Annali di Matematica pura ed applicata (IV), Vol. CLXXVIII (2000), pp. 143-174

Dynamic Programming for the Stochastic Burgers Equation (*). GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE

- We solve a control problem for the stochastic Burgers equation using the dynamic programming approach. The cost functional involves exponentially growing functions and the analog of the kinetic energy; the case of a distributed parameter control is considered. The Hamilton-Jacobi equation is solved by a compactness method and a-priori estimates are obtained thanks to the regularizing properties of the transition semigroup associated to the stochastic Burgers equation; a fixed point argument does not seem to apply here.

Abstract.

1. - I n t r o d u c t i o n .

The stochastic Burgers equation can be considered as a simplified and interesting model for the study of turbulence phenomena [4]. It has been suggested that, in order to have a better understanding of the important problem of the control of turbulence, a first step can be to consider this equation (see [5]). Following this strategy, our aim is to study optimal control problems for this equation and to develop mathematical tools which are powerful enough so that they can be used for the Navier-Stokes equations. These present additional difficulties and will be treated in a subsequent article. As in a first paper on this subject, we consider distributed parameter controls. The controlled Burgers equation is

~x

a(X 2)

dX = ( - ~ + ----~ + Bz) dt + V~dW ,

for ~ e ( 0 , 1), t e [ 0 , T],

(1.1)

X(~, 0) = x(~),

for ~ e (0, 1),

X(O,t)=X(1, t)=O,

for t e [ 0 , T].

Here B is a linear operator, W is a cylindrical Wiener process on a stochastic basis (~9, ~, F, (~t)tE [0, T]) - - in other words, d W / d t is the space-time white noise --, Q is a

(*) Entrata in Redazione il 10 dicembre 1998. Indirizzo degli AA.: GIUSEPPEDA PRATO:Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy. E-mail: [email protected]; ARNAUDDEBUSSCHE:CNRS et Universit~ de Paris Sud, 91405 Orsay, France.

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covariance operator and z denotes the control which is taken in the set ~ R ---- {Z ~L2(ff2 • [0, T] • [0, 1]), Iz(., t) IL2(0, 1) < R, P a.s.}. By analogy with turbulence, the aim is to find a control z which minimizes the quantity

- ~ (., t) L2(o,1)dt, representing the kinetic energy. Hence it seems natural

to consider a cost functional defined as follows

J(z) = E

(/t

-z:: (. , t)

c

L2(0, 1)

)

+ [z(t) 12L2(O,1)dt + cf(X(. , T) ) ,

where r is a given functional on L 2(0, 1). It is not difficult to see that this cost functional is always infinite unless we assume that Q is a trace class operator. Indeed, without this condition, the solution X to (1.1) does not have a square integrable derivative. We wish to study this control problem by the dynamic programming approach and we are led to the resolution of a Hamilton-Jacobi-Bellman equation in the infinite dimensional space H = L2(0, 1). The particular case B = V ~ corresponding to a noise acting on the control has been treated in [7]. In this work, a Hopf transform could be used, giving an explicit solution of the Hamilton-Jacobi-Bellman equation, and the existence of a unique optimal control given by the closed loop equation could be proved. In this work, we wish to consider the more natural case B = I. All our results can easily be extended to the case of a bounded operator B. Formally, if u: [ 0, T] • H - o R is the solution of the following Hamilton-JacobiBellman equation

(1.2)

{ Dtu=1Tr

2

( 32x

~(x2)

~

)

(Qu~)+ - ~ + ---~,u~ - F(u~) + I -~

2

L2(0, 1)

u(x, O) = q~(x), where ( . , . ) stands for the L 2(0, 1) inner product and the Hamiltonian F is given by

tx

-2 jpj2 if IpJ

F(p) =

1Re ff IPl > R IPlR- 2

then for any z, the following identity holds T

(1.3)

J(z)=u(T,x)+ 1 E ~ Iz(t,')+ux(T-t,X(t,'))lL2(O, 1) l~o

-y.([u~(T - t, X(t, "))1 - R) dt ,

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with x(a) = 0 for a ~<0 and z(a) = a 2 for a/> 0. It follows that the optimal control is given by z*(~, t) = - D p F ( u x ( T - t, X*(t, .)))(~),

(1.4)

where X* is the optimal state given by the solution of the closed loop equation

(1.5)

dX* = ( ~ X * / ~2

+

O(X* )2 3~

)

D p F ( u x ( T - t, X*(t, .))) dt + V ~ d W ,

X*(0) = x . Hence, our program is to get a smooth solution of (1.2) so that a unique solution to (1.5) can be found. Then the control problem is completly solved since we deduce the existence and uniqueness of the optimal control which is given by (1.4). Equation (1.2) is a Hamilton-Jacobi equation in a Hilbert space. Such equations have already been considered in several works (see [2], [3], [6], [7], [8], [12], [13], [14], [15], [16], [17]). However none of the available results apply here. This is due to the non regular nonlinearity of the Burgers equation and to the singular term [gX/3~[2L2(O,1). Only in [7] a similar case has been treated. But as already mentioned, in this work, a special form of the Hamiltonian was considered so that a Hopf transformation could be used. One way to solve (1.2) is to introduce the transition semigroup (Pt)t >Io associated to the Burgers equation. It is defined by:

Ptq)(x)=E[q)(Y(t))],

cfleBb(H), x e H ,

t>~O,

where Bb (H) is the space of all borelian bounded real functions on H and Y is the solution to the uncontrolled Burgers equation

6~ y dY=(-~+ (1.6)

~y2

a---~)dt+h/QdW,

Y(~, 0) = x(~),

for ~ E ( 0 , 1 ) ,

t E [ 0 , T],

for ~ e (0, 1),

Y(O,t)=Y(1, t)=O,

for t ~ [0, T].

Then we can write the mild form of (1.3): t

(1.7)

t

u(t, .) = Pt cf - [ Pt -8 F(u~ (s,.) ) ds + [ Pt -8 g ds . o

o

Usually, this kind of integral equation is solved thanks to a fLxed point argument in the space of bounded and differentiable functions C~(H). This requires a smoothing property on the semigroup (Pt)t >~o. In general it is required that Pt maps bounded continuous functions into C1 (H) so that the loss of a derivative in the first integral term can be compensated. Such a result is not known for the Burgers equation. However we have shown in [8] that the semigroup (Pt)t >-o associated to (2.9) do satisfy a smoothing property under an additional assumption on Q which amounts to require that it is not

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too degenerate. However, this property is weaker than the one mentionned above in the sense that Pt maps bounded continuous functions into differentiable functions with exponential growth and we are unable to use a fLxed point argument here. Hence, we proceed differently. We consider approximated problems obtained by Galerkin approximations of (1.1). The corresponding Hamilton-Jacobi equation is easy to solve and a bound on the solution in Cb(H) is easily obtained. When we try to use the result of [8] on (1.7), we see that the smoothing propery of the transition semigroup do compensate the loss of one derivative. However, this introduces an exponential factor. The trick here is to use more smoothing on Pt and to compensate the exponential factor thanks to an interpolation inequality. Thus we are able to prove an a-priori estimate in a suitable space of smooth functions with exponential growth. These estimates enable us to construct a C 2 solution of (1.2) by a compactness argument. Then, thanks to the smoothness of this solution, it is not difficult to get the existence and uniqueness of a solution to (1.5) and thus we have solved our control problem. In fact, we will treat a slightly more general problem in which the cost functional contains exponentially growing terms. We want to emphasize that one of the main ingredients in our argument is the smoothing property of the transition semigroup. This property is not valid for the Navier-Stokes equation. However, a weaker form of this property holds and our proof can be adapted with additional technical difficulty. This will be treated in a forthcoming paper. The article is organized as follows. In section 2, we set the notations and state our results. Then, we introduce the approximated problems in section 3 and prove a-priori estimates on the sequence of approximate solution in section 4. Section 5 is devoted to the construction of u solution to (1.2) and section 6 to the existence of an optimal state solution to (1.5).

2. - N o t a t i o n s

and main result.

Let H = L2(O, 1) be endowed with the usual norm and inner product denoted by I" I and (., -) respectively. Let also H k(0, 1 ), k e N be the Sobolev space of all functions in H whose derivatives up to the order k belong to H and H 1(0, 1) the subspace of H i ( 0 , 1) of all functions vanishing at 0 and 1. We define the selfadjoint negative operator A = D ~ , with domain D ( A ) = = H 2( 0, 1 ) A Ho1(0, 1 ). We have D(( - A)1/2 ) = Hol ( 0, 1 ), and we denote its norm and inner product by [IxH =

I(-A)l/2xl,

((x, y)) = ( ( - A ) l / 2 x , ( - A ) l / 2 y ) ,

x, y ~ H l ( O ,

1),

respectively. It is well known that A has an orthonormal basis of eigenvectors (ek)k~N given by ek(~) = V ~ s i n k z ~ ,

kEN,

they correspond to the eigenvalues ,~k = - k 2z2,

kEN.

~E [0, 1];

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Here and below, De (resp. Dr) denotes the derivative with respect to the space variable ~ 9 (0, 1) (resp. the time variable t 9 [0, T]). For any m 9 I'm is the orthogonal projector onto the space spanned by {el, ..., e~}. We also define the nonlinear operator

B(x) =D~(x2),

x9

1),

and consider a linear symmetric nonnegative operator Q which is assumed to be of trace class, and a cylindrical Wiener process W on H associated to a stochastic basis (tg, ~,, {~}t~>0, F). The space L ~ ( ~ x [0, T]; H) consists of all square integrable and adapted processes with values in L2(0, T; H). We also need to define some spaces of functions from H to R with exponential growth. For k 9 a9 1) and E > 0 we set

C:+~(H) = {%0 9

+oo},

Yr>0, r>0

where By = {x e l l : Ixl ~
~9

r>0

For e = 0, we use the notation Cok + ~(H) = C~ + ~(H) and [. Ih + ,, 0 = [" [h + ,. The derivative of a function ~fl on H will be denoted by D~p or by ~ ~. The cost function of our control problem defined below involves two functions, g and ~. We assume that there exists 7 1> 0 such that:

q) 9 C~

(2.1)

for some 7 I> 0,

and that g = g 1 + g 2 with (2.2)

g'

9

C~ (H),

and g~ defined on D ( ( - A ) 1/2) is of class C 2 and there exists a constant % such that for any x, h 9 1/2) Ig2(x) l ~<%(llxll2 + 1),

(2.3)

IDg (x)hl -< cg(Jlxll + 1)JJhJJ, [D 2g2(x)(h, h) I <~cgllhll2.

Moreover, we assume that g and ~0 are bounded below. The control problem we want to study is to find z 9163 Iz(t)] ~
x [0, T], H) such that

1 Iz(s)12) ds+ ~(X(T))], J(z) = E [ J ( g(X(s) ) + -~

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where X is the solution to the controlled Burgers equation

dX = (AX + B(X) + z) dt + V ~ dW , (2.4)

X(0) = x ,

and x e H is given. In all the article, R is a fixed positive number as well as T. As proved in [9], equation (2.4) has a unique solution with trajectories in L ~ (0, T; H) N L2(0, T; H i ( 0 , 1)). If we use the dynamic programming approach this control problem can be solved by considering the Hamilton-Jacobi-Bellman equation:

(2.5)

f Dtu= 1Tr(Qux~)+(Ax+B(x),u~)-F(ux)+g(x), 2

u(x, O) = q~(x) ,

t ~ [ 0 , T], x ~ H

xeH ,

where the Hamiltonian F is given by

f1

IP] e

F(p) =

(2.6)

ff IP] ~
1R2 ]p]R- 2

ff ]p] > R .

Assuming that we can find a solution u to (2.5), the optimal control z * is given by the formula

z*(t) = - D p r ( u ~ ( T - t, X*(t) )) ,

(2.7)

where X* is the optimal state given by the solution of the closed loop equation

dX* = (AX* + B(X* ) - Dp F(u~ (T - t, X * (t) ))) dt + V ~ d W , (2.8)

X*(0) = x.

We introduce the transition semigroup (Pt)t >~o associated to the Burgers equation. It is defined by:

Ptq~(x)=E[cf(Y(t))],

cf~Bb(H),

xeH,

t>~O,

where Bb (H) is the space of all borelian bounded real functions on H and Y is the solution to the uncontrolled Burgers equation

d Y = (AY+ B(Y))dt + V ~ d W , (2.9)

Y(0) = x .

In fact, as follows from Proposition 4.1, Ptcf can be defined for any C e C ~ with < ~o = ~2/2]]QIL(H).

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Then we can write the mild form of (2.5): t

(2.10)

t

u(t, ") = Pt~ - ~Pt_sF(u~(s, "))ds + f P t - s g d s . o

o

DEFINITION 2.1. - We say that u is a mild solution of (2.5) if u 9 C([0, T] • H), for any t > O, u(t, .) 9 for some ~ 9 [0, So], and (2.10) holds for t e [0, T]. As mentionned in the introduction, we are not able to use a fixed point argument to find a mild solution u to (2.10). However, we use (2.10) to get a-priori estimates on approximated solutions and we need a smoothing property of the transition semigroup. This kind of property have been studied in [8] and holds under the additional assumption on Q:

IQ-1/2xl ~cl(-A)~/2xl,

(2.11)

x9

for some c > 0 and f l 9 1). Then, using the a-priori estimates obtained in this way, we obtain the following result. THEOREM 2.2. - A s s u m e that (2.1), (2.2), (2.3), (2.1) hold, that r < e o = z 2 / 2 IIQII~(H), and that g l 9 CI(H). Then, there exists a mild solution u of the Hamilton-Jacobi equation. Then, to solve completely the control problem, it remains to prove that the fondamental identity (1.3) holds and to obtain a solution to the closed loop equation. This will be done in section 6 where we prove our second result. THEOREM 2.3. - Under the same assumptions as in Theorem 2.2, for any control z 9 9 L~(g2 x [0, T]; H) such that Iz(t) [ <<.R, P a.s., the following identity holds: T

S(z) = u(x, T) + 1 E ~ Iz(t) + u : ( T - t, X(t) )12 - ~ ( l u : ( T - t, X(t) )] - R) dt , o

where z(a) = Ofor a <~0 and z(a) = a2 for a >~0 and X satisfies (2.4). Furthermore, the closed loop equation (2.8) has a unique solution X * , so that there exists a unique optimal control which is given by (2.7). REMARK2.4. - It is possible to construct a mild solution without the assumption g i E e C~ (H). However, this assumption is necessary to prove that the derivative of u is locally Lipschitz and this is essential in the proof of the next result. In fact, under the assumptions of Theorem 2.2, we can prove that u(t, .) is a C 2 function for t > 0 and that it is a strict solution of the Hamilton-Jacobi equation in the sense that for x 9 D(A), u(., x) belongs to CI([0, T]) and for t 9 [0, T], x 9 eq. (2.5) is fulfilled.

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REMARK2.5. - Our results easily extends to other cost functional. The main point is that the Hamiltonian F needs to be a Lipschitz function.

3. - G a l e r k i n a p p r o x i m a t i o n s a n d p r e l i m i n a r i e s .

Let us fix a positive integer m and introduce the following mappings 1

m2r 2

fro(x)- 2 m + x 2'

~ore(x) = ~o

xeR,

(o)

and

m+lxl ~

Bm(x)=DJm(x),

gin(x) = g

re+Ix

xePmH,

12 ,

x ~ PmH .

We also need the operator

Q~ = Pm QPm. The approximated control problem is: to find zmeL~(~ • [0, T], PmH) such that Izm(t) l <.R, a.s. minimizing

(3.1)

Jm(zm) = E

1

)

]

gm (Xm(s)) + ~ Izm(s)] 2 ds + cfm(Xm(T)) ,

where Xm satisfies

dXm = (AXm + Bm(Xm) + zm)dt + ~rQ-m~dW,

(3.2)

xm(o) = xm,

and xm e Pm H. Since ~m, g~ are bounded, it is not difficult to prove that there exists a unique solution um of the approximated Hamilton-Jacobi equation 1

{ Dtu m = 2Tr(QmD2Um + (Ax + Bin(x), Dum)) , (3.3)

urn(x, O) = (pro(x),

-F(Dum) + gin(x), xePmH.

t e [0, T], x e P ~ H ,

t>0,

Moreover um e Cb([0, T], PmH) and, for any t > 0, urn(t, ")~ C~(PmH). Also um satisfies the mild form of (3.33): t

(3.4)

t pm

urn(t, " ) = P ~ e f m - fP~_~F(Dum(s, "))ds + f t_~gmds o

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where (P~)tuo is the transition semigroup associated to the approximated Burgers equation: P ~ ~(x) = E[q~(Ym(t))],

qgeBb(PmH), t>~O, x e H ,

and Y~ is the solution of

{ dYm = (AY,~ + Bm(Y,~) ) dt + V ~ d W ,

(3.5)

Ym(O) = x .

The following estimate can be proved as in [8]. In this result, as well as in the remaining of the article, c, C, ci, c('), are constants depending on the data of the problem, A, Q, fl, g, ~o, R, T, and sometimes on its arguments but not on m. The same symbol can be used for different constants. PROPOSITION 3.1. - Assume that Q is a nuclear operator, then for any m e N we

have: i) for all k ~ N,

\E(

sup lYre(t)[2+

ofllY (")ll2e

"
ii) for all ~ <~eo=z2/2IIQIIr(H) and any t e [0, T],

We also need some information on the first and second differential of Y and of its Galerkin approximation with respect to the initial data. For h e l l , we define ~]h and Ch as the solutions of

Dtrl h =A~I h + D~(yrIh), (3.6)

rib(O) = h E H ,

and

(3.7)

Dr ~h = A~h + D~(y~h) + D$ ((yh)2), ~h(O) = O.

It can be checked that ~]~ and ~h are indeed the first and second differential of Ywith respect to x in the direction h. Also we define ~]~ and ~ as the first and the second derivatives of Ym(t) with re-

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spect to x 9 PmH in the direction of h ~ PmH. They are solutions of D ~ h -_ A ~ h + P ~ D ~ ( f ; ( Y m ) ~ % ) ,

(3.8)

~I~(0) = h e P , ~ U ,

and (3.9)

D t ~ h = A ~ mh + p . ~ D ~ ( f ~ (, y m ) ~ % ) + p m D

~( f U ,, Y m ) ( ~ mh ) )2,

~%(o) = o .

Then using the same method as in [8], we have the following estimate. LEMMA 3.2. - A s s u m e that (2.11) holds. For any 5 > O, k e n such that for any h 9 m9 x9 and t 9 [0, T]

El sup

-4-

E

[lI,A(s)ll ds

there exists C(5, k)

+

o"

+tt~-I

E f IQml/2r]~(s)IUds

< C(~,

k) e ~l~'z Ihl ~,

o

and

+ t t~-I E

IQml/2~h(s) 12ds

~
Of course, from this lemma similar estimates on ~k and ~a follow. Among the results of convergence of the Galerkin approximations, we will need the following one. LEMMA 3.3. - Let Ym be the solution of (3.5) with initial datum Pm x, x 9 H. Then (Ym),,~N converges to Y, the solution of (2.9), in L 2 ( ~ • [0, T]; H i ( 0 , 1)), in L2(Y2, C([0, T]; H)), and in C([0, T]; H) almost surely. Moreover sup I~/p~h- ~/pmhIL~(~• [0, T];H~(0,1)) -->0"

IhJ =1

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153

t PROOF. -

t

WA

-~

Let us set y,~=Ym-WA, m and y = Y - W A ,

IeA(t-s)Vr'QdW(s), and e ~ = y ~ - y o

with

WA, m=feA(t-s)V~mdW(s) ,

then

o

Dte.~ = Aem + Bm(Y~) - D~(y2) ,

and, since (Pro - I) e.~ = (I - Pro) Y, 1 , , , , - D t ]e~ ]2 + lLemll2 = D~ 2

mY~

y2

_ D~

mY~

'

'

)

( I - Pm) y . J

We estimate the right hand side above using first an integration by part and HSlder inequality and then the elementary inequality ( m x a / m + x ~) <~ ( 1 / 2 V m ) x s, the Sobolev embeddings H1/a(0, 1 ) r 1) and H1/4(0, 1 ) c L a ( 0 , 1) and an interpolation inequality. We obtain:

)

m+----~--y2m +IY~+YIL~(O,1)I(I-Pm)WAI lie/I] + 9 +liFo + ~ l l ~ IL ' , (0,1) ~<

C -

-

m

IYmI~HI~(o,1) + cIYm + Yl~(0,1) I(I-- Pm) WAI z + 1 +cllYm§ Yll~ lem 12+ ~llemll 2.

Similarly, we have for the second term

~ m + Y~ , ( I - Pm) y <<.cllY.~lll(l- Pm) yl . We apply the Gronwall lemma and get ft ~:f~ ft( ]em(t) 12 + jllemll2ds<~e _lly~+~d~ c iym[51~(o, 1) +

+ c l Ym + YI ~ | (o, 1) ](I - P~) WA 12 + clIY.~ II1(I - Pro) Y l ) ds. It is not difficult to prove that (Ym)m~N is bounded in L| T ; H ) and in L 2 (0, T; H I (0, 1 )) almost surely so that by interpolation inequalities and Agmon's inequality we deduce the almost sure convergence of em to 0, and thus of Y~ to Y, in C(0, T; H). Also taking the supremum in t and the expectation of this inequality, using Proposition (3.1), we deduce the convergence in L2(Q • [0, T];//o1(0, 1)), and in L2(t2, C([0, T]; H)).

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GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

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To prove the second part of the lemma, we set for h e H , [hi = 1,

and write

ld 2 dt

- --I~

I ~ + I1~11 ~ =

(D~(fm(Ym) ~ ) , ( P m -- 1) ~ )

- ( ( f ~ ( Y ~ ) - Ym) tl Pmh, D~s~)

_((ym _ y ) ~ h , D ~ ) - (Y~, D~s~). After similar manipulations as above, we obtain t

I~(t)

[~ +

fll~ll 2ds
J(I

o

[L~(O,X)llYml]+

o

+[l~hlllY~lL~(O,,)l(I-Pm)~P~hI

C

1)[riP-alz + + - - I YmlL~(O, 4 m

Ir/~"h IZds. + l Y e - YIL-(O,,) 2 We now take the expectation, use again Agmon's inequality and deduce the result thanks to Proposition 3.1 and the first part of the lemma. 9 t

With this result, we can establish the convergence of the term [

Pt m - s g mzd s

in 3.4.

o

The other terms require much more work and will be treated in next sections. LEMr4a 3.4. - For any x e H and t e [0, T], t

t

f ptm_s gmz ds(Pm x) ...., f pt_~ gZ ds(x), o

o t

t

DfPtm8 gmz ds(Pm x) -->D fPt_~g2ds(x). 0

o

GIUSEPPE

PROOF.

f

-

Dynamic programming, e t c .

DA PRATO - ARNAUD DEBUSSCHE:

155

L e t x e H and t 9 [0, T], then by (2.3)

t

t

f

0

g de(x)--

0

= fE

g2

))

mYra(t-s) m + l Y m ( t - s ) l2 - g 2 ( y ( t - s ) )

0

<.c

ds<"

(llY.dt-s)ll2+llY(t-s)ll2+ l ) d s

m + I Y m ( t - s ) l2

- Y(t-s)

;

ds

It is easy to see that by L e m m a 3.3 and Proposition 3.1, it goes to zero. Concerning the t

differentials, we first remark that, as easily checked, D ~ Pt-~ g2ds(x) is well defined and belongs to H . Thus it is sufficient to prove that 0

f

t

f

t

D Pt~_~ g2mds(Pmx) - ProD Pt-~g 2ds(x) --o0 0

0

for any x 9 H and t e [ 0, T]. L e t h 9 H , then

(or

P 2 ~ g~ds(Pmx) - P r o D

;

Pt_sg2ds(x), P ~ h

0

)

=

t

= ] E ( D g 2 ( y m ( t - sl).~]P-h(t - s) - Dg2(y(t - s)).~Pmh(t - s ) ) d s . 0 The supremum over h e H of this quantity can be estimated by (2.3), Proposition 3.1, L e m m a 3.2 and Lemma 3.3, and it can be seen that it goes to zero, which finishes the proof. 9

4. - A - p r i o r i

estimates.

We now derive a-priori estimates on u~ solution of (3.3). We first show a smoothing property of (p ~t )t~>o in the spaces C~k + a (P~H). We have the following result which slightly generalizes the result of [8]. PROPOSITION 4.1. - Assume that (2.11) holds and let 7 < So, a 9 [0, 1], 5 > 0. Then P ~ can be extended to C~ and there exists C(7, a, 6) such that, for any q~9 l+a 9 C~ H) and t 9 T], P[~qJeC~+6(PmH) and

I P ~ ] I += r+o ~
156

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

PROOF.

-

Dynamic programming, etc.

By the Bismut-Elworthy formula, see [1], [11], we have

t

q~(Ym(t)) (Qml/2~lh(s), dW(s)) .

Let us choose k such that y(k + 1/k) <~eo, then using the HSlder inequality, Proposition 3.1 and Lemma 3.2

1

(DPf~cf(x), h) <<.--[ Ir

I((tf

y(k+

E

h (Qm-1/2 ~]m(S), dW(s))

)k)]l,k <~

o


~
]P~ cf(x) ] << . C(?,) ]~P]o, re rl~l~. The result follows for a = 0 . Let ~/ 9 CI+~(PmH), we write

(D2P[~f(x) h, h)=D

[1( )j

E ~p(Y.~(t) (Q~1/27?~(s), dW(s))

)1 =

f

) )

1 E ((D~p(Y.~(t)), t/~(t)) (Q.71/2 ~]hm(8) ' dW(s)) + t 0 +

IF.( f t

~P(Ym(t)) (Qml/2~h(8), dW(s))

and deduce by similar arguments as above

](D2p~n~fl(x) h, h)] <~C(r, 5)1~fl]l,7+~t-(1 +/~)/2e(r+2~)lx12 [hi 4 + +C(y, 5)]~p]o, 7+~t-(1+~)/2e(r+2t)]x]2 ]h] 4. It follows IPt~fl]2, r +2~ ~
Dynamic programming, etc.

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

157

Using the semigroup property of Pt~, we have ]ptm cfl ]2, v+2~ ~< C(1 ~, (~) t -(1 +~) [(P[0, r, which is the result for a = 1 (note that 5 > 0 is arbitrary). The case a 9 (0, 1) follows easily by using Lemma 4.2 below. 9 LEMMA 4.2. - Let 0 <~~ 2, 7 2 and 0 <~Sl, s2. Then for any ~ 9 [0, 1] and q9 in C~'~(PmH) N C~ (Pro H), we have ~ 9 C~ (Pro H ) with 7 = ,~71 + ( 1 - ,~) 7 2, s = ,~sl + + (1 - ,~)s2. Moreover IS2, Y2" PROOF.

-

Let r > 0, it is well known that if ~o

9

CSl(Br)N C~2(B,),

~ C(~' 81' 82)ICP]~Sl(Br)IQ9 ICSZ(Br)l-)"

[r

with C()~, s~, s~) independent of m and r (this is easily seen by a dilatation argument). Thus e -772 1~21CS(Br)~ C(,~, Sl, sz)(e-~1r2 Ir

and the result follows.

I (p ICS2(Br))1-2,

9

We will get an a-priori estimate on um from the equation (3.4) and using that u~ is associated to a control problem. We start with the estimate of the last term in (3.4). From now on, a is fixed in (0, 1) and is chosen to satisfy (4.1)

t9

(1 + a)(1 +fi) < 2 .

LEMMA 4.3. - Let ~ < ~ o , T], m 9

5 >0

and assume (2.2), (2.3), (2.11). Then for any

t

p mt_sgmds 9 Cr+ l+a~ (P,~H) o

and t

f sgmd I

1+a,7+6

o

<~C(a, 7, 6)(Igl]o,y + %) 9

I f moreover g 1 9 C~ (H), then t

I Pt~-~gmds 9 C~+ ~(PmH) o

GIUSEPPE DA PRATO ARNAUD DEBUSSCHE: Dynamic programming, etc.

158

-

and t

Li~.~.d. I PROOF.

.~c~ ~,,1,1 ~+c.~

Using Proposition 4.1, we have

-

t

t

IPt_=gmds 0"

<.C(a, ~,r))lg~lo, rf(t-s)-(l+a)(l+Z)/2ds<~C(a, )',5)lgl]o,r, 1+a'7+5

0

by (4.1). Similarly, if g i 9 C1 (H) t

t

[Pt_,g~ds

<<-C(7,5)lglll, rf(t-s)-(l+~)/2ds<~C(~',(~)lgZll, r. 2'7+6

0

We now give the corresponding estimate for g~. We have, for any x ell, t 9 9 [0, T] t

t

IJ":-'~ <'=I -- 0s "=~'~<"-"<=>>)'<'= where Y= is the solution of (3.5). By (2.3) and Proposition 3.1-i), we deduce: t

Pt_=g,~ds <-

).

Moreover, for h e H

) (/

(/

)

(IY=(s)l + 1)(l~(s)l + ClYm(s)lll'7~(s) I)ds ,

D P ~ =g=ds, 2 h <~cgE

and by Lemma 3.2 and Proposition 3.1

(s t

)

D Pt-~gmds, m 2 h ~
Thus we obtain t Pt - s

0

gmds

<~C(5) %. 1'7+6

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

Similarly, we have:

((i) D2

Dynamic programming, etc.

)(t

p ~ ~g~ds 2 h, h = E

IDg~(Ym(s)). ~ ( s ) +

159

)

D2g~(Ym(s))(~(s), y~(s))ds ,

and, proceeding as above, t

is 2 I Pt-sgmds

"o

~
2,?+5

We conclude the proof thanks to Lemma 4.2. We are now able to derive an estimate on urn. PROPOSITION 4.4. - A s s u m e that (2.13), (2.14), (2.15) and (2.29) hold and that ~ < s o. Then for any 5 > 0 and a satisfying (4.1) there exists C(a, y, 5) depending also on the data F, q~, g, T, ... such that for any m ~ N: s u p t (1+")(1+z)/2 t ~ [o, T]

lure(t,

.) I I + a , y + ( t ~ C(a, •, (~).

PROOF. Since u~ is the solution of the Hamilton-Jacobi equation (3.4), we know that it is the value function to the approximate control problem: -

urn(T, x) = inf{Jm(z,~): Izm I eL~(~9 • [0, T]; P ~ H ) , Izm(t) l <.R} <-Jm(O).

Moreover by (2.1), (2.2), (2.3)

J~(O) = E

gm(Ym(s))ds+qgm(Ym(T)

<.

~
<. C(r)(e ~1~12+ Ixl z + 1)

by Proposition 3.1. Thus, since ~ and g are bounded below lure( T, ") Io, ~ <~c(7).

Considering a control problem on [0, t] instead of [0, T] we obtain this estimate with T

160

GIUSEPPE D A PRATO -

ARNAUD DEBUSSCHE: Dynamic

programming, etc.

replaced by any t 9 [0, T] and

(4.2)

sup

te [0, T]

lure(t, .)1o,7 ~< c(r).

We now infer from (3.4) that

t (4.3)

lUre(t, ")Ii+a,y+5 ~ IP[a(f m ll +a,r+o + f IP~-*F(DUm (s' ") )[l +a,y+ad8 +

0 t

+ JP~-sgmdS[l+a,r+~ By Propositions 4.1 and L e m m a 4.3: t

(4.4)

[P~n~m]l+a'$+6~-

~IP~n-sgmds[

l+a,y+6

~< c(y, a, d)[t

- ( 1 + a ) ( 1 +fl)/2 1~010, Y +

[gl [O,r+Cg].

Also by Propositions 4.1, ff y + (5/1 + a) < So, t

f [P~_~F(Dum(s, 0

"))11

+a, y+~ds <~ t

<~c(r, a, 6) f It - s[-(l+a)(l+fl)/2 [F(Dum(S ' "))[o, r+(5/l +a) d8 . 0 From (2.6) and using Lemma 4.2

(4.5)

IF(Dum( s, "))10, ~+ (xl +a)<<-Rlum(s, ")II,~+(xl+a)~< C(~?, (~, a)lUre(8, ")

a/l+a 0, y

lUm(8, .)ll/l+a l+a,y+~

and, by (4.1),(4.2)

t f IP~-*F(Dum(s' "))11+~,r+a ds <<0 t C(~, a, (~)~ It--8 I-(l+a)(l+fl)/2 lUre( 8 , .)]~o.l+a [Um(8, .) ,

11 +/ la+, ~a + 5

t,4~ ~~

0

~
sup lure(s, ")]l/l+al+a,y + ~ t~ [0, T]

1 ~

"~"

sup ]urn( s, ")]1§176y§ + c(r, a, 5).

r e [ 0 , T]

By this estimate, putting (4.4) in (4.3), we obtain the result for 5 < (~0 - y)(1 + a), which implies the result for any 6 > 0. 9

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

PROPOSITION 4.5.

tion g 1 ~ C~1 then

-

Dynamic programming, etc.

Under the same assumptions as in Proposition 4.4, i f in addisup t 1+~ ]Um(t , ")12, r + ~ c ( ~ ,

5).

t~ [0, T]

PROOF.

-

161

We first consider the case 1 + 2 a < 2, then t + o

l + 2 a , y+2~ t

+ f IPF-=F(Dum(s, .))11+2a,7+zsds. o

The first and the second terms are estimated by Proposition 4.1 and L e m m a 4.3. F o r the third term, we use again Proposition 4.1 t

f lP~-=F(Dum( s, ")) I1 +2a, y +25 d s

~

0 t c(~], a , 5 ) f ] t

- s I - ( l +a)(l +fl)/2

IF(Du,~(s, .))i,,y+ods

<~

o

t

~< c(~, a , 5) ] It - s I -(1 + a)(1 +Z)/2 lug(s, " ) I I + a , y + s d 8 ~ c(~], a , 5 ) o

thanks to Proposition 4.4. If I + 2 a < 2, this argument can be iterated and after a finite n u m b e r of steps, the result follows. Clearly, the case 1 + 2 a ~>2, is proved in the same way. 9 We will need one more estimate on u,~; indeed in our compactness argument, an estimate on the differential of u~ in a space D ( ( - A ) =) (which is compactly embedded in H ) is necessary. In fact, due to the presence of gm 2 , we do not have such an estimate on um but only on v~ defined by t Vm ~-" Um --

f P=t - s g m=ds.

o

PROPOSITION 4.6. - Under the same assumptions as in Proposition 4.4, let e > 0 be such that fl + 2 e < 1. Then there exists c( a , y, e) and k( e ) such that

sup t (1+/~+2~)/2 I(-A)=Dv,~(t, ")lo,~+k(=)<- c(y, e). t~ [0, T]

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DA PRATO - ARNAUD DEBUSSCHE: Dynamic programming, etc.

GIUSEPPE

PROOF. We use the following generalization of Proposition 4.1 whose proof is given in appendix. -

LEMMA 4.7. - L e t s > 0 be such that fl + 2 ~ < 1, and let c(y, s) and k(s) such that for any q~ 9 C ~

l( -A)~DP~q~lo, y+k(~) <~c(r, s)]q~lo, rt

y < s o.

Then there exists

-(1 +fl +2e)/2

Then we write t

vm(t, .) = P~q~ + f P~_~[F(Du~(s, .)) + g~] d s , 0

and choosing ~ such that y + (5/1 + a) < E0,

[( -A)~Dvm( t, ") [o, r +k(~)+ (o/1+~) <- [( - A ) ~ D P ~ (P[o, r + ~(~)+ (o/1+ a) "~t

+ f [(-AyDP~_~F(Dum(s,

.))Io, y+k(~)+(xl+,)ds+

0 t +

[

f(-A)

e

m

1

DPt_sg m Io, ~,+k(~)+(xl+,~)ds <~c(E, ~, 5, a).

0 t

9 [~0 0,y+(5/1 +a) b't -(l+fl+2D/2a•

f(t-s)-(l+~§

")) Io,y+(o/l§

Igilo,y§

0

The result follows since I" 1 0 , $ + ( 8 / 1 + a ) ~

['[O,y

and by (4.2),(4.5) and Proposition 4.4

[F(Dum ( s, .) ) [o, ~ + (5/1 + a)

5. - P r o o f o f T h e o r e m

~ C(~/, (~, C~) S - (1 + fl)/2.

2.2.

We construct a solution u of equation (2.5) using a compactness argument and the apriori estimates derived in section 4.

Step 1: Pointwise convergence of a subsequence of (Um)m~N. We consider the translated function t

vm =um - | P ~ - s g ~ d s 0

GIUSEPPE D A PRATO - ARNAUD DEBUSSCHE:

Dynamic programming, etc.

163

and extend vm on [0, T] • H by setting

Vm(t,x)=vm(t, Pmx),

xeH,

t e [ 0 , T].

By Proposition 4.5 and Lemma 4.3 we have sup t 1+~ [vm(t, ")Iz,~+~<<.c(r, 6)

(5.1)

te [0, T]

for 5 > 0 and by Proposition 4.6 sup t (1 +~+2~)/~ I( -A)~Dvm( t, ") 10, ~+k(~)~
(5.2)

t~[0, T]

for s > 0 such that fl + 2 e < 1. Moreover, by (3.3) we have

(5.3)

Dt vm

1 T r (QmDZv,O + (Ax + Bin(x), Dvm) - F(Dum) + g~(x)

for x ~ P m H , t~ [0, T]. For l e N , we set

Kz=Bp~H(O,l)={xePtH, v~ = v,,, IK,• t~,, rl,

Ixl <.l},

1

vL=-[,

V~=PlDv~lx,•

We infer from (5.1), (5.3), that for any l, (5.4)

sup

(x, t) ~Kl x [~t, T]

IDtv~(x, t) l < C(7, 5, 1).

Let x e K t , s, t e [0, vt]

IVy(x, t) - V~(x, s)] = IDv~(x, t) - Dvt~(x, s) I <~ Iv~(., t) - v~(. , S) ]CI(Kt) <. <~C(1) Ivy(., t) - v~(., s)I~C/~(K,)IVy(., t) -- V~(., S)I b/~(K,). So that by (5.1), (5.4).

IVy(x, t) - V~(x, s) I ~
IVY(x, t ) - Vt~(y, t) I <<-]Dvm(x, t ) - D v m ( y , t) I <<.c(r, 5, l ) I x - Yl. This proves that ( V t ~ ) ~ i is equicontinuous on K~ • [vt, T]. By (5.2) and AscoliArzela Theorem, there exists V t continuous from Kt • [rt, T] to P1H and a subsequence (V~(m)),~N such that Y~t(m) ~ y l

in C(Kt • [rt, T]; PtH). By a diagonal extraction, we can construct a subsequence mk

164

GIUSEPPE DA PRATO ARNAUD DEBUSSCHE: Dynamic programming, etc. -

such that ylmk ----)y 1 in C(Kt• [v,, T]; PtH) for a n y / e N . Clearly, for ll ~
['g/l' T]*

Let (x, t ) e (IUNPLH) • (0, T], then there exists/o such that x e K ~ and t i> v~. Since for l >I lo, V~k(x, t) = P~DVmk(X, t), we have from (5.2)

I(--A)~V~k(X, t) I <-C(~, 5, lo). It follows

I ( - A ) ~ V t ( x, t) l <- c(~, 5, 10). Hence, we can define V(x, t) in D ( ( - A ) ~) by

Pt V(x, t) = Vt(x, t) ,

1 >1lo,

obtaining V defined on ,E~ , ( [ J P I H ) • (0, T] which is the pointwise strong limit in H of V l. Moreover, since by (5.1) for x, y e P ~ H with Ixl, lYl ~0,

IV~k(x, t ) - Vtmk(y, t) I <- C(t, r) l x - y l

,

we know, letting ink, l--) ~ , that V is continuous on ( ~NPt H) • (0, T] so that it can be extended uniquely to H • (0, T]. l Writing, for x e H, t > 0, 1 I> max { Ix ], 1/t },

IDvmk(x, t) - V(x, t) I <- IDvmk(Pt x, t) - Dvm~(X, t) I + + I ( I - P~)nvmk(Plx, t) I + IV~k(Pzx, t ) - V~(Plx, t) I + + ] ( I - P l ) V(Ptx, t) I + IV(Ptx, t) - V(x, t) I , we easily deduce from (5.1), (5.2) that for any x e H ,

t>0

Dv~k(x, t)--)V(x, t) in H . Similarly, we can define a function v on H • (0, T], such that for x E H , t > O,

vine(x, t)---)v(x, t). Finally, we have from (5.1), for x, h ~ H , t > 0 with Ixl ~ r and Ihl ~<1:

Ivmk(x + h, t) - vmk(x, t) - DVmk(X, t)'h I <~C(t, r, ~, 5) Ihl 2

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

Dynamic programming, e t c .

165

and letting mk--~ ~ , we deduce

Iv(x + h, t) - v ( x ,

t) - V(x,

t).h I <. C(t, r, r, 6)Lhl 2.

This proves that v is differentiable and Dv(x, t) = V(x, t). By Lemma 3.4 we deduce that t

u = v + |Pt_sg2ds ,I

o

is continuous on [0, T] • H , differentiable with respect to x and

(5.5)

u,~(t, x)----)u(t, x)

and

Dumk ( t , x ) - ~ D u ( t , x)

(5.6)

for any t, x e (0, T] • H . We set u(0, x) = ~0(x) and as easily checked the convergence in (5.5) holds for any t e [0, T] and x ~ H .

Step 2: Passage to the limit. Let us note that u(t,. ) e C1+ ~(H) for t > 0 and from (5.1) and Lemma 4.3 it follows that sup t 1+~ ]Du(t, x ) - D u ( t ,

(5.7)

y)] ~ c(•, 6) e (y+o)r2 I x - y[

t 9 [0, T]

for any x, y e By. We now prove that u is a mild solution of (2.5). For any k we have t

(5.8)

Umk( t, PinkX ) = p[~k cf mk(Pm~x ) + f P~2~ F(Dumk ( s, 9)) ds (Pink x ) + o t

t

f pmk ,1 ,.,2 + j--t_8~,~kds(Pmkx)+ jf p~k t-sym~ds(Pmk x)' o

for x e H , t e [ 0 , T]. t The fourth term converges to J write o

fPt-8g2ds

P~kcfm~(Pmkx) = E(~o (

o

by Lemma 3.4. For the first term we

mkYm~(t) m~+

IYmk(t)l 2

We know that

Ym~(t) ----)Y(t),

a.s. in

H

)) "

166

G I U S E P P E D A PRATO - ARNAUD DEBUSSCHE:

Dynamic programming, etc.

hence, since cf is continuous

qD

---) qJ(Y( t ) ) ,

m~ + IY,#t)12

a.s. in H .

Moreover

mkY~(t) ) me + IY.,(t) so that by Proposition 3.1-ii) it defines a bounded sequence in L~~

It follows

Pt~* ~ ~ ( P ~ x) --) E(~(Y(t))) = Pt q~(x) when ink--* ~ . Similarly we prove that t

t

mk I f Pt_,g,~ds(P,~x)"'~ f Pt-sg'ds(x). o

o

Finally, for s > 0,

IDumk (s, Ym~(t - s) ) - Du(s, Y(t - s) )] <~ ]Du,~k (s, Y,~(t - s) ) - Dumk (s, Y ( t - s) ) I + + IDu,~(s, Y(t - s)) - Du(s, Y(t - s)) I , the first term goes to zero by Proposition 4.5, the boundedness of (Y,~k(t - S))k~N in H , and Lemma 3.3. The second goes also to zero by (5.6). We deduce

F(Dumk (s, Ymk(t- s) ))----) r(Du(s, Y ( t - s) )) for any s > 0, almost surely w e t2. Moreover

IF(Du~k (s, Y ~ ( t - s) ) ) I <<'RIDu~(s, Y ~ ( t - s)) I <<. <~R[um~(s, Ymk(t - s)) 11, ~,+5 e ( r + t )

lYm~(t-s)[2 ~ c ( y , (~) R 8 - ( l +fl)/2 e(r+t)lYmk(t-s)12

by (4.2),(5.1) and Lemma 4.2. Since y < e o , we can choose 5 such that y + 5 < c o and deduce that F(Dum~(s, Y,~k(t-s))) is bounded in LP([0, T]• with 1 > p > m i n { 2 / ( 1 + +fl), e0/(y + 5)}. This implies t

t

f P~2sF(Dumk(s, .)) d s = E_[ F(Du~k (s, Ymk(t- s) )) ds--> o

o t

E [ l (Vu(s, o

t

r(t-

= [ o

)) d s

GIUSEPPE D A PRATO - ARNAUD DEBUSSCHE:

Dynamic programming, e t c .

167

We can now let k--* ~ in (5.8) and obtain t

t

u(t, x) = Ptq~(x) - f Pt_sF(Du(s, "))ds(x)+ ~ Pt_sgds(x) o

o

for x e H , t~ [0, T]. It remains to note that the right hand side is in C([0, T]; H) so that this is also true for u. 9

6. - P r o o f o f T h e o r e m 2.3.

Let x e H , z e L 2 ( ~ • [0, T]; H) such that Iz(t) I ~
Jm(zm) = urn(t, x) + T

+ l E f ,zm(t) +

t , Xm(t)) 12 - Z( IDu,~( T - t, Xm(t)) I - R) dt.

o

We use the following lemma. LEMMA 6.1. - The sequence (Xm)me N converges to X, solution of (2.4) almost surely in L2([0, T]; H i ( 0 , 1)) A C([0, T]; H). Moreover Jm(zm) converges to J(z). PROOF. The almost sure convergence in L2([0, T]; H i ( 0 , 1)) N C([0, T]; H) is proved as for Lemma 3.3. By Ito formula, we have -

T

(6.2)

T

1E]X,~(T)]2+F~l~,~(t)]12dt=F~](z~(t),Xm(t))dt+

1TrQmT.

2

2

o

o

We deduce that (Xm)m~N is bounded in L~(~9 L2(~9; H), it follows easily that (6.3)

•

[0, T]; H~(0, 1)) and (Xm(T))m~N in

Xm-"->X in L 2 ( 9 • [0, T]; H01(0, 1)), Xm(T)-*X(T) in L2(Q; H ) ,

weakly. Since (Zm)mE N converges strongly to z in L2(t9 • [0, T]; H), we deduce that T the right hand side of (6.2) converges to Ef(z(t), X(t)) dt + _1TrQT. 0

2

168

Dynamic programming, etc.

GIUSEPPE D A PRATO - ARNAUD DEBUSSCHE:

And by Ito formula applied to IXI 2, it follows T

1

2

T

E I X m ( T ) I~ + E f I~m(t)ll2dt--> 1 E IX(T)12 + E ~ I~(t)ll 2 dt 2 o

o

proving that that the convergences in (6.3) hold in the strong sense. We now proceed as in [8], Proposition 2.2 and use Ito formula for e ~Ix~12. Since Izm(t) l ~
for e < e o.

Then, using the same arguments as in step 2 in section 5, we easily prove the convergence of Jm(zm). " We have I D u ~ ( T - t, Xm(t)) - D u ( T - t, X ( t ) ) ] <~ <~ ]Dum(T - t, Xm(t)) - D u m ( T - t, X ( t ) ) I + ] D u ~ ( T - t, X ( t ) ) - D u ( T - t, X ( t ) ) ]. F o r t e [0, T), we know from Proposition 4.5 and L e m m a 6.1 that the first term goes to zero almost surely. So does the second by (5.6). We deduce that P a.s. and any t e [0, T) Izm(t) + Du,~(T - t, X ~ ( t ) ) Is - •( IDum(T - t, Xm(t)) I - R ) --~ Iz(t) + D u ( T - t, X ( t ) ) 12 - Z( IDu( T - t, X ( t ) ) I - R ) . Moreover (6.4)

I]zm(t)+Du,~(T-t,

Xm(t))12-z(IDu,~(T-t,X,~(t))]

-R)

I

<- 4 R I D u m ( T - t, X ~ ( t ) ) I + R 2 and by Lemma 3.4, Proposition 4.5 and (4.2) I D u , ~ ( T - t, Xm(t)) I <. c(~, 5 ) ( T - t ) - ( l +t~)/2e (~ +6)lXm(t)12 so that the left hand side of (6.4) is bounded in LP(t9 • [0, T] with p > min {e0/(~ + + 5), 2/(1 +fl)} > 1 and this implies T

E ~ Iz,~(t) + D u m ( T - t, Xm(t)) I~ - Z( IDum( T - t, Xm(t)) ] - R ) dt---~ o T

--> E f Iz(t) + D u ( T - t, X ( t ) ) 12 - ~( ID u ( T - t, X ( t ) ) I - R ) d t . o

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE:

Dynamic programming, etc.

169

By Lemma 6.1 and (5.5), we deduce T

J(z)=u(t,x)+

-1E~ Iz(t)+Du(T-t,X(t))12-y.(IDu(T-t,X(t))l 2

-R)dt

o

Let us now consider the closed loop equation (2.8). We first prove uniqueness of solutions. PROPOSITION 6.2. - There exists at most one solution to equation (2.8) with trajectories in L ~ ( 0 , T; H) AL2(0, T; H~(0, 1)). PROOF.

-

Let X1, X~ be two solutions and X = X1 - X 2 . We have

d Z = A X + B(X1) - B(X2) - DpF(u~(T - t, X1 (t))) + D p F ( u ~ ( T - t, X2(t))) and, after classical computations, and using (5.7), 1 d - --Ixl~+ 2 dt

I~r 2 = (B(X1) - B ( X 2 ) , X) -

-(DpF(u~(T-

t, Xl(t))) - D p F ( u x ( T - t ,

X2(t)), X) <~

I~12 + c(l~l II4/~ + I~x~ll4/~) IXl 2 + Re(v, 6) e(~+~)M~(T - t) -(1 +~> IXl ~ with M = max{ IX1 IL|

IX2 IL~(O,T;H)}, and by Gronwall lemma

IXl 2 ~ e c(l~Xillt/8+ I~X2114/~-t-Rc(~, ) 5)e(r+~)M2fl(T- t) -(~) IX(O) 12

for t e [0, T). Since X(0) = 0, we deduce that X(t) = 0 on [0, T). Moreover, the assumptions of the Proposition classically imply that X is in C([0, T]; H) so that X(t) = 0 on [0, T]. 9 We prove existence by compactness. It is not difficult to show that there exists a unique solution X* to the closed loop equation associated to the approximated control problem: dX* = (AX*~ + Bm(X*m) - D p F ( D u m ( T - t, X*~(t)) )) dt + V ~ d W

,

X*(O) = Pmx. Moreover, since DpF(p) < . R ,

VpeH ,

we can prove that (X*(t))m~N is almost surely bounded in L2([0, T]; H~(O, 1)) A A C([0, T]; H). It is then standard to deduce that a subsequence converges to X* solution to (2.24). Then, by Proposition 6.2, we know that the whole sequence converges and that X* is an adapted process. 9

170

GIUSEPPE DA PRATO ARNAUD DEBUSSCHE: Dynamic programming, etc. -

A. - P r o o f o f L e m m a

4.7.

We split the proof of this result in several lemmas. In this section we use the notation

Ixl8 = l ( - A ) ~ x l ,

seR,

xeD(-A)~).

Also Y~ is the solution of (3.5) and for ~ > 0, h e P m H . We set r/~ h = ~/(m -A)'h, i.e. y~h is the solution of (3.8) with initial datum rl~ ~ h (0) = ( - A y h . LEMMA AA. - For any 1/2 > s > 0 , there exists C(e) such that for x, h e P ~ H , t e [ 0 , T], t

I~aeh (t )1%

+

t

e,h(s) I~/z-,ds < eC<')JIIYm<~>l~a~Ihl e. Ir]m

0

PROOF.

-

Let e > 0 and take the inner product of (3.8) with ( -A)-Z'--",/mh'.

1

-Dt I~ h Iz-~+ I~ h Ilm-, = (D~(f(n(Y~)rlm'~,h), ( - A ) -2~ rl~~ ~)= 2

= - ( f ~ (, Y m ) r l m~,h , D$( _ A ) - 2 , rl~h). Since

Ifm(Ym) IL|

1) ~< clYm IL~(O, 1) ~<

clIYmlI,

by the Sobolev embedding theorem and

ID~(-A)-Z~I~h l <~clrl~ h II/e-e, we have -((f~(Y,~)r/~h), D ~ ( - A ) - 2 ~ r l ~ h) <~CIIY~nlllrl~h l Irl~ h 11/2-2~. Now, by interpolation among the spaces D((-A)8), we have

for any s e [ - s , 1 / 2 - e) and 1 2 1

< c(~)llYmll ~ I ~ h I% + ~ I ~ h 1~/2-~. An application of the GronwaU lemma finishes the proof.

9

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE: Dynamic programming, etc.

A.2. - For any

LEMMA

K>0,

there exists t*(K) >0 such that, for any

171

meN,

E e K J liY'(s)ii2ds) <~ e (l':/2+e~

PROOF.

-

By Ito formula we have

t

12 IYm(t)12+I I]Ym(s)i]2ds= 1lxl

t

+ JI (Ym(s), v ~ d W ( s ) ) +

o

1TrQm2t.

o

Let

K2 t

t

Z m ( t ) = K f (Ym(s), v ~ d W ( s ) ) -

-~- ~. I V ~ = Y , ( ~ ) i ~ d s

o

o

then

de zm = KeZm(Ym(s), V ~ d W ( s ) ) and E(e z~(t)) = E(e z~(~ = 1. We write t

t

eKIIIYm(~)ll2ds~ eK/2Ixl2eK2/a~IV'~ Y~(~)12d~eZ,~(t)e (1/2)TrQmt o

o

t ~< 1 e K/2Ixl~f e (K2/2)tlIQII~H)IY.(8)12dse zm(t)e (lfz) TrQmt ' t

o

by Jensen inequality. Thus ff (K2/2) tlIQII~(H)<-so, i.e. ff t <<.t*(g) pectation and obtain, b y Proposition 3.1-ii)

= z2/K21tQII~(H)),

we can take the ex-

E(eKfilY~(s)ll2d*) ~e((K/2)+(K2/2)lIQIl~n)t)l~12e(1/2)TrQt. In particular, for t = t* (K), we obtain the result. COROLLARY A.3. and t(s, k) > 0 such

9

For any ((1 - fl)/2) > ~ > 0 and k e N , there exists cl(s, that for h, x e P m H and t e t ( s , k) k/2)2/k

Qm

q~ ~J I

ds

<
k)eC2(~)lxl2 ihl2t1-2~-~.

k), c2(~)

172

GIUSEPPE DA PRATO ARNAUDDEBUSSCHE: Dynamic programming, etc. -

PROOF. - By (2.11), and (A.1)

IQmi/2r]~h I ~ IQ-1/2rI~h I ~ Cl7]~ h 1fl/2~ C(~, ,S) I~s~ h Ii_-e2s-fl ]~]~h 2e+fl I / 2 - e" It follows t

t

0

0

<~c(fl, e)t 1-2~-~ sup s~[0.

T]

o t

<~c(fl, e) eC(~)~I]y~(~)ll2d~Iht2t 1-2~-~ 0

by Lemma A.1. Let t(s, k) = t*(kC@)/2), then for t e [0, t(e, k)]

]Qm

r]• (s)12ds

<~C(fl, e)[E(e~C(~)J IIYm(s)H2d~

tl-2~-Z]hl 2<< .

<~C(fl, e) e(C(~)/2+~~

1-2~-~ Ih] 2.

9

We now prove Lemma 4.7. Let e > 0 be such that fl + 2e < 1, k such that ~((k + + 1)/k) <. e o and t(e, k) given in Corollary A.3. By the Bismut-Elworthy formula, we have for x, h e P , ~ H and t e [0, T]

( ( - A ) ~ D P ~ rp(x), h)= (DP~ p(x), ( -A)~ h) = 1E

( /

(Q~l/2rl~h(s), dW(s) )

0

and, arguing as in the proof of Proposition 4.1 1 [ ( / ) k / 2 ] l / k ((-A)~DP~cf(x), h) <<.C(~, k)-~ I(plo, re y1~'2 E IQm-~/2..~,h~o~ qm ~Jl 2ds

Therefore for t 9 [0, t(e, k)]

((-A)~Dp~n cp(x), h) <~C($, k)Iq~lo, ye (c(~)/2+7)1~1~Ihlt -(1+~+ 2~)/2

)

GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE: Dynamic programming, etc.

173

F o r t > t(s, k) we write

(( - A ) ~ D P ~ q~(x), h) = ( ( - A ) DPt(~,k)Pt-~(~, k) q~(x), h) <~

<~c(~,~,,k)[ p ~t_7(~,k)q~lo,re (c(e)/2+r)lxl2~(~, k) -(~ +,6 + 2c)/2 Ihl, and use

IPtcplo,~,
9

REFERENCES [1] J. M. B I S M U T , Large deviations and the MaUiavin Calculus, Birkh~iuser, Basel, Boston, Berlin, 1984. [ 2 ] P. CANNARSA - G. DA PRATO, Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal., 90 (1990), pp. 2747. [3] P. CANNARSA- G. DA PRATO,Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, in: Stochastic partial differential equations and applications (G. Da Prate - L. Tubaro Eds.), pp. 72-85, Pitman Research Notes in Mathematics Series n. 268, 1992. [4] D. H. CHAMBERS - R. J. ADRIAN - P. MOIN - D. S. STEWART - H. J. S U N G , Karhunuen-Loeve expansion of Burgers model of turbulence, Phys. Fluids (31), p. 2573, 1988. [5] H. CHOI - R. TEMAM - P. MOIN - J. KIM, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), pp. 509-543. [6] G. DA PRATO,Some results on Bellman equation in Hilbert spaces, SIAM J. Control and Optimization, 23, 1 (1985), pp. 61-71. [ 7 ] G. DA PRATO - A . D E B U S S C H E , Control of the stochastic Burgers model of turbulence, SIAM J. Control Optimiz, 37, No. 4 (1999), pp. 1123-1149. [8] G. DA PRATO - A. DEBUSSCHE, Differentiability of the transition semigroup of stochastic Burgers equation, Rend. Acc. Naz. Lincei, s. 9, v. 9 (1998), pp. 267-277. [9] G. DA PRATO - A. DEBUSSCHE - R. TEMAM, Stochastic Burgers equation, NoDEA (1994), pp. 389-402. [10] G. DA PRATO - J. ZABCZYK,Stochastic Evolution Equations in Infinite Dimensions, Cambridge University Press, 1992. [11] K. D. E L W O R T I - I Y , Stochastic flows on Riemannian manifolds, in: Diffusion processes and related problems in analysis, Vol. II (M. A. Pinsky and V. Wihstutz, eds.), pp. 33-72, Birkh~iuser, 1992. [12] F. GozzI, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Commun. in partial differential equations, 20 (5-6) (1995), pp. 775-826. [13] F. GozzI, Global Regular Solutions of Second Order Hamilton-Jacobi Equations in Hilbert spaces with locally Lipschitz nonlinearities, J. Math. Anal. Appl., 198 (1996), pp. 399443. [14] F. GozzI - E. RouY, Regular solutions of second order stationary Hamilton-Jacobi equations, J. Differential Equations, 130 (1996), pp. 201-234. [15] F. GozzI - E. RouY - A. SWIECH, Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control, SIAM J. Control and Optimis, to appear.

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GIUSEPPE DA PRATO - ARNAUD DEBUSSCHE: Dynamic programming, etc.

[16] P. L. LIONS, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions, Part I: The case of bounded stochastic evolution, Acta Math., 161 (1988), pp. 243-278; Part II: Optimal control of Zakai's equation, in: Stochastic partial differential equations and applications (G. Da Prato - L. Tubaro eds.) Lecture Notes in Mathematics No. 1390, Springer-Verlag, pp. 147-170, 1990; Part III: Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal. 86 (1991), pp. 1-18. [17] A. SWIECH, Viscosity solutions of fuUy nonlinear partial differential equations with ~unbounded,, terms in infinite dimensions, Ph.D. Thesis, University of California at Santa Barbara, 1993.