve ~lurnber h. We assume that

(2) Thew e m t s a posztzve constant u such that

{ w ( t k ) - ~ ( t k - ~ ) } ~ <_ u 2 h ; k = 1 ,..,. v. (ii) l'here exlsts a positive constant c such that

for all real numbers a,, b k , j , k = 1, . . ,n with it 5 IV

It is easy to check that if to is a function of bounded variation on [O,T] with total variation q ( u : ) . then w satisfies Assumption 2.3. In order to apply the result; of Section 3 to the stochastic case , we need the following lenuna .

Lemma 2.2. Let (a, A, P ) be a probabzlzty space wzth Jlltratzon Ft and let ru be a Ft-standard Wzener process on the space . Suppose that E{a: + h:} < m ,then w satzsjes Assumptzon 2.3 wzth probabzlzty 1 .

Proof. 1) Since w is a Wiener process on the space , we have

E { [ w ( t k ) - ~ ( t ~ - ~ ) ] ~ } = u 2 ( t k - t k - l ) = a z h

for some positive constant u and

E { w ( t k ) w ( t 3 ) } = 0 ; j # k .

Set

and we have

E{ l 6; - u 2 h 1') = E{tGir;lk] - u4h2.

From the definition of stochastic integrals (Doob[7] , p.427-434 ) , we have the formula

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N-PERSON STOCHASTIC INCLUSIONS. I1

(2.4)

Thus,

Therefore

and hence

with probability 1 . 2) Again by the same argument as above , we get

We have

We have applied 2.4 and 2.5 in the above calculations . Therefore

with probability 1 . 0

In the proof of the above lemma , the hypothesis that o is a constant , is crucial. The Holder inequality

{ E ( f 2 . 1 2 ) } 2 I E ( f 4 ) E ( 1 ) becomes an equality when f (w; 1 , s ) = w(w; t ) - w(w; s ) and it is a consequence of the formula,used in the construction of the stochastic integral when a is a constant.

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1036 BUI AN TON

In this section we shall consider the deternlinistic case of 1.1. The main result of the section is the following theorem.

Theorem 3.1. Let G, C be as in Assun~ption 2.1 and let w be a continuous function on [O, T] satisfying Assumption 2.3 . Then for any given

{[; u,; a,v) E K x U

Similarly for v and for w.

Lemma 3.1. Suppose all the hypotheses of Theorem 3.1 are satisfied . Let 16, i) be single-valued Lipschitz coiltii~uous ntappings of H x U into H . Suppose that

l l ~ ( y ; U N H + l l i ( y ; U H H I C{1 + I I Y I I H + IIuIIc(o,T;u))

Then for each n , there exists a solution yn of 3.2 . Moreover

lI~iL(t)lI2H I '(1 + IlEIlL + l l ~ ~ l I ~ ( o , ~ . ~ / , ) + l l ~ J ~ I l ? Z ( o , ~ , u ~ - l ~ }

uizth

yh ( t ) = yn for t E [ ( n - l ) h , n h ) ; n = 1, ., N .

C zs andependent of n , u,, n,v, w . Proof. The existence of a solution yn of 3.2 is well known . We shall now establish the estimate of the lemma .

We have

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N-PERSON STOCHASTIC INCLUSIONS. I1 1037

Since ( A ( y k ) , y k ) ~ 2 0 , we obtain by taking into account our hypotheses on G , L

k=l The different constants C are all independent of n.

From the discrete analog of the Gronwall lemma , we obtain

I \Y"/~; 5 C{l + 11 ' !11$ + I I1 l~ l l i (O ,T;Ul ) + I Ia~vI I i (O ,T;U~- l ) } The lemma is now an immediate consequence of the above estimate. 0

Lemma 3.2. Suppose all the hypotheses of Theorem 3.1 are satisfied and let yh be as i n L e m m a 3.1. T h e n

The c o n s t a i ~ t C is independent of h

Proof. We have from 3.2

l l y k - y k - l 1 1 ~ 2 + h ( A ( y k ) , yk - y k - ' ) ~ = h ( ~ ( ~ ~ ; U S ; a j u k ) , yk - y k - l ) H

+ ( L ( y k ; U S ; a 3 v k ) , yk - y k - l ) H ( ~ k - W k - ' )

But

( A ( ~ ~ ) , yk - y k - l ) ~ > l K ( y k ) - I ~ ( ~ ~ - ~ ) > 0

Therefore

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BUI AN TON

\.Ve get , by using hypothesis (i) of Assumption 2.3

5 Y * - yk-'IIL c C { l + MIL + I I u , I I & ~ , T ; ~ , ~ + I I ~ i ~ I I t ( o , r ; a ~ - l l ) k=1

The lemma is now an immediate consequence of the above estimate .

and similarly for vh.

Lemma 3.3. S ~ p p o s e all the hypotheses of Theorem 3.1 are satzsfied and let yh be as zn Lemina 3.1 . Then

{ y h ; G ( Y ~ , u h , ~ ; r 3 u h ) ; L (yh ; fLh,]; ~ 1 ~ ) ) { Y ; G ( Y ; l l j ; T ~ V ) , i ( ~ ; u ] ; K ~ U ) )

z n

( ~ ~ ( 0 , T ; H ) n (LW(O, T ; H))weak.} x ( L 2 ( 0 , T : H ) ) e p n i , as h -r O+

Moreover

I Y IzI C { 1 + IIEIILH i- I I U ~ I I & O , T , I ~ ) + / I ~ ~ ~ ~ I I ~ C ( O T L ~ ~ - ~ ~ )

luzth y ( t ) E K for almost all t E [0, TI.

Proof. From the continuity of the L2(0 , T ; H)-norm ,we have

with yh ( s ) = 0 for J outside of [O,T] Thus .

(3.3) I Y I I 125 C i l + l l E l l & + l l"~112Cco.~;~, i , , + i I n u l l Z C ( o , ~ ; ~ ~ f - i ) l

It follows from Lemma 2.1 and from the above estimate that

By hypotheses ,both c, i are Lipschitz continuous in the aprropriate spaces and it is now clear that

Lemma 3.4. Let y k be as in Lemma 3.1 . Then

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N-PERSON STOCHASTIC INCLUSIONS. 11

C is a constant independent of h , n.

Proof. 1) From 3.2 , we have

2 hllA(~k)~li 5 C h ( l + 1/(11?f + IIUJ(\$(O,T;~) + I l~jvl(C(o,T;U~-1))

+ (L(yk ; US; r j u k ) , A(yk))~2irk

2) Taking the summation from k = 1 to k = n and then taking the square the two sides . we obtain

Applying hypothesis (ii) of Assumption 2.3 and we have

i.

I Cho2 C lIA(Y"1l;,lli(Yk; u:'; r,vk)l12H k=l

I t follows from 3.4 , the above estimate and from the estimates of Lemma 3.1 that

( 3 . 5 ) + Ch 2 I lA(Yk)lk k=1

The different constants C are all independent of h, n. 3) Since (a - c)' 2 0 , we obtain from 3.5

Therefore

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L e m m a 3.5. Suppose all the hypotheses of Theorem 3.1 and of Lemma 3.1 are sat- isfied . Then there exists y ( t ) 6 K a.e. on [O,T] with A ( y ) € 81r (y ) such that

y ( t ) + A ( y ( s ) ) d s = [ + 41 ~ ( y ; u,; nJu )ds + 1 &(y; u,; nJu )dw

Moreover

1 Y 12 + I I ~ Y ) I I L ~ , T : H ~ 5 C{1 + \\t\\i + ~ \ u ~ \ ~ ~ ( o . ~ ; r S ) + I I ~ ~ U I I ~ C ( ~ . T ; U N - ~ ~ } ~

P~oo j . 1 ) From 3.2 , we get

We have . by taking subsequences

Hut A(yh) E aIK(yh ) and since yh -+ y in LZ(O. T ; H ) , we obtain from a standard argument of the theory of monotone operators that x = A ( y ) .

3) Since y k E k' , we have y,,(t) E k' for almost all t and since li: is a closed convex subset of H , we deduce that y(t) is in k' for almost all t .

All the other assertions are direct consequences of Lemma 3.1 , 3.2 and 3.3.

P r o o f of T h e o r e m 3.1.1) Since Q, C are U.S.C. set-valued mappings of L2(0,T: H ) x U into the closed convex subsets of LZ(O, T; H) , we may applj ~e approximate sr- lection theorem . There exist single-valued mappings Lipschitz continuous mapping.; G,, L , of L 2 ( 0 , T ; H) x U into L 2 ( 0 , T ; H) such that

the range of G , is contained in the convex hull of the range of Q . 0 Graph G, c Graph L7 + &(ball radius 1 about the graph ). Similarly for L F . 2) From Lemma 3.4 , we have y, such that

We have, by taking subsequences

y, -+ y in L2(0 , T ; H ) n (L"(0, T; H))weak*

It is clear that y ( t ) E K for almost all t E [0, TI .

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N-PERSON STOCHASTIC INCLUSIONS. I1 1041

Since 9 is an u.s.c. set-valued mapping of L2(0, T ; H ) x U into the closed convex subsets of L2(0,T; H ) , its graph is closed a i d therefore also weakly closed . It follows from our hypotheses on G that

G(yE; ~ l j ; r j v ) + G(y; uj; r j v ) in (L2 (0 ,T ; H))weak with G E G

On the other hand , we have

GE(z ; ~ j ; X ~ V ) C G ( z ; uj; T , V ) +&(unit ball about 9) . Therefore

G,(y,; uj; r j v ) + G(y; u,; Tjv) in (L2(0 ,T; H)),,,k.

Similarly for L,(y,; u,; r j v ) . Hence

(1' G ( ~ ; u,; njv)ds; L(y; uj; r j v ) d u ~ )

in (IJZ(O, T ; H))Leak for almost all t . 1'

3) We have

(\A( .; Y ~ ) ~ ~ L ~ ( o , T : H ) 5 C Since A is maximal monotone in L2(0,T; H ) and y , + y in LZ(O, T ; H ) , we get

The estimate of the theorem is now an immediate consequence of that of Lemma 3.5 .

The theorem is proved .

4. O P E N LOOP FOR INCLUSIONS WITH SINGULAR INPUTS

In this section we shall establish the existence of an open loop equilibrium of 2.2 -2.3.

The main result of the section is the following theorem . Theorem 4.1. Suppose d l tht- hypolheses of Theorem 3.1 are satrsfied and suppose that Assumptzon 2.2 zs verzfied . Then there exzsts

{ i ; ii) E R ( [ ; i L j ; r j i i ) x U

such that

J3( t ; c; i L J ; r jC) 5 J j ( t ; y; vj; a j S ) ; j = I , ., N

for all y E R(<; v,; ~ j i i ) and all v E U. R ( [ ; uj; r j v ) denotes the non-empty set of all solutions o f 2 2 with controls {u,; x,v}

First we have the following lemma.

Lemma 4.1. Suppose all the hypotheses of Theorem 4.1 are satisfied . Then for any given (4; u,, T,U} E IC x 24 , there exists jj E R ( [ ; u,; n,v) such that

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1042 BUI AN TON

j J ( [ ; u,; K , U ) = d { J , ( E ; y; u,; K,U) : v y E R(E; u,: x J v ) j

(4.1) = J J ( [ ; y: u,; r J u )

Pvoof. Let { y n } be a minimizing sequence of the optin~ization problem 4.1 with

From the estimates of Theorem 3.1 , we have

I yrL I z I c with C independent of n . Let n -+ cc then u e obtain a subsequence, which we denote again by yn such

tha t

yn + d in L 2 ( 0 , T : H) n (Lm(O,T; H) )meak - ; i 2.

The same proof as in that of Lemma 3.4 and of Theorem 3.1 shows that j E R((: ZL,; x,u).

Let n + m in 4.2 , then with our hypotheses on f,. g, , it is clear that

.?,(E; u,; ~ , v ) = JJ(E; d ; u,; x,v)

The lemma is proved . 0

Let

with j, as in Lemma 4.1

L e m m a 4.2. Suppose all the hypotheses of Theorem 4.1 arr satzsfied . Then there exzsts u* E U such that

Proof. 1 ) Let {u'" be a minimizing sequence of the optimization problem 4.4 with

Suppose that u'" u* in C ( 0 , T; U ) , then u* E U since U is a compact convex subset of C ( 0 , T; C r ) .With the control {u;; r , v ) , we obtain by applying Lemma 4.1

j j(E; ujn; K J V ) = J j ( l ; Y;; ujn; x j ~ )

From the estimates of Theorem 3.1 , we get by taking subsequences which we denote again by y;

yjn -+ y; in LZ(O, T ; H ) n ( L m ( 0 , T ; H))weak.

A proof exactly as in that of Lemma 3.4 gives y; E R([; u;; 5 v ) . With our hypotheses on f,, g, , we deduce that

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N-PERSON STOCHASTIC INCLUSIONS. I1

j j ( ( ; u;; n j v ) 5 J j ( [ ; y;; u;; n j v )

5 liminf j j ( ( ; u?; r j v ) .

2 ) Suppose that

U J ; X ~ V ) I -6 + ~ ~ ( 4 ; x:; 6;; T ~ V )

for a given 6 > 0 and for any 47 -+ U; in U with

Then , as in the first part we get

j l ( [ ; u;; T ~ V ) 5 -6 + J ~ ( ( ; 2; u;; S ~ V )

with x E R([; uj ; x jv) . Since

R((; u;; a lv ) = n {R((; u;; a j v ) : U ; E B,(u;)} C

we obtain , by taking the illfirnuin over R([; uj ; n j v )

j i ( ( ; u;; T ~ V ) = l i m j 3 ( ( ; u?; r j v ) = J j ( [ ; Y ; ; u;; T ~ V )

Therefore N

d ( ~ ) = C j j ( ( ; u;; T ~ v ) . ]=I

The lemma is proved 0

Let

(4.5) D ( v ) = {u* : u' as in Lemma 4.2) = {u* : *(u*; v ) 5 Q(u; v ) , V u E U }

Lemma 4.3. Suppose all the hypotheses of Theorem 4.1 are satisfied . Then D ( v ) is a non-empty closed subset of U . Proof. The set D ( v ) is non-empty by Lemma 4.2 . A proof as in that lemma shows that the set is closed .We shall not reproduce it . Lemma 4.4. Suppose all the hypotheses of Theorem 4.1 are satisfied . Then there exzsts a unique 4 E U such that

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1044 BUI AN TON

(4.6) g,(E; 4,; x,v) = a, = inf{!j,([; u,; x,v) : Vu E D ( v ) } ; 1 = l , . . , N .

Plaof. Let u ' q e a minimizing sequence of 4.5 with

and un -+ u in C ( 0 , T ; [I). Since un E D ( v ) and the set D ( v ) is closed , we get 11 E D ( v ) . . With our hypotheses on g,, , we have

gj(E; u,"; ~ j v ) -+ g,(E; u j ; ~ j u )

and hence gl ( ( ; u ] ; x,v) = aj It remains to show that the solution is unique . Suppose that 4 , f i are two elements of D ( v ) with

g j ( E ; f i j ; ~ j v ) = g j ( E ; f i j ; ~ I v ) = in f{g, ([; u,; T , V ) : Vu E D ( v ) }

By hypothesk , g,(E; .; x lv ) are 1.s.c. on LZ(O,T; U,) and hence the subdifferen- tiais as,([; .; x l v ) exist . We have

T T L { g J ( ( ; w l ; T J V ) - g J ( [ ; 'J; 2 L ( ~ ( ' 1 ) > w~ - G 3 ) ~ r j d t

for all w, E LZ(O,T; U,) with ~ ( t , ) E dg,(E; .; x,v).

Since C, is a minimum of g , over D ( v ) , we have

It is clear then that

l T ( X ( d J ) 3 ' j - ' J ) H ~ ' = O

Similarly

Therefore

By hypothesis , d g j ( ( ; .; x,v) is strictly maximal monotone and hence G I = * . U ] ; 1 = 1, ..,AT.

The lemma is proved . Let 7 be the nonlinear mapping of U into U given by

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where Q is the unique element of D ( u ) with

In view of Lemma 4.4 , the mapping 7 is well-defined.

L e m m a 4.5. Suppose all the hypotheses of Theorem 4.1 are satisfied and let 7 be as in 4.7 ,then 7 has a fixed point .

P~oo f . Since U is a compact convex subset of C ( 0 , T ; U ) , t o apply the Schauder fixed point theorem to the mapping 7 it suffices to show that 7 is continuous .

1) Let u" E U and let 7 u " = Qn . Then it follows from Theorem 3.1 that there exists y," E R([; Q;! .rr3vn).

We obtain by takmg subsequences

i 11

L ~ ( o , T ; H ) n ( ~ " ( 0 , T ; H) )wenk . x C ( O , T ; U ) .

for j = I , . . N. A proof as in that of Lemma 3.4 gives yj E R(E; u j ; ~ , v ) . 2 ) Recalling the definition of Q , we get by taking into account our hypotheses on

f 3 I 93

WF now show that C is in D ( I I ) . First we note that D ( u ) = nn D ( v n ) . Thus if 4 is in D(un) for n 2 no , then

Thus ,

for all 4 E D ( v ) . Therefore Q E D ( u ) and 7 u = Q. The lemma is now an immediate consequence of the Schauder fixed point theo-

rem 13

Proof of Theorem 4.1. Since 7 has a fixed point ii E U , we get from the definition of the mapping 7

Q ( i ; 1) 5 Q ( v ; 6 ) for all v E U It then follows from the definition of Q that

Take u = (wk, skG) in the above inequality and we get

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j k ( ( ; f i k ; a k 6 ) 5 . j k ( f ; w k ; a&) for all wk E Uk

BUI AN TON

Repeating the process N - 1 times and we get the theorem .

The main results of the paper are the following theorems . T h e o r e m 5.1. Let ( R , A , P ) be a probability space , w be a F-s tandard Wiener pmcess on the space and suppose that the measure P is regular in the interior . Suppose that all the hypotheses of Theorem 3.1 are satisfied . Then for any given

{ E ; U } E L , ~ ( R , A, P ; H ) x L ~ ( R , A, P ; C ( O , T ; u)) with { ( ( w ) ; u ( . ; w ) ) E K x U a.s. , there exists a solution y of 1.1 with

The set of ali solutions of 1.1 is denoted by R,to([; u,; n,v).

T h e o r e m 5.2. Suppose all the hypotheses of Theorems 4.1 and 5.1 are verzfied Then there exists 6 E L2(61; A, P ; U ) and Q E R,t,((; C,; T,G) such that

E { J , ( [ ; i ; 6,; n,C)} 5 E { J , ( ( ; y; v,; n 3 C ) } ; 1 = I . . . A'

for all y E R,t,([; v,; x,fi) and all 21, E L 2 ( R , A, P ; U j ) .

To prove T h e o r ~ m 5.1 , we shall apply Von-Neumann theorem on sections of set ~ a l u e d mappings wi th closed graphs and closed images . L e m m a 5.1. Suppose all the hypoth~ses of Theorem 3.1 are satzsjied . Then for any gzuen {(; u,, n ,v)} E h' x U , the set-valued rnappzng R ( [ ; u,; n , v ) } of H x C ( 0 . T ; U) znto L2(0, T ; H ) has non-empty closed zmages . The graph of R ( ( ; u,; x,v) zs closed.

Proof. From Theorem Theorem 3.1 , we know that R(E; uj; n j v ) has a non-empty irnage in L2(0 , T; H ) . We now show that the graph of R is closed .

Suppose that

{ln; u:; a3vn) --+ { E ; u:; a J v )

in H x C(0,T; I J ) with [" K and {u;,a3un} E U . Since k: , U are closed convex subsets of H and of C ( 0 , T; U) , we have (4 ; u,, n,v) E

K x U . Let yn E R(En; u;; a 3 v n ) . From Theorem 3.1 , we have

C is a constant independent of n . Thus we have a subsequence, denoted again by yn such that

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with {G, L ) E 8, L . Since 8, L are u.s.c. set-valued mappings of L2(0, T ; H ) x C ( 0 , T ; U ) into the

closed convex subsets of L2(0 ,T; H) , their graphs are closed and hence weakly closed . Therefore

With yn -r y in L2(0 ,T; H ) and A(yn) -+ cP weakly in L2(0 ,T; H ) , we get by a standard argument of the theory of monotone operators @ = A(y) .

Therefore y E R ( ( ; uj; a j v ) and the graph is closed . ,

It remains to show that the image of R((; uj; ajv) is closed in L 2 ( 0 , T ; H ) . The proof is almost identical to the above arguments and we shall not reproduce it .

Proof of Theorem 5.1. 1 ) It follows from Lemna 2.2 that w satisfies Assumption 2.3 with probability 1 . It follows from Von Neumann theorem and from Lemma 5.1 that R ( ( ; uj; a j v ) has a section which is universally Radon measurable. We shall call such section R([; u,; s j v ) .

Let A be the mapping

2) Since P is assumed to be regular in the interior , there exists a compact set Kn c R such that

1 P ( R - K n ) 5 -.

n Without loss of generality we may assume that I(,, is an increasing sequence .

Since A is a random variable , the restriction An of A to I(, is continuous. The Radon measure P induces on I(, a Radon measure Pn and thus , An(Pn) is a Radon measure on H.

With R ( ( ; 11,; a,v) being universally Radon measurable , R((; uj; s j v ) o A is 1'-111casura.ble of I<,, ill L"(O,1'; 11) .

3) Let ~ ( w ) = R ( ( ; uj; a j v ) o A(w)

and let yn(w) = y(w) if w E IC, ; yn(w) = 0 if w 4 Kn.

Then yn as mappings of R into L2(0 ,T; H ) are measurable and

yn(w) -4 y(w) VW E R - UnKn i.e. a.s w E R

since P(UnKn) = 1. 4) We have

y(w) = R ( ( ; uj; x j v ) o A(w)

E R((; uj; a j u ) . Dow

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BUI AN TON

The theorem is proved . To prove Theorem 5.2 we shall need the following lemma

L e m m a 5.2. Supposr all the hypotheses of Theorem 4.1 are satisfied . Let M be the set-valued mapping

M ( [ ) = { { c , G} : Q E R(E; G,; .rrj i)}

whe~e J,(<: i ; 6,; nJG) = inf{J,(E; u,; n , ~ ) : V u E U).

Then M zs a set-valued mappzng of K C H znto the closed subsets of L2 (0 , T ; H ) x C ( 0 , T ; U ) ) wzth closed graph . Proof. 1 ) It is clear from Lemma 4.1 that the set iZ/1 is non-empty . We now show that the graph of M is closed.

Let E n E K with tn + E in H. Suppose that { y n , u n ) E M(En) and that

{ y n , un } -+ { i , C } in L2(0, T ; H ) x C ( 0 , T ; U).

Since yn E R(En; uj"; n j u n ) , it follows from the estimates of Theorem 3.1 that

By taking subsequences ,we obtain as in the proof of Lemma 5.1

{ 4 y n ) , G ( y n ; u n ) , L ( y n ; u n ) } + ( 4 6 1 , G ( i : f i ) , L ( i ;

in (Lm(O, T ; H ) ) w e a p x ( L 2 ( 0 , T ; H))Leal, . Moreover i E R(E; GI; x,fi) With our hypotheses on j,, gl , we get

J l ( t n ; yn; u;; n,ujn) + J , ([ ; i; 12,; T , G ) .

2) We now show that

J,(E; 9; Gj; n1&) = i n f { J j ( E ; y; u j ; n , u ) : V u E U). Since {y'" u"} E M(En) , we have by definition

vn E U ; xn E R ( [ " ; v:; ?r1vn)

such that

J,(En; yn; u:; n3un) 5 Jj(En; xn; v;; n1vn).

From the estimates of Theorem 3.1 we get 1 xn Jz< C. As before , by taking subsequences we have

{ x n , v n } + { r , v} in L2(0 , T ; H) x C ( 0 , T ; U )

with x E R(E; vj; .rrjv). Moreover

J j ( t n ; xn; v;l; ?rjvfL) + J j ( t ; x ; v j ; n,v)

Conibining with the first part , we get

J,(E; i ; G,; x,G) 5 J , ( E ; xi v,; n p )

3 ) Let t 6 E B s ( [ ) where B s ( ( ) is a ball of radius 5 in H about the point 6 . Dow

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N-PERSON STOCHASTIC INCLUSIONS. 11

Then from the previous part we get

J,([; &; 6 j ; K ~ G ) 5 E $ JJ(&; 56; v3; K ~ U )

for all xg E R(E6; vj; s jv) . Hence

J,([; 9; 6;; T,O) 5 inf{J,([s; 2 6 ; vj; K ~ V ) :

V x6 E R(&; vj; nju) ; Vv E U) 5 inf{Ji([; x; v,; K ~ U ) : Vx E R([; wj; R ~ V ) ,

vv E U}

It follows that

Therefore the graph of M is closed . 4) Now an almost identical proof shows that the image of M is closed in L2(0, T ; H ) x

C ( 0 T ; L ) 0

Proof of Theorem 5.2. From Lemma 5.2 and from Von Neumann theorem , we know that M has a universally Radon measurable section. Let A be the mapping of R -+ Iil given by

=

and set r = M o A .

Repeating the proof of Theorem 5.1 and we get the stated result .

6. OPTIMAL ALLOCATION O F RESOURCES IN T H E FIGHT AGAINST DRUG USE

In this section we shall apply Theorem 5.2 to the study of an optimal allocation of resources in the fight against drug use. Mathematical models of the drug market have been studied by Caulkin [4] ,by Dawid and Feichtinger [6] and by others in a special issue of Mathematical and Computer Modelling [5] . While most of the models are descriptive,the one introduced by Dawid and Feichtinger involved an optimization problem leading to a 2-person differential game and an open loop Nash equilibriurn. For specific functiorial forms and with 2-player, explicit calculations in the deterministic case are possible and a careful analysis of the problem was made by Dawid and Feichtinger .

As an application of Theorem 5.2 we shall consider the case of N-player with functional forms closer to the standard diffusion dynamics and with a white nolse effect . The model seems new .

Let f j be the population size of the country and let y ( t ) be the stock of drug users at the time t , 0 5 y ( t ) 5 f j ; t E [O,T]. The change in the stock of drug users is governed by : (i) the activities of the drug dealers , (ii) the death of drug users and (iii) the rehabilitation of the addicts .

We shall assume that Dow

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1050 BUI AN TON

r The dynamics of the stock variable is a diffusion one which is modelled by the logistic growth function y (y - y) .

r Let udealer denote the efforts of the drug dealers in attracting new costumers and in selling drugs with 0 5 ud,,,,, 5 Ed,,l,,. We shall assume that the marginal effect of the efforts of the drug dealers decrease with an increase in the level of efforts and shall model the growth term in the system dynamics by l/lldenie?-(~(y).

r 'l'he population ot add~c t s decreases with a death rate D and is reduced by medical rehabilitation . The government has two ways of keeping the stock of drug users down .

The first strategy is through medical treatment . The effect of the treatment depends on the allocated government budget on drug fighting. The governmmt effort through medical rehabilitation is measured by the control variable um,d .We shall model the effect of the medical treatment by the set-valued mapping

r ( y ; u ) = 6 4 ~ ) if umed 2 UL = X & Z S ( y ) if 0 5 L - d < U m d < ~,,d ; E [O, 11 = 0 if 0 5 um,d I Cm,d.

Thus when the budget allocated to the rehabilitation program is below certain level , we are assuming that the effect of the program is negligible and it has a significant impact only when the budget is above a level uLed. We also assumed that the marginal effect of the government effort decreases with the level of effort .

We shall denote by

The second strategy that the government can use in the fight against drug users is through law enforcements : destruction of crops , seizure raids , custom inspections

r 7 . 1 he government effort is measured through the control variable uenf,,, = u3 and obviously depends on the budget allocation and is modeled by the set-valued mapping

We assume that the effect is negligible when the budget and therefore the effort is below a critical level and has a maximum impact only when the budget is a t the level u : , , ~ ~ , , ~ Since those law enforcements measures have to be carried out in a random fashion in order to have any impact , we shall consider t the stochastic term Ldw where w is a Wiener process to model the impact of law enforcements measures .

We shall consider the stocl~astic inclusion

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y ( ; 0 ) = J ; E E N,yl. Throughout the section we shall take

H = R , U, = R ; j = 1 , 2 , 3 ; X: = [0, y]

and we shall make R into a Hilbert space with the usual inner product and norm

U is a compact convex subset of ( C ( 0 , T ; R ) ) 3

Lemma 6.1. Let a , -0, y be continuous positive convex functions with

for all y. . Let R , A , P ) be a probabilzty space and suppose that the probabzlzty measure P zs r~gular zn the znterzor . Let w be a p-s tandard Wzener process on the space . Then for each

{ J ; Udralerr U m e d , uenjorc} E L 2 ( f l , A, P ; H ) X ( L 2 ( a , A> ~ ( O I T ; R ) ) l 3

with { t i udealerr U m e d , ~ e n f o r c } E X U

a.e. on [O,T] and a.s. there exists y E L 2 ( R , A, P : Lm(O,T; W ) ) , solution of 6.3 . Moreover

Proof. It is trivial to check that -9, L given by 6.1-6.2 are u.s.c. set-valued map- pings of L2(0 , T; H ) x U into the closed convex subsets of L2(0 , T ; H ) . Now a direct application of Theorem 4.1 gives the stated result .

We shall associate with 6.3 cost functionals based on the following assumptions . 0 The income function g ~ ~ ~ l e T ( y ) of the drug dealers is a positive convex increasing

function. The cost function gi$,, to the drug dealers (besides their fixed costs) are due

mainly to the law enforcement actions and to their own competitions . Without police actions and without any competition among the drug dealers , the

cost to the dealer group will be zero , i.e.

The other obvious observation is that if the group is inactive ,then the cost is zero , 1.e.

We shall assume that the cost to the dealer will increase linearly with the effort of the law enforcement agencies .

With the above underlying assumptions we propose to use for cost functions gSf;tl,, ,the functions

where uen,,,,(y) > 0 measures the efficiency of the law enforcement actions , udeoler(y) is the effect of the group competition among the dealers and (1 - A)u3 is the effort of the various law enforcement agencies .

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We shall assume that venjOTc(y) , vd,,le,(y) are of the forms

where v l , vz are two positive constants . Let

The government costs are of three different types . The medical costs due to long term medical illness caused by the use of drugs

and the medical rehabilitation costs. The loss of tax revenues to the lack of productivity of the drug addicts .

We shall assume that both costs increase with the drug addict population and take the form

where B is a positive continuous function on [O,T]. The cost of the law enforcements is assumed to be

gs(y; u l ; U Z ; u3) = & ( Y ) u ~

where S(y ) is a stritcly convex function of y..

Lemma 6.2. Let g,; j = 1,2 ,3 be as above . Then g, satisfies Assumptzon 2.2

Proof. The proof is trivial .

Let

Applying Theorem 5.2 , we obtain the following result.

Theorem 6.1. Suppose all the hypotheses of Lemma 6.1 are satisfied an,d let J, be as in (6.4) . Then for any given J E L 2 ( R , A , P ; H ) with ( E K a.s. , there exists

4 E U ; j j E R,t,(J; 4,; n j 4 )

with

j j E L 2 ( R , A, P ; Lm(O, T ; H ) ) ; E X: a.e. on [0, T1,a.s.

Moreover

E { J j ( E ; 4 j ; n,C)) 5 E { J 3 ( J ; Y ; uj; r j Q ) )

for all y E R,to(J; 0,; r j 4 ) , all u, E Uj ; j = 1 ,2 ,3 .

Thus there exists a control

= (4deale?, k e d , 4enfmc) E U Dow

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which will minimize the medical costs and maximize the impact of the law enforce- ments effort and reduce the income of the drug dealers to a minimum.

1. J.P.Aubin, Mathematical methods of game and economrc theoy,North Holland, 1979. 2. A.Bensoussan ,Results on Stochastic Navter-Stokes Equations, Control in partial differential

equations. Lecture notes in pure and applied mathematics. 165 , Marcel Dekker ,1994,pp.ll-21. 3. A.Bensoussan and R.Temam, Equations stochastiques du type Nauier-Stokes, J . Functional Anal-

ysis 1 3 (1973),195-222 4. J.P.Caulkins,Local drug markets response to focused police enforcement, Operations Research,

4 1 (1993) ,848-863. 5. J.P.Caulkins , Editor.Mathematica1 models of drug markets and drug policy, Math Com-

put.Modelling 17(1993),1-115. 6. H.Dawid and G. Feichtinger ,Optimal allocation of drug control efforts: a dafferenttal game

analysts,J,Optimization Theory and Applications .91(1996) ,279-297. 7. J.L.Doob,Stochastic Processes,John Wiley ,1953. 8. Bui An Ton, Open loop equilrbrium strategy for qutast-uartational inequalzties and for constratned

non-cooperatwe games, Numerical Functional Analysis and Optimization 7(1996),1053-1091. 9. Bui An Ton,Optimal control for N-p~rson dzfferenttal stochastzc incluszons,Stochastic Analysis

and Applications.17 ,6, (1999) ,911-935. 10. Bui An 'Ton ,Stochastic Navter-Stokes equations, International J.Applied Math.l ,(1999)

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