Viscosity Solutions of Dynamic-Programming ... - Springer Link

Arch. Rational Mech. Anal. 163 (2002) 295–327 Digital Object Identifier (DOI) 10.1007/s002050200203

Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations ´ ¸ ch Fausto Gozzi, S. S. Sritharan & Andrzej Swie Communicated by L. C. Evans

Abstract In this paper we study the infinite-dimensional Hamilton-Jacobi equation associated with the optimal feedback control of viscous hydrodynamics. We resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution.

1. Introduction Optimal control of Navier-Stokes equations has many applications in traditional aerodynamic and hydrodynamic drag reduction, lift enhancement, turbulence control and combustion control. Data assimilation in meteorology and oceanography involves the task of finding best initial data and unknown boundary and body forces by optimal-control methodology [40]. During the past decade several fundamental advances have been made in developing an organized mathematical theory of this subject [16, 19, 32]. One of the open problems identified in [33] was the global unique solvability of the infinite-dimensional Hamilton-Jacobi equation associated with the feedback synthesis for Navier-Stokes equations. The problem is resolved in this paper. In this respect, this paper is at the crossroads of two theoretical developments which have taken place in the past several years, namely flow-control theory and the theory of viscosity solutions for infinite-dimensional Hamilton-Jacobi equations. Local solvability for the Hamilton-Jacobi equation associated with the feedback control 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 (in the semigroup formulation) of a semilinear Hamilton-Jacobi equation associated with the stochastic control of two-dimensional Navier-Stokes equations is given in [12] where the existence of smooth solutions is proved in some

296

´ ¸ ch Fausto Gozzi, S. S. Sritharan & Andrzej Swie

cases. For the impulse-control problem, the Hamilton-Jacobi equation becomes a quasi-variational inequality and this is again resolved in a mild sense in [26]. The viscosity-solution method which provides another notion of a generalized solution was first introduced in the context of finite-dimensional Hamilton-Jacobi equations by Crandall & Lions [9] and was generalized to infinite dimensions by a number of authors. 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 reader to [2, 3, 6, 23, 25, 35, 36], [10, Part VI] and the references therein. This paper deals with Hamilton-Jacobi-Bellman (HJB) equations associated with optimal control of Navier-Stokes equations in two-dimensional domains under homogeneous boundary conditions. Periodic boundary conditions can also be treated by our methods but we do not do it here as this case is easier. The approach presented here allows natural extensions to second order HJB equations of this type. We think that modifications of the methods developed here can be applied to prove comparison results for such equations. However, connecting these second order HJB equations to optimal control of stochastic Navier-Stokes equations seems to be a challenging and difficult problem (see [12]). Finally we mention that after this paper had been submitted we learned about a manuscript [31] that deals with abstract infinite-dimensional Hamilton-Jacobi equations and uses a different notion of viscosity solution. The results of [31] can be applied to some HJB equations coming from the optimal control of Navier-Stokes equations. In the rest of this section we will describe the mathematical problem. In Section 2 we establish various continuous-dependence theorems for the controlled NavierStokes equation that are needed in the paper. In Section 3 we describe a suitable notion of the viscosity solution used in this paper. In Section 4 we prove the uniqueness theorem for these viscosity solutions for time-independent and time-dependent Hamilton-Jacobi equations. Section 5 is devoted to the application of these abstract PDE (partial differential equation) results to an optimal-control problem. In Section 5.1 we show that the value function has the required continuity properties, and in Section 5.2 we prove that it is indeed a viscosity solution in the sense of the definition given in Section 3. In Section 5.3 we briefly discuss the applicability of our results. Let us now describe a typical optimal-control problem for the abstract NavierStokes equation [32, Chapter I]. We will assume throughout that the the kinematic viscosity is equal to 1. Let ⊂ IR2 be an open and bounded set with smooth boundary. Let x ∈ D ; IR2 , div x = 0 in L2 ; IR2 V = the closure of x ∈ D ; IR2 , div x = 0 in H 1 ; IR2 ,

H = the closure of

and let PH be the orthogonal projection in L2 ; IR2 onto H. Define Ax = −PH x, and B(x, y) = PH [(x · ∇)y]. Let t 0 be the initial time and T t be the terminal time. Our state variable is the velocity field X : [t, T ] × → IR2 that

Optimal Control of the Two-Dimensional Navier-Stokes Equations

297

satisfies the equation dX(s) = −AX(s) − B (X(s), X(s)) + f (s, a(s)) ds X(t) = x ∈ H,

in (t, T ] × H,

(1.1)

where f : [0, T ] × → V , is a complete metric space (the control set), and a(·) : [0, T ] → is a measurable function which plays the role of a control strategy. We will denote the set of control strategies by U. The minimization, over all controls a(·) ∈ U, of a cost functional T l(s, X(s), a(s))ds + g(X(T )) J (t, x; a) = t

formally leads to the Hamilton-Jacobi-Bellman (HJB) partial differential equation in (0, T ) × H ut −Ax + B(x, x), Du + inf a∈ {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) = inf J (t, x; a). a(·)∈U

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 versions of (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.

2. Preliminaries on the two-dimensional controlled Navier-Stokes equation 2.1. Abstract spaces and the Stokes operator Throughout the paper will be of IR2 with smooth an open and bounded subset 2 m,p m,p boundary. We denote by W ; IR (or simply by W () ) the Sobolev space of order 0 m ∈ IR and power p 1 of functions with values in IR2 (which 2 can be seen as a product space W m,p ; IR2 = W m,p (; IR) ). The norm of x ∈ W m,p ; IR2 will be denoted by xm,p . We will use the notation Lp for

298


W 0,p . Moreover, when p = 2, we will write H m for W m,2 . We will also be using the negative Sobolev spaces H −m . The space H can be alternatively defined by 1 H = x ∈ L2 ; IR2 , div x = 0 in D ; IR2 , x · n = 0 in H − 2 (∂) , where n is the outward normal to the boundary, see [37, Chapter 1]. The operator A = −PH with the domain

D (A) = H 2 ; IR2 ∩ V

is called the Stokes operator. It is well known that A is positive definite, self-adjoint, A−1 is compact, and A generates an analytic semigroup. For γ 0 we denote by γ γ Vγ the domain of A 2 , D(A 2 ), equipped with the norm γ

xγ = A 2 x0,2 .

(2.5)

For γ < 0 the space Vγ is defined as the completion of H under the norm (2.5). If γ > − 21 the norm of Vγ is equivalent to the norm of H γ (see [16, Lemma 4.5]). Moreover the space V1 coincides with V. Identifying H with its dual, the space V−γ is the dual of Vγ for γ > 0. We will also be using the customary notation V for the dual of V and the duality pairing between V and V will be denoted by ·, ·. The same symbol will also be used 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 remainder of the paper. – Sobolev-imbedding type inequalities: If m 0, mp 2 and p q then W m,p () → Lq (), i.e. , x0,q C xm,p

for

2p 2−mp

x ∈ W m,p () ,

(noting that when mp = 2 the imbedding holds for all q < +∞). Combining the above with of norms of Vγ and H γ , we find that for γ ∈ (0, 1]

the equivalence 2 (q ∈ [2, +∞) if γ = 1) Vγ → Lq (), i.e., and q ∈ 2, 1−γ

x0,q C xγ

for

x ∈ Vγ .

(2.6)

– Gagliardo-Nirenberg type inequality: 1

1

x0,4 C x02 x12

for

x ∈ V.

(2.7)

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

Sγ z0 CSz0 z0

1−γ

.

(2.8)


299

2.2. The Euler operator (inertia term) B We define the trilinear form b(·, ·, ·) : V × V × V → IR as b (x, y, z) = 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 already used in the Navier-Stokes equation (1.1). We will write B (x) for B (x, x) when no confusion arises. This operator can be extended in different topologies. By the incompressibility condition 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 Hölder inequality, for p, q > 1, p1 +

1 q

+

= 1,

1 2

|b (x, y, z)| C x0,p y1 z0,q .

(2.10)

2. By (2.7) we obtain 1

1

1

1

|b (x, y, z)| C x0,4 y1 z0,4 C x02 x12 y1 z02 z12

(2.11)

which gives, when x = z, |b (x, y, x)| C x20,4 y1 C x0 x1 y1 .

(2.12)

2.3. The controlled Navier-Stokes equation: existence, uniqueness 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]). Definition 1. (i) A weak solution of (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 almost every 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 (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 ).


300

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). We will often denote it simply by X(·) when there is no possibility of confusion. We first recall the standard existence and uniqueness results [37] for the Navier-Stokes equation (1.1). Theorem 1. Let f : [0, T ]× → V−1 be bounded and continuous. Let 0 t T , and let a(·) ∈ U. (i) Given an initial datum x ∈ H, there exists a unique weak solution X(·; t, x, a) of the state equation (1.1). Moreover, the following estimate holds: T T f(s, a(s))2−1 ds. (2.13) sup X(s)20 + X(s)21 ds x20 + t

tsT

t

(ii) Given an initial datum x ∈ V and assuming moreover that f : [0, T ] × → H is bounded and continuous, there exists a unique strong solution X(·; t, x, a) of (1.1). Moreover, the following estimate holds: T sup X(s)21 + AX(s)20 ds t

tsT

x21 exp (E(t, T )) + where

E(t, τ ) := C

t

τ

T t

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

X(r)20 X(r)21 dr

C sup

trτ

X(r)20

τ t

(2.14)

X(s)21 ds.

The next proposition gathers continuous-dependence estimates for solutions of equation (1.1). The negative norm estimates given here generalize the results given in [32, Chapter I]). Proposition 1. Let f : [0, T ] × → V−1 be bounded and continuous. Then: (i) Let 21 α 1. There exist constants C > 0, depending only on the indicated variables and independent of a(·) ∈ U, such that for all initial conditions x, y ∈ H, X(s) − Y (s)−α

C ||x||0 , ||y||0 , sup

t

a(·)∈U

t

T

X(s) − Y (s)21−α

T

C ||x||0 , ||y||0 , sup

a(·)∈U

T t

f(r, a(r))2−1 dr

x − y−α ,

(2.15)

x − y2−α ,

(2.16)

f(r, a(r))2−1 dr


||X(s) − x||−α

C ||x||0 , sup

a(·)∈U

T t

301

f(r, a(r))2−1 dr

1

(s − t) 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 the strategies a(·) ∈ U such that ||X(s) − x||0 C ||x||1 ,

sup

s∈[t,T ],a∈

1

f (s, a)2−1 (s − t) 2 ,

(2.19)

and if in addition f : [0, T ] × → 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. (i) We first prove (2.15). Let X(s) and Y (s) be the solutions 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−α Z(s) we get, for s ∈ [t, T ],

X (s), A−α Z (s) = −AX(s) − B(X(s)) + f (s, a(s)) , A−α Z (s) , (2.21) X(t) = x,

Y (s), A−α Z (s) = −AY (s) − B(Y (s)) + f (s, a(s)) , A−α Z (s) , Y (t) = y.

(2.22)

Subtracting the resulting equalities, we obtain

Z (s), A−α Z (s) = − AZ(s), A−α Z (s) + B(Y (s)) − B(X(s)), A−α Z (s) , Z(t) = x − y, (2.23) so α 1−α 1 d A− 2 Z(s)20 + A 2 Z(s)20 2 ds = −b X(s), X(s), A−α Z (s) + b Y (s), Y (s), A−α Z (s) . Now, by linearity, adding and subtracting b Y (s), X(s), A−α Z (s) ,

(2.24)

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


302

By Hölder inequality we have (see (2.10)) |b(Z(s), X(s), A−α Z(s))| CZ(s)0, 2 X(s)1 A−α Z(s)0, α

2 1−α

.

Sobolev imbeddings give us (see (2.6)) Z(s)0, 2 CZ(s)1−α α

and A−α Z(s)0,

2 1−α

C A−α Z(s)α = CZ(s)−α .

Therefore we obtain |b(Z(s), X(s), A−α Z(s))| CZ(s)1−α X(s)1 Z(s)−α 41 Z(s)21−α + CX(s)21 Z(s)2−α (2.25) for some constant C > 0. The second term is a little more difficult to estimate. Similarly, we have |b(Y (s), Z(s), A−α Z(s))| = |b(Y (s), A−α Z(s), Z(s))| CY (s)0, 2 A−α Z(s)1 Z(s)0, 2

α

1−α

= CY (s)0,

2 1−α

A

1 2 −α

Z(s)0 Z(s)0, 2 . α

Estimates (2.6) and (2.8) give Y (s)0,

α

2 1−α

CA 2 Y (s)0 = C(A 2 )α Y (s)0 CY (s)α1 Y (s)1−α 0 , 1

α

A 2 −α Z(s)0 = (A 2 )1−α (A− 2 Z(s))0 CA 1

1

Z(s)0, 2 CA α

1−α 2

1−α 2

α

−2 Z(s)1−α Z(s)α0 , 0 A

Z(s)0 ,

and, by (2.13), sup

s∈[t,T ]

Y (s)1−α 0

y20 +

T t

f(s, , a(s))2−1 ds

1−α 2

.

Therefore collecting these estimates and using Young’s inequality we obtain |b(Y (s), Z(s), A−α Z(s))| CY (s)α1 Y (s)1−α 0 A CY (s)α1 A

2 1 4 Z(s)1−α

1−α 2

1−α 2

α

−2 Z(s)2−α Z(s)α0 0 A α

−2 Z(s)2−α Z(s)α0 0 A

(2.26)

+ CY (s)21 Z(s)2−α

Using (2.25) and (2.26) in (2.24) produces 1 d 1 1−α Z(s)2−α + A 2 Z(s)2 C(X(s)21 + Y (s)21 )Z(s)2−α . (2.27) 2 ds 2


303

Gronwall’s inequality and (2.14) now yield s

X(s) − Y (s)2−α = Z(s)2−α eC t (X(r)1 +Y (r)1 )dr x − y2−α T 2 f(r, ||x|| ||y|| a(r))−1 dr x − y2−α C 0, 0, 2

2

t

(2.28) which gives us (2.15). Then putting (2.28) into (2.27) we get (2.16). To prove (2.17) we set 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−α Y (s), we get 1 d Y (s)2−α = − A1−α X(s), Y (s) (2.30) 2 ds

−b X (s) , X (s) , A−α Y (s) + f (s, a (s)) , A−α Y (s) . Since α

1 2

we have, using also (2.13),

|A1−α X(s), Y (s)| A1−α X (s) 0 Y (s)0 CX(s)1 . Moreover by (2.12) and (2.13) b X (s) , X (s) , A−α Y (s) C X (s)0 A−α Y (s)1 X (s)1 CX(s)1 Y (s)0 X(s)0 CX(s)1 . Finally

f (s, a (s)) , A−α Y (s) f (s, a (s))−1 A−α Y (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 d Y (s)2−α C X(s)1 + f (s, a (s))−1 . 2 ds Integrating and using the Cauchy-Schwarz inequality, we then get X(s) − x2−α 

C

s

t

1 X(r)21 dr

2

+

T t

f (r, a (r)) 2−1 ds

21

  (s − t) 2

1

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

s∈[t,t+ε],a(·)∈U

X(s; t, x, a) − x0 → 0

as ε → 0.


304

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

X(sn ; t, x, an ) − x0 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 inf X(sn ; t, x, an )0 x0 , n→∞

and from the energy inequality (2.13) we can conclude that lim sup X(sn ; t, x, an )0 x0 . n→∞

Combining these two results gives convergence of the norms and this, along with the weak convergence in H, yields X(sn ; t, x, an ) → x strongly in H , which is a contradiction. (ii) Let X (s) = X (s; t, x, a) be the weak solution of (1.1) with initial conditions x at time t. Set as before Y (s) = X (s) − x so that Y (s) satisfies (2.29). Taking the inner product of (2.29) with Y (s) we obtain 1 d Y (s)20 = − AX(s), Y (s) − b (X (s) , X (s) , Y (s)) 2 ds + f (s, a (s)) , Y (s) = − AY (s), Y (s) − Ax, Y (s)

(2.31)

−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 d Y (s)20 + Y (s)21 = − Ax, Y (s) − b (Y (s) , X (s) , Y (s)) (2.32) 2 ds +b (x, x, Y (s)) + f (s, a (s)) , Y (s) . Now we have

1 1 |Ax, Y (s)| = A 2 x, A 2 Y (s) x1 Y (s)1

1 8

Y (s)21 + C.

(2.33)

Moreover, by (2.12), |b (Y (s) , X (s) , Y (s))| C Y (s)0 Y (s)1 X (s)1 18 Y (s)21 + C Y (s)20 X (s)21

(2.34)


305

and |b (x, x, Y (s))| C x0 x1 Y (s)1

1 8

Y (s)21 + C,

(2.35)

and finally |f (s, a (s)) , Y (s)| f (s, a (s))−1 Y (s)1 18 Y (s)21 + C f (s, a (s))2−1 .

(2.36)

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

1 1 d Y (s)20 + Y (s)21 C 1 + Y (s)20 X (s)21 + f (s, a (s))2−1 . 2 ds 2 Therefore it follows by Gronwall’s inequality that s s 2 2 2 f (r, a (r))−1 dr eC t X(r)1 dr , X (s) − x0 C (s − t) + t

from which our claim (2.19) easily follows. Moreover we note that, setting ω1 (r − t) = X (s) − x20 , we also have s s f (r, a (r))2−1 dr X(r) − x21 dr C (s − t) + t t s +C ω1 (r − t) X (r)21 dr. t

The proof of (2.20) is the same as the proof of estimate (2.18) in part (i) once we know that lim sups→t X (s) 0 x0 , which follows from the assumption that f is bounded in H 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 2. A function ψ is a test function for equation (1.3) if ψ = ϕ+δ(t) ||x||20 , where 1

• ϕ ∈ C 1 ((0, T ) × H), A 2 Dϕ is continuous, and ϕ is weakly sequentially lowersemicontinuous, • δ ∈ C 1 ((0, T )) is such that δ > 0 on (0, T ). Definition 3. A function ψ is a test function for equation (1.4) if ψ = ϕ + δ ||x||20 , where 1

• ϕ ∈ C 1 (H), A 2 Dϕ is continuous, and ϕ is weakly sequentially lower-semicontinuous, • δ > 0.


306

In the definition below we assume that F : [0, T ] × V × V → IR. Definition 4. 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 (respectively u + 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 supersolution. Definition 5. 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 has a local minimum) at x then x ∈ V and λu(x) + Ax + B(x, x), Dψ(x) + F (x, Dψ(x)) 0 (respectively λu(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 supersolution. The somehow arbitrary choice of the function · 20 as part of the test function was made for simplicity, and because in this paper we only deal with linearly growing solutions. 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 imply the uniqueness of solutions of (1.3) and (1.4) under certain conditions. We will assume the following hypothesis. Hypothesis 1. The Hamiltonian F : [0, T ] × Vβ × V → IR, and there exist a modulus of continuity ω, a constant β in the range 0 < β < 21 , and moduli ωR such that, for every R > 0, (4.37) |F (t, x, p) − F (t, y, p)| ωR x − yβ + ω x − yβ p0 if x0 , y0 R, |F (t, x, p) − F (t, x, q)| ω ((1 + x0 )p − q1 ) , |F (t, x, p) − F (s, x, p)| ωR (|t − s|) , if x0 , y0 , p0 R.

(4.38) (4.39)


307

Theorem 2. Let Hypothesis 1 hold. Let u, v : (0, T ) × H → IR be respectively a viscosity subsolution, and a viscosity supersolution of (1.3). Let u(t, x), −v(t, x), |g(x)| C(1 + x0 ) and let u(t, ·), v(t, ·), g be locally Lipschitz continuous in ·β−1 norm on bounded subsets of H, uniformly for t in closed subsets of (0, T ). Let moreover (i) limt↑T (u(t, x) − g(x))+ = 0 (4.40) (ii) limt↑T (v(t, x) − g(x))− = 0 uniformly on bounded subsets of H. Then u v. Theorem 3. Let Hypothesis 1 hold. Let u, v : H → IR be respectively a viscosity subsolution, and a viscosity supersolution of (1.4). Let u, −v be bounded from above and be Lipschitz continuous in · β−1 norm on bounded subsets of H. Then u v. Proof of Theorem 3. Set α = 1 − β and assume for simplicity that λ = 1. For ε, δ > 0 we consider the function 7(x, y, ε, δ) := u(x) − v(y) −

x−y2−α − δ(x20 + y20 ). 2ε

Because of the continuity assumptions on u, v and the fact that A−1 is compact, the function above is weakly sequentially upper-semicontinuous and so it attains a global maximum at some points x¯ , y¯ ∈ V (by the definition of viscosity solution). For a fixed δ we will show (as for instance in [22]) that the points x¯ and y¯ are bounded independently of ε, lim sup lim sup δ(¯x20 + ¯y20 ) = 0, δ→0

ε→0

(4.41)

and lim sup ε→0

¯x − y¯ 2−α =0 2ε

for fixed δ.

(4.42)

(Please notice that we have reversed the order in which we pass to the limits in the above expressions compared with [22].) To see this we set m1 (ε, δ) = sup 7(x, y, ε, δ), x,y∈H

m2 (δ) = sup x,y∈H

u(x) − v(y) − δ x20 + y20

and note that we have m = lim m2 (δ) and m2 (δ) = lim m1 (ε, δ). δ↓0

ε↓0


308

Now, m1 (ε, δ) = 7(¯x, y¯ , ε, δ) = u(¯x) − v(¯y) −

¯x−¯y2−α − δ(¯x20 + ¯y20 ) 2ε

and, for fixed δ, m1 (ε, δ) +

¯x−¯y2−α ¯x−¯y2−α = u(¯x) − v(¯y) − − δ(¯x20 + ¯y20 ) 4ε 4ε m1 (2ε, δ).

Thus ¯x−¯y2−α m1 (2ε, δ) − m1 (ε, δ). 4ε This gives (4.42). Similarly, ¯x−¯y2−α δ δ m1 (ε, δ) + (¯x20 + ¯y20 ) = u(¯x) − v(¯y) − − (¯x20 + ¯y20 ) 2ε 2 2 δ m1 ε, 2 which gives δ δ (¯x20 + ¯y20 ) m1 (ε, δ) − m1 ε, , 2 2 from which we obtain (4.41). Using the definition of viscosity solution and b(x, x, x) = 0, we now have 1 u(¯x) + 2δA¯x, x¯ + A¯x, A−α (¯x − y¯ ) ε 1 1 + b(¯x, x¯ , A−α (¯x − y¯ )) + F x¯ , A−α (¯x − y¯ ) + 2δ x¯ 0 ε ε and 1 v(¯y) − 2δA¯y, y¯ + A¯y, A−α (¯x − y¯ ) ε 1 1 + b(¯y, y¯ , A−α (¯x − y¯ )) + F y¯ , A−α (¯x − y¯ ) − 2δ y¯ 0. ε ε Combining these two inequalities we obtain 1 u(¯x) − v(¯y) + 2δ(¯x21 + ¯y21 ) + ¯x − y¯ 21−α ε 1 −α + (b(¯x, x¯ , A (¯x − y¯ )) − b(¯y, y¯ , A−α (¯x − y¯ ))) ε 1 1 + F x¯ , A−α (¯x − y¯ ) + 2δ x¯ − F y¯ , A−α (¯x − y¯ ) − 2δ y¯ 0. ε ε

(4.43)


309

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 estimate the first one. We have, by Hölder inequality, 1 |b(¯x − y¯ , x¯ , A−α (¯x − y¯ ))| ε 1 = |b(¯x − y¯ , A−α (¯x − y¯ ), x¯ )| ε 1 c ¯x0,q A−α+ 2 (¯x − y¯ )0,2p ¯x − y¯ 0,2p , ε where we took p and q such that

1 p

(4.44)

+ q1 = 1, and later p will be sufficiently small.

To continue we will use inequality (2.8). We choose τ such that 0 < τ < α − 21 . To estimate A−α+ 2 (¯x − y¯ )0,2p we first notice that if p is sufficiently close to 1, Sobolev imbedding (2.6) guarantees that 1

τ

A−α+ 2 (¯x − y¯ )0,2p CA−α+ 2 + 2 (¯x − y¯ )0 . 1

1

α

We now set S = A 2 , and z = A− 2 (¯x − y¯ ) in (2.8). Then 1

τ

A−α+ 2 + 2 (¯x − y¯ )0 = S1−α+τ z0 CSz1−α+τ zα−τ 0 0 1

= CA

1−α 2

(¯x − y¯ )1−α+τ ¯x − y¯ α−τ −α . 0

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 = A then obtain

1−α 2

¯x − y¯ τ (1−α) = Sτ z0 CSzτ0 z1−τ = CA 0

and z = x¯ − y¯ in (2.8). We 1−α 2

(¯x − y¯ )τ0 ¯x − y¯ 1−τ 0 . (4.45)

Therefore, plugging (4.45) and (4.45) into (4.44), and estimating further ¯x0,q C¯x1 we get 1 |b(¯x − y¯ , x¯ , A−α (¯x − y¯ ))| ε 1−α C ¯x1 A 2 (¯x − y¯ )1−α+2τ ¯x − y¯ α−τ x − y¯ 1−τ −α ¯ 0 0 ε


310

which, upon using ¯x − y¯ 0 CA

1−α 2

(¯x − y¯ )0 , yields

¯x − y¯ 1−α ¯x − y¯ α−τ 1 |b(¯x − y¯ , x¯ , A−α (¯x − y¯ ))| C¯x1 . √ √ −α ¯x − y¯ 1−α+τ 0 ε ε ε (4.46) We now notice that 7(¯x, y¯ , ε, δ) {7(¯x, x¯ , ε, δ), 7(¯y, y¯ , ε, δ)} .

(4.47)

We use the fact that u and v are locally Lipschitz continuous in · −α norm and ¯x0 , ¯y0 Rδ independently of ε for a fixed δ to deduce from (4.47) that ¯x − y¯ 2−α Kδ ¯x − y¯ −α ε for some Kδ > 0. This implies that 1

2 ¯x − y¯ −α Kδ . √ ε

Therefore 1

2 ¯x − y¯ α−τ ¯x − y¯ −α α− 1 −τ α− 1 −τ ¯x − y¯ −α 2 Kδ ¯x − y¯ −α 2 → 0 √ −α = √ ε ε

as ε → 0 by (4.42) since α −

1 2

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

¯x − y¯ 1−α 1 |b(¯x − y¯ , x¯ , A−α (¯x − y¯ ))| ¯x1 σ1 (ε, δ) √ ε ε δ¯x21 +

¯x − y¯ 21−α 2ε

σ2 (ε, δ)

(4.48)

for some local moduli σ1 and σ2 . Similarly we obtain ¯x − y¯ 21−α 1 |b(¯y, x¯ − y¯ , A−α (¯x − y¯ ))| δ¯y21 + σ2 (ε, δ). ε 2ε

(4.49)

We can now proceed with estimating (4.43). Using (4.48), (4.49), and Hypothesis 1 in (4.43) (notice that β = 1 − α) it follows that 1 1 u(¯x) − v(¯y) + ¯x − y¯ 21−α − ¯x − y¯ 21−α σ2 (ε, δ) + δ(¯x21 + ¯y21 ) ε ε ¯x − y¯ −α − ωRδ (¯x − y¯ 1−α ) − ω ¯x − y¯ 1−α ε ω(δ(1 + ¯x0 )¯x1 ) + ω(δ(1 + ¯y0 )¯y1 ). (4.50)


311

For fixed µ, δ > 0 we now choose Cµ and Cµ,δ so that ω(s) µ + Cµ s and ωRδ (s) µ + Cµ,δ s. Then ω(δ(1 + ¯x0 )¯x1 ) + ω(δ(1 + ¯y0 )¯y1 ) 2µ + Cµ δ ((1 + ¯x0 )¯x1 + (1 + ¯y0 )¯y1 ) 2µ + δ(¯x21 + ¯y21 ) + Dµ δ 1 + ¯x20 + ¯y20 .

(4.51)

Moreover,

¯x − y¯ −α ωRδ (¯x − y¯ 1−α ) + ω ¯x − y¯ 1−α ε ¯x − y¯ −α 2µ + Cµ ¯x − y¯ 1−α + Cµ,δ ¯x − y¯ 1−α ε ¯x − y¯ 21−α ¯x − y¯ 2−α 2µ + + Cµ,δ ¯x − y¯ 1−α + 2Cµ2 . 2ε ε

(4.52)

Putting (4.51) and (4.52) into (4.50), and using (4.41) and (4.42), we therefore obtain ¯x − y¯ 21−α 1 u(¯x) − v(¯y) + − σ2 (ε, δ) − Cµ,δ ¯x − y¯ 1−α 2 ε 4µ + σ3 (ε, δ; µ) for some function σ3 such that for every fixed µ lim sup lim sup σ3 (ε, δ; µ) = 0. δ→0

We now observe that

(4.53)

ε→0

lim inf

ε→0 r>0

r2 − Cµ,δ r 4ε

= 0.

(4.54)

This yields u(¯x) − v(¯y) 4µ + σ4 (ε, δ; µ)

(4.55)

for some function σ4 satisfying (4.53). Finally (4.55) yields u(x) − v(x) − 2δx20 u(¯x) − v(¯y) 4µ + σ4 (ε, δ; µ). If we now let ε → 0, δ → 0 and then µ → 0, we find that u v. Proof of Theorem 2. Given µ > 0, define vµ (t, x) = v(t, x) +

µ . t

(uµ )t − Ax + B(x, x), Duµ + F (x, Duµ )

µ T2

uµ (t, x) = u(t, x) −

µ , t

Then uµ and vµ satisfy respectively


312

and (vµ )t − Ax + B(x, x), Dvµ + F (x, Dvµ ) −

µ . T2

Let Cµ be such that ω(s) 2Tµ 2 + Cµ s, and let Kµ = 2Cµ2 . For ε, δ, γ > 0, and 0 < Tδ < T , we consider the function uµ (t, x) − vµ (s, y) −

x−y2−α (t − s)2 − δeKµ (T −t) x20 − δeKµ (T −s) y20 − 2ε 2γ

on (0, Tδ ] × H. This function has a global maximum at t¯, s¯ , x¯ , y¯ , where 0 < t¯, s¯ , and x¯ , y¯ are bounded independently of ε for a fixed δ. Moreover x¯ , y¯ ∈ V. Similarly to the stationary case we have lim sup lim sup lim sup δ(¯x20 + ¯y20 ) = 0, δ→0

ε→0

(4.56)

γ →0

lim sup lim sup ε→0

γ →0

x¯ − y¯ 2−α =0 2ε

for fixed δ,

(4.57)

for fixed δ, ε.

(4.58)

and lim sup γ →0

(t¯ − s¯ )2 =0 2γ

If u is not less than or equal to 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 using the definition of viscosity solution we obtain t¯ − s¯ 1 − δKµ eKµ (T −t¯) ¯x2 − 2δeKµ (T −t¯) A¯x, x¯ − A¯x, A−α (¯x − y¯ ) γ ε 1 1 µ − b(¯x, x¯ , A−α (¯x − y¯ )) + F t¯, x¯ , A−α (¯x − y¯ ) + 2δeKµ (T −t¯) x¯ 2 ε ε T and t¯ − s¯ 1 + δKµ eKµ (T −¯s ) ¯y2 + 2δeKµ (T −¯s ) A¯y, y¯ − A¯y, A−α (¯x − y¯ ) γ ε µ 1 1 − b(¯y, y¯ , A−α (¯x − y¯ )) + F s¯ , y¯ , A−α (¯x − y¯ ) − 2δeKµ (T −¯s ) y¯ − 2 . ε ε T Combining these two inequalities, we find that Kµ δ eKµ (T −t¯) ¯x20 + eKµ (T −¯s ) ¯y20 + 2δ eKµ (T −t¯) ¯x21 + eKµ (T −¯s ) ¯y21 1 1 1 + ¯x − y¯ 21−α + b(¯x, x¯ , A−α (¯x − y¯ )) − b(¯y, y¯ , A−α (¯x − y¯ )) ε ε ε 1 −α + F s¯ , y¯ , A (¯x − y¯ ) − 2δeKµ (T −¯s ) y¯ ε 1 −α − F t¯, x¯ , A (¯x − y¯ ) + 2δeKµ (T −t¯) x¯ ε µ −2 2. T

(4.59)


313

We now estimate F t¯, x¯ , 1 A−α (¯x − y¯ ) + 2δeKµ (T −t¯) x¯ − F t¯, x¯ , 1 A−α (¯x − y¯ ) ε ε 1 1 + F s¯ , y¯ , A−α (¯x − y¯ ) − 2δeKµ (T −¯s ) y¯ − F s¯ , y¯ , A−α (¯x − y¯ ) ε ε Kµ (T −t¯) Kµ (T −¯s ) ω (1 + ¯x0 )2δe ¯x1 + ω (1 + ¯y0 )2δe ¯y1 µ 2 + Cµ (1 + ¯x0 )2δeKµ (T −t¯) ¯x1 T +Cµ (1 + ¯y0 )2δeKµ (T −¯s ) ¯y1 µ 2 + 2δCµ2 2 + eKµ (T −t¯) ¯x20 + eKµ (T −¯s ) ¯y20 T +δ eKµ (T −t¯) ¯x21 + eKµ (T −¯s ) ¯y21 .

Using this and the fact that Kµ = 2Cµ2 in (4.59), we therefore obtain 1 δ(¯x21 + ¯y21 ) + ¯x − y¯ 21−α ε 1 1 + b(¯x, x¯ , A−α (¯x − y¯ )) − b(¯y, y¯ , A−α (¯x − y¯ )) ε ε 1 −α 1 µ + F s¯ , y¯ , A (¯x − y¯ ) − F t¯, x¯ , A−α (¯x − y¯ ) − 2 + σ1 (δ, µ) (4.60) ε ε T for some local modulus σ1 . The rest of the proof follows that of the stationary case. We first let γ → 0 and use (4.39) to eliminate the dependence on t¯ and s¯ above, and then estimate all the terms as in the proof of Theorem 3. After doing this we obtain ¯x − y¯ 21−α µ 1 − σ2 (ε, δ) − Dµ,δ ¯x − y¯ 1−α − 2 + σ3 (γ , ε, δ; µ) 2 ε 2T for some constant Dµ,δ , a local modulus σ2 , and a function σ3 such that, for every fixed µ, lim sup lim sup lim sup σ3 (γ , ε, δ; µ) = 0. δ→0

ε→0

γ →0

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

5. Optimal control problem for the Navier-Stokes equation In this section we study an optimal-control problem for the Navier-Stokes equation. We only consider the finite-horizon problem. Similar results can be obtained

314


for the infinite-horizon problem by the same methods. We recall that in the finitehorizon case for a fixed T > 0 and 0 t < T we try to minimize the cost functional T l (s, X (s; t, x, a) , a(s)) ds + g (X (T ; t, x, a)), (5.61) J (t, x; a (·)) = t

where X (·; t, x, a) is the solution of (1.1). We assume the following. Hypothesis 2. There exist a constant 0 < β < 21 , 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 × and, for t, s ∈ [0, T ], x, y ∈ Vβ , and a ∈ , |l(t, x, a) − l(s, y, a)| σR (|t − s|) + LR x − yβ if x0 , y0 R, (5.62) |l (s, x, a)| C(1 + x1 );

(5.63)

(ii) g : H → IR is such that |g(x) − g(y)| LR x − yβ−1 if x0 , y0 R, |g(x)| C (1 + x0 ) ;

(5.64) (5.65)

(iii) f : [0, T ] × → H is bounded, continuous, and f(·, a) is uniformly continuous, uniformly for a ∈ . Let us briefly discuss the applicability of our control problem. Note that if we take as a separable Banach space then the following example fits Hypothesis 2: f(t, a) = κ(a )Na, where κ(·) ∈ C ∞,0 (IR+ ) here acts like an engineering constraint on the control actuation, and the control operator N ∈L( ; H) is a linear continuous operator representing spatial localization of the control. Here C ∞,0 (IR+ ) denotes smooth functions which identically vanish outside a finite interval [0, R]. For conducting fluids (even slightly conducting fluids such as salt water) such controllers can be realized through the action of Lorentz force [27, Chapter 10], [8, Chapter IV]. Since the Lorentz force is given by j ×B where j is the current and B is the magnetic field, spatial localization of such distributed control forces can be realized by choosing a suitable shape and location for the magnets. This is implemented for example in a boundary-layer-stabilization experiment in [21]. A cost functional of the form T β J (t, x; a (·)) = A 2 (X(s) − Xd (s))0 ds t

+

t

T

λ(a(s) )a(s)2 ds + A

β−1 2

X(T )0


315

with λ(·) ∈ C ∞,0 (IR+ ) being a discount factor and Xd (t) either a desired field or a known smooth field satisfies Hypothesis 2. We note that if the control is bounded (control set is bounded) then the discount factors κ(·) and λ(·) are not needed. 1 Although it is weaker than the enstrophy norm A 2 X(s)0 , the above integrand β A 2 (X(s) − Xd (s))0 still amounts to minimizing spatial turbulence structures contained in the deviation field X(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 forces due to environmental effects which cannot be completely incorporated into the model due to the complexity of the actual problem. In these applications Xd would represent the extrapolated measurements and the unknown time-dependent forces are regarded as controls with possible spatial localization and hence fit within the above abstract formulation. The integrand we chose above is in fact stronger than the L2 -norm integrands used in the literature on data assimilation of geophysics. We will prove that the value function for the problem, V(t, x) = inf J (t, x; a (·)) a(·)∈U

(5.66)

is the unique viscosity solution of the HJB equation ut − Ax + B(x, x), Du + inf a∈ {f(t, a), Du + l(t, x, a)} = 0 u(T , x) = g(x)

in (0, T ) × H, in H.

(5.67)

The lemma below is easy and we omit the proof. Lemma 1. Under the assumptions of Hypothesis 2 the Hamiltonian F given by F (t, x, p) = inf {f(t, a), p + l(t, x, a)} a∈

is continuous on [0, T ] × Vβ × V and it satisfies (4.37)–(4.39). 5.1. Properties of the value function The continuity properties of the value function are described by the following result. Proposition 2. Let Hypothesis 2 hold. Let α = 1 − β. Then for every R > 0 there exists a constant CR such that the value function V defined by (5.66) satisfies 1 (5.68) |V(t1 , x) − V(t2 , y)| CR |t1 − t2 | 2 + x − y−α for t1 , t2 ∈ [0, T ] and x0 , y0 R. Moreover |V(t, x)| C(1 + x0 ).

(5.69)

316


Proof. We will first prove the local Lipschitz continuity of V(t, ·). Let 1 2 M = MR = R 2 + T K 2 , where K is such that supt∈[0,T ],a∈ f(t, a)−1 K. It then follows from (2.13), (2.15) and (2.16) that for x0 , y0 R we have |V(t, x) − V(t, y)| sup |J (t, x; a (·)) − J (t, y; a (·)) | a(·)∈U

sup

T

|l(s, X(s), a(s)) − l(s, Y (s), a(s))|ds

t

a(·)∈U

+ |g(X(T )) − g(Y (T ))| LM sup

T

X(s) − Y (s)β ds + X(T ) − Y (T )−α

t

a(·)∈U

CR x − y−α . Estimate (5.69) is obvious from (2.13), (5.63), and (5.65). It remains to prove the local Hölder continuity of V in t. Let 0 t1 t2 T . Then |V(t1 , x) − V(t2 , x)| sup |J (t1 , x; a (·)) − J (t2 , x; a (·)) | a(·)∈U

sup

a(·)∈U

t2 t1

+ sup

a(·)∈U

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

|l (s, X (s; t1 , x, a(·)) , a(s))

t2

− l(s, X (s; t2 , x, a(·)) , a(s))|ds + sup |g (X (T ; t1 , x, a(·))) − g (X (T ; t2 , x, a(·))) | a(·)∈U

so that, writing X1 (s) for X (s; t1 , x, a(·)) and X2 (s) for X (s; t2 , x, a(·)) , and using Hypothesis 2 and (2.13), |V(t1 , x) − V(t2 , x)| sup

a(·)∈U

t2 t1

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

+ C(x0 ) sup

a(·)∈U

T

t2

||X1 (s) − X2 (s)||β ds

+ ||X1 (T ) − X2 (T )||−α .


317

We now observe that, for every a(·) ∈ U, t2 t2 |l (s, X1 (s), a(s)) |ds C (1 + ||X1 (s)||1 ) ds t1

t1

C (t2 − t1 )

1 2

t2 t1

1 + ||X1 (s)||21

ds

21

1

C(x0 ) (t2 − t1 ) 2 , and since X1 (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) T 2 2 ||X1 (s) − X2 (s)||β ds + ||X1 (T ) − X2 (T )||−α sup a(·)∈U

t2

1

C(x0 )|(t2 − t1 ) 2 . Combining the last two inequalities we therefore obtain 1

|V(t1 , x) − V(t2 , x)| C(x0 )(t2 − t1 ) 2 . Remark 1. It is clear from the proof that to obtain Proposition 2 we do not have to assume that f has values in H. It is enough to assume that the values of f are bounded in V−1 . 5.2. Existence of solutions of HJB equations Let us begin with the Bellman principle of optimality for Navier-Stokes equations [33, 32]. Proposition 3. For every 0 t τ T and x ∈ H, τ V(t, x) = inf l (s, X (s; t, x, a(·)) , a(s)) ds + V (τ, X (τ ; t, x, a(·))) . a(·)∈U

t

Theorem 4. Assume that Hypothesis 2 is true. Then the value function V is the unique viscosity solution of the HJB equation (5.67) that satisfies (5.68) and (5.69). Proof. The uniqueness part follows from Theorem 2 and Lemma 1. Here we will only prove that V is a viscosity solution and in fact only that the value function is a viscosity supersolution. The subsolution part is very similar and in fact easier. We set B (x) = B (x, x) throughout this proof. Let ψ (t, x) = ϕ (t, x) + δ(t) ||x||20 be a test function. Let V + (ϕ + δ ||·||20 ) have a local minimum at (t0 , x0 ) ∈ (0, T ) × H.


318

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 . (5.70) 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 ) + ε > 2

t0 +ε

t0

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

Then, by (5.70), ε − 2

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 then dividing both sides by ε, we obtain ε−

1 ε

t0 +ε

t0

−

1 ε

+

1 ε

+

1 ε

−

1 ε

+

1 ε

−

1 ε

−

1 ε

l (s, Xε (s) , aε (s)) ds

t0 +ε

t0 t0 +ε t0 t0 +ε t0 t0 +ε t0 t0 +ε t0 t0 +ε t0 t0 +ε t0

ϕt (s, Xε (s)) ds AXε (s) , Dϕ (s, Xε (s)) ds B (Xε (s)) , Dϕ (s, Xε (s)) ds f (s, aε (s)) , Dϕ (s, Xε (s)) ds δ(s) AXε (s) , Xε (s) ds δ(s) f (s, aε (s)) , Xε (s) ds δ (s) ||Xε (s)||20 ds.

(5.71)

Now, setting λ = inf t∈[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 t0 +ε ||Xε (s)||21 ds ε t0 1 t0 +ε l (s, Xε (s) , aε (s)) ds ε− ε t0 1 t0 +ε ϕt (s, Xε (s)) ds + ε t0 1 t0 +ε AXε (s) , Dϕ (s, Xε (s)) ds − ε t0 1 t0 +ε B (Xε (s)) , Dϕ (s, Xε (s)) ds − ε t0 1 t0 +ε f (s, aε (s)) , Dϕ (s, Xε (s)) ds + ε t0 t0 +ε 1 δ (s) ||Xε (s)||20 ds + ε t0 1 t0 +ε δ(s) f (s, aε (s)) , Xε (s) ds. + ε t0

319

(5.72)

The assumptions on l yield t0 +ε 1 1 t0 +ε X l X ds , a ds C 1 + (s, (s) (s)) (s) ε ε ε 1 ε ε t0 t0 t0 +ε λ 1 Xε (s)21 ds. (5.73) C+ · 8 ε t0 Moreover the assumptions placed on ϕ and (2.18) yield A1/2 Dϕ (s, Xε (s)) 0 C(x0 0 )

(5.74)

for ε < ε0 for some fixed ε0 > 0. Therefore AXε (s) , Dϕ (s, Xε (s)) = A 21 Xε (s) , A 21 Dϕ (s, Xε (s)) 1

1

A 2 Xε (s) 0 A 2 Dϕ (s, Xε (s)) 0 = C(x0 0 ) Xε (s)1 λ Xε (s)21 . 8 Regarding the third term, we have, using (2.12), (2.13) and (5.74), B (Xε (s)) , Dϕ (s, Xε (s)) = |b (Xε (s) , Xε (s) , Dϕ (s, Xε (s)))| C+

C ||Xε (s)||0 ||Xε (s)||1 ||Dϕ (s, Xε (s))||1 C(x0 0 ) Xε (s)1 C+

λ Xε (s)21 . 8

320


Finally | f (s, aε (s)) , Xε (s) |, | f (s, aε (s)) , Dϕ (s, Xε (s)) | C(x0 0 ).

(5.75)

Therefore, using inequalities (5.73)–(5.75) above and the fact that ϕt (s, Xε (s)) and δ (s) are bounded independently of ε on [t0 , t0 + ε0 ] for some ε0 > 0, we obtain from (5.72) λ ε

t0 +ε

t0

||Xε (s)||21 ds C

for some constant C independent of ε. We now take ε = X s; t0 , x0 , a1/n (·) . Inequality (5.76) gives then n

t0 +1/n

t0

(5.76) 1 n

and set Xn (s) =

Xn (s)21 ds C

so that, along a sequence tn ∈ t0 , t0 + n1 , Xn (tn )21 C, and thus along a subsequence, still denoted by tn , we have Xn (tn ) A x¯ 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 the weak limit in H it follows that x0 = x¯ ∈ V. Step 2. The supersolution inequality. We go back to inequality (5.72). By (2.13) (or (2.14)) we have Xε (s)0 C(x0 0 ) = R for every ε and s ∈ [t0 , T ]. Therefore using (5.62) and (2.20) we obtain t0 +ε 1 , x , a X , a ds − l ds [l (s, (s) (s)) (t (s))] ε ε 0 0 ε ε t0 t0 +ε 1 σR (|s − t0 |) + LR ||Xε (s) − x0 ||β ds ω1 (ε) ε t0 for some modulus ω1 . We will be using ω1 to denote various moduli throughout the rest of the proof. Similarly, by the continuity of ϕt , we obtain t0 +ε 1 ϕt (s, Xε (s)) ds − ϕt (t0 , x0 ) ε t0 1 t0 +ε ωt0 ,x0 (|s − t0 | + ||Xε (s) − x0 ||0 ) ds ω1 (ε). ε t0


321

Moreover t0 +ε 1 AXε (s) , Dϕ (s, Xε (s)) − Ax0 , Dϕ (t0 , x0 ) ds ε t0 t0 +ε 1 A (Xε (s) − x0 ) , Dϕ (s, Xε (s)) ds ε t0 t0 +ε Ax0 , Dϕ (s, Xε (s)) − Dϕ (t0 , x0 ) ds . + t0

It now follows from (5.74) and (2.20) that |A (Xε (s) − x0 ) , Dϕ (s, Xε (s))| 1 1 = A 2 (Xε (s) − x0 ) , A 2 Dϕ (s, Xε (s)) 1

Xε (s) − x0 1 A 2 Dϕ (s, Xε (s)) 0 C(x0 0 )Xε (s) − x0 1 ω1 (ε). 1

By the continuity of A 2 Dϕ we also get Ax0 , Dϕ (s, Xε (s)) − Dϕ (t0 , x0 ) 1 1 = A 2 x0 , A 2 [Dϕ (s, Xε (s)) − Dϕ (t0 , x0 )] x0 1 ωϕ,t0 ,x0 (|s − t0 | + Xε (s) − x0 0 ) ω1 (ε). Next we have t0 +ε 1 B B , Dϕ X − , Dϕ ds , x (X (s)) (s, (s)) (x ) (t ) ε ε 0 0 0 ε t0 t0 +ε 1 |b (Xε (s) , Xε (s) , Dϕ (s, Xε (s))) − b (x0 , x0 , Dϕ (t0 , x0 ))| ds = ε t0 t0 +ε 1 |b (Xε (s) − x0 , Xε (s) , Dϕ (s, Xε (s)))| ε t0 + |b (x0 , Xε (s) − x0 , Dϕ (s, Xε (s)))| ds t0 +ε |b (x0 , x0 , Dϕ (s, Xε (s)) − Dϕ (t0 , x0 ))| ds . + t0


322

Now observe that, by (2.11), |b (Xε (s) − x0 , Xε (s) , Dϕ (s, Xε (s)))| CXε (s) − x0 1 Xε (s) 1 Dϕ (s, Xε (s)) 1 , |b (x0 , Xε (s) − x0 , Dϕ (s, Xε (s)))| CXε (s) − x0 1 x0 1 Dϕ (s, Xε (s)) 1 and, by (2.12) |b (x0 , x0 , Dϕ (s, Xε (s)) − Dϕ (t0 , x0 ))| Cx0 0 x0 1 Dϕ (s, Xε (s)) − Dϕ (t0 , x0 ) 1 Cx0 0 x0 1 ωϕ,t0 ,x0 (|s − t0 | + Xε (s) − x0 0 ) . It then follows from (5.74) and (2.20) that t0 +ε 1 ω1 (ε) . B B , x , Dϕ X − , Dϕ ds (X (s)) (s, (s)) (x ) (t ) ε ε 0 0 0 ε t0 Also t0 +ε 1 [f (s, aε (s)) , Dϕ (s, Xε (s)) − f (t0 , aε (s)) , Dϕ (t0 , x0 )] ds ε t0 1 t0 +ε |f (s, aε (s)) , Dϕ (s, Xε (s)) − Dϕ (t0 , x0 )| ds ε t0 1 t0 +ε |f (s, aε (s)) − f (t0 , aε (s)) , Dϕ (t0 , x0 )| ds + ε t0 1 t0 +ε f (s, aε (s))0 Dϕ (s, Xε (s)) − Dϕ (t0 , x0 )0 ds ε t0 1 t0 +ε ωf (|s − t0 |) Dϕ (t0 , x0 )0 ds + ε t0 ω1 (ε) . Finally the estimates of the terms containing δ. We have t0 +ε 2 ω1 (ε) Ax AX ] (s), X (s) − δ(t ) , x ds [δ(s) ε ε 0 0 0 ε t0 2 + ε

1 1 1 1 δ(s) A 2 Xε (s), A 2 (Xε (s) − x0 ) + A 2 (Xε (s) − x0 ), A 2 x0 ds

t0 +ε

t0

C ω1 (ε) + ε ω1 (ε)

t0 +ε

t0

[Xε (s)1 Xε (s) − x0 1 + x0 1 Xε (s) − x0 1 ] ds


323

estimating as before. Similarly we obtain t0 +ε 1 [δ(s) f (s, aε (s)) , Xε (s) − δ(t0 ) f (t0 , aε (s)) , x0 ]ds ω1 (ε) ε t0 and

t0 +ε 1 2 2 δ (s) ||Xε (s)||0 ds − δ (t0 ) ||x0 ||0 ω1 (ε) . ε t0

Using all these estimates in (5.72) yields 1 t0 +ε ω1 (ε) − l (t0 , x0 , aε (s)) ds ε t0 − ψt (t0 , x0 ) + Ax0 , Dψ (t0 , x0 ) 1 t0 +ε f (t0 , aε (s)) , Dψ (t0 , x0 ) ds. + B (x0 ) , Dϕ (t0 , x0 ) − ε t0 so that − ψt (t0 , x0 ) + Ax0 + B (x0 ) , Dψ (t, x0 ) 1 t0 +ε + [f (t0 , aε (s)) , −Dψ (t0 , x0 ) + l (t0 , x0 , aε (s))] ds ω1 (ε) . ε t0 The claim now follows upon taking first the infimum over a ∈ inside the integral and then letting ε → 0. 5.3. Applications and possible further developments The characterization of the value function given by Theorem 4 can be considered as a starting point for the dynamic-programming analysis of the optimal-control problem considered in this paper. It opens up several possibile directions for future research. We outline some of these issues below: 1. One such direction is obtaining verification theorems in the context of viscosity solutions. Verification theorems use the HJB equation to give sufficient (and/or necessary) conditions for optimality. They are natural and easy to prove if the value function is a smooth solution of the HJB equation (see, e.g., [15]). In the nonsmooth case such optimality conditions have been proved in the finite-dimensional case (see, e.g., [5], [39, Section 5.3]). For the dynamic programming of NavierStokes equation, such a verification theorem is proved in [33, 13], [32, Chapter I, Theorem 1.3.12] with a weaker definition of solution. Verification theorems and optimal synthesis results are also proved for an abstract infinite-dimensional problem in [39, Section 6.5] however these results are rather weak. Moreover the abstract problem considered there cannot be applied to problems with unbounded nonlinearities. Completion of the verification theory for the control of Navier-Stokes equations would require some new ideas, in particular the introduction of appropriate generalized sub- and supergradients that would be consistent with our definition

324


of viscosity solution. Optimality conditions provided by the theory could then be used to construct optimal feedback controls, at least theoretically. 2. Another important direction is the construction of ε-optimal controls and trajectories, and numerical approximations. There exist several numerical methods appropriate for solving optimal-control problems for infinite-dimensional systems (in particular for problems governed by nonlinear PDE). One of such promising and recently popular methods is the POD (proper orthogonal decomposition) method (see for instance [24, 29, 30]). It is a reduced order method that is based on projecting the control system onto finite-dimensional subspaces that are spanned by basis elements that contain some characteristics of the expected solution. It also produces finite-dimensional approximations of optimal controls. We are not aware of any applications of these methods to HJB equations in Hilbert spaces. 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 discrete dynamic programming. The existence and uniqueness results proved in this paper lay a groundwork for such an attempt. We refer the reader to [5] for an overview of such results in finite dimensions. A different approach to numerical approximation and construction of ε-optimal controls may use the specific nature of our problem, more precisely the weak continuity of the viscosity solution. In the light of this, we can try to approximate (uniformly on bounded sets) the unique viscosity solution (and therefore the value function) of the infinite-dimensional HJB equation by viscosity solutions of suitable finite-dimensional HJB equations connected to finite dimensional control problems. Procedures of this type have been used to prove existence of solutions for other infinite-dimensional problems [10, Part IV], [34]. Known numerical methods for finite-dimensional HJB equations and dynamic-programming techniques could then be used to produce ε-optimal controls and trajectories. Of course the case of Navier-Stokes equations is different from the problems considered in [10, 34] but the idea is worth pursuing. Sequential quadratic programming is a very promising method where the nonlinearity in the Navier-Stokes equation is approximated by Newton’s method to obtain a series of linear 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 sequence of approximate problems which fall within the realm of linear-control theory and can be solved effectively by well-established methods. An interesting theoretical question would then be to establish the convergence of such a sequence of approximate solutions to the unique viscosity solution proved in this paper. 3. Optimal control with state constraint is an important component of flow control as it allows us to deal with the task of controlling interesting regions in the flow domain where we have concentrated turbulence activity, see for example [14]. In the Hamilton-Jacobi framework this will correspond to an initial-boundary value problem (boundary value problem for the stationary case) which would be a substantial additional step that needs to be addressed in the future. 4. As we pointed out in the introduction, unique solution of the Hamilton-Jacobi equation corresponding to the dynamic programming of the Navier-Stokes equations have been established for a neighborhood of the origin in [4] using different


325

methods. The uniqueness of viscosity solution proved in this paper may be used to extend this local result to a global one. 5. Finally we would like to mention that we believe that our ideas and methods can be extended to handle the important case of control of three-dimensional Navier-Stokes equations. The essential reason for the analysis to be limited to two dimensions was the lack of a global-solvability theorem and the related continuousdependence theorems for the Navier-Stokes equations in three dimensions. The form of the Hamilton-Jacobi equations however is independent of the spatial dimensions. ´ ¸ ch was partially supported by NSF grant DMS 0098565. Acknowledgements. A. Swie

References 1. V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Anal. 31, (1998) 15–31. 2. V. Barbu & G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces, Research Notes in Math. vol. 86, Pitman, Boston, MA 1983. 3. V. Barbu & G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces; Variational and semigroup approaches, Ann. Mat. Pura Appl. 4 142, (1985) 303–349. 4. V. Barbu & S. S. Sritharan, H∞ -Control theory of fluid dynamics, Proc. R. Soc. Lond., Ser. A 454, (1998) 3009–3033. 5. M. Bardi, & I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 1997. 6. P. Cannarsa, F. Gozzi & H. M. Soner, A dynamic programming approach to nonlinear boundary control problems of parabolic type, J. Funct. Anal. 117, (1992) 25–61. 7. P. Cannarsa & M. E. Tessitore, Infinite dimensional Hamilton–Jacobi equations and Dirichlet boundary control problems of parabolic type, SIAM J. Control Optim. 34, (1996) 1831–1847. 8. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc., New York, 1981. 9. M. G. Crandall & P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, (1983) 1–42. 10. M. G. Crandall & P. L. Lions, Hamilton–Jacobi equations in infinite dimensions. Part IV: Hamiltonians with unbounded linear terms, J. Funct. Anal. 90, (1990) 237–283. Part V: Unbounded linear terms and B–continuous solutions, J. Funct. Anal. 97, (1991) 417– 465. Part VI: Nonlinear A and Tataru’s method refined, in Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991), pp. 51–89, Lecture Notes in Pure and Appl. Math., 155, Dekker, New York, 1994. Part VII: The HJB equation is not always satisfied, J. Funct. Anal. 125, (1994) 111–148. 11. R. Daley, Atmospheric Data Analysis, Cambridge University Press, Cambridge, 1991. 12. G. Da Prato & A. Debussche, Dynamic programming for the stochastic NavierStokes equations, Math. Model. Numer. Anal. 34, (2000) 459–475. 13. H. O. Fattorini & S. S. Sritharan, Necessary and Sufficient Conditions for Optimal Controls in Viscous Flow, Proc. Roy. Soc. Edinburgh, Series A, 124, (1994) 211–251. 14. H. O. Fattorini & S. S. Sritharan, Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion, Appl. Math. Optim. 38, (1998) 159–192. 15. W. H. Fleming & H. M. Soner, Controlled Markov processes and viscosity solutions, Springer-Verlag, Berlin, New-York, 1993.

326


16. A. V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications, Translations of Mathematical Monographs, Vol. 187, American Mathematical Society, 1999. ´ ¸ ch, Second order Hamilton-Jacobi equations in Hilbert 17. F. Gozzi, E. Rouy, & A. Swie spaces and stochastic boundary control, SIAM J. Control Optim. 38, (2000) 400–430. ´ ¸ ch, Hamilton-Jacobi-Bellman equations for the optimal control of 18. F. Gozzi & A. Swie the Duncan-Mortensen-Zakai equation, J. Funct. Anal. 172 (2000) 466–510. 19. M. D. Gunzburger, Editor, Flow Control, Springer-Verlag, New York, 1995. 20. M. Heinkenschloss, Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes flow, Appl. Optim. 15, (1998) 178–203. 21. C. Henoch & J. Stace, Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force, Phys. Fluids 7, (1995) 1371–1383. 22. H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal. 105, (1992) 301–341. 23. M. Kocan & P. Soravia, A viscosity approach to infinite-dimensional Hamilton-Jacobi equations arising in optimal control with state constraints, SIAM J. Control Optim. 36, (1998) 1348–1375. 24. K. Kunisch & S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102, (1999) 345–371. 25. X. Li & J. Yong, Optimal control theory for infinite-dimensional systems, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1995. 26. J. L. Menaldi & S. S. Sritharan, Impulse control of stochastic Navier-Stokes equation, to appear in Nonlinear Anal. 27. H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, New York, 1978. 28. S. S. Ravindran, Numerical Approximation of Optimal flow Control by SQP Methods, in Optimal Control of Viscous Flow, S. S. Sritharan, Editor, SIAM, Philadelphia, 1998, pp. 181–198. 29. S. S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Internat. J. Numer. Methods Fluids 34, (2000) 425–448. 30. S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput. 15, (2000) 457–478. 31. K. Shimano, A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces, preprint, 2001. 32. S. S. Sritharan, Editor, Optimal Control of Viscous Flow, SIAM, Philadelphia, 1998. 33. S. S. Sritharan, Dynamic programming of the Navier-Stokes equation, Systems and Control Letters 16, (1991), 299–307. ´ ¸ ch, Unbounded second order partial differential equations in infinite34. A. Swie dimensional Hilbert spaces, Comm. Partial Differential Equations 19, (1994) 1999– 2036. 35. D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl. 163, (1992) 345–392. 36. D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach, J. Differential Equations 111, (1994) 123–146. 37. R. Temam, The Navier-Stokes Equations, North Holland, 1977. 38. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983.


327

39. J. Yong & X.Y. Zhou, Stochastic Control: Hamiltonian Systems and HJB equations, Springer-Verlag, Berlin, New-York, 1999. 40. C. Wunsch, The Ocean Circulation Inverse Problem, Cambridge University Press, 1996. Dipartimento di Matematica per le Decisioni Economiche, Finanziarie e Assicurative Università di Roma “La Sapienza” Via del Castro Laurenziano 9, 00161 Roma, Italy and Space & Naval Warfare Systems Center, Code D73H San Diego, CA 92152-5001, USA and School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, USA (Accepted January 30, 2002) c Springer-Verlag (2002) Published online June 4, 2002 –