Exact controllability in projections for three-dimensional Navier-Stokes equations

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www.elsevier.com/locate/anihpc

Exact controllability in projections for three-dimensional Navier–Stokes equations

Contrôlabilité exacte en projections pour les équations de Navier–Stokes tridimensionnelles

Armen Shirikyan

Laboratoire de Mathématiques, Université de Paris-Sud XI, Bâtiment 425, 91405 Orsay Cedex, France Received 27 December 2005; accepted 7 April 2006

Available online 20 September 2006

Abstract

The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus inR³. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution for any initial function and a large class of external forces.

Résumé

L’article est consacré à l’étude de propriétés de contrôlabilité pour les équations de Navier–Stokes 3D dans un domaine borné.

On établit une condition suffisante sous laquelle le problème en question est exactement contrôlable en toute projection de dimension finie. Notre condition suffisante est vérifiée pour tout tore dansR³. Les démonstrations sont basées sur un développement d’une approche générale introduite par Agrachev and Sarychev dans le cas 2D. Comme une conséquence simple du résultat de contrôlabilité, on montre que le problème de Cauchy pour le système de Navier–Stokes 3D possède une unique solution forte pour toute donnée initiale et une grande classe de forces extérieures.

MSC:35Q30; 93B05; 93C20

Keywords:Exact controllability in projections; 3D Navier–Stokes system; Agrachev–Sarychev method

E-mail address:Armen.Shirikyan@math.u-psud.fr (A. Shirikyan).

doi:10.1016/j.anihpc.2006.04.002

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0. Introduction

Let us consider the three-dimensional Navier–Stokes (NS) system

˙

u+(u,∇)u−νu+ ∇p=f (t, x), divu=0, (0.1)

where the space variables x =(x1, x2, x3) belong to a three-dimensional torus T³⊂R³, ν >0 is the viscosity, u=(u1, u2, u3)andp are unknown velocity field and pressure, and f (t, x) is an external force. Suppose that f is represented in the form

f (t, x)=h(t, x)+η(t, x), (0.2)

wherehis a given function and ηis a control taking on values in afinite-dimensional subspaceE⊂L²(T³,R³).

Eqs. (0.1), (0.2) are supplemented with the initial condition

u(0)=u0, (0.3)

whereu0∈H¹(T³,R³)is a divergence-free vector field. Let us denote byH the space of functionsu∈L²(T³,R³) such that divu=0 onT³. We fix an arbitrary subspaceF⊂H and denote byP_F:H→H the orthogonal projection ontoF. Problem (0.1), (0.2) is said to becontrollable in a time T >0 for the projection to F if for any initial functionu₀and anyuˆ∈F there exists an infinitely smooth controlη:[0, T] →Esuch that (0.1)–(0.3) has a unique strong solutionu(t;η), which satisfies the relation

P_Fu(T;η)= ˆu. (0.4)

One of the main results of this paper says that if the spaceEis sufficiently large, then for anyT >0 andν >0 and any finite-dimensional subspaceF ⊂Hproblem (0.1), (0.2) is controllable in timeT for the projection toF.

A general approach for studying controllability of PDEs in finite-dimensional projections was introduced by Agrachev and Sarychev in the landmark article [1] (see also [2]). They considered the 2D NS system on a torus and proved that it is controllable for the projection to any finite-dimensional spaceF, with a control function taking on values in a fixed subspaceE. We emphasise that the time of controlT can be chosen arbitrarily small, and the control spaceEdoes not depend onνandT.¹The Agrachev–Sarychev approach is based on the concept ofsolid controlla- bility(cf. Definition 2.6 of the present paper). They construct explicitly an increasing sequence of finite-dimensional subspaces{E_k}k0such thatE₀=E, and the following two properties hold.

(i) There is an integerN1 such that the NS system is solidly controllable by anE_N-valued control.

(ii) If the NS system is solidly controllable by anEk-valued control for some integerk1, then it is solidly controllable by anEk−1-valued control.

These two assertions imply the required result.

In this paper, we take a slightly different viewpoint based on uniform approximate controllability.²Namely, we shall say that the NS system (0.1), (0.2) isuniformly approximately controllable (UAC) if for any constantε >0, any initial functionu0, and any compact subsetK of the phase space there is a continuous mappingΨ fromK to the space ofE-valued controls such that for every uˆ∈K problem (0.1)–(0.3) with η=Ψ (u)ˆ has a unique strong solutionu(t;η), which satisfies the inequality

u(T;η)− ˆu< ε, (0.5)

where·denotes theL²-norm. It turns out that assertions (i) and (ii) remain valid for the 3D NS system if we replace the solid controllability by uniform approximate controllability (cf. [16]). Hence, we prove that ifEis sufficiently large, then problem (0.1), (0.2) is UAC by anE-valued control. The required result on exact controllability in finite- dimensional projections is a simple consequence of the above property. Indeed, letB_F(R)be the closed ball inF of radiusRcentred at origin and letK=B_F(R). In this case, it follows from (0.5) that

P_Fu(T;Ψ (u))ˆ − ˆu< ε for anyuˆ∈B_F(R). (0.6)

1 It is shown in [2] that the 2D Euler and Navier–Stokes equations are controllable by a control of dimension four.

2 Note that the concept of uniform approximate controllability is implicitly present in the Agrachev–Sarychev argument [1].

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The functionΦ:uˆ→^PFu(T , Ψ (u))ˆ is continuous fromBF(R) toF. Using the Brouwer fixed point theorem and inequality (0.6), it is easy to show (see Proposition 1.1) thatΦ(B_F(R))contains the ballB_F(R−ε). SinceR >0 is arbitrary, we conclude that (0.4) holds for anyuˆ∈F and an appropriateE-valued control functionη.

In conclusion, we note that the problem of controllability and stabilisation for the Navier–Stokes and Euler equations was in the focus of attention of many researchers; for instance, see the papers [10,5–7,15,11,8,9,14,3,12,13,4]

and references therein. However, the powerful techniques developed in those papers do not apply to the present setting because of the specific type of control we are interested in.

The paper is organised as follows. In Section 1, we recall a simple sufficient condition for surjectivity of a continuous mapping in a finite-dimensional space and formulate two perturbative results on unique solvability of NS-type equations. Section 2 contains the formulations of the main results of this paper. We also discuss some corollaries on solid controllability in finite-dimensional projection and the Cauchy problem for the 3D NS system. The proofs are presented in Section 3. Finally, in Appendix A, we prove an auxiliary result used in Section 3.

Notation.We denote byR₊the half-line[0,+∞)and byJ_T the interval[0, T]. Ifs1 andr0 are some integers, then we setJT(r, s)= [tr, tr+1), wheretr=rT /s. LetJ⊂R₊be a closed interval, letD⊂R³be a bounded domain, letXbe a Banach space with a norm · , and letKbe a metric space. We shall use the following functional spaces.

• L^p(J, X)is the space of measurable functionsu:J→Xwith finite norm uL^p(J,X):=

J

u(t )^p

Xdt 1/p

, (0.7)

where · Xstands for the norm inX. Ifp= ∞, then (0.7) is replaced by uL^∞(J,X):=ess sup

t∈J

u(t )

X. We shall writeL^p(J )instead ofL^p(J,R).

• L^p_loc(R₊, X) is the space of functions u:R₊→X whose restriction to any finite interval J ⊂R₊ belongs toL^p(J, X).

• C^k(J, X)is the space of continuous functionsu:J→Xthat arektimes continuously differentiable. In the case k=0, we shall writeC(J, X).

• C(K, X)is the space of continuous functionsu:K→X. IfX=R, then we writeC(K).

• H^s(D,R³)is the space of vector functions(u1, u2, u3)whose components belong to the Sobolev space of orders.

In the cases=0, it coincides with the Lebesgue spaceL²(D,R³).

• H,V,U, andXT are standard functional spaces arising in the theory of Navier–Stokes equations; they are defined in Section 1.2.

1. Preliminaries

1.1. Image of continuous mappings

Let F be a finite-dimensional vector space with a norm · F, letBF(R) be the closed ball inF of radius R centred at origin, and letΦ:B_F(R)→F be a continuous mapping. The following result is a simple consequence of the Brouwer theorem.

Proposition 1.1.Suppose there is a constantε∈(0, R)such that Φ(u)−u

Fε for anyu∈BF(R). (1.1)

ThenΦ(B_F(R))⊃B_F(R−ε).

Proof. Let us fix any pointuˆ∈B_F(R−ε)and consider the continuous mapping Ψ:B_F(R)→F, Ψ (u)= ˆu−Φ(u)+u.

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It follows from (1.1) thatΨ (BF(R))⊂BF(R). Therefore, by the Brouwer theorem (e.g., see Section 1.19 in [17]), Ψ has a fixed pointu₀∈B_F(R). Direct verification shows thatΦ(u₀)= ˆu. Thus, any pointuˆ∈B_F(R−ε)has a preimage, and we obtain the required inclusion. 2

1.2. Strong solutions of Navier–Stokes type equations

We first introduce some standard functional spaces arising in the theory of 3D Navier–Stokes (NS) equations. Let H=

u∈L² D,R³

: divu=0 inD,(u,n)|∂D=0 ,

wherenis the outward unit normal to ∂D, and let Π be the orthogonal projection inL²(D,R³)onto the closed subspaceH. We denote byH^s =H^s(D,R³)the space of vector functionsu=(u₁, u₂, u₃)with components in the Sobolev class of ordersand byH₀^s(D,R³),s >1/2, the space of functionsu∈H^s vanishing on∂D. Let · sbe the usual norm inH^s. In the cases=0, we write · . Define the spaces

V =H₀¹ D,R³

∩H, U=H² D,R³

∩V and endow them with natural norms.

It is well known (e.g., see [18]) that the NS system is equivalent to the following evolution equation inH:

˙

u+νLu+B(u)=f (t ), (1.2)

whereL= −Π is the Stokes operator andB(u)=Π{(u,∇)u}is the bilinear form resulting from the nonlinear term in the original system. LetE⊂Ube a finite-dimensional vector space and letE^⊥be its orthogonal complement inH. Denote byP=^PE andQ=^QE the orthogonal projections inH onto the subspacesEandE^⊥, respectively.

Along with (1.2), consider the Cauchy problem

˙

w+νL_Ew+^Q

B(w)+B(v, w)+B(w, v)

=f (t ), (1.3)

w(0)=w₀, (1.4)

whereLE=^QL,B(v, w)=Π{(v,∇)w}, andv∈L⁴(JT, V )andf∈L²(JT, E^⊥)are given functions. We set XT =C(J_T, V )∩L²(J_T, U ), XT(E)=C

J_T, V ∩E^⊥

∩L²

J_T, U∩E^⊥ . The following result is established in [16, Section 1.4] (see Theorem 1.8).

Proposition 1.2.For any ν >0andR >0 there are positive constantsεandC such that the following assertions hold.

(i) Letvˆ∈L⁴(J_T, H¹),fˆ∈L²(J_T, E^⊥), andw₀∈V ∩E^⊥be some functions such that problem(1.3), (1.4)with v= ˆv,f = ˆf,w₀=w₀has a solutionw∈XT(E). Suppose that

ˆv_L⁴_(J_T_,H¹₎R, ˆf_L²_(J_T_,E⊥)R, w_X_T R.

Then, for any triple(v, f, w₀)satisfying the inequalities

v− ˆvL⁴(J_T,H¹)ε, f − ˆfL²(J_T,E^⊥)ε, w0−w0V ε, (1.5) problem(1.3), (1.4)has a unique solutionw∈XT(E).

(ii) Let

R:L⁴ J_T, H¹

×L²

J_T, E^⊥

×

V∩E^⊥

→XT(E)

be an operator that is defined on the set of functions(v, f, w₀)satisfying(1.5)and takes each triple(v, f, w₀)to the solutionw∈XT(E)of (1.3), (1.4). ThenRis uniformly Lipschitz continuous, and its Lipschitz constant does not exceedC.

We now consider Eq. (1.3) in whichEis a finite-dimensional vector space spanned by some eigenfunctions of the Stokes operatorL. Namely, let{e_j}be a complete set of normalised eigenfunctions forL, letH_Nbe the vector span

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of{ej,1jN}, and letH_N^⊥be the orthogonal complement ofHNin the spaceH. We denote byP_N andQ_Nthe orthogonal projections inH onto the subspacesH_N andH_N^⊥, respectively.

Let us consider the equation

˙

w+νL_N(w+v)+^QNB(w+v)=f (t ), (1.6)

whereLN=^QNL.

Proposition 1.3.For anyR >0andν >0there is an integerN01and a constantC >0such that the following assertions hold.

(i) LetNN₀be an integer and let functionsv∈XT,f ∈L²(J_T, H_N^⊥), andw₀∈H_N^⊥∩V satisfy the inequalities

v_X_T R, fL²(J_T,H )R, w₀V R. (1.7)

Then problem(1.6), (1.4)has a unique solutionw∈XT(H_N).

(ii) Let S be an operator that takes each triple (v, f, w₀)satisfying (1.7)to the solution w∈XT(H_N) of (1.6), (1.4). ThenSis uniformly Lipschitz continuous in the corresponding spaces, and its Lipschitz constant does not exceedC.

Proof. The existence and uniqueness of solution is established in [16] (see Proposition 1.10). The proof of (ii) is rather standard, and we only outline it.

Letw¹, w²∈XT(H_N)be two solutions of (1.6), (1.4) that correspond to some triples(vⁱ, fⁱ, w₀ⁱ),i=1,2. Then the functionw=w¹−w²∈XT(H_N)is a solution of the problem

˙

w+νL_Nw=g(t ), w(0)=w¹₀−w₀², where we set

g(t )=

f¹−f²

−νL_N

v¹−v²

−^QN

B

w¹+v¹

−B

w²+v² .

Repeating literally the argument used in Step 2 of the proof of Proposition 1.10 in [16], we show that ifNis sufficiently large, then

w_XT C₁f¹−f²

L²(JT,H )+v¹−v²

L²(JT,U )+w₀¹−w²₀

V

+1

2w_XT, (1.8)

whereC₁>0 is a constant depending onlyT,R, andν. Inequality (1.8) implies the required result. 2 2. Main results

2.1. Exact controllability in observed projections Consider the controlled Navier–Stokes (NS) equations

˙

u+νLu+B(u)=h(t )+η(t ), (2.1)

u(0)=u0, (2.2)

whereh∈L²_loc(R₊, H )andu₀∈V are given functions andηis a control function with range in a finite-dimensional vector space E⊂U. For any h∈L²(J_T, H ),u0∈V, and T >0, we denote by Θ_T(h, u0)the set of functions η∈L²(JT, H )for which problem (2.1), (2.2) has a unique solution u∈XT. It follows from Proposition 1.2 with E= {0}andv≡0 thatΘT(h, u0)is an open subset ofL²(JT, H ).

Let us fix a constantT >0, a finite-dimensional spaceF ⊂H, and a projectionP_F:H→H ontoF.

Definition 2.1.Eq. (2.1) withη∈L²(J_T, E)is said to beP_F-controllable in timeT if for anyu₀∈V anduˆ∈F there isη∈Θ_T(h, u₀)∩L²(J_T, E)such that

P_Fu(T )= ˆu, (2.3)

whereu∈XT denotes the solution of (2.1), (2.2).

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To formulate the main result of this paper, we introduce some notation. For any finite-dimensional subspaceE⊂U, we denote byF(E)the largest vector spaceG⊂Usuch that any elementη₁∈Gis representable in the form

η₁=η− k

j=1

α_jB ζ^j

,

whereη, ζ¹, . . . , ζ^k∈Eare some vectors andα₁, . . . , α_k are non-negative constants. SinceBis a quadratic operator, we see thatF(E)⊂Uis a well-defined finite-dimensional subspace containingE. For a finite-dimensional subspace E⊂U, we set

E₀=E, E_k=F(E_k₋₁) fork1, E_∞= ∞ k=1

E_k. (2.4)

The following theorem is the main result of this paper.

Theorem 2.2.Leth∈L²_loc(R₊, H )and letE⊂Ube a finite-dimensional subspace such thatE_∞is dense inH. Then for anyT >0, any finite-dimensional subspaceF ⊂H, and any projectionP_F:H→H ontoF the Navier–Stokes system(2.1)withη∈L²(J_T, E)isP_F-controllable in timeT. Moreover, the control functionηcan be chosen from the spaceC^∞(J_T, E).

In the case of a general bounded domain, it is difficult to check whetherE_∞is dense inH. However, Theorem 2.2 remains valid for the NS equation (2.1) on a 3D torus, and it is shown in [16, Section 2.3] that³ ifE⊃HN for a sufficiently largeN1, thenE_∞contains all the eigenfunctions ofL. Thus, we obtain the following result.

Corollary 2.3.LetT³be a torus inR³. Then there is an integerN1such that if the control spaceEcontainsH_N, then for any constantsν >0andT >0, any functionh∈L²_loc(R₊, H ), any finite-dimensional subspaceF ⊂H, and any projectionP_F:H→H ontoF the Navier–Stokes system(2.1)onT³withη∈L²(J_T, E)isP_F-controllable in timeT, and the control functionηcan be chosen from the spaceC^∞(J_T, E).

The proof of Theorem 2.2 is based on a property of uniform approximate controllability for (2.1). That concept is of independent interest and is discussed in the next subsection.

2.2. Uniform approximate controllability

Let us fix anyT >0 andh∈L²(J_T, H )and denote byR(u₀, η)an operator that is defined on the set D(R)=

(u₀, η)∈V×L²(J_T, H ): η∈Θ_T(h, u₀)

and takes each pair(u0, η)∈D(R)to the solutionu∈XT of problem (2.1), (2.2). Proposition 1.2 withE= {0}and v≡0 implies thatD(R)is an open subset ofV ×L²(J_T, H ), andRis locally Lipschitz continuous onD(R). For anyt∈JT, we denote byRt(u0, η)the restriction ofR(u0, η)to the timet.

LetX⊂L²(JT, H )be an arbitrary vector space, not necessarily closed. We endowXwith the norm ofL²(JT, H ).

Definition 2.4.Eq. (2.1) with η∈X is said to beuniformly approximately controllable in time T if for any initial pointu₀∈V, any compact setK⊂V, and anyε >0 there is a continuous function

Ψ:K→X∩Θ_T(h, u₀) such that

sup

ˆ u∈K

RT

u0, Ψ (u)ˆ

− ˆu

V < ε. (2.5)

3 Recall thatH_Ndenotes the vector space spanned by the firstNeigenfunctions of the Stokes operatorL.

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The following result shows that, under the conditions of Theorem 2.2, Eq. (2.1) is uniformly approximately controllable (UAC).

Theorem 2.5.Leth∈L²_loc(R+, H )and letE⊂Ube a finite-dimensional subspace such thatE_∞is dense inH. Then for anyT >0andν >0the Navier–Stokes system(2.1)withη∈C^∞(J_T, E)is UAC in timeT.

Theorem 2.5 will be established in Section 3. Here we show that the exact controllability in a projection is a simple consequence of UAC; in the next subsection, we deduce some corollaries from Theorems 2.2 and 2.5.

Proof of Theorem 2.2. Let us fix a timeT >0, an initial pointu₀∈V, and a projectionP_F:H→Honto a finite- dimensional subspaceF ⊂H. Recall thatB_F(R)stands for the closed ball in F of radiusR centred at origin and denote byCthe norm ofP_F:H→H. Let us fix anyR > Cand chooseδ >0 so small that

sup

ˆ u∈B_F(R)

e⁻^δLuˆ− ˆu1

2. (2.6)

Denote byKthe image ofB_F(R)under e⁻^δL. This is a compact subset ofV, and by Theorem 2.5, there is a continuous mapping⁴

Ψ:K→C^∞(J_T, E)∩Θ_T(h, u₀) such that

sup

ˆ v∈K

RT

u₀, Ψ (v)ˆ

− ˆv

V<1

2. (2.7)

It follows (2.6) and (2.7) that sup

ˆ u∈B_F(R)

RT

u₀, Ψ e⁻^δLuˆ

− ˆu<1.

Therefore the continuous mapping Φ:BF(R)→F, uˆ→^PFRT

u0, Ψ e⁻^δLuˆ

,

satisfies inequality (1.1) withε=C. Hence, by Proposition 1.1, we haveΦ(BF(R))⊃BF(R−C). In particular, it follows that for anyuˆ∈BF(R−C)there isη∈C^∞(JT, E)∩ΘT(h, u0)such that^PFRT(u0, η)= ˆu. SinceR > C is arbitrary, we obtain the conclusion of Theorem 2.2. 2

2.3. Solid controllability and Cauchy problem for the NS system

In this subsection, we establish some corollaries of Theorems 2.2 and 2.5. Let E⊂U and F ⊂H be finite- dimensional subspaces, letP_F:H→H be a projection ontoF, and letT >0 be a constant.

Definition 2.6.The control system (2.1) withη∈L²(J_T, E) is said to besolidlyP_F-controllable in timeT if for any R >0 and u₀∈V there is a constant ε >0 and a compact set C⊂L²(J_T, E)∩Θ_T(h, u₀)such that, for any continuous mappingS:C→F satisfying the inequality

sup

η∈C

S(η)−^PFRT(u₀, η)ε, (2.8)

we haveS(C)⊃B_F(R).

Proposition 2.7.Under the conditions of Theorem2.2, for anyT >0, any finite-dimensional subspaceF ⊂H, and any projectionP_F:H→H ontoF, Eq.(2.1)withη∈L²(J_T, E)is solidlyP_F-controllable in timeT >0.

4 Recall that the control spaceC^∞(J_T, E)∩Θ_T(h, u₀)is endowed with the metric generated by the norm inL²(J_T, H ).

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Proof. Let us fix any constantR >0, functionu0∈V, and subspaceF ⊂H. As was shown in the proof of Theo- rem 2.2 (see Section 2.2), there is a continuous mappingΨ:B_F(R+2)→L²(J_T, E)∩Θ_T(h, u₀)such that

sup

ˆ

u∈BF(R+2)

P_FRT

u0, Ψ (u)ˆ

− ˆu1. (2.9)

Let us setC=Ψ (B_F(R+2)). Since dimF <∞andΨ is continuous, we conclude thatC is a compact subset of L²(J_T, E)∩Θ_T(h, u0). LetS:C→F be an arbitrary continuous mapping such that (2.8) holds withε=1. Then it follows from (2.9) that the mappingS◦Ψ:BF(R+2)→F satisfies the inequality

sup

ˆ

u∈BF(R+2)

S◦Ψ (u)ˆ − ˆu

V 2.

Applying Proposition 1.1, we see thatS◦Ψ (B_F(R+2))⊃B_F(R). It follows thatS(C)⊃B_F(R). SinceR >0 was arbitrary, this completes the proof of Proposition 2.7. 2

We now show that the control functionηin Theorem 2.2 can be taken from a finite-dimensional subspace. Namely, we have the following result.

Proposition 2.8. Suppose that the conditions of Theorem 2.2 are fulfilled, and let X be a vector space dense inL²(J_T, E). Then for any positive constants T and R, any initial function u₀∈V, any subspaceF ⊂H with dimF <∞, and any projection^PF:H→H ontoF, there is a ballBin a finite-dimensional subspaceY ⊂Xsuch that

P_FRT(u₀, B)⊃B_F(R). (2.10)

In particular, we can takeX=C^∞(J_T, E).

Proof. By Proposition 2.7, Eq. (2.1) with η∈L²(J_T, E) is solidly P_F-controllable in time T. Let ε >0 and C⊂L²(J_T, E)∩Θ_T(h, u₀)be the corresponding constant and compact set entering Definition 2.6. It follows from Proposition 1.2 withE= {0}andv≡0 thatΘ_T(h, u₀)is an open subset ofL²(J_T, E). Therefore there isδ >0 such that

O_δ(C)=

η∈L²(J_T, E): dist(η,C)δ ⊂Θ_T(h, u₀), where we set

dist(η,C)=inf

ζ∈Cη−ζL²(J_T,H ).

Furthermore, sinceXis dense inL²(JT, E), we can find a finite-dimensional subspaceY ⊂Xsuch that sup

η∈CP_Yη−ηL²(J_T,E)δ, (2.11)

whereP_Y denotes the orthogonal projection inL²(J_T, E)ontoY. It follows that

P_YC⊂O_δ(C)⊂Θ_T(h, u₀). (2.12)

By Proposition 1.2, the operatorR(u₀,·):Θ_T(h, u₀)→XT is locally Lipschitz continuous. Therefore, takingδ >0 sufficiently small, we deduce from (2.11) and (2.12) that

sup

η∈C

RT(u0, PYη)−RT(u0, η)

V ε

C,

whereCis the norm of^PF regarded as an operator fromV toH. Thus, the mappingS(η)=^PFRT(u0, PYη)satisfies (2.8). Hence, by Proposition 2.7, we have^PFRT(u0, PYC)⊃BF(R). It remains to note thatPYCis contained in a ball of the finite-dimensional spaceY ⊂X. 2

We now consider the Cauchy problem for the NS equation (1.2). LetG⊂H be a closed vector space. For any u₀∈V,T >0, andν >0, letΞ_{T ,ν}(G, u₀)be the set of functionsf ∈L²(J_T, G)for which problem (1.2), (2.2) has a unique solutionu∈XT. IfE⊂Gis a closed subspace, then we denote byGEthe orthogonal complement ofE inGand byQ(T , G, E)the orthogonal projection inL²(J_T, G)onto the subspaceL²(J_T, GE).

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Proposition 2.9.LetE⊂Ube a finite-dimensional subspace such thatE_∞is dense inHand letG⊂H be a closed subspace containingE. ThenΞ_{T ,ν}(G, u₀)is a non-empty open subset ofL²(J_T, G)such that

Q(T , G, E)Ξ_{T ,ν}(G, u₀)=L²(J_T, GE) for anyT >0,ν >0,u₀∈V .

Proof. The fact thatΞ_{T ,ν}(G, u₀)is open follows immediately from Proposition 1.2. The other claims of the proposition are equivalent to the following property: for any h∈ L²(J_T, GE) there is η∈L²(J_T, E) such that h+η∈Ξ_{T ,ν}(G, u₀). This is a straightforward consequence of Theorem 2.2. 2

3. Proof of Theorem 2.5

3.1. Scheme of the proof

LetEbe a finite-dimensional vector space and letE1=F(E)(see (2.4)). Along with Eq. (2.1), consider two other control systems:

˙ u+νL

u+ζ (t ) +B

u+ζ (t )

=h(t )+η(t ), (3.1)

˙

u+νLu+B(u)=h(t )+η₁(t ). (3.2)

Hereηandζ areE-valued controls andη₁is anE₁-valued control. Let us fix a constantε >0, an initial pointu₀∈V, a compact set K⊂V, and a vector spaceX⊂L²(J, H ). Eq. (2.1) withη∈X is said to be uniformly(ε, u₀,K)- controllableif there is a continuous mapping

Ψ:K→X∩Θ_T(h, u₀)

such that (2.5) holds. In what follows, ifε,u₀, andK are fixed in advance, then the above property will be called uniformε-controllability.

The concept of uniformε-controllability for (3.1) is defined in a similar way. Namely, letΘ_T(h, u0)be the set of pairs(η, ζ )∈L²(JT, H )×L⁴(JT, H²)for which problem (3.1), (2.2) has a unique solutionu∈XT and letRbe an operator that is defined on the set

D R

=

(u0, η, ζ )∈V ×L²(JT, H )×L⁴ JT, H²

: (η, ζ )∈ΘT(h, u0)

and takes each triple(u₀, η, ζ )∈D(R)to the solutionu∈XT of (3.1), (2.2). Rewriting Eq. (3.1) in the form

˙

u+νLu+B(u)+B(u, ζ )+B(ζ, u)=h(t )+η−νLζ−B(ζ )

and applying Proposition 1.2 withE= {0}, we see thatD(R)is an open subset ofV×L²(J_T, H )×L⁴(J_T, H²), and the operatorRis locally Lipschitz continuous onD(R).

Now letX⊂L²(J, H )×L⁴(J, H²)be a vector space, not necessarily closed. Eq. (3.1) with(η, ζ )∈Xis said to beuniformly(ε, u₀,K)-controllableif there is a continuous mapping

Ψ:K→X∩Θ_T(h, u₀) such that

sup

ˆ u∈K

RT

u0,Ψ ( u)ˆ

− ˆu

V < ε, (3.3)

whereRt(u0, η, ζ )denotes the restriction ofR(u0, η, ζ )to the timet.

The proof of Theorem 2.5 is based on the following three propositions (cf. Propositions 3.1 and 3.2 and Section 2.2 in [16]). Let us fix a constantε >0, an initial pointu₀∈V, and a compact subsetK⊂V.

Proposition 3.1 (Extension principle). Let E⊂U be a finite-dimensional vector space. Then Eq. (2.1) with η∈ C^∞(J_T, E)is uniformlyε-controllable if and only if so is Eq.(3.1)with(η, ζ )∈C^∞(J_T, E×E).

Proposition 3.2(Convexification principle). LetE⊂U be a finite-dimensional subspace and letE₁=F(E). Then Eq.(3.1)with(η, ζ )∈C^∞(J_T, E×E)is uniformlyε-controllable if and only if so is Eq.(3.2)withη₁∈C^∞(J_T, E₁).

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Proposition 3.3.Let E⊂U be a finite-dimensional vector space such that E_∞ is dense in H. Then there is an integerk1depending onε,u₀, andKsuch that Eq.(2.1)withη∈C^∞(J_T, E_k)is uniformlyε-controllable.

If Propositions 3.1–3.3 are established, then for anyε >0,u₀∈V, andK⊂V we first use Proposition 3.3 to find an integerk1 such that Eq. (2.1) withη∈C^∞(J_T, E_k)is uniformly(ε, u₀,K)-controllable. Combining this property with Propositions 3.2 and 3.1 in whichE=E_k₋₁, we conclude that Eq. (2.1) withη∈C^∞(J_T, E_k₋₁)is uniformly(ε, u₀,K)-controllable. Repeating this argumentk−1 times, we see that the same property is true for Eq. (2.1) withη∈C^∞(J_T, E). Sinceε,u₀, andKare arbitrary, this completes the proof of Theorem 2.5.

To prove the above propositions, we repeat the scheme used in [16] (see Sections 2.2, 3.2, and 3.3). The important point now is that we have to follow carefully the dependence of controls on the final stateu. The proofs of Propo-ˆ sitions 3.1–3.3 are carried out in next three subsections. Here we formulate a lemma on uniformε-controllability; it will be used in Sections 3.2–3.4. As before, we fix a constantε >0, an initial pointu₀∈V, and a compact setK⊂V. Lemma 3.4.LetX, Y⊂L²(J_T, H )be vector spaces such thatXis contained in the closure ofY and Eq.(2.1)withη∈ Xis uniformlyε-controllable. Then there is a finite-dimensional subspaceY₀⊂Y such that Eq.(2.1)withη∈Y₀is uniformlyε-controllable.

To prove this lemma, it suffices to repeat the argument used in the proof of Proposition 2.8; we shall not dwell on it. Also note that an analogue of Lemma 3.4 is true for Eq. (3.1).

3.2. Extension principle: proof of Proposition 3.1

We need to show that if Eq. (3.1) with(η, ζ )∈C^∞(J_T, E×E)is uniformlyε-controllable, then so is Eq. (2.1) withη∈C^∞(J_T, E). SinceC^∞(J_T, E)is dense inL²(J_T, E), in view of Lemma 3.4, it suffices to establish that property for Eq. (2.1) withη∈L²(J_T, E).

Recall thatPandQstand for the orthogonal projection inH onto the subspacesEandE^⊥, respectively. Let Ψ:K→C^∞(J_T, E×E)∩Θ_T(h, u0), Ψ ( u)ˆ =

η(t,u), ζ (t,ˆ u)ˆ ,

be an operator for which (3.3) holds. We choose any sequence of functionsϕ_k∈C^∞(R)with the following properties:

0ϕk(t )1 for allt∈R, (3.4)

ϕ_k(t )=0 fort0 andtT , (3.5)

ϕk(t )=1 for 1/ktT −1/k. (3.6)

We now define a sequence of continuous mappingsΨ_k:K→L²(J_T, E)by the following rule:

• for anyuˆ∈Kandk1, set vk(t,u)ˆ =ϕk(t )ζ (t,u)ˆ +^PRt

u0,Ψ ( u)ˆ

, t∈JT; (3.7)

• denote byw_k(·,u)ˆ ∈XT(E)the solution of the problem⁵

˙

w+νL_Ew+^Q

B(w)+B(v_k, w)+B(w, v_k)

=f_k(t,u),ˆ

w(0)=^Qu₀, (3.8)

wherev_k=v_k(t,u)ˆ and fk(t,u)ˆ =^Q

h(t )−B

vk(t,u)ˆ

−νLvk(t,u)ˆ

; (3.9)

• denoteu_k(t,u)ˆ =v_k(t,u)ˆ +w_k(t,u)ˆ and defineΨ_k by the formula Ψ_k(u)ˆ =η_k(t,u)ˆ := ˙v_k+^P

νLu_k+B(u_k)−h

. (3.10)

5 We shall show that such a solution exists fork1.

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We claim that for sufficiently largek1 the functionΨk is well defined and continuous and satisfies (2.5). Indeed, let us write

v(t,u)ˆ =^PRt

u0,Ψ ( u)ˆ

, w(t,u)ˆ =^QRt

u0,Ψ ( u)ˆ . Thenw(·,u)ˆ ∈XT(E)is a solution of the problem

˙

w+νLEw+^Q

B(w)+B(v+ζ, w)+B(w, v+ζ )

=f (t,u),ˆ

w(0)=^Qu₀, (3.11)

wherev=v(t,u)ˆ andf (t,u)ˆ =^Q(h−B(v+ζ )−νL(v+ζ )). We wish to consider (3.8) as a perturbation of (3.11).

SinceKis compact andR(u₀,Ψ ( ·)):K→XT is continuous, we have sup

ˆ u∈K

v(·,u)ˆ

XT +w(·,u)ˆ

XT

<∞. (3.12)

It follows from (3.4), (3.6), and (3.7) that sup

ˆ u∈K

vk(·,u)ˆ −

v(·,u)ˆ +ζ (·,u)ˆ

L⁴(JT,U )→0 ask→ ∞. (3.13)

Combining this with standard estimates for the nonlinear term and the fact that dimE <∞, we conclude that (cf. (3.8) in [16])

sup

ˆ u∈K

f_k(·,u)ˆ −f (·,u)ˆ

L²(J_T,H )→0 ask→ ∞. (3.14)

Proposition 1.2 and relations (3.12)–(3.14) imply that there is an integerk₀1 such that, for anykk₀anduˆ∈K, problem (3.8) has a unique solutionw_k(·,u)ˆ ∈XT(E). Moreover, the functionuˆ→w_k(·,u)ˆ is continuous fromK toXT(E). It follows from (3.10) that the operatorΨ_kis well defined and continuous forkk₀.

Let us show thatΨ_ksatisfies (2.5) fork1. Since the resolving operator associated with (3.11) is locally Lipschitz continuous (see Proposition 1.2), for anyuˆ∈Kandkk0, we have

w_k(·,u)ˆ −w(·,u)ˆ

XT Cf_k(·,u)ˆ −f (·,u)ˆ

L²(J_T,H )+v_k(·,u)ˆ −v(·,u)ˆ

L⁴(J_T,V )

, whereC >0 does not depend onkandu. Combining this inequality with (3.13) and (3.14), we deriveˆ

sup

ˆ u∈K

w_k(·,u)ˆ −w(·,u)ˆ

XT →0 ask→ ∞. (3.15)

Now note that, in view of (3.5) and (3.7), we have RT

u₀, Ψ_k(u)ˆ

− ˆu

V RT

u₀, Ψ_k(u)ˆ

−RT

u₀,Ψ ( u)ˆ

V+RT

u₀,Ψ ( u)ˆ

− ˆu

V

w_k(·,u)ˆ −w(·,u)ˆ

V +RT

u₀,Ψ ( u)ˆ

− ˆu

V.

Taking the supremum overuˆ∈Kand using (3.3) and (3.15), we see thatΨksatisfies (2.5) for sufficiently largekk0. The proof of Proposition 3.1 is complete.

3.3. Convexification principle: proof of Proposition 3.2

We need to prove that if Eq. (3.2) with η1∈C^∞(JT, E1)is uniformlyε-controllable, then so is Eq. (3.1) with (η, ζ )∈L²(J_T, E)×L⁴(J_T, E); the converse assertion is obvious in view of Proposition 3.1. Let us outline the main idea.

LetΨ₁:K→L²(J_T, E₁)be a continuous mapping such that ˆ

ε:=sup

ˆ u∈K

RT

u₀, Ψ₁(u)ˆ

− ˆu

V < ε. (3.16)

By definition, the function u1(t,u)ˆ =Rt(u0, Ψ1(u))ˆ satisfies Eq. (3.2) in which η1=η1(·,u)ˆ :=Ψ1(u). We wishˆ to approximate u1(t,u)ˆ by a solution u(t,u)ˆ of Eq. (3.1) with some functions η(·,u), ζ (ˆ ·,u)ˆ ∈L^∞(JT, E). This approximation should be such that

sup

ˆ u∈K

u T ,u)ˆ

−u1(T ,u)ˆ

V ε− ˆε, (3.17)

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and the mappinguˆ→(η(·,u), ζ (ˆ ·,u))ˆ is continuous as an operator fromKto the spaceL²(JT, E)×L⁴(JT, E).

To constructu(t,u), one could try to apply the argument used in Section 3.3 of [16] for approximating individualˆ solutions. Unfortunately, it does not work because it is difficult to ensure that the resulting control functionsηandζ continuously depend onu. To overcome this difficulty, we first approximateˆ η1(t,u)ˆ by a family of piecewise constant controlsη˜1(t,u)ˆ with range in the convex envelope of a finite set not depending onuˆ(cf. Section 12.3 in [1]). We next repeat the scheme of [16] to construct an approximation for solutionsu˜1(t,u)ˆ corresponding toη˜1(t,u). A difficultˆ point of the proof is to follow the dependence of the control functions onu. In what follows, we shall omit the tildeˆ from the notation.

The realisation of the above scheme is divided into several steps. We begin with a generalisation of the concept of uniform approximate controllability.

Step1. LetA= {η^l₁, l=1, . . . , m} ⊂E1 be a finite set. For any integers1, denote by Ps(JT, A)the set of functionsη₁∈L²(J_T, E₁)satisfying the following properties:

• there are non-negative functionsϕ_l∈L^∞(J_T),l=1. . . , m, such that m

l=1

ϕ_l(t )=1, η₁(t )= m

l=1

ϕ_l(t )η₁^l for 0t < T;

• the functionsϕ_lare representable in the form ϕ_l(t )=

s−1

r=0

c_lrI_r,s(t ) for 0t < T ,

wherec_lr0 are some constants andI_r,sdenotes the indicator function of the intervalJ_T(r, s)= [t_r, t_r₊1)with t_r =rT /s.

The setP_s(J_T, A)is endowed with the metric d_P(η₁, ζ₁)=

m

l=1

ϕ_l−ψ_lL^∞(J_T), η₁, ζ₁∈P_s(J_T, A),

where{ϕ_l}and{ψ_l}are the families of functions corresponding toη₁andζ₁, respectively.

Recall that we have fixed a constantε >0, an initial pointu₀∈V, and a compact setK⊂V. We shall say that Eq. (3.2) withη₁∈P_s(J_T, A)isuniformlyε-controllableif there is a continuous⁶mappingΨ₁:K→P_s(J_T, A)such that Ψ₁(u)ˆ ∈Θ_T(h, u₀)for any uˆ∈K, and (3.16) holds. A proof of the following lemma is based on a standard argument of the control theory and is given in Appendix A.

Lemma 3.5.Let us assume that Eq.(3.2)withη₁∈C^∞(J_T, E₁)is uniformlyε-controllable. Then there is a finite set A= {η^l₁, l=1, . . . , m} ⊂E₁and an integers1such that Eq.(3.2)withη∈P_s(J_T, A)is uniformlyε-controllable.

LetΨ1:K→Ps(JT, A)be the function constructed in Lemma 3.5. We write Ψ₁(u)ˆ =η₁(t,u)ˆ =

m

l=1

ϕ_l(t,u)ηˆ ^l₁.

The definition of the spaceP_s(J_T, A)and of its metric imply that the functionsϕ_lhave the form ϕ_l(t,u)ˆ =

s−1

r=0

c_lr(u)Iˆ _r,s(t ), (3.18)

6 We emphasise that, in contrast to Definition 2.4 in which the vector spaceXis endowed with the norm ofL²(J_T, H ), the mappingΨ₁is required to be continuous with respect to the topology ofPs(J_T, A), which is stronger than that ofL²(J_T, H ).