Appl Math Optim 3273-97 ( 1 9 5 )

O 1995 Springer-Verlag New

York Inc.

Sensitivity, Controllability, and Necessary Conditions of Optimal Control Problems Governed by Integral Equations

Centre de recherche s i r les tra ports (C.R.T.), and Dbartement d'intormatique et recherche op+iratic~nnelle, Vniversitt de Mclntreal, C.P.6128, succ. A,

Montrkal, Canada H3C 157

Abstract. Infinite-dimensional perturbations in all constrairits of an optimal control problem governed by a Volterra integral equation with the presence of a state constraint arc considered. These perturbations give rise to a value function, whose analysis through the prordmal normal technique provides sensitivity, controllability. and even necessary conditions for the basic problem. Actually all information about the value Function is contained in Clarke's normal cone of its epigraph, which can be characterized by the proximal normal formula. key Words. Optimal control, State constraint, Volterra integral equation, Necessary conditions, Canuollability, Sensitivity, Value function, Proximal normal analysis, Proximal analysis. AMS Classificalion.

1

1

49J22, 49K22, 93B05, 93W6, Y3B35.

Before we present the principal material which is used in the forthcoming sections, ns we give some basic nutation. We denote by T the i n ~ c m [O, l 11 and by . ~ F ' ( TRn) ,

.+

the space of absolutely continuous functions x : T R n. which have derivatives i in thc space P ~ ( T , Rn).An absolutely continuous function is called simply an "arc." H denotes any real Hilbert space except if it is specified, with inner product { . , .) and the standard corresponding norm denoted by II.II. R denotes a h e d compact set from R m endowed with the Borelian u-algebra, C(n) denotes the space of con-

*o

Giuws & p ~denotes j &a space of pmW&ty &don measure defud on 0, ad Z1(T, C(O)j denotes the spaoe of &IWS of functions f: T x n + R,It, u l - flt,u), such that fi,u ) t Lekgue meamable, fit,. ) is in Cia), and q, ,lf(.,uJ is in glCT, It). The norm in this last spa= is given by Ilf llwlt~,qon:- ITma,, nrfil, ull df.c*+(I 1 denom the space of m t i v e Radon meamves defined on I, a cbse whet of T,If f i s a d function dehed on I& we dcfme its epigraph as the set of paink which are on or above the graph of f ; that is, the set epi f = {fx, r) s H x R: f i x ) 5 r). In the following subaectims we review the main features !hat are useful in the sukqucnt treatment concern@ proximal normal mdysk, Clarke's generaked dds,and controk. For more defails on thme thee subject$ the reader k brvited.to wm'tt 151, [61,and C141.

~~

Proximal mdy& started with the notion of perpemliculars introduced by CIarke in the fmite-dimensioaal[6],which has given rise to a formula for the pneFalized gradient of the d h m c e function from a c l o d set C, dd,(.). The iact that the n o d cone of C is generated by ddc(.) involve3 an hnprhnt m n t r which b called the "prwdmal normal vsctm." w p p e we ate adasad subset C h m B and a point x in C. Then a vector g is said to be a proKimal normal =tor to C at x if a canstant a > 0 &its such that

w,

-

or*quirdently, dc( g/2u + x ) Il/Zu)IIg 1. W e denote by PN,lx) the set of all proximal n d vectors to C at x, Ylrhieb h a c o w tone always containing 0. bequality (F1) say that tht point x m i a h b s the functional - (5, c ) + w Jlc- xllZ over C, ad, in d probluq this wrtivn lea& to €he study of uwbb a d i q optinhen problems, whicb are often &dad, The tsstntial role of the proximal inequality (1.0 is to a m dl pertuhd w d t s in the objective. functional of the auxihy multing problem. Cwsequetltly, we can state, for a m p l e , n e w s s q conditions fat the unpmtmhd problem -taking dl - p t w h d constraints by just knowing the neixs&q mt~diomfor the m e p d l e m without the presence d sat, it h an easy exercise these constraints. Note that when, ia addition, C Is a to prove that inquati$ (1.1) is s h p W to the following:

The normd cone of C at x E C, NcG),can be obtained from progimal norm& by a quence of elementary algebrac and topological operations via the EoWq proximal n o d formula:

,

,

ammr this question, suppase that we haw given a problem Wed 9 and we perturb it bJC a cmah pametea a! to get another *ern, say ad, with a 0 1 =9. Let Nrr) be the value of @(a) (which may be equal to f 4 and d d e r the epigraphofthe function V. 1ttofte.n thecast that ~ n ~ a l to ~ epi V are reIatcd to multipllcrs of ,the a u x h y problem determined by the proximal inequality (1.1) in some fashim, and so the normal cone NN (0,HQ)) ia related to multipliers accordingto the pmximd n o d fornula (1.2). nerefore if it happens tbat N,,(O, Y(O)) is not reduced to 0,then m e wnclusio~son existence and necessary conditions, e k , can be c o h e d . Thh program is d N 1out in Won 2,whm w& $hi@a value fumiim related to the bmic problem d this p a p sbtd in the neat &on. and Fmmula (92)was proved b t by Clarke 161 in the fifift&ensiond -then mended bp B o d and Strojwas 121in the Wte-dimensional wtting. In the p a r t i a h mse of a lBb& space such as in (1,2),L~ewenEll] has given a simple proof based -nth@ on $ e a m e ~ d properties in H i b spa-. ~ Proxhd formula (1.21 has h m marry applications in sewitrid subjects, especially in the finite dimslsiwal we hi tke l d t 6 yew bx some of these applic~tionssee [41-[8l,[lo], [I21 1151, and 1161 We apply this brinuYa an m f ucmiom in -ions 4 and 5 when we study mme d u e h d m related to the basic p x o t r b We mai& b CI&'s & i d mm N&) the uppr ~miIXw.~uous W.S.CJ normal cone &(xY genW& wTkheforms, whi& is d e W by

This normal cone coincides wi& Chuke'sder mild ~

~

o on C. n

s

Clarke's G e n e m M Cuk&

Let f: H 4 R U (+a)be alawersemiaontiauws(ls.c)fuaclion andlet x ~ H b e a point at which f is finite. Then epi f 1 a dosed subset of H x R and (x, f i x ) ) E epi f. We define the generalid gradient and the asymptotic gmeralimd gradient at X,

af~xl,amf(x)w follows:

Then df(x) is a weakclosed convex s u b t of H (which may be empty) and d m f ( x )is a w e a k 4 d convex cone always containing 0. Note that in the infinrltedime~onal HiIbert space, it is possible that every subquence of a weakliy convergent sequence appearing in (1.2), cowerg= to 0 and in this case afIx) * @ In the W&nsional space this h a t i o n does not occur, because wery sequence of w i t vectors contains a subsequence cowerghg to a nonzero vector.

r

$

$he set whose When f is I d l y L W t z isx3we p m e that $fix) is support function is denoted by f Yx;. ), d e d "Clarke's dbctiond -the" df atxaxrdgiwby

In fhig

4;.

afld is bwdd an#.Pus it is w d l y cmpacL

we say wt

k4@w (in.Clarke's 3e-t if it@@t uwd ~ o o a l derivativt fib;.) exists and ooiacidcs with f O(x;.). We say that f bas a strict derivative at x d w a M by D$x) if

'-

4BJ(xl,v)

L

-

lim Y+XJ

LO

'

fir.+tu, -f(y) 1

exi& Eor every u in H. SufWeat donditions for f: to be mgda.m.&ctJy ctihmtiable are given in 13J. Q t ~ . mde p r n ~ & , a f ~ * s g e ~ @ d w ~ f , , $ , , b t w a fmdomslike f abw,!et Gib.G~ . Wclad O subset$M R",and (x,, @ E G, #,Cz, m n we have h e follawiw

- -

Ii) +f2Xx9 C; afid + afi(x1. (ii) d ( q f X d n aflx) hr any real o u m k p. Ciii) & xc,(xl, xg MC$xl) X NcJxzl. (iv) ~fhi ~ b a i m d s n a i o n h lthen , NC(x(x) 0 *r E int(C). Iv) if H is 6nite4bmional, then IdfI.1as a ndlifuudion is u.rc. at x.

-

Proxi& subgradients conststUte another mute leading to ClatWs 4culus with mom rc6nmenL l h m&e.r WRin thk -h is'falind in we-y that ~ i s a ~ s u ~ r p v^i d etd d~e d t~h &f& r~~ e u~~ > O ~ d fyoinr d l ant@hbdidd&~h~b

[a

We denah by d"flx) the wt of all proximdsubgradie~tsof f at x. A d t that we will: need in the proof of firollaxy 5.1 is fornufated in the following theorem lsee Theorem 3.6 of E9D.

JI I

We define a relaxed mntml as a aappinp v : T -. prtdR) which associates, to each a pmhbility Radon mcaurc v(t) dcfincd on i2 ~ u c hthat, for all f r

- I,,

I ET,

P'IT, CXfl )I. thc mapping r / ( I , u) d v l t Xu) i% Cebcsguc mchwrablc. Notc h a t every ordinary control rdr) can bc idcntificd with a rclaxrd mntrol. namely, the Dirac measurc &@,, co~ccnmtcdat ti(t 1 for cach r e T . Notc also that wc can scr each r e 1 d control as a linear functional an 2'0, C(R13 defined by

Wc dcmt the set of all classes I h l o k k g u c masure) of r e l a d mtmls by 9.If is a funrrion hlonang fa YV.(?".C(flN, then the notation f i r . 14r)Iis thc

same aq

1, f r t , u r d s 4 ~ X t d .

'h natural topolagim ~ norm lopam defined by

If v ll

he dcfincd in the

J I

space SF. The hrsl m e is the dual

sup

Iv(f)l-c~~;spl~lr)Xfl),

Pf"~fn.r,rrr.'1

ic T

I=

where 1 v(rH dcnotcs thc total variation of !hc mcasurc 14). The second onc is the w'-topology whose wnvcrgcncc k defined by

~;-v(w'S

iff { u , . f ) + ( ~ ~ , f ) ,

V~EP'(T,C(R)).

tt is demanstratcd in 1141 that G ' is eonvex, and that, with this topology, 3 is m p a c t a n d sequentially mmpilct and is the clmurc of fU, the class of ordinary control%;thnt is, thc set of all rncasurablc functions ti{-) such that dr) E R n.c.

in T .

2 The Aaslc RoMrm Wc arc given the funclion~f: R m X Rn R F: T X R" X Rm R 4: T X T X Rm x R" R*,and 8: T x Rm4 R,a subser I of T , and two s u k t s C*, and C, of -)

Rm.Thc pr&tcm which m n s i . in minimizing the functional

under the following constraints.

is termed the original problem and Iabcled 55 To l x faithful to tradition, we take thc state x of 9' to hc an a&olutt!y corthuous Function and thc control td a IEnLmgue) measurahlc function. In gcncral, this kind of pmbltra docs not admit a wluzion even whcn thc dala behave iccly. Thir failure is due cssenlially to the wncornpctncs of the scl of adrnisiblc controls. However, we mn asserl rxiaencc in slomc particular ELUCqee pp. 91-96 of [ISD. Ncvcrthclm thc laxa at ion of the problem in ahc sense af Warga ensurcs the cxislcncc of am1 facilitates mnside~ahly thc study of n v wnd~tiomand a m i a r d value functions (sce 1151). Wc &fine the relared problem 4 mrresponding ro P by

subject to:

Thc tol&ng

x(t1

-

1'

~ ( 0)I- d ( f *,T, x(s1, v ( s ) l & , 0

h y p t hews mmain in form t h r o u ~ h this t p3p:

is pintly L i p b i a : of rank K,. (M2) C,, I1 arc compac! s e k C , and J nrc c l w d . (H3) ?'be funcrbn F is mu1mrabIe in 1 , mnlinuour in x, and mntinuously differentiable in x: the function 4 i% measurable in s, continuous in ( x , u ) , c~ntinumsIyditfcfctcntiable in r and x. and 4#,iF motinuouq as a function of (r,s,.r, u), g is mcasunhlc in f and continuously diffcrenriaMe in x.

(HI)f

IH4) For cach compacr ~;ctr in Rn x RR, a furwfion such that IF(t,x,*)I+[F,(/.x.u): 0, i t f 4 x ) ~ b f ' ( xand ) &x) A M ' I X L In fact, lhese IWO wfs of rnultiplicm arc linked in somc manner. We &fine, for cach I r_ 0, the linear operalor 0, (R1, R ', q', 4'): ,or7;: x S ( 1 ) x R* x R" -r fi. ahcrc S ( I ) la the Hilbtat space given by

-

S(I)

- 2~

E

-

C"I1):

dplr3

-

6 E Y21~.~)3

£ ( l ) d t lor wmc

~p~+pI}:-~tlst for2 }d~p l ~ f ~ m t i t t l dd ~t ~~

and

k tk w m as M except that fl: i~replaed by

{f)-t~(tldt,

in the definition of H

sirch rhat

Then wc have

and it i s cay ro wc that, for cach x

E

~d,lg,),

Note [bat M U ( x )= (I)) iff,for cach C4,. E l , E 2 + f , ) and 0, d m s nut depend on ICI. 12).It is c a q tu ~

t

-

M"Cx). f o " 6.1 0 that t P, is a one-twnc linear

6

operator and in addition is bicontinuous. We de6nc th {two mnm

Then wing the pruxinlsl normal formula (1.2) with C -- epi V and rbc tkfinition of MY E) wc have, from LEmma 4.2, ff,, ,IO, Y(O))c w rl Sincc 1hc rwcrse incluqion ir a l w a ~imc by thc dcfinitiuns of aI/(l)S and d'V(0) ((1.3). (1.411, KC can thu~ mitt

a&'

hs pointed out in 181, if i t happens that A+,' ,CO, 1ICO)) + {O), then (1.4) implies that point, which m a n s that cirhcr hfl(T) nr M DIX)/{O) ir noncmpty. This fact, or, cquivalcntly,

N u W contairu a m m m

-

solution to 5 has tither a nontrivial multiplier with A 0 or el= a multiplier (which may be equal to 0) with A 1. This last formula suggests dividing the mllcctjon of multiplierr related ro admkssiblc arm inlo two ateprics;: t k wt of nvrmel mulriplitw ~ ' b i d , ( . G )and ) the sct of abnormal rnuliiplierr i ~ o ( ~ d , ( $ ) b , which contain M ' ( 1 ) and A I ~ Z ) ,~~pcctivrty. Thtse o h e m t i o n 9 suwmt thc folkwing definitions of some normality conccpls.

-

says that wmc

(1) An dmirsiblc slate x for 4 is said to k mmal if MQd.r)= (0). (2) & Is mid 10 be xlronpJy normal if each r E ~ d , ( $ ) is normal, that is. M ' ( A ~ , ( ~ I ) (01. (3) 2 t'is said to bt normal if cach x E Z is mm91,that b, ~ " ( = {01. 9

-

Fmm ( 4 3 , we scc that nurmalip can al-w bc c q r e d in tern~sof thc multiplier set Mn.I t is clear tbar 12) 4 (3). In parricular, if x E ~ d , ( & ) is such that he litate constraint is inactive along x , that is. p(r, xlrN < O a.e. in I and xIO) or .r(1) is an intcrior p i n t fo Cb, CI.rqxctiwly, thcn x murs be normal. Ohic follows from (i). (iii), anti Iv) of Lemma 4.1 by using Gronw;1Hk lemma.) Thcreforc, if Cn = R A or C1 Rb and - 1, problcrn 5 is strunph. normal. k s p i r c the facl that normalily is a prop~r!ywhich ir satjsficd in almst all problems of intcrcst, and which was nrgltcrrd in the past, if it; impanent to tnf it in any nsc, kcarm i t intcmncs in ihe fimt place when we study sensitivity and mn~ollabilitplor cxamplc. To bc more prcciw, wc state, after 1 k following tkorcm, some rtsulhmnccrninp; t h e two p i n k

-

-

Thtottm 4.1 (Ncces~aryConditions). S ~ r p p s rthat V(0) 4 +*. Thtn &n Jras a sdzdfmn t x , v 3 ro whicl~rirrtrt. c o m p r d g E f l s ; l , a ~iculurA E (0, 11, a rwnrtqprir Y m m f m p on 7,nnd IHW tvcton. 5, E 4 ,f(X(O). XI l)), c1 E d,,fldfl), ,r( 1)) such that w rkC fdI0n.m cor~fitianr;

lil

q(0)E At,

+ flCp(xC0)),

(This Imt condition ir c s c ~ i a lh u s e , for A = 0,q 0).

-

0, condftiwrs IiHiv) arc rriciafb

$~tl$hl'for p

(a;,a:,a;, a;. L:') Pmof. Given a qucncc ((A, t;. ti. ti, - A ' ) in PN mmtrging weakly to (6.. t,. L,, C,, - I ) with (o:, o;.;:or. I.') converging to 10,0,O, 0, V(Ol}in H, thcn Irrnma 4.1 asem that there is a scqucnce x, c Em,such that for wme p, E&:, [,' E d,,f(x,(O). x & l N ( j t, 21, concl~iomla), Cs), hold, We now define constant%m , , arcs q,, and measures F~via

-

-

and

Fmrn (4,6), (4.81, and (4.10), w deduce that each p, is a mnncgativc Radon masure with a drnsity in &I a d s u p p r t d on I with s 1. Since is bounded, then it contains a subsequence, which we &I not mlabcl, converging weakly-' to a wrtain nonnegative mcasurc + supporlcd on 1, with p ( I ) lirn, j ~ ~ (N lo)w , ( i,) iv bounded and then, along pfrhaps a further s u k q u c n c t , i,-. A E 10, 11. AFin Lcrnma 4.2,wc can prwc that ( 9 , )contains a subscqucr#rc which we do not relabcl, converging uniformly to some 4 IE .@," a d q, 4 4 weakly in 2"'. D i d i n # (4,6) acrom by nt, and taking fhc limit as i m. lead5 to

,,

from which we dcduce the nontrividity of the multiplier ( q , p, i). By scaling a m by i, when @ O, wc obtain (~lntlusion(v). By using Propition 3,l(a), w t can p to a subsquenet of {(x,, v,)] if nccesary such !hat ( r , v ) E ~ d ( & ~cxins ) as in Lemma 4.l(a) and so J ( x , v) 2 V(0). I-lowever, tk 1.s.c. of Y m r 1hr origin implies that, for i large enough, V l a , ) < Y(O) + l/i and then VCO) 2 lim, na,) lim, I ( x , , v , ) J ( J , v ) . (x, Y ) E Z. Going now to the limit in conclusiom (4.10)=-44.13$,we ohtain conditions I i M v ) of thc thcortm, cxcepr that in (iv) we can only confirm !hat tht support of !he measure. k in I whcn g is not ntcrwrily us.c, in f . 'nc next stcp is to establish that the support of p is in fact the wt of "activc" state constraint rime {I E E : g ( t , At)) 0) when g is u.s.c. in I . Sllpposc for i k mumcot that x is the unique solution of 4, an m u m p i o n which can rcmwed as vva?r done nt the end of thc proof of 'Iheorcrn 3.6 of 181. In the limiting argument considcrcd atmvc, we a n note that, for any f i x 4 integer k > 0.we an a w r t that

-

for i suff~ientlylarge. (This is due to the uniform convcrgcncc of p ( t , x,(r)) to g(r, AI)). 1 This incites us to m s l d e r rhc problem gkwhich is identical to 6, eept that the closed set 1 is r e p l a d by the clovcd sct Ik Ibecausr: g(., x) is u,s.c). Thcn we repcat what ar done st the beginning of the proof, where, instcad of (4.101, we haye CLI: is s u p p r I d an I,, and then we proxed exactly as in tht proof of Theorem 3.6 of 18L

ScnsYirlty and Coatsollability ICnalysls

5.

This sccrion is &voted to the study d ~cnsitivity~ n d contmllahjlity malylri* with r c v t to thc value fumion V . 'To begin with, we have thc follmin~basic result. Thcoma 5.1 (Semilivity). I A VIO)
0 exists wch that

E-(JFnt

Fit

52,

l antaims a

mkqucncc, which m do not mlahl, such that Ix. vS E ~dE#,(l,l) exists with r, converging uniformly to x , v, mwrging weakly-" to v, and lim,, ,,J ( x , , v , ) N x . v ) . I3y thc continuity of the di~tanccfunction and the dominated convergence t h m r n , we concludc that dC9,,$xIO)) Q and \ , f ~ ( r , x ( r ) l 3 a J ( f ) )dl* 0. which is equivalent ro saying that x(9) E C, t a, and ~ ( r .r( , + a,(t 1 % 0 a.e. in I. So (x, u ) E ~ d ( 4 and~then )

-

-

J ( x , Y S 2 J ( x , , va 1.

-

-

(4.4)

-

-

We nmv rkfim 2 , E fl, hy zkl) x,(r) + ( 1 rX y, x,(O)) ls E T ) in such that r,lll) y, E C, + a,,r,Il) x,(l) E C , + oy.and Ilz, - x,ll, 5 a,, Vi. Lct N > 0 be hcd such that I , 2 Z 2 K f + n~flt,)and N' > 0 w h that J l t , , P , ) + (2Kf -I- IIKIlr7)uH > J ( x v ) , V i 2 N',Thcn, if h' 2 N * ,wc can wdlc

-

a way

-

hen@ a mntrrdfction with (5.4). The cast N s N' Ls freatcd exactly as kfm.This achiwcs fhc proof of thc lcmma and thc pmpmition.

Remmrk 52. hll he development cunddcred before w u l d slill w o k .without major changcs if we had mnridered, imstead of rhc H i h r t spa= I 1 defined at the hginnirtg of Sedion 3, the follflwing one: H

-

{ ( a oair , al,a,): no E

IJ,, a,+ aoCQ)E A , %

HzSand a> IIJJ*

-

whcre Ho (a,E dP:: drp E HI). H,is 3 dosed subspace of P:, find H,,H, arc c l w d r u b c t s of Rn o n d Y :.rcspcliVCIy. The inner p d w ! in H is defincd by

whcre inncr p d u c t s in subspaots H,,H,, H, arc restricted inner p r d u d s d spwcs containing Ihcrn. With t h i ~choitc of H, wc mn cansider cach pcrlurbarion of /3.la), O.lcS,13.ld1, or (3.leS sepratcly by taking A H , IO). H 3 (0);H, H, (01. 11, (0); 111 HJ = (01, A (01; snd A 111 (01. A, = (01, mpcrive@. The qxrnlor 0, delined just befort formula 142) can be redcfincd in each apprqriale cast. Komlity ptayr a substantial role in sensitiviry and m!mllability. The following rtrults show how nomaliry inrcrvcm to givc important conscqucnccs.

-

-

-

-

-- - -

-

bf. S u m that the m~lltis false. Then a sequence la,) in H mnvcrpinp s~onglyto 0. a sequcm (x,, P, E. Z,,, and nomro vcc~orsI ([, ti, 6; E M "( x , 1 Ltt ( p , . fi,. {;, 5;) E A'"(x,)c ~ r r r ~ p a n dtos ( € 6 , 5;. ti, 6 ; ) via 8,. As we have done in the ptmf of 7heortm 4.1. we deline

>

[i,

and

and wc go to a subcquelxle of I i ) for which q E f l : , fi E C* * CS) cxbt such that g, convcrga uniformly to q, ji, conveqeau~akly-*la h a n d 4 l,, €,, €, € I ) E M n ( x ) is given juqt for wme x E ?; with ( f , , t 2 ,1,) = OO(q, +) Ilhc dcfinithn of bcfore (4.2)E.In addition, we haw at thc limit in (5.5) P PIE) + q l,lf which implies that ( q , fir rr (O,O) and then M O I E) c (01, a cun!mdiction with r he normality ot do.

(,,.

-

a

P r o p l t h 53. If

gmir rtwmnl, d m M V Em1 owl -1'IT,) ant h t n d d .

< ch2)

-

If . l l Em) ( were nor bunded, then ( p,, p,, 5, in .rl(Z, 1 should crist such that [p,ll, -+ .).m a d I!4:113J 4 -COY (with d p , ( J ) t ; I l ) d f l . NOWf ~ l j k n e e ~ a a l yth m e proccd~rrcns in tht prmf of thc pmvbus propition, wc can find g E ,6: ii , E C a t ( l ) such that 1 i( I ) + llqllw and ( I , , 4,. t:, 0 ):- Pa ( q , +) E M O I L ) . So M " ( Z , 1 + {O) and , 4 .k not normal (a cnntradiction). G pllmj,

-

V T ( O : Q ) S wp fl

sup

V' (0; a) 5 iaf

rr X

I / , ( o ; u )2

n.hm

R

-

sup

(O,(F,

C ~ V

El,

Xz), a),

f p . p . ~ ~ . l ~ ) ~ J ~ < a ~

inf =t

((.a)

T f r h r '(XI

inf

(fl,(p, P I

x ~x , ~ ) a), ,

t p " ~ . l ~ . < * * ~ ~ ~ z l

(ao.a,,n,, Q,).i?

(l,, €,,

62,

t.11E H-

&U

In prlic~rlor,if, and is *en iy

-

fur cur11 st

M ' ( x ) ir mdfltr~ito a sindculn (€*), then

YCO;.1

o,( pA, Crx, G, 5;) and in addiiion - V is w p ~ l n nt r urn. # 8 , ( p , k t 1 ,g2), tlr~nV I ~ o s l ~ t S r r i r n t i ! ~ M-n, if Z I x ) and 5 ' I ~ F OD,NO) = F. C o ~ ffor ~a sntall , enw@ (in rlfc norm of F1 1, u.e h a ~ tilo v fulIuw~~tq nppmimatim with $'

-

-

Vda) = UO)I

+ C 8 1 1 ~ . ~ . 5Lt z1) , Q ) .

-

'CZ) n d V(N) iy an immrdistc comequcnce of hf. The f m u l a dV(O1 = Theorem 4.1, whilc the formula dxV(0) {O) is a cclnwqutncc of [I I], Propxition 10) and N iif c k d in H, 4.2, and (4.4) a i m Ib prove that. when aV(0) {OJ. in addition tu thc nornlality, V is L j ~ h i ~ mat 0,wc use Propaition 1.1. h r & n g 10 the lattcr, we haw just for~mvcthat. for n E kl, d'V(a1 is uniformiy bounrlcd in a neighborhad of zero. This 1st stnterncnt is lhc conlcnI of thc proposition h h . For rhc proof of thc rest of statcmcnts, wc mfer the reader tn pp. 242-24 of lhl orpp. 108-111 of IlSL a

-

hf, WG p m fhc inclusion d' V( a) C .W 'lH, P for cach a in wmc neighbrh a d of 0, N, and then the lcmrna f o l h from Propodion 5 3 Irt & E drMmE. Thcn, fm mmc a > 0, for all n J near a (more precisely fla' - o l < 8/2,wElh Pa king n o r m a l a n d V(-)is finite within 4 :- S!JRS. we havc

Lr!

-

(x", 6 Z-, so I/(cr) - J ( x ' , u'). For any. function x E .@: w a r xb. G' n t a r v g . r,, E Co near c:, C, war C. (where c; ~'(0)- a,.c; = xA(l) C Y ~ )and, . far any r E P nmr r' i- -#if, x'II))- o,tC~),we dcf~nc IF")

YE

a; := x(0)

- c,,

V ) E ~ 6 ( & , . ) and thcn Y ( r r l ) 5 J ( x , to). Note that irlequality 15.8) is !be same as inquality (4.1) cxmpt that in the prc.wnt onc A 1 a d rhc Term a t a ' - ~ 1 5a% k n t in (5.81 and in addition (5.X) is valid onIy for t h a' war a.

So Cx.

-

Now if we go through thc p m f of h r n a 4.2 with inequaliv I5RS instcad OF (4.1). then rwrything iv the mmc with h I and the tcrm o(flx,, x , ) + x, - Jix', v'))' if not i ~ l u d c din the expression of f: Consequently, we can finc Ip, H 11,. cl) E ~ ' ( r '(the ) definition of J T ~giwn S jvrt a f c r Ihc proof of Lcmma 4-21such that 6 = I (,+, IE1, tJ) O , ( p , p, {,, EJ. So I F M'IT,) and ~ h cproof in finishd. 0

-

el.

-

Rrmsrk .U. Suppose that problem 5,is mrmal. l k n , a m d i n g to Corollary 5.1, I/ is Lipschitz ncsr 0 and one hu dEJ(0) c n { @ l [ ~ " X ) ] n dv(0)). Sinm 8,: .dF: x S ( I 1 x R n x Rn 11 is a linear one-toanc and himntinuws operator. it then follow that Q;kV(O) = $ . P ' ( z ) r"~ 8ibV0)). On thc h i s of this last formula, we Gin ask tke follow in^ question: 1s there a function c: 11, :- f l ;X S(IS'Rn x RR -r R LJ {+ 53) such that d P ( 0 ) Oi h?V(O) and i s lipwlritz near zero? The a n w r is "ycs" and in addition this function is ~rniquc.Wc define V e t l F - I , where 6; dcnotcs the ttdjoint opcrator to 0 , defined by 08;: A -c I f , iq (a, ol P ) ~ ~V,a_6 11, VB G HI,It is clear !hat such that (@:a, Liipschia neat xm.Lct us shaw zhnt ;IIV ( n ) satisfies t hc rcquircd equality. For t h i ~it S U ~ S - to F sttow that the fwa ndcs of this cqwlity haw Ihc .same support function, Ihc: s o p p r t of rhc risht-hand sitlc is givcn by

-

4

-

-

-

-

sup ( o ; - ' u * # ) , C P r w01

(by the definition of thc geacraliztd directional derivative and ~ h linearity c of d:-yl. Howcver, wc h o w that thk lant function. it., CV * 8; - 'lo(@-).is r h t support of the Icft-hand sick of thc equality to prove. So the equality is satisfied and we hsve the fonnula

T b j x tclls us !hat multipliers figuring in nrccssary conditions (1Shcorrm 4.1) a r t in fact rclatcd via formula ( 5 9 ) to the new value function 9, which acts on elements of the Hjlhert space 11, (-#??: x S ( I ) x R n x RRf.

4, 4II

-

miF - &=

- .*.1111 . 1 l 1I R

-

97

2 1 . M . B o r n i n a n d H . M ~ P r o x i m a l ~ a n d h ~ ~ e r f ~ ~ I n ~ a c b mace. I: '&om. Omad. J. Math,38 hM1431-452. 4

apace, 1: A p p h m c h d . J. Math, 2 (1W,428-472 F. H.Clatke, Pwnubed o@mi mtrol problems, IEW T h m Aytcumt. C o n W 31 (19861, S35-542

a of Dgn-c and Moasm& Ophhtioat, Regional Cmfamce Striea SUM FWDpa, io ~pglied 6. F.H.Clarte, ~ d o andnMo-aotb Analpi6 Umuh in A p p M M a t h m a w $JAM, Pbiladtlphta, PA, 1990. 7, P . H . U w b a n d P . D . ~ n . % ~ f u a c t i o n i n ~ c o n t r o l : ~ ~ , ~ itJr and tImt-opWty, SEPLM J. Cbnfml Optim,24 (19%6), 243-263.

5.

. t

F. B Qarke, M

rw.

F.H.drukeadP.RW~~m~control:A~mtudy& m d SiAM J. Control Optind, 25 U967x 1440-45& 9. F. H.CLarkc, R J. Sttm, aad P. R WokmH, S u b p d h t sibwia for mwo!onicity and he Lipdda wdtio& Repria 1992 lad J. Gatwin, The genep d h t of a marginal hmaiomd in mathtmathl p r 0 k . g ~ ~ Matb. O p - Rtu,41 I1979k 45S-463. 11. P. D. Tncwcm. 'Zhe wwmal formula in fWbert space, N&ar Ahal., 11 (1981X m-995. Unkmiad de Monk&, 1987. 12 N. lbk&Adyic hmiamle EI optimimtb, fill. 13. R T. RdMeW, Eneoeaons of mbgmdbb d d t u witti a j p l h t to ~ Noalinear An?J, 9 (1985), 665698. 1 J. Warp. O @ h d Comliol of Differsntial and F u m i o d Epatba& hw& N w

~~

Y a 1972 15. ~ ~ - o p t i m h f a n * ~ ~ a r r m ~ ~ p r d r s ~ ~ ~ ~ ~ ~ . ~ b e r j * U l l i w d 6 de MOlwd, 1991, 16, A Y ~ ~ a n d c p n t d I a b ~ d ~ g d b y i n f e g F a l ~ v i a ~ Chmd 1. M&, W 11993l,1104-112Q. 17. kYmRelasredqW&-mbyh&equa*J.Math.M A@, 175[11t;l993l,126-142.