VISCOSITY SOLUTIONS OF DYNAMICPROGRAMMING EQUATIONS FOR THE

OPTIMAL CONTROL OF 2-DNAVIER-STOKES EQUATIONS

Fausto GozziDipartimento di Matematica per le Decisioni

Economiche, Finanziarie e Assicurative

Universita di Roma “La Sapienza”Via del Castro Laurenziano 9, 00161 Roma, Italy

S. S. Sritharan

Space & Naval Warfare Systems Center, Code D73HSan Diego, CA 92152-5001, U.S.A.

Andrzej Swiech ∗

School of Mathematics, Georgia Institute of TechnologyAtlanta, GA 30332, U.S.A.

Abstract

In this paper we study the infinite dimensional Hamilton-Jacobi equation as-sociated with the optimal feedback control of viscous hydrodynamics. We resolvethe global unique solvability problem of this equation by showing that the valuefunction is the unique viscosity solution.

Key words: Optimal control of fluids, Navier-Stokes equations, Hamilton-Jacobi equa-tions, viscosity solutions.

2000 Mathematics Subject Classification: 49L25, 35R15, 49L20.

∗A. Swiech was supported by NSF grant DMS 0098565.

1

1 Introduction

Optimal control of Navier-Stokes equations has many applications in traditional aero-dynamic and hydrodynamic drag reduction, lift enhancement, turbulence control andcombustion control. Data assimilation in meteorology and oceanography involves the taskof finding best initial data and unknown boundary and body forces by optimal controlmethodology [40]. During the past decade several fundamental advances have been madein developing an organized mathematical theory to this subject [16, 19, 32]. One of theopen problems identified in [33] was the global unique solvability of the infinite dimen-sional Hamilton-Jacobi equation associated with the feedback synthesis for Navier-Stokesequation. This problem is resolved in this paper. In this respect, this paper is at the crossroads between two theoretical developments in the past several years, namely flow con-trol theory and the theory of viscosity solutions for infinite dimensional Hamilton-Jacobiequations. Local solvability for the Hamilton-Jacobi equation associated to the feedbackcontrol of Navier-Stokes equation has been achieved in the past in the context of H∞-control theory [4] and time periodic control [1]. Global solvability for mild solutions (inthe semigroup formulation) of a semilinear Hamilton-Jacobi equation associated with thestochastic control of 2-d Navier-Stokes equations is given in [12] where the existence ofsmooth solutions is proven in some cases. For the impulse control problem Hamilton-Jacobiequation becomes a quasi-variational inequality and this is again resolved in a mild sensein [26]. Viscosity solution method which provides another notion of a generalized solutionwas first introduced in the context of finite dimensional Hamilton-Jacobi equations by M.G. Crandall and P. L. Lions [9] and was generalized to infinite dimensions by a number ofauthors. Of particular relevance to this paper are [10, Parts IV,V,VII] and [7, 17, 18, 22].For other literature on infinite-dimensional Hamilton-Jacobi equations we refer the readerto [2, 3, 6, 23, 25, 35, 36], [10, PartVI] and the references therein. This paper deals withHamilton-Jacobi-Bellman (HJB) equations associated to optimal control of Navier-Stokesequations in two dimensional domains under homogeneous boundary conditions. Periodicboundary conditions can also be treated by our methods but we do not do it here as thiscase is easier.

The approach presented here allows natural extensions to second order HJB equationsof this type. We think that modifications of the methods developed here can be appliedto prove comparison results for such equations. However connecting these second orderHJB equations to optimal control of stochastic Navier-Stokes equations seems to be anchallenging and difficult problem (see [12]).

Finally we mention that after this paper had been submitted we have learnt about amanuscript [31] that deals with abstract infinite dimensional Hamilton-Jacobi equationsand uses a different notion of viscosity solution. The results of [31] can be applied to someHJB equations coming from the optimal control of Navier-Stokes equations.

In the rest of this section we will describe the mathematical problem. In section 2we establish various continuous dependence theorems for the controlled Navier-Stokesequation that are needed in the paper. In section 3 we describe a suitable notion ofthe viscosity solution used in this paper. In section 4 we prove the uniqueness theoremfor these viscosity solutions for time independent and time dependent Hamilton-Jacobi

2

equations. Section 5 is devoted to the application of these abstract PDE results to anoptimal control problem. In section 5.1 we show that the value function has the requiredcontinuity properties, and in section 5.2 we prove that it is indeed a viscosity solutionin the sense of the definition given in section 3. In section 5.3 we briefly discuss theapplicability of our results.

Let us now describe a typical optimal control problem for the abstract Navier-Stokesequation [32, Ch. I]. We will assume throughout that the the kinematic viscosity is equalto 1. Let Ω ⊂ R2 be an open and bounded set with smooth boundary. Let

H = the closure ofx ∈ D (Ω; R2

), div x = 0

in L2

(Ω; R2

)V = the closure of

x ∈ D (Ω; R2

), div x = 0

in H1

(Ω; R2

),

and let PH be the orthogonal projection in L2 (Ω; R2) onto H. Denote by Ax = −PH∆x,and by B(x,y) = PH [(x · ∇)y]. Let t ≥ 0 be the initial time and T ≥ t be the terminaltime. Our state variable is the velocity field X : [t, T ]×Ω → R2 that satisfies the equation

dX(s)

ds= −AX(s)−B (X(s), X(s)) + f (s, a(s)) in (t, T ]×H,

X(t) = x ∈ H,

(1.1)

where f : [0, T ] × Θ → V′, Θ is a complete metric space (the control set), and a(·) :[0, T ] → Θ is a measurable function which plays the role of a control strategy. We willdenote the set of control strategies by U . The minimization, over all controls a(·) ∈ U , ofa cost functional

J(t,x; a) =

∫ T

t

l(s,X(s), a(s))ds+ g(X(T ))

formally leads to the Hamilton-Jacobi-Bellman (HJB) partial differential equation in(0, T )×H

ut − 〈Ax + B(x,x), Du〉+ infa∈Θ

〈f(t, a), Du〉+ l(t,x, a) = 0 in (0, T )×H

u(T,x) = g(x) in H.

(1.2)

This equation should be satisfied by the value function

V(t,x) = infa(·)∈U

J(t,x; a).

Equation (1.2) falls into a general class of infinite dimensional Hamilton-Jacobi equations

ut − 〈Ax + B(x,x), Du〉+ F (t,x, Du) = 0 in (0, T )×H

u(T,x) = g(x) in H(1.3)

that will be the object of our study. Likewise we will be investigating stationary versionsof (1.3),

λu+ 〈Ax +B(x,x), Du〉+ F (x, Du) = 0 in H, (1.4)

that are related to the infinite horizon optimal control problems.

3

2 Preliminaries on the 2-D controlled Navier-Stokes

equation

2.1 Abstract spaces and the Stokes operator

Throughout the paper Ω will be an open and bounded subset of R2 with smooth boundary.We denote byWm,p (Ω; R2) (or simply byWm,p (Ω) ) the Sobolev space of order 0 ≤ m ∈ Rand power p ≥ 1 of functions with values in R2 (which can be seen as a product spaceWm,p (Ω; R2) = [Wm,p (Ω; R)]2). The norm of x ∈Wm,p (Ω; R2) will be denoted by ‖x‖m,p.

We will use the notation Lp for W 0,p. Moreover, when p = 2, we will write Hm for Wm,2.We will also be using the negative Sobolev spaces H−m. The space H can be alternativelydefined by

H =x ∈ L2

(Ω; R2

), div x = 0 in D′ (Ω; R2

), x · n = 0 in H− 1

2 (∂Ω),

where n is the outward normal to the boundary, see [37, Ch.1]. The operator

A = −PH∆

with the domainD (A) = H2

(Ω; R2

) ∩V

is called the Stokes operator. It is well known known that A is positive definite, self-adjoint, A−1 is compact, and A generates an analytic semigroup. For γ ≥ 0 we denote byVγ the domain of A

γ2 , D(A

γ2 ), equipped with the norm

‖x‖γ = ‖A γ2 x‖0,2. (2.5)

For γ < 0 the space Vγ is defined as the completion of H under the norm (2.5). Ifγ > −1/2 the norm of Vγ is equivalent to the norm of Hγ (see [16, Lemma 4.5]). Moreoverthe space V1 coincides with V. Identifying H with its dual, the space V−γ is the dual ofVγ for γ > 0. We will also be using the customary notation V′ for the dual of V and theduality pairing between V′ and V will be denoted by 〈·, ·〉. The same symbol will also beused to denote the inner product in H if both entries are in H. Finally the operators Aγ

are isometries between V2δ = and V2(δ−γ) for each real γ and δ. We recall below Sobolev

imbeddings and other inequalities that we will need in the sequel.

• Sobolev imbedding type inequalities: If m ≥ 0, mp ≤ 2 and p ≤ q ≤ 2p2−mp

then

Wm,p (Ω) → Lq (Ω), i.e.

‖x‖0,q ≤ C ‖x‖m,p , for x ∈Wm,p (Ω) ,

(reminding that when mp = 2 then the imbedding holds for all q < +∞). Combiningthe above with the equivalence of norms of Vγ and Hγ, we obtain that for γ ∈ (0, 1]

and q ∈[2, 2

1−γ

](q ∈ [2,+∞) if γ = 1) Vγ → Lq (Ω), i.e.

‖x‖0,q ≤ C ‖x‖γ , for x ∈ Vγ. (2.6)

4

• Gagliardo Nirenberg type inequality:

‖x‖0,4 ≤ C ‖x‖1/20 ‖x‖1/2

1 , for x ∈ V. (2.7)

• Interpolation inequality: Recall that if an operator S generates an analytic semigroupthen there exists a constant C such that for every x ∈ D(S) and 0 ≤ γ ≤ 1

||Sγz||0 ≤ C||Sz||γ0 ||z||1−γ0 . (2.8)

2.2 The Euler operator (inertia term) B

We define the trilinear form b(·, ·, ·) : V ×V ×V → R as

b (x,y, z) =

∫Ω

z(ξ) · (x(ξ) · ∇ξ)y(ξ) dξ

and the bilinear operator B(·, ·) : V ×V → V′ as

〈B(x,y), z〉 = b (x,y, z)

for all z ∈ V. This is just another way to introduce the operator B that we have alreadyused in the Navier-Stokes equation (1.1). We will write B (x) for B (x,x) when no confu-sion arises. This operator can be extended in different topologies. By the incompressibilitycondition we have

b (x,y,y) = 0, b (x,y, z) = −b (x, z,y) . (2.9)

Moreover we have the following estimates for b(x,y, z), with x,y, z ∈ V :

1. By Holder inequality, for p, q > 1, 1p

+ 1q

+ 12

= 1

|b (x,y, z)| ≤ C ‖x‖0,p ‖y‖1 ‖z‖0,q . (2.10)

2. By (2.7) we obtain

|b (x,y, z)| ≤ C ‖x‖0,4 ‖y‖1 ‖z‖0,4 ≤ C ‖x‖1/20 ‖x‖1/2

1 ‖y‖1 ‖z‖1/20 ‖z‖1/2

1 (2.11)

which gives when x = z

|b (x,y,x)| ≤ C ‖x‖20,4 ‖y‖1 ≤ C ‖x‖0 ‖x‖1 ‖y‖1 . (2.12)

2.3 The controlled Navier-Stokes equation: existence, unique-ness and continuous dependence

In this section we derive various estimates for solutions of the Navier-Stokes equation(1.1). We begin with the definition of solution (see for instance [37, 38]).

5

Definition 2.1 (i) A weak solution of the equation (1.1) is a map

X ∈ C([t, T ];H) ∩ L2(t, T ;V) with X ′ ∈ L2(t, T ;V−1)

such that (1.1) is satisfied as an equality in V−1 i.e. for every z ∈ V and almostevery s ∈ (t, T ), we have

〈X ′(s), z〉 = 〈−AX(s)−B(X(s), X(s)) + f (s, a(s)) , z〉

(ii) A strong solution of the equation (1.1) is a map

X ∈ C([t, T ];V) ∩ L2(t, T ;D(A)) with X ′ ∈ L2(t, T ;H)

such that (1.1) is satisfied as an equality in H for almost every s ∈ (t, T ).

Given 0 ≤ t ≤ T , an admissible control strategy a(·) ∈ U and an initial datum x ∈ H,a weak (or strong) solution of the state equation (1.1) will be denoted by X(·; t,x, a). Wewill often denote it simply by X(·) when there is no possibility of confusion. We first recallthe standard existence and uniqueness results [37] about the Navier-Stokes equation (1.1).

Theorem 2.2 Let f : [0, T ]×Θ → V−1 be bounded and continuous. Let 0 ≤ t ≤ T , andlet a(·) ∈ U .

(i) Given an initial datum x ∈ H there exists a unique weak solution X(·; t,x, a) of thestate equation (1.1). Moreover we have the estimate

supt≤s≤T

||X(s)||20 +

∫ T

t

||X(s)||21ds ≤ ‖x‖20 +

∫ T

t

‖f(s, , a(s))‖2−1 ds. (2.13)

(ii) Given an initial datum x ∈ V and assuming moreover that f : [0, T ] ×Θ → H isbounded and continuous there exists a unique strong solution X(·; t,x, a) of equation(1.1). Moreover we have the estimate

supt≤s≤T

||X(s)||21 +

∫ T

t

||AX(s)||20ds

≤ ‖x‖21 exp (E(t, T )) +

∫ T

t

‖f(s, , a(s))‖20 exp (E(s, T )) ds (2.14)

where

E(t, τ) := C

∫ τ

t

||X(r)||20||X(r)||21dr ≤ C supt≤r≤τ

||X(r)||20∫ τ

t

||X(s)||21ds.

The next proposition gathers continuous dependence estimates for solutions of equa-tion (1.1). The negative norm estimates given here generalize the results given in [32, Ch.I]).

6

Proposition 2.3 Let f : [0, T ]×Θ → V−1 be bounded and continuous. Then

(i) Let 1/2 ≤ α ≤ 1. There exist constants C > 0, depending only on the indicatedvariables and independent of a(·) ∈ U , such that for every initial conditions x,y ∈ H

||X(s)− Y (s)||−α ≤ C

(||x||0 , ||y||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)||x− y||−α,

(2.15)∫ T

t

||X(s)− Y (s)||21−α ≤ C

(||x||0 , ||y||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)||x− y||2−α,

(2.16)

||X(s)− x||−α ≤ C

(||x||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)(s− t)1/4 , (2.17)

and there is a modulus σ, independent of the strategies a(·) ∈ U , such that

||X(s)− x||0 ≤ σ(s− t). (2.18)

(ii) For every initial condition x ∈ V there exists a constant C > 0 independent of thestrategies a(·) ∈ U such that

||X(s)− x||0 ≤ C

(||x||1 , sup

s∈[t,T ],a∈Θ

‖f (s, a)‖2−1

)(s− t)

12 , (2.19)

and if in addition f : [0, T ]×Θ → H is bounded and continuous then

||X(s)− x||1 ≤ σ(s− t) (2.20)

for some modulus σ independent of the strategies a(·) ∈ U .

Proof. Proof of (i). We first prove (2.15). Let X(s) and Y (s) be respectively thesolutions of (1.1) with initial conditions x and y respectively. Let Z(s) = X(s) − Y (s).Then taking the inner products of (1.1) for X(s) and Y (s) with A−αZ(s) (i.e. using thedefinition of weak solution since A−αZ(s) ∈ V) we get, for s ∈ [t, T ], 〈X ′(s),A−αZ (s)〉 = 〈−AX(s)−B(X(s)) + f (s, a(s)) ,A−αZ (s)〉

X(t) = x,(2.21)

〈Y ′(s),A−αZ (s)〉 = 〈−AY (s)−B(Y (s)) + f (s, a(s)) ,A−αZ (s)〉Y (t) = y.

(2.22)

Subtracting the resulting equalities we obtain 〈Z ′(s),A−αZ (s)〉 = −〈AZ(s),A−αZ (s)〉+ 〈B(Y (s))−B(X(s)),A−αZ (s)〉Z(t) = x− y,

(2.23)

7

so

1

2

d

ds||A−α

2Z(s)||20+||A1−α

2 Z(s)||20 = −b (X(s), X(s),A−αZ (s))+b(Y (s), Y (s),A−αZ (s)

).

(2.24)Now, by linearity, adding and subtracting b (Y (s), X(s),A−αZ (s)),

−b (X(s), X(s),A−αZ (s))

+ b(Y (s), Y (s),A−αZ (s)

)= −b(Z(s), X(s),A−αZ(s))− b(Y (s), Z(s),A−αZ(s)).

By Holder inequality we have (see (2.10))

|b(Z(s), X(s),A−αZ(s))| ≤ C||Z(s)||0, 2α||X(s)||1||A−αZ(s)||0, 2

1−α.

Sobolev imbeddings give us (see (2.6))

||Z(s)||0, 2α≤ C||Z(s)||1−α, and ||A−αZ(s)||0, 2

1−α≤ C ||A−αZ(s)||α = C||Z(s)||−α.

Therefore we obtain that

|b(Z(s), X(s),A−αZ(s))| ≤ C||Z(s)||1−α||X(s)||1||Z(s)||−α ≤ 1

4||Z(s)||21−α+C||X(s)||21||Z(s)||2−α

(2.25)for some constant C > 0. The second term is a little more difficult to estimate. Similarlyas before we have

|b(Y (s), Z(s),A−αZ(s))| = |b(Y (s),A−αZ(s), Z(s))|≤ C||Y (s)||0, 2

1−α||A−αZ(s)||1||Z(s)||0, 2

α

= C||Y (s)||0, 21−α

||A 12−αZ(s)||0||Z(s)||0, 2

α.

Estimates (2.6) and (2.8) give

||Y (s)||0, 21−α

≤ C||Aα2 Y (s)||0 = C||(A 1

2 )αY (s)||0 ≤ C||Y (s)||α1 ||Y (s)||1−α0 ,

||A 12−αZ(s)||0 = ||(A 1

2 )1−α(A−α

2Z(s)) ||0 ≤ C||A 1−α

2 Z(s)||1−α0 ||A−α

2Z(s)||α0 ,||Z(s)||0, 2

α≤ C||A 1−α

2 Z(s)||0.and, by (2.13)

sups∈[t,T ]

||Y (s)||1−α0 ≤

(‖y‖2

0 +

∫ T

t

‖f(s, , a(s))‖2−1 ds

) 1−α2

.

Therefore collecting these estimates and using Young’s inequality we obtain

|b(Y (s), Z(s),A−αZ(s))| ≤ C||Y (s)||α1 ||Y (s)||1−α0 ||A 1−α

2 Z(s)||2−α0 ||A−α

2Z(s)||α0≤ C||Y (s)||α1 ||A

1−α2 Z(s)||2−α

0 ||A−α2Z(s)||α0 (2.26)

≤ 1

4||Z(s)||21−α + C||Y (s)||21||Z(s)||2−α

8

Using (2.25) and (2.26) in (2.24) produces

1

2

d

ds||Z(s)||2−α +

1

2||A 1−α

2 Z(s)||2 ≤ C(||X(s)||21 + ||Y (s)||21)||Z(s)||2−α. (2.27)

Gronwall’s inequality and (2.14) now yield

||X(s)− Y (s)||2−α = ||Z(s)||2−α ≤ eC∫ st (||X(r)||21+||Y (r)||21)dr||x− y||2−α

≤ C(||x||0 , ||y||0 ,

∫ T

t‖f(r, a(r))‖2

−1 dr)||x− y||2−α

(2.28)

which gives us (2.15). Then putting (2.28) into (2.27) we get (2.16). To prove (2.17) weset Y (s) = X (s)− x so that

Y ′ (s) = X ′ (s) = −AX (s)−B (X (s)) + f (s, a (s)) (2.29)

Taking the inner product of (2.29) with A−αY (s) we get

1

2

d

ds||Y (s)||2−α = − ⟨A1−αX(s), Y (s)

⟩(2.30)

−b (X (s) , X (s) ,A−αY (s))

+⟨f (s, a (s)) ,A−αY (s)

⟩.

Since α ≥ 1/2 we have, using also (2.13),∣∣⟨A1−αX(s), Y (s)⟩∣∣ ≤ ∥∥A1−αX (s)

∥∥0‖Y (s)‖0 ≤ C||X(s)||1.

Moreover by (2.12) and (2.13)∣∣b (X (s) , X (s) ,A−αY (s))∣∣ ≤ C ‖X (s)‖0 ||A−αY (s)||1 ‖X (s)‖1

≤ C||X(s)||1||Y (s)||0||X(s)||0 ≤ C||X(s)||1.Finally∣∣⟨f (s, a (s)) ,A−αY (s)

⟩∣∣ ≤ ‖f (s, a (s))‖−1

∥∥A−αY (s)∥∥

1

≤ C ‖f (s, a (s))‖−1 ||Y (s)||0 ≤ C ‖f (s, a (s))‖−1 .

Putting these inequalities in (2.30) produces

1

2

d

ds||Y (s)||2−α ≤ C

(||X(s)||1 + ‖f (s, a (s))‖−1

).

Integrating and using the Cauchy-Schwartz inequality we then get

||X(s)− x||2−α ≤ C

((∫ s

t

||X(r)||21dr) 1

2

+

(∫ T

t

‖f (r, a (r))‖2−1 ds

) 12

)(s− t)

12

and the claim follows. To prove the fourth estimate we argue by contradiction. We wantto show that

sups∈[t,t+ε],a(·)∈U

‖X(s; t,x, a)− x‖0 → 0 as ε→ 0.

9

Assume that this is false. Then there exist sequences sn, an(·) such that

sn → t, ‖X(sn; t,x, an)− x‖0 ≥ ε.

However by the boundedness of ‖X(sn; t,x, an)‖0 and the third estimate of (i) we have

X(sn; t,x, an) → x strongly in V−α and weakly in H.

Moreover weak convergence in H implies that

lim infn→∞

‖X(sn; t,x, an)‖0 ≥ ‖x‖0,

and from the energy inequality (2.13) we can conclude that

lim supn→∞

‖X(sn; t,x, an)‖0 ≤ ‖x‖0.

Combining these two results gives convergence of the norms and this, along with the weakconvergence in H, yields that X(sn; t,x, an) → x strongly in H which is a contradiction.Proof of (ii) Let X (s) = X (s; t,x, a) be the weak solution of (1.1) with initial conditionsx at time t. Set as before Y (s) = X (s)−x so that Y (s) satisfies (2.29). Taking the innerproduct of (2.29) with Y (s) we obtain

1

2

d

ds||Y (s)||20 = −〈AX(s), Y (s)〉 − b (X (s) , X (s) , Y (s))

+ 〈f (s, a (s)) , Y (s)〉 (2.31)

= −〈AY (s), Y (s)〉 − 〈Ax, Y (s)〉−b (X (s) , X (s) , Y (s)) + 〈f (s, a (s)) , Y (s)〉 .

Now by the tri-linearity of b(·, ·, ·) and the orthogonality relations (2.9)

b (X (s) , X (s) , Y (s)) = b (Y (s) , X (s) , Y (s)) + b (x, X (s) , Y (s))

= b (Y (s) , X (s) , Y (s)) + b (x,x, Y (s)) ,

which yields, when plugged into (2.31),

1

2

d

ds||Y (s)||20 + ‖Y (s)‖2

1 = −〈Ax, Y (s)〉 − b (Y (s) , X (s) , Y (s)) (2.32)

+b (x,x, Y (s)) + 〈f (s, a (s)) , Y (s)〉 .Now we have

|〈Ax, Y (s)〉| = ∣∣⟨A1/2x,A1/2Y (s)⟩∣∣ ≤ ‖x‖1 ‖Y (s)‖1 ≤

1

8‖Y (s)‖2

1 + C. (2.33)

Moreover, by (2.12),

|b (Y (s) , X (s) , Y (s))| ≤ C ‖Y (s)‖0 ‖Y (s)‖1 ‖X (s)‖1 ≤1

8‖Y (s)‖2

1+C ‖Y (s)‖20 ‖X (s)‖2

1

(2.34)

10

and

|b (x,x, Y (s))| ≤ C ‖x‖0 ‖x‖1 ‖Y (s)‖1 ≤1

8‖Y (s)‖2

1 + C, (2.35)

and finally

|〈f (s, a (s)) , Y (s)〉| ≤ ‖f (s, a (s))‖−1 ‖Y (s)‖1 ≤1

8‖Y (s)‖2

1 + C ‖f (s, a (s))‖2−1 . (2.36)

Using (2.33)–(2.36) in (2.32) produces

1

2

d

ds||Y (s)||20 +

1

2||Y (s)||21 ≤ C

[1 + ‖Y (s)‖2

0 ‖X (s)‖21 + ‖f (s, a (s))‖2

−1

].

Therefore it follows by Gronwall’s inequality that

‖X (s)− x‖20 ≤ C

[(s− t) +

∫ s

t

‖f (r, a (r))‖2−1 dr

]eC

∫ st ‖X(r)‖21dr

from which our claim (2.19) easily follows. Moreover we note that, setting ω1 (r − t) =‖X (s)− x‖2

0, we also have∫ s

t

||X(r)− x||21dr ≤ C

[(s− t) +

∫ s

t

‖f (r, a (r))‖2−1 dr

]+ C

∫ s

t

ω1 (r − t) ‖X (r)‖21 dr.

The proof of (2.20) is the same as in case (i) (estimate (2.18)) once we know thatlim sups→t ‖X (s) ‖0 ≤ ‖x‖0 which follows from the assumption that f is bounded inH and from (2.14).

3 Viscosity solutions of the HJB equation

The definition of a viscosity solution that we propose here has its predecessors in [10] and[7, 17, 18]. We just have to use appropriate test functions.

Definition 3.1 A function ψ is a test function for equation (1.3) if ψ = ϕ + δ(t) ||x||20,where

• ϕ ∈ C1 ((0, T )×H), A12Dϕ is continuous, and ϕ is weakly sequentially lower-

semicontinuous,

• δ ∈ C1 ((0, T )) is such that δ > 0 on (0, T ).

Definition 3.2 A function ψ is a test function for equation (1.4) if ψ = ϕ + δ ||x||20,where

• ϕ ∈ C1 (H), A12Dϕ is continuous, and ϕ is weakly sequentially lower-semicontinuous,

• δ > 0.

11

In the definition below we assume that F : [0, T ]×V ×V → R.

Definition 3.3 A function u ∈ C ((0, T )×H) is a viscosity subsolution (supersolution)of (1.3) if for every test function ψ, whenever u − ψ has a local maximum (respectivelyu+ ψ has a local minimum) at (t,x) then x ∈ V and

ψt(t,x)− 〈Ax +B(x,x), Dψ(t,x)〉+ F (t,x, Dψ(t,x)) ≥ 0

(respectively

− (ψt(t,x)− 〈Ax +B(x,x), Dψ(t,x)〉) + F (t,x,−Dψ(t,x)) ≤ 0.)

A function is a viscosity solution if it is both a viscosity subsolution and a viscosity su-persolution.

Definition 3.4 A function u ∈ C(H) is a viscosity subsolution (supersolution) of (1.4)if for every test function ψ, whenever u− ψ has a local maximum (respectively u+ ψ hasa local minimum) at x then x ∈ V and

λu(x) + 〈Ax + B(x,x), Dψ(x)〉+ F (x, Dψ(x)) ≤ 0

(respectivelyλu(x)− 〈Ax + B(x,x), Dψ(x)〉+ F (x,−Dψ(x)) ≥ 0.)

A function is a viscosity solution if it is both a viscosity subsolution and a viscosity su-persolution.

The somehow arbitrary choice of the function ‖ · ‖20 as part of the test function was

made because of simplicity and since in this paper we only deal with linearly growingsolutions. Other radial functions can be chosen to handle more general cases.

4 Comparison principles

In this section we prove comparison principles for viscosity solutions that in turn implythe uniqueness of solutions of (1.3) and (1.4) under certain conditions. We will assumethe following hypothesis.

Hypothesis 4.1 F : [0, T ] × Vβ × V → R and there exist a modulus of continuity ω,0 < β < 1

2, and moduli ωR such that for every R > 0 we have

|F (t,x,p)−F (t,y,p)| ≤ ωR (||x− y||β)+ω (||x− y||β||p||0) , if ‖x‖0, ‖y‖0 ≤ R, (4.37)

|F (t,x,p)− F (t,x,q)| ≤ ω ((1 + ||x||0)||p− q||1) , (4.38)

|F (t,x,p)− F (s,x,p)| ≤ ωR (|t− s|) , if ‖x‖0, ‖y‖0, ‖p‖0 ≤ R. (4.39)

12

Theorem 4.2 Let Hypothesis 4.1 hold. Let u, v : (0, T ) × H → IR be respectively aviscosity subsolution, and a viscosity supersolution of (1.3). Let

u(t,x), −v(t,x), |g(x)| ≤ C(1 + ||x||0)and let u(t, ·), v(t, ·), g be locally Lipschitz continuous in || · ||β−1 norm on bounded subsetsof H, uniformly for t in closed subsets of (0, T ). Let moreover

(i) limt↑T (u(t,x)− g(x))+ = 0(ii) limt↑T (v(t,x)− g(x))− = 0

(4.40)

uniformly on bounded subsets of H. Then u ≤ v.

Theorem 4.3 Let Hypothesis 4.1 hold. Let u, v : H → IR be respectively a viscositysubsolution, and a viscosity supersolution of (1.4). Let u,−v be bounded from above andbe Lipschitz continuous in || · ||β−1 norm on bounded subsets of H. Then u ≤ v.

Proof of Theorem 4.3. Set α = 1−β and assume for simplicity that λ = 1. For ε, δ > 0we consider the function

Φ(x,y, ε, δ) := u(x)− v(y)− ||x−y||2−α

2ε− δ(||x||20 + ||y||20).

Because of the continuity assumptions on u, v and the fact that A−1 is compact, thefunction above is weakly sequentially upper-semicontinuous and so it attains a globalmaximum at some points x, y ∈ V (by the definition of viscosity solution). For a fixed δwe will show (as for instance in [22]) that the points x and y are bounded independentlyof ε,

lim supδ→0

lim supε→0

δ(||x||20 + ||y||20) = 0, (4.41)

and

lim supε→0

||x− y||2−α

2ε= 0 for fixed δ. (4.42)

(Please notice that we have reversed the order in which we pass to the limits in the aboveexpressions compared with [22]). To see this we set

m1(ε, δ) = supx,y∈H

Φ(x,y, ε, δ),

m2(δ) = supx,y∈H

u(x)− v(y)− δ

(||x||20 + ||y||20)

and note that we have

m = limδ↓0

m2(δ) and m2(δ) = limε↓0

m1(ε, δ).

Now,

m1(ε, δ) = Φ(x, y, ε, δ) = u(x)− v(y)− ||x−y||2−α

2ε− δ(||x||20 + ||y||20)

13

and for fixed δ,

m1(ε, δ) +||x−y||2−α

4ε=u(x)− v(y)− ||x−y||2−α

4ε− δ(||x||20 + ||y||20) ≤ m1(2ε, δ).

Thus

||x−y||2−α

4ε≤m1(2ε, δ)−m1(ε, δ).

This gives (4.42). Similarly,

m1(ε, δ) +δ

2(||x||20 + ||y||20) = u(x)− v(y)− ||x−y||2−α

2ε− δ

2(||x||20 + ||y||20) ≤ m1(ε,

δ

2)

which givesδ

2(||x||20 + ||y||20) ≤ m1(ε, δ)−m1(ε,

δ

2),

from which we obtain (4.41). Using the definition of viscosity solution and b(x,x,x) = 0we now have

u(x) + 2δ〈Ax, x〉 +1

ε〈Ax,A−α(x− y)〉

+1

εb(x, x,A−α(x− y)) + F (x,

1

εA−α(x− y) + 2δx) ≤ 0

and

v(y)− 2δ〈Ay, y〉 +1

ε〈Ay,A−α(x− y)〉

+1

εb(y, y,A−α(x− y)) + F (y,

1

εA−α(x− y)− 2δy) ≥ 0.

Combining these two inequalities we obtain

u(x)− v(y) + 2δ(||x||21 + ||y||21) +1

ε||x− y||21−α

+1

ε(b(x, x,A−α(x− y))− b(y, y,A−α(x− y)))

+F (x,1

εA−α(x− y) + 2δx)− F (y,

1

εA−α(x− y)− 2δy) ≤ 0. (4.43)

We are now going to estimate the crucial second line in (4.43). We have

b(x, x,A−α(x−y))−b(y, y,A−α(x−y)) = b(x−y, x,A−α(x−y))+b(y, x−y,A−α(x−y)).

Both terms above can be estimated similarly and so we will only show how to estimatethe first one. We have by Holder inequality

1

ε|b(x− y, x,A−α(x− y))| =

1

ε|b(x− y,A−α(x− y), x)|

≤ c

ε||x||0,q||A−α+ 1

2 (x− y)||0,2p||x− y||0,2p, (4.44)

14

where we took p and q such that 1/p+ 1/q = 1, and later p will be sufficiently small. Tocontinue we will use inequality (2.8). We choose τ such that 0 < τ < α− 1

2. To estimate

||A−α+ 12 (x − y)||0,2p we first notice that if p is sufficiently close to 1, Sobolev imbedding

(2.6) guarantee that

||A−α+ 12 (x− y)||0,2p ≤ C||A−α+ 1

2+ τ

2 (x− y)||0.

We now set S = A12 , and z = A−α

2 (x− y) in (2.8). Then

||A−α+ 12+ τ

2 (x− y)||0 = ||S1−α+τz||0≤ C||Sz||1−α+τ

0 ||z||α−τ0 = C||A 1−α

2 (x− y)||1−α+τ0 ||x− y||α−τ

−α .

To estimate ||x− y||0,2p we again use the Sobolev imbedding

||x− y||0,2p ≤ C||x− y||τ(1−α)

that holds if p is small enough, and then set S = A1−α

2 and z = x− y in (2.8). We thenobtain

||x− y||τ(1−α) = ||Sτz||0 ≤ C||Sz||τ0||z||1−τ0 = C||A 1−α

2 (x− y)||τ0||x− y||1−τ0 . (4.45)

Therefore, plugging (4.45) and (4.45) into (4.44), and estimating further

||x||0,q ≤ C||x||1we get

1

ε|b(x− y, x,A−α(x− y))| ≤ C

ε||x||1||A 1−α

2 (x− y)||1−α+2τ0 ||x− y||α−τ

−α ||x− y||1−τ0

which, upon using ||x− y||0 ≤ C||A 1−α2 (x− y)||0, yields

1

ε|b(x− y, x,A−α(x− y))| ≤ C||x||1 ||x− y||1−α√

ε

||x− y||α−τ−α√

ε||x− y||1−α+τ

0 . (4.46)

We now notice that

Φ(x,y, ε, δ) ≥ Φ(x,x, ε, δ),Φ(y,y, ε, δ) . (4.47)

We use the fact that u and v are locally Lipschitz continuous in || · ||−α norm and‖x‖0, ‖y||0 ≤ Rδ independently of ε for a fixed δ to deduce from (4.47) that

||x− y||2−α

ε≤ Kδ||x− y||−α

for some Kδ > 0. This implies that

||x− y||12−α√

ε≤√Kδ.

15

Therefore

||x− y||α−τ−α√

ε=||x− y||

12−α√

ε||x− y||α−

12−τ

−α ≤√Kδ||x− y||α−

12−τ

−α → 0

as ε→ 0 by (4.42) since α− 12

+ τ > 0. Using this in (4.46) we thus obtain

1

ε|b(x− y, x,A−α(x− y))| ≤ ||x||1 ||x− y||1−α√

εσ1(ε, δ)

≤ δ||x||21 +||x− y||21−α

2εσ2(ε, δ) (4.48)

for some local moduli σ1 and σ2. Similarly we obtain

1

ε|b(y, x− y,A−α(x− y))| ≤ δ||y||21 +

||x− y||21−α

2εσ2(ε, δ). (4.49)

We can now proceed with estimating (4.43). Using (4.48), (4.49), and Hypothesis 4.1 in(4.43) (notice that β = 1− α) it follows that

u(x)− v(y) +1

ε||x− y||21−α −

1

ε||x− y||21−ασ2(ε, δ) + δ(||x||21 + ||y||21)

− ωRδ(||x− y||1−α)− ω

(||x− y||1−α

||x− y||−α

ε

)≤ ω(δ(1 + ||x||0)||x||1) + ω(δ(1 + ||y||0)||y||1). (4.50)

For fixed µ, δ > 0 we now choose Cµ and Cµ,δ so that ω(s) ≤ µ + Cµs and ωRδ(s) ≤

µ+ Cµ,δs. Then

ω(δ(1 + ||x||0)||x||1) + ω(δ(1 + ||y||0)||y||1)≤ 2µ+ Cµδ ((1 + ||x||0)||x||1 + (1 + ||y||0)||y||1)≤ 2µ+ δ(||x||21 + ||y||21) +Dµδ

(1 + ||x||20 + ||y||20

). (4.51)

Moreover

ωRδ(||x− y||1−α) + ω

(||x− y||1−α

||x− y||−α

ε

)

≤ 2µ+ Cµ||x− y||1−α||x− y||−α

ε+ Cµ,δ||x− y||1−α

≤ 2µ+||x− y||21−α

2ε+ Cµ,δ||x− y||1−α + 2C2

µ

||x− y||2−α

ε. (4.52)

Putting (4.51) and (4.52) into (4.50), and using (4.41) and (4.42) we therefore obtain

u(x)− v(y) +

(1

2− σ2(ε, δ)

) ||x− y||21−α

ε− Cµ,δ||x− y||1−α ≤ 4µ+ σ3(ε, δ;µ)

16

for some function σ3 such that for every fixed µ

lim supδ→0

lim supε→0

σ3(ε, δ;µ) = 0. (4.53)

We now observe that

limε→0

infr>0

(r2

4ε− Cµ,δr

)= 0. (4.54)

This yieldsu(x)− v(y) ≤ 4µ+ σ4(ε, δ;µ) (4.55)

for some function σ4 satisfying (4.53). Finally (4.55) yields that

u(x)− v(x)− 2δ||x||20 ≤ u(x)− v(y) ≤ 4µ+ σ4(ε, δ;µ).

If we now let ε→ 0, δ → 0 and then µ→ 0 we obtain that u ≤ v.

Proof of Theorem 4.2. Given µ > 0, define

uµ(t,x) = u(t,x)− µ

t, vµ(t,x) = v(t,x) +

µ

t.

Then uµ and vµ satisfy respectively

(uµ)t − 〈Ax +B(x,x), Duµ〉+ F (x, Duµ) ≥ µ

T 2

and(vµ)t − 〈Ax +B(x,x), Dvµ〉+ F (x, Dvµ) ≤ − µ

T 2.

Let Cµ be such that ω(s) ≤ µ2T 2 +Cµs, and let Kµ = 2C2

µ. For ε, δ, γ > 0, and 0 < Tδ < T,we consider the function

uµ(t,x)− vµ(s,y)− ||x−y||2−α

2ε− δeKµ(T−t)||x||20 − δeKµ(T−s)||y||20 −

(t− s)2

2γ

on (0, Tδ] × H. This function has a global maximum at t, s, x, y, where 0 < t, s, andx, y are bounded independently of ε for a fixed δ. Moreover x, y ∈ V. Similarly to thestationary case we have

lim supδ→0

lim supε→0

lim supγ→0

δ(||x||20 + ||y||20) = 0, (4.56)

lim supε→0

lim supγ→0

||x− y||2−α

2ε= 0 for fixed δ, (4.57)

and

lim supγ→0

(t− s)2

2γ= 0 for fixed δ, ε. (4.58)

17

If u v it then follows from (4.57), (4.58) and (4.40) that for small µ and δ, and for Tδ

sufficiently close to T we have t, s < Tδ if γ and ε are sufficiently small. Therefore usingthe definition of viscosity solution we obtain

t− s

γ− δKµe

Kµ(T−t)||x||2 − 2δeKµ(T−t)〈Ax, x〉 − 1

ε〈Ax,A−α(x− y)〉

−1

εb(x, x,A−α(x− y)) + F (t, x,

1

εA−α(x− y) + 2δeKµ(T−t)x) ≥ µ

T 2(4.59)

andt− s

γ+ δKµe

Kµ(T−s)||y||2 + 2δeKµ(T−s)〈Ay, y〉 − 1

ε〈Ay,A−α(x− y)〉

−1

εb(y, y,A−α(x− y)) + F (s, y,

1

εA−α(x− y)− 2δeKµ(T−s)y) ≤ − µ

T 2. (4.60)

Combining these two inequalities it follows that

Kµδ(eKµ(T−t)||x||20 + eKµ(T−s)||y||20

)+ 2δ

(eKµ(T−t)||x||21 + eKµ(T−s)||y||21

)

+1

ε||x− y||21−α +

1

εb(x, x,A−α(x− y))− 1

εb(y, y,A−α(x− y))

+F (s, y,1

εA−α(x− y)− 2δeKµ(T−s)y)− F (t, x,

1

εA−α(x− y) + 2δeKµ(T−t)x) ≤ −2

µ

T 2.

(4.61)We now estimate

|F (t, x,1

εA−α(x− y) + 2δeKµ(T−t)x)− F (t, x,

1

εA−α(x− y))|

+|F (s, y,1

εA−α(x− y)− 2δeKµ(T−s)y)− F (s, y,

1

εA−α(x− y))|

≤ ω((1 + ||x||0)2δeKµ(T−t)||x||1) + ω((1 + ||y||0)2δeKµ(T−s)||y||1)≤ µ

T 2+ Cµ((1 + ||x||0)2δeKµ(T−t)||x||1) + Cµ((1 + ||y||0)2δeKµ(T−s)||y||1)

≤ µ

T 2+ 2δC2

µ

(2 + eKµ(T−t)||x||20 + eKµ(T−s)||y||20

)+ δ

(eKµ(T−t)||x||21 + eKµ(T−s)||y||21

).

Using this and the fact that Kµ = 2C2µ in (4.61) we therefore obtain

δ(||x||21 + ||y||21) +1

ε||x− y||21−α

+1

εb(x, x,A−α(x− y))− 1

εb(y, y,A−α(x− y))

+F (s, y,1

εA−α(x− y))− F (t, x,

1

εA−α(x− y)) ≤ − µ

T 2+ σ1(δ, µ) (4.62)

18

for some local modulus σ1. The rest of the proof follows that of the stationary case. Wefirst let γ → 0 and use (4.39) to eliminate the dependence on t and s above, and thenestimate all the terms as in the proof of Theorem 4.3. After doing this we obtain(

1

2− σ2(ε, δ)

) ||x− y||21−α

ε−Dµ,δ||x− y||1−α ≤ − µ

2T 2+ σ3(γ, ε, δ;µ)

for some constant Dµ,δ, a local modulus σ2, and a function σ3 such that for every fixed µ

lim supδ→0

lim supε→0

lim supγ→0

σ3(γ, ε, δ;µ) = 0.

This inequality, upon employing (4.54), yields a contradiction if we let γ → 0, ε→ 0 andthen δ → 0.

5 Optimal control problem for the Navier-Stokes equa-

tion

In this section we study an optimal control problem for the Navier-Stokes equation. Weonly consider the finite horizon problem. Similar results can be obtained for the infinitehorizon problem by the same methods. We recall that in the finite horizon case for a fixedT > 0 and 0 ≤ t < T we try to minimize the cost functional

J (t,x; a (·)) =

∫ T

t

l (s,X (s; t,x, a) , a(s)) ds+ g (X (T ; t,x, a)) , (5.63)

where X (·; t,x, a) is the solution of (1.1). We assume the following.

Hypothesis 5.1 There exist a constant 0 < β < 12, an absolute constant C > 0, and

families of moduli σR and constants LR for R > 0 such that

(i) l is continuous on [0, T ] ×Vβ ×Θ and for t, s ∈ [0, T ],x,y ∈ Vβ, and a ∈ Θ wehave

|l(t,x, a)− l(s,y, a)| ≤ σR(|t− s|) + LR‖x− y‖β if ‖x‖0, ‖y‖0 ≤ R, (5.64)

|l (s,x, a)| ≤ C(1 + ‖x‖1); (5.65)

(ii) g : H → R is such that

|g(x)− g(y)| ≤ LR‖x− y‖β−1 if ‖x‖0, ‖y‖0 ≤ R, (5.66)

|g(x)| ≤ C (1 + ‖x‖0) ; (5.67)

(iii) f : [0, T ] × Θ → H is bounded, continuous, and f(·, a) is uniformly continuous,uniformly for a ∈ Θ.

19

Let us briefly discuss the applicability of our control problem. Note that if we takeΘ as a separable Banach space then the following example fits Hypothesis 5.1:

f(t, a) = κ(||a||Θ)Na,

where κ(·) ∈ C∞,0(R+) here acts like an engineering constraint on the control actuationand the control operator N ∈L(Θ;H) is a linear continuous operator representing spatiallocalization of the control. Here C∞,0(R+) denotes smooth functions which identicallyvanish outside a finite interval [0, R]. For conducting fluids (even slightly conducting fluidssuch as salt water) such controllers can be realized through the action of Lorentz force[27, Ch. 10], [8, Ch. IV]. Since the Lorentz force is given by j ×B where j is the currentand B is the magnetic field, spatial localization of such distributed control force can berealized by choosing a suitable shape and location for the magnets. This is implementedfor example in a boundary layer stabilization experiment in [21]. A cost functional of theform

J (t,x; a (·)) =

∫ T

t

||Aβ2 (X(s)−Xd(s))||0ds+

∫ T

t

λ(||a(s)||Θ)||a(s)||2Θds+ ||Aβ−12 X(T )||0

with λ(·) ∈ C∞,0(R+) being a discount factor and Xd(t) either a desired field or a knownsmooth field satisfies Hypothesis 5.1. We note that if the control is bounded (controlset is bounded) then the discount factors κ(·) and λ(·) are not needed. Although it is

weaker than the enstrophy norm ||A1/2X(s)||0, the above integrand ||Aβ2 (X(s)−Xd(s))||0

still amounts to minimizing spatial turbulence structures contained in the deviation fieldX(t)−Xd(t) and hence is useful in practice. In data assimilation problems in meteorology[11] and oceanography [40] a major step is to estimate the unknown distributed forcesdue to environmental effects which cannot be completely incorporated into the modeldue to the complexity of the actual problem. In these applications Xd would representthe extrapolated measurements and the unknown time dependent forces are regardedas controls with possible spatial localization and hence fit within the above abstractformulation. The integrand we chose above is in fact stronger than the L2-norm integrandsused in the literature on data assimilation of geophysics.

We will prove that the value function for the problem,

V(t,x) = infa(·)∈U

J (t,x; a (·)) . (5.68)

is the unique viscosity solution of the HJB equationut − 〈Ax +B(x,x), Du〉+ inf

a∈Θ〈f(t, a), Du〉+ l(t,x, a) = 0 in (0, T )×H,

u(T,x) = g(x) in H.

(5.69)The lemma below is easy and we omit the proof.

Lemma 5.2 Under the assumptions of Hypothesis 5.1 the Hamiltonian F given by

F (t,x,p) = infa∈Θ

〈f(t, a),p〉 + l(t,x, a)is continuous on [0, T ]×Vβ ×V and it satisfies (4.37), (4.38), (4.39).

20

5.1 Properties of the value function

The continuity properties of the value function are described by the following result.

Proposition 5.3 Let Hypothesis 5.1 hold. Let α = 1 − β. Then for every R > 0 thereexists a constant CR such that the value function V defined by (5.68) satisfies

|V(t1,x)− V(t2,y)| ≤ CR

(|t1 − t2| 12 + ‖x− y‖−α

)(5.70)

for t1, t2 ∈ [0, T ] and ||x||0, ||y||0 ≤ R. Moreover

|V(t,x)| ≤ C(1 + ||x||0). (5.71)

Proof. We will first prove the local Lipschitz continuity of V(t, ·). Let

M = MR =(R2 + TK2

) 12 ,

where K is such that supt∈[0,T ],a∈Θ ||f(t, a)||−1 ≤ K. It then follows from (2.13), (2.15)and (2.16) that for ||x||0, ||y||0 ≤ R we have

|V(t,x)− V(t,y)| ≤ supa(·)∈U

|J (t,x; a (·))− J (t,y; a (·)) |

≤ supa(·)∈U

(∫ T

t

|l(s,X(s), a(s))− l(s, Y (s), a(s))|ds+ |g(X(T ))− g(Y (T ))|)

≤ LM supa(·)∈U

(∫ T

t

||X(s)− Y (s)||β ds+ ||X(T )− Y (T )||−α

)≤ CR||x− y||−α.

Estimate (5.71) is obvious from (2.13), (5.65), and (5.67). It remains to prove the localHolder continuity of V in t. Let 0 ≤ t1 ≤ t2 ≤ T . Then

|V(t1,x)− V(t2,x)| ≤ supa(·)∈U

|J (t1,x; a (·))− J (t2,y; a (·)) |

≤ supa(·)∈U

∫ t2

t1

|l (s,X (s; t1,x, a(·)) , a(s)) |ds

+ supa(·)∈U

∫ T

t2

|l (s,X (s; t1,x, a(·)) , a(s))− l (s,X (s; t2,x, a(·)) , a(s))| ds

+ supa(·)∈U

|g (X (T ; t1,x, a(·)))− g (X (T ; t2,x, a(·))) |

so that, writing X1 (s) for X (s; t1,x, a(·)) and X2 (s) for X (s; t2,x, a(·)) , and using Hy-pothesis 5.1 and (2.13),

|V(t1,x)− V(t2,x)| ≤ supa(·)∈U

∫ t2

t1

|l (s,X1(s), a(s))| ds

21

+C(||x||0) supa(·)∈U

(∫ T

t2

||X1 (s)−X2 (s)||β ds+ ||X1 (T )−X2 (T )||−α

).

We now observe that for every a(·) ∈ U∫ t2

t1

|l (s,X1(s), a(s)) |ds ≤ C

∫ t2

t1

(1 + ||X1(s)||1) ds

≤ C (t2 − t1)12

(∫ t2

t1

(1 + ||X1(s)||21

)ds

) 12

≤ C(||x||0) (t2 − t1)12 ,

and sinceX1 (s) = X (s; t1,x, a(·)) = X (s; t2, X(t2; t1,x, a(·)), a(·)) ,

we have by (2.15), (2.16) and (2.17)

supa(·)∈U

(∫ T

t2

||X1 (s)−X2 (s)||2β ds+ ||X1 (T )−X2 (T )||2−α

)≤ C(||x||0)|(t2 − t1)

12 .

Combining the last two inequalities we therefore obtain

|V(t1,x)− V(t2,x)| ≤ C(||x||0)(t2 − t1)12 .

Remark 5.4 It is clear from the proof that to obtain Proposition 5.3 we do not have toassume that f has values in H. It is enough to assume that the values of f are boundedin V−1.

5.2 Existence of solutions of HJB equations

Let us begin with Bellman principle of optimality for Navier-Stokes equation [33, 32].

Proposition 5.5 For every 0 ≤ t ≤ τ ≤ T and x ∈ H we have

V(t,x) = infa(·)∈U

∫ τ

t

l (s,X (s; t,x, a(·)) , a(s)) ds+ V (τ,X (τ ; t,x, a(·))).

Theorem 5.6 Assume that Hypothesis 5.1 is true. Then the value function V is theunique viscosity solution of the HJB equation (5.69) that satisfies (5.70) and (5.71).

Proof. The uniqueness part follows from Theorem 4.2 and Lemma 5.2. Here wewill only prove that V is a viscosity solution and in fact only that the value functionis a viscosity supersolution. The subsolution part is very similar and in fact easier. Weset B (x) = B (x,x) throughout this proof. Let ψ (t,x) = ϕ (t,x) + δ(t) ||x||20 be a testfunction. Let V + (ϕ+ δ ||·||20) have a local minimum at (t0,x0) ∈ (0, T )×H.

22

Step 1. We prove that x0 ∈ V. For every (t,x) ∈ (0, T )×H

V (t,x)− V (t0,x0) ≥ −ϕ (t,x) + ϕ (t0,x0)− δ(t) ||x||20 + δ(t0) ||x0||20 . (5.72)

By the dynamic programming principle for every ε > 0 there exists aε (·) ∈ U such that,writing Xε (s) for X (s; t0,x0, aε(·)) , we have

V (t0,x0) + ε2 >

∫ t0+ε

t0

l (s,Xε (s) , aε(s)) ds+ V (t0 + ε,Xε (t0 + ε)) .

Then, by (5.72),

ε2 −∫ t0+ε

t0

l (s,Xε (s) , aε(s)) ds ≥ V (t0 + ε,Xε (t0 + ε))− V (t0,x0)

≥ −ϕ (t0 + ε,Xε (t0 + ε)) + ϕ (t0,x0)− δ(t0 + ε) ||Xε (t0 + ε)||20 + δ(t0) ||x0||20 .Using the chain rule and the orthogonality relation (2.9) in the above inequality, and thendividing both sides by ε we obtain

ε− 1

ε

∫ t0+ε

t0

l (s,Xε (s) , aε (s)) ds

≥ −1

ε

∫ t0+ε

t0

ϕt (s,Xε (s)) ds+1

ε

∫ t0+ε

t0

〈AXε (s) , Dϕ (s,Xε (s))〉 ds

+1

ε

∫ t0+ε

t0

〈B (Xε (s)) , Dϕ (s,Xε (s))〉 ds

−1

ε

∫ t0+ε

t0

〈f (s, aε (s)) , Dϕ (s,Xε (s))〉 ds+1

ε

∫ t0+ε

t0

δ(s) 〈AXε (s) , Xε (s)〉 ds

−1

ε

∫ t0+ε

t0

δ(s) 〈f (s, aε (s)) , Xε (s)〉 ds− 1

ε

∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds. (5.73)

Now, setting λ = inft∈[t0,t0+ε0] δ (t) for some fixed ε0 > 0, we have for ε < ε0

1

ε

∫ t0+ε

t0

δ(s) 〈AXε (s) , Xε (s)〉 ds ≥ λ

ε

∫ t0+ε

t0

||Xε (s)||21 ds.

Therefore it follows that

λ

ε

∫ t0+ε

t0

||Xε (s)||21 ds ≤ ε− 1

ε

∫ t0+ε

t0

l (s,Xε (s) , aε (s)) ds (5.74)

+1

ε

∫ t0+ε

t0

ϕt (s,Xε (s)) ds− 1

ε

∫ t0+ε

t0

〈AXε (s) , Dϕ (s,Xε (s))〉 ds

−1

ε

∫ t0+ε

t0

〈B (Xε (s)) , Dϕ (s,Xε (s))〉 ds

+1

ε

∫ t0+ε

t0

〈f (s, aε (s)) , Dϕ (s,Xε (s))〉 ds

+1

ε

∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds+1

ε

∫ t0+ε

t0

δ(s) 〈f (s, aε (s)) , Xε (s)〉 ds.

23

The assumptions on l yield∣∣∣∣1ε∫ t0+ε

t0

l (s,Xε (s) , aε (s)) ds

∣∣∣∣ ≤ C

[1 +

1

ε

∫ t0+ε

t0

‖Xε (s)‖1 ds

]

≤ C +λ

8· 1

ε

∫ t0+ε

t0

‖Xε (s)‖21 . (5.75)

Moreover the assumptions placed on ϕ and (2.18) yield that∥∥A1/2Dϕ (s,Xε (s))∥∥

0≤ C(||x0‖0) (5.76)

for ε < ε0 for some fixed ε0 > 0. Therefore

|〈AXε (s) , Dϕ (s,Xε (s))〉| = | ⟨A1/2Xε (s) ,A1/2Dϕ (s,Xε (s))⟩ |

≤ ∣∣∣∣A1/2Xε (s)∣∣∣∣

0

∥∥A1/2Dϕ (s,Xε (s))∥∥

0

= C(||x0‖0) ‖Xε (s)‖1 ≤ C +λ

8‖Xε (s)‖2

1

Regarding the third term we have, using (2.12), (2.13) and (5.76),

|〈B (Xε (s)) , Dϕ (s,Xε (s))〉| = |b (Xε (s) , Xε (s) , Dϕ (s,Xε (s)))|≤ C ||Xε (s)||0 ||Xε (s)||1 ||Dϕ (s,Xε (s))||1 ≤ C(||x0‖0) ‖Xε (s)‖1 ≤ C +

λ

8‖Xε (s)‖2

1 .

Finally

| 〈f (s, aε (s)) , Xε (s)〉 |, | 〈f (s, aε (s)) , Dϕ (s,Xε (s))〉 | ≤ C(‖x0‖0). (5.77)

Therefore, using the inequalities (5.75)–(5.77) above and the fact that ϕt(s,Xε(s)) andδ′(s) are bounded independently of ε on [t0, t0 +ε0] for some ε0 > 0, we obtain from (5.74)

λ

ε

∫ t0+ε

t0

||Xε (s)||21 ds ≤ C (5.78)

for some constant C independent of ε. We now take ε = 1/n and setXn (s) = X(s; t0,x0, a1/n(·)).

Inequality (5.78) gives then

n

∫ t0+1/n

t0

||Xn (s)||21 ds ≤ C

so that, along a sequence tn ∈ (t0, t0 + 1/n) ,

||Xn (tn)||21 ≤ C,

and thus along a subsequence, still denoted by tn, we have

Xn (tn) x

24

weakly in V for some x ∈ V. This also clearly implies weak convergence in H. However,by (2.18), Xn (tn) → x0 strongly (and weakly) in H. Therefore, by the uniqueness of theweak limit in H it follows that x0 = x ∈ V.

Step 2. The supersolution inequality. We go back to inequality (5.74). By (2.13) (or(2.14)) we have ‖Xε(s)‖0 ≤ C(‖x0‖0) = R for every ε and s ∈ [t0, T ]. Therefore using(5.64) and (2.20) we obtain∣∣∣∣1ε

∫ t0+ε

t0

[l (s,Xε (s) , aε (s)) ds− l (t0,x0, aε (s))] ds

∣∣∣∣≤ 1

ε

∫ t0+ε

t0

(σR(|s− t0|) + LR ||Xε (s)− x0||β

)ds ≤ ω1(ε)

for some modulus ω1. We will be using ω1 to denote various moduli throughout the restof the proof. Similarly, by the continuity of ϕt, we obtain∣∣∣∣1ε∫ t0+ε

t0

ϕt (s,Xε (s)) ds− ϕt (t0,x0)

∣∣∣∣ ≤ 1

ε

∫ t0+ε

t0

ωt0,x0 (|s− t0|+ ||Xε (s)− x0||0) ds ≤ ω1(ε).

Moreover ∣∣∣∣1ε∫ t0+ε

t0

[〈AXε (s) , Dϕ (s,Xε (s))〉 − 〈Ax0, Dϕ (t0,x0)〉] ds∣∣∣∣

≤ 1

ε

[∫ t0+ε

t0

|〈A (Xε (s)− x0) , Dϕ (s,Xε (s))〉| ds

+

∫ t0+ε

t0

|〈Ax0, Dϕ (s,Xε (s))−Dϕ (t0,x0)〉| ds].

It now follows from (5.76) and (2.20) that

|〈A (Xε (s)− x0) , Dϕ (s,Xε (s))〉| =∣∣⟨A1/2 (Xε (s)− x0) ,A

1/2Dϕ (s,Xε (s))⟩

0

∣∣≤ ||Xε (s)− x0||1

∥∥A1/2Dϕ (s,Xε (s))∥∥

0

≤ C(‖x0‖0) ||Xε (s)− x0||1 ≤ ω1(ε).

By the continuity of A1/2Dϕ we also get

|〈Ax0, Dϕ (s,Xε (s))−Dϕ (t0,x0)〉|=

∣∣⟨A1/2x0,A1/2 [Dϕ (s,Xε (s))−Dϕ (t0,x0)]

⟩∣∣≤ ||x0||1 ωϕ,t0,x0 (|s− t0|+ ||Xε (s)− x0||0) ≤ ω1(ε).

Next we have∣∣∣∣1ε∫ t0+ε

t0

[〈B (Xε (s)) , Dϕ (s,Xε (s))〉 − 〈B (x0) , Dϕ (t0,x0)〉] ds∣∣∣∣

=1

ε

∫ t0+ε

t0

|b (Xε (s) , Xε (s) , Dϕ (s,Xε (s)))− b (x0,x0, Dϕ (t0,x0))| ds

25

≤ 1

ε

[∫ t0+ε

t0

|b (Xε (s)− x0, Xε (s) , Dϕ (s,Xε (s)))|+ |b (x0, Xε (s)− x0, Dϕ (s,Xε (s)))| ds

+

∫ t0+ε

t0

|b (x0,x0, Dϕ (s,Xε (s))−Dϕ (t0,x0))| ds].

Now observe that by (2.11)

|b (Xε (s)− x0, Xε (s) , Dϕ (s,Xε (s)))| ≤ C‖Xε (s)− x0‖1‖Xε (s) ‖1‖Dϕ (s,Xε (s)) ‖1,

|b (x0, Xε (s)− x0, Dϕ (s,Xε (s)))| ≤ C‖Xε (s)− x0‖1‖x0‖1‖Dϕ (s,Xε (s)) ‖1,

and, by (2.12)

|b (x0,x0, Dϕ (s,Xε (s))−Dϕ (t0,x0))| ≤ C‖x0‖0‖x0‖1‖Dϕ (s,Xε (s))−Dϕ (t0,x0) ‖1

≤ C‖x0‖0‖x0‖1ωϕ,t0,x0 (|s− t0|+ ||Xε (s)− x0||0) .It then follows from (5.76) and (2.20) that∣∣∣∣1ε

∫ t0+ε

t0

[〈B (Xε (s)) , Dϕ (s,Xε (s))〉 − 〈B (x0) , Dϕ (t0,x0)〉] ds∣∣∣∣ ≤ ω1 (ε) .

Also ∣∣∣∣1ε∫ t0+ε

t0

[〈f (s, aε (s)) , Dϕ (s,Xε (s))〉 − 〈f (t0, aε (s)) , Dϕ (t0,x0)〉] ds∣∣∣∣

≤ 1

ε

∫ t0+ε

t0

|〈f (s, aε (s)) , Dϕ (s,Xε (s))−Dϕ (t0,x0)〉| ds

+1

ε

∫ t0+ε

t0

|〈f (s, aε (s))− f (t0, aε (s)) , Dϕ (t0,x0)〉| ds

≤ 1

ε

∫ t0+ε

t0

‖f (s, aε (s))‖0 ‖Dϕ (s,Xε (s))−Dϕ (t0,x0)‖0 ds

+1

ε

∫ t0+ε

t0

ωf (|s− t0|) ‖Dϕ (t0,x0)‖0 ds

≤ ω1 (ε) .

Finally the estimates of the terms containing δ. We have∣∣∣∣2ε∫ t0+ε

t0

[δ(s) 〈AXε(s), Xε(s)〉 − δ(t0) 〈Ax0,x0〉] ds∣∣∣∣

≤ ω1(ε) +2

ε

∫ t0+ε

t0

δ(s)[∣∣∣⟨A 1

2Xε(s),A12 (Xε(s)− x0)

⟩∣∣∣+ ∣∣∣⟨A 12 (Xε(s)− x0),A

12 x0

⟩∣∣∣] ds≤ ω1(ε) +

C

ε

∫ t0+ε

t0

[‖Xε(s)‖1‖Xε(s)− x0‖1 + ‖x0‖1‖Xε(s)− x0‖1] ds ≤ ω1(ε)

26

estimating as before. Similarly we obtain∣∣∣∣1ε∫ t0+ε

t0

δ(s) 〈f (s, aε (s)) , Xε (s)〉 ds− δ(t0) 〈f (t0, aε (t0)) ,x0〉∣∣∣∣ ≤ ω1 (ε)

and ∣∣∣∣1ε∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds− δ′ (t0) ||x0||20∣∣∣∣ ≤ ω1 (ε) .

Using all these estimates in (5.74) yields

ω1 (ε)− 1

ε

∫ t0+ε

t0

l (t0,x0, aε (s)) ds

≥ −ϕt(t0,x0) + 〈Ax0, Dϕ (t0,x0)〉

+ 〈B (x0) , Dϕ (t0,x0)〉 − 1

ε

∫ t0+ε

t0

〈f (t0, aε (s)) , Dϕ (t0,x0)〉 ds

−δ(t0) 〈f (t0, aε (t0)) ,x0〉+ 2δ(t0) 〈Ax0,x0〉 − δ′ (t0) ||x0||20so that

−ϕt(t0,x0)− δ′ (t0) ||x0||20 + 〈Ax0 +B (x0) , Dϕ (t,x0)〉 − δ(t0) 〈f (t0, aε (t0)) ,x0〉

+1

ε

∫ t0+ε

t0

[〈f (t0, aε (s)) ,−Dϕ (t0,x0)〉+ l (t0,x0, aε (s))] ds ≤ ω1 (ε) .

The claim now follows upon taking first the infimum over a ∈ Θ inside the integral andthen letting ε→ 0.

5.3 Applications and possible further developments

The characterization of the value function given by Theorem 5.6 can be considered asa starting point for the dynamic programming analysis of the optimal control problemconsidered in this paper. It opens up several possibile directions for future research. Weoutline some of these issues below:

1. One such direction is obtaining verification theorems in the context of viscositysolutions. Verification theorems use the HJB equation to give sufficient (and/or neces-sary) conditions for optimality. They are natural and easy to prove if the value functionis a smooth solution of the HJB equation (see e.g. [15]). In the nonsmooth case suchoptimality conditions have been proved in the finite dimensional case (see e.g. [5], [39,Section 5.3]). For the dynamic programming of Navier-Stokes equation, such a verifica-tion theorem is proved in [33], [13], [32, Ch. I, Theorem 1.3.12] with a weaker definitionof solution. Verification theorems and optimal synthesis results are also proved for anabstract infinite dimensional problem in [39, Section 6.5] however these results are ratherweak. Moreover the abstract problem considered there cannot be applied to problems

27

with unbounded nonlinearities. Completion of the verification theory for the control ofNavier-Stokes equations would require some new ideas, in particular the introduction ofappropriate generalized sub- and supergradients that would be consistent with our defini-tion of viscosity solution. Optimality conditions provided by the theory can then be usedto construct optimal feedback controls, at least theoretically.

2. Another important direction is the construction of ε-optimal controls and trajecto-ries, and numerical approximations. There exist several numerical methods appropriate forsolving optimal control problems for infinite dimensional systems (in particular for prob-lems governed by nonlinear PDE). One of such promising and recently popular methodsis the POD (proper orthogonal decomposition) method (see for instance [24, 29, 30]). It isa reduced order method that is based on projecting the control system onto finite dimen-sional subspaces that are spanned by basis elements that contain some characteristics ofthe expected solution. It also produces finite dimensional approximations of optimal con-trols. We are not aware of any applications of these methods to HJB equations in Hilbertspaces. This field is wide open. However some results in this direction seem possible.One is a construction of an abstract approximation scheme that is connected to discreteDynamic Programming. The existence and uniqueness results proved in this paper lay agroundwork for such an attempt. We refer the reader to [5] for an overview of such resultsin finite dimensions. A different approach to numerical approximation and constructionof ε-optimal controls may use the specific nature of our problem, more precisely the weakcontinuity of the viscosity solution. In light of this one can try to approximate (uniformlyon bounded sets) the unique viscosity solution (and therefore the value function) of theinfinite dimensional HJB equation by viscosity solutions of suitable finite dimensionalHJB equations connected to finite dimensional control problems. Procedures of this typehave been used to prove existence of solutions for other infinite dimensional problems[10, Part IV], [34]. One could then use known numerical methods for finite dimensionalHJB equations and dynamic programming techniques to produce ε-optimal controls andtrajectories. Of course the case of Navier-Stokes equations is different from the problemsconsidered in [10, 34] but the idea is worth pursuing.

Sequential quadratic programming is a very promising method where the nonlinearityin the Navier-Stokes equation is approximated by Newton’s method to obtain a series oflinear problems [20, 28]. An approximation of this type applied to the nonlinear B(x,x)term in the Hamilton-Jacobi equation would lead (for a quadratic cost) to a sequenceof approximate problems which fall within the realm of linear control theory and can besolved effectively by well established methods. An interesting theoretical question wouldthen be to establish the convergence of such a sequence of approximate solutions to theunique viscosity solution proved in this paper.

3. Optimal control with state constraint is an important component of flow controlas it allows us to deal with the task of controlling interesting regions in the flow domainwhere we have concentrated turbulence activity, for example [14]. In the Hamilton-Jacobiframework this will correspond to an initial-boundary value problem (boundary valueproblem for the stationary case) which would be a substantial additional step that needsto be addressed in the future.

4. As we pointed out in the introduction, unique solution of the Hamilton-Jacobi

28

equation corresponding to the dynamic programming of the Navier-Stokes equations havebeen established for a neighborhood of the origin in [4] using different methods. Theuniqueness of viscosity solution proved in this paper may be used to extend this localresult to a global one.

5. Finally we would like to mention that we believe that our ideas and methods canbe extended to handle the important case of control of three dimensional Navier-Stokesequations. The essential reason for the analysis to be limited to two dimensions was thelack of global solvability theorem and the related continuous dependence theorems for theNavier-Stokes equations in three dimensions. The form of the Hamilton-Jacobi equationshowever is independent of the spatial dimensions.

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