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Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

  • Irena Lasiecka , Buddhika Priyasad and Roberto Triggiani ORCID logo EMAIL logo

Abstract

We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } of controls localized on { Γ ~ , ω } . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ of the boundary Γ = Ω . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } localized on { Γ ~ , ω } . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ( Ω ) for d = 3 ) and a corresponding Besov space for the thermal component, q > d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

MSC 2010: 35K05; 35Q30; 35R30; 93C20; 93S15

Dedicated to Mikhail Klibanov


Award Identifier / Grant number: DMS-1713506

Award Identifier / Grant number: 668998

Funding statement: The research of I. Lasiecka and R. Triggiani was partially supported by the National Science Foundation under Grant DMS-1713506. The research of B. Priyasad was partially supported by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program.

Appendix A Some auxiliary results for the Stokes and Oseen operators. Analytic semigroup generation, maximal regularity, domains of fractional powers

In this section, we collect some known results used in the paper. As a prerequisite of the present Appendix A, we make reference to Definition 1.2, equations (1.7)–(1.11) and Remark 1.2.

A.1 The Stokes and Oseen operators generate a strongly continuous analytic semigroup on L σ q ( Ω ) , 1 < q <

Theorem A.1

Let d 2 , 1 < q < , and let Ω be a bounded domain in R d of class C 3 .

  1. Then the Stokes operator - A q = P q Δ in (1.14), repeated here as

    (A.1) - A q ψ = P q Δ ψ , ψ D ( A q ) = W 2 , q ( Ω ) W 0 1 , q ( Ω ) L σ q ( Ω ) ,

    generates an s.c. analytic semigroup e - A q t on L σ q ( Ω ) . See [32] and the review paper [35, Theorem 2.8.5, p. 17].

  2. The Oseen operator A q in (1.17),

    (A.2) A q = - ( ν A q + A o , q ) , D ( A q ) = D ( A q ) L σ q ( Ω ) ,

    generates an s.c. analytic semigroup e A q t on L σ q ( Ω ) . This follows as A o , q is relatively bounded with respect to A q 1 / 2 , see (1.15); thus a standard theorem on perturbation of an analytic semigroup generator applies [69, Corollary 2.4, p. 81].

  3. We have

    { 0 ρ ( A q ) = the resolvent set of the Stokes operator A q , A q - 1 : L σ q ( Ω ) L σ q ( Ω ) is compact .

  4. The s.c. analytic Stokes semigroup e - A q t is uniformly stable on L σ q ( Ω ) ; there exist constants M 1 , δ > 0 (possibly depending on 𝑞) such that

    (A.3) e - A q t L ( L σ q ( Ω ) ) M e - δ t , t > 0 .

A.2 Domains of fractional powers D ( A q α ) , 0 < α < 1 of the Stokes operator A q on L σ q ( Ω ) , 1 < q <

We elaborate on (1.16).

Theorem A.2

For the domains of fractional powers D ( A q α ) , 0 < α < 1 , of the Stokes operator A q in (A.1) = (1.14), the following complex interpolation relation holds true, see [33] and [35, Theorem 2.8.5, p. 18]:

[ D ( A q ) , L σ q ( Ω ) ] 1 - α = D ( A q α ) , 0 < α < 1 , 1 < q < ;

in particular,

[ D ( A q ) , L σ q ( Ω ) ] 1 / 2 = D ( A q 1 / 2 ) W 0 1 , q ( Ω ) L σ q ( Ω ) .

Thus, on the space D ( A q 1 / 2 ) , the norms L q ( Ω ) and L q ( Ω ) are related via the Poincaré inequality.

A.3 The Stokes operator - A q and the Oseen operator A q , 1 < q < , generate s.c. analytic semigroups on the Besov space, from (1.11)

They are defined by (A.4)

(A.4a) ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p = { g B q , p 2 - 2 / p ( Ω ) : div g = 0 , g | Γ = 0 } if 1 q < 2 - 2 p < 2 ,
(A.4b) ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p = { g B q , p 2 - 2 / p ( Ω ) : div g = 0 , g ν | Γ = 0 } B ~ q , p 2 - 2 / p ( Ω ) if  0 < 2 - 2 p < 1 q .
Theorem A.1 states that the Stokes operator - A q generates an s.c. analytic semigroup on the space L σ q ( Ω ) , 1 < q < , hence on the space D ( A q ) in (1.14) = (A.1), with norm D ( A q ) = A q L σ q ( Ω ) as 0 ρ ( A q ) . Then one obtains that the Stokes operator - A q generates an s.c. analytic semigroup on the real interpolation spaces in (A.4). Next, the Oseen operator A q = - ( ν A q + A o , q ) in (A.2) = (1.17) likewise generates an s.c. analytic semigroup e A q t on L σ q ( Ω ) since A o , q is relatively bounded with respect to A q 1 / 2 , as A o , q A q - 1 / 2 is bounded on L σ q ( Ω ) . Moreover, A q generates an s.c. analytic semigroup on D ( A q ) = D ( A q ) (equivalent norms). Hence A q generates an s.c. analytic semigroup on the real interpolation space of (A.4). Here, below, however, we shall formally state the result only in the case 2 - 2 p < 1 q , i.e., 1 < p < 2 q 2 q - 1 , in the space B ~ q , p 2 - 2 / p ( Ω ) , as this does not contain B.C., Remark 1.2. The objective of the present paper is precisely to obtain stabilization results on spaces that do not recognize B.C.

Theorem A.3

Let 1 < q < , 1 < p < 2 q 2 q - 1 .

  1. The Stokes operator - A q in (A.1) = (1.14) generates an s.c. analytic semigroup e - A q t on the space B ~ q , p 2 - 2 / p ( Ω ) defined in (1.11) = (A.4b) which moreover is uniformly stable, as in (A.3),

    e - A q t L ( B ~ q , p 2 - 2 / p ( Ω ) ) M e - δ t , t > 0 .

  2. The Oseen operator A q in (A.2) = (1.17) generates an s.c. analytic semigroup e A q t on the space B ~ q , p 2 - 2 / p ( Ω ) in (1.11) = (A.4).

A.4 Space of maximal L p regularity on L σ q ( Ω ) of the Stokes operator - A q , 1 < p < , 1 < q < , up to T =

We return to the dynamic Stokes problem in { φ ( t , x ) , π ( t , x ) } , (A.5)

(A.5a) φ t - Δ φ + π = F in ( 0 , T ] × Ω Q ,
(A.5b) div φ 0 in Q ,
(A.5c) φ | Σ 0 in ( 0 , T ] × Γ Σ ,
(A.5d) φ | t = 0 = φ 0 in Ω ,
rewritten in abstract form, after applying the Helmholtz projection P q to (A.5a) and recalling A q in (A.1) = (1.14) as

(A.6) φ + A q φ = F σ P q F , φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p .

Next, we introduce the space of maximal regularity for { φ , φ } as [35, p. 2; Theorem 2.8.5 (iii), p. 17], [31, p. 1404-5], with 𝑇 up to ∞,

(A.7) X ~ p , q , σ T = L p ( 0 , T ; D ( A q ) ) W 1 , p ( 0 , T ; L σ q ( Ω ) )

(recall (A.1) = (1.14) for D ( A q ) ) and the corresponding space for the pressure as

Y ~ p , q T = L p ( 0 , T ; W ^ 1 , q ( Ω ) ) , W ^ 1 , q ( Ω ) = W 1 , q ( Ω ) / R .

The following embedding, also called trace theorem, holds true, see [5, Theorem 4.10.2, p. 180, BUC for T = ], [71]:

(A.8) X ~ p , q , σ T X ~ p , q T L p ( 0 , T ; W 2 , q ( Ω ) ) W 1 , p ( 0 , T ; L q ( Ω ) ) C ( [ 0 , T ] ; B q , p 2 - 2 / p ( Ω ) ) .

For a function 𝒈 such that div g 0 , g | Γ = 0 , we have g X ~ p , q T g X ~ p , q , σ T .

The solution of equation (A.6) is

(A.9) φ ( t ) = e - A q t φ 0 + 0 t e - A q ( t - s ) F σ ( τ ) d τ .

The following is the celebrated result on maximal regularity on L σ q ( Ω ) of the Stokes problem due originally to Solonnikov [77, 78, 79, 80, 81] reported in [35, Theorem 2.8.5 (iii) and Theorem 2.10.1, p. 24, for φ 0 = 0 ], [74], [31, Proposition 4.1, p. 1405].

Theorem A.4

Let 1 < p , q < , T . With reference to problem (A.5) = (A.6), assume

(A.10) F σ L p ( 0 , T ; L σ q ( Ω ) ) , φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p .

Then there exists a unique solution φ X ~ p , q , σ T , π Y ~ p , q T to the dynamic Stokes problem (A.5) or (A.6), continuously on the data; there exist constants C 0 , C 1 independent of T , F σ , φ 0 such that, via (A.8),

(A.11) C 0 φ C ( [ 0 , T ] ; B q , p 2 - 2 / p ( Ω ) ) φ X ~ p , q , σ T + π Y ~ p , q T φ L p ( 0 , T ; L σ q ( Ω ) ) + A q φ L p ( 0 , T ; L σ q ( Ω ) ) + π Y ~ p , q T C 1 { F σ L p ( 0 , T ; L σ q ( Ω ) ) + φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p } .

In particular, the following statements hold.

  1. With reference to the variation-of-parameter formula (A.9) of problem (A.6) arising from Stokes problem (A.5), we have, recalling (A.7), that the map

    F σ 0 t e - A q ( t - τ ) F σ ( τ ) d τ is continuous ,
    L p ( 0 , T ; L σ q ( Ω ) ) X ~ p , q , σ T L p ( 0 , T ; D ( A q ) ) W 1 , p ( 0 , T ; L σ q ( Ω ) ) .

  2. The s.c. analytic semigroup e - A q t generated by the Stokes operator - A q , see (A.1) = (1.14), on the space ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p , see the statement below (A.4), satisfies

    e - A q t is continuous , ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p X ~ p , q , σ T L p ( 0 , T ; D ( A q ) ) W 1 , p ( 0 , T ; L σ q ( Ω ) ) .

    In particular, via (A.4b), for future use, for 1 < q < , 1 < p < 2 q 2 q - 1 , the s.c. analytic semigroup e - A q t on the space B ~ q , p 2 - 2 / p ( Ω ) satisfies

    e - A q t is continuous , B ~ q , p 2 - 2 / p ( Ω ) X ~ p , q , σ T .

  3. Moreover, for future use, for 1 < q < , 1 < p < 2 q 2 q - 1 , then (A.11) specializes to

    φ X ~ p , q , σ T + π Y ~ p , q T C { F σ L p ( 0 , T ; L σ q ( Ω ) ) + φ 0 B ~ q , p 2 - 2 / p ( Ω ) } .

A.5 Maximal L p regularity on L σ q ( Ω ) of the Oseen operator A q , 1 < p < , 1 < q < , up to T <

We next transfer the maximal regularity of the Stokes operator ( - A q ) on L σ q ( Ω ) asserted in Theorem A.4 into the maximal regularity of the Oseen operator A q = - ν A q - A o , q in (A.2) exactly on the same space X ~ p , q , σ T defined in (A.7), however, only up to T < .

Thus, consider the dynamic Oseen problem in { φ ( t , x ) , π ( t , x ) } with equilibrium solution y e , see (1.2), (A.12)

(A.12a) φ t - Δ φ + L e ( φ ) + π = F in ( 0 , T ] × Ω Q ,
(A.12b) div φ 0 in Q ,
(A.12c) φ | Σ 0 in ( 0 , T ] × Γ Σ ,
(A.12d) φ | t = 0 = φ 0 in Ω ,

L e ( φ ) = ( y e . ) φ + ( φ . ) y e ,

rewritten in abstract form, after applying the Helmholtz projector P q to (A.12a) and recalling A q in (A.2), as

(A.13) φ t = A q φ + P q F = - ν A q φ - A o , q φ + F σ , φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p ,

whose solution is, via (1.18) = (A.2),

φ ( t ) = e A q t φ 0 + 0 t e A q ( t - τ ) F σ ( τ ) d τ ,
φ ( t ) = e - ν A q t φ 0 + 0 t e - ν A q ( t - τ ) F σ ( τ ) d τ - 0 t e - ν A q ( t - τ ) A o , q φ ( τ ) d τ .

Theorem A.5

Let 1 < p , q < , 0 < T < . Assume, as in (A.10),

F σ L p ( 0 , T ; L σ q ( Ω ) ) , φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p ,

where D ( A q ) = D ( A q ) , see (A.2) = (1.18). Then there exists a unique solution φ X ~ p , q , σ T , π Y ~ p , q T of the dynamic Oseen problem (A.12), continuously on the data, that is, there exist constants C 0 , C 1 independent of F σ , φ 0 such that

C 0 φ C ( [ 0 , T ] ; B q , p 2 - 2 / p ( Ω ) ) φ X ~ p , q , σ T + π Y ~ p , q T φ L p ( 0 , T ; L q ( Ω ) ) + A q φ L p ( 0 , T ; L q ( Ω ) ) + π Y ~ p , q T C T { F σ L p ( 0 , T ; L σ q ( Ω ) ) + φ 0 ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p } ,

where T < . Equivalently, for 1 < p , q < , the following statements hold.

  1. The map

    F σ 0 t e A q ( t - τ ) F σ ( τ ) d τ is continuous , L p ( 0 , T ; L σ q ( Ω ) ) L p ( 0 , T ; D ( A q ) = D ( A q ) ) ,

    where then, automatically, see (A.13), L p ( 0 , T ; L σ q ( Ω ) ) W 1 , p ( 0 , T ; L σ q ( Ω ) ) , and ultimately, via (A.7),

    L p ( 0 , T ; L σ q ( Ω ) ) X ~ p , q , σ T L p ( 0 , T ; D ( A q ) ) W 1 , p ( 0 , T ; L σ q ( Ω ) ) .

  2. The s.c. analytic semigroup e A q t generated by the Oseen operator A q , see (A.2) = (1.18), on the space ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p satisfies, for 1 < p , q < ,

    e A q t is continuous , ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p L p ( 0 , T ; D ( A q ) = D ( A q ) ) ,

    and hence, automatically, by (A.7),

    e A q t is continuous , ( L σ q ( Ω ) , D ( A q ) ) 1 - 1 / p , p X ~ p , q , σ T .

    In particular, for future use, for 1 < q < , 1 < p < 2 q 2 q - 1 , we have that the s.c. analytic semigroup e A q t on the space B ~ q , p 2 - 2 / p ( Ω ) satisfies

    e A q t is continuous , B ~ q , p 2 - 2 / p ( Ω ) L p ( 0 , T ; D ( A q ) = D ( A q ) ) , T < ,

    and hence, automatically,

    e A q t is continuous , B ~ q , p 2 - 2 / p ( Ω ) X ~ p , q , σ T , T < .

A proof is given in [56, Appendix B].

Appendix B Material in support of the proof of Theorem 4.1. The required UCP and D * B q * f in (4.16)

B.1 The required UCP

We return to the operator A q in (1.33),

(B.1) A q = [ A q - C γ - C θ e - B q ] : W σ q ( Ω ) = L σ q ( Ω ) × L q ( Ω ) D ( A q ) = D ( A q ) × D ( B q ) = ( W 2 , q ( Ω ) W 0 1 , q ( Ω ) L σ q ( Ω ) ) × ( W 2 , q ( Ω ) W 0 1 , q ( Ω ) ) W σ q ( Ω ) .

With Φ = [ φ , ψ ] , the PDE version of A q Φ = λ Φ is

(B.2) { - ν Δ φ + L e ( φ ) + π - γ ψ e d = λ φ in Ω , - κ Δ ψ - y e ψ + φ θ e = λ ψ in Ω , div φ = 0 in Ω , φ = 0 , ψ = 0 on Γ .

Several UCPs for over-determined versions of the eigenproblem (B.2) are given in [89, 87]. However, establishing in Theorem 4.1 controllability of the finite-dimensional projected problem (4.21) or (4.27) via verification of the Kalman rank condition (4.29) involves the following UCP for the adjoint problem. First, recall from (4.3) that the adjoint A q * or A q in (B.1) is

A q * = [ A q * - C θ e * - C γ * - B q * ] : W σ q ( Ω ) = L σ q ( Ω ) × L q ( Ω ) D ( A q * ) = D ( A q * ) × D ( B q * ) = ( W 2 , q ( Ω ) W 0 1 , q ( Ω ) L σ q ( Ω ) ) × ( W 2 , q ( Ω ) W 0 1 , q ( Ω ) ) W σ q ( Ω ) .

With Φ * = [ φ * , ψ * ] , the PDE version of A q * Φ * = λ Φ * is

(B.3) { - ν Δ φ * + L e * ( φ * ) + ψ * θ e + π = λ φ * in Ω , - κ Δ ψ * - y e ψ * - γ φ * e d = λ ψ * in Ω , div φ * = 0 in Ω , φ * = 0 , ψ * = 0 on Γ .

The UCP invoked in the proof of Theorem 4.1 is given by the following theorem.

Theorem B.1

Theorem B.1 ([89, Theorem 5])

Let { φ , ψ , π } [ W 2 , q ( Ω ) L σ q ( Ω ) ] × W 2 , q ( Ω ) , π W 1 , q ( Ω ) be a solution of the dual problem (B.4)

(B.4a) - ν Δ φ * + L e * ( φ * ) + ψ * θ e + π = λ φ * in Ω ,
(B.4b) - κ Δ ψ * - y e ψ * - γ φ * e d = λ ψ * in Ω ,
(B.4c) div φ * = 0 in Ω
(B.4d) { φ * ( 1 ) , , φ * ( d - 1 ) } = 0 , ψ * = 0 on ω ,
with over-determination in (B.4d). Then φ * = 0 , ψ * = 0 , π = const . in Ω.∎

Proof of D * B * f = - κ f ν | Γ , f D ( B q * ) , in (4.16), essentially by Green’s formula

(1) Let 1 < q < , and define

(B.5) B o , q h = y e h , L q ( Ω ) D ( B o , q ) = W 0 1 , q ( Ω ) L q ( Ω ) .

Then

(B.6) B o , q * h = - y e f , L q ( Ω ) D ( B o , q * ) = W 0 1 , q ( Ω ) L q ( Ω ) .

In fact, recalling y e | Γ 0 and div y e 0 in Ω from (1.2c)–(1.2d), we compute, in the duality pairing L q ( Ω ) , L q ( Ω ) , with h D ( B o , q ) , f D ( B o , q * ) , by (B.5),

B o , q h , f = Ω f y e h d Ω = Γ h f y e ν d Γ - Ω h div ( f y e ) d Ω = - Ω h y e f d Ω = h , B o , q * f Ω

(2) We return to the operator B q in (1.18), with h D ( B q ) ,

(B.7) B q h = - κ Δ h + B o , q h so that B q * h = - κ Δ f + B o , q * f , f D ( B q * ) .

We shall show that

(B.8) D * B * f = - κ f ν | Γ , f D ( B q * ) = W 2 , q ( Ω ) W 0 1 , q ( Ω ) .

In fact, by (B.6) and (B.7), we compute with v L 2 ( Γ ) , recalling the definition of 𝐷 in (1.23a),

D * B q * f , v Γ = B q * f , D v Ω = Δ f , - κ D v Ω + f , B o , q * D v Ω = f , ( - κ Δ - y e ) D v Ω + Γ f ν ( - κ D v ) d Γ + Γ f ( κ D v ν ) d Γ = - κ f ν , v Γ for all v L q ( Γ ) ,

and (B.8) is established. ∎

Appendix C Validation of the Kalman controllability conditions (4.29) with fluid vectors { u 1 , , u i } , i K , i = 1 , , M , having only ( d - 1 ) components

For the sake of the clarity, we shall consider separately the cases d = 2 and d = 3 .

Case d = 2

Express the 2-dimensional vectors u i and φ i j * of Section 4 in terms of their two components,

(C.1) u i = [ u i ( 1 ) , u i ( 2 ) ] , φ i j * = [ φ i j * ( 1 ) , φ i j * ( 2 ) ] , i = 1 , , M , j = 1 , , i .

Here, we shall recall the matrix W i in (4.22), but we shall replace the matrix U i in (4.23) with the following matrix:

U i ( 2 ) = [ u 1 ( 2 ) , φ i 1 * ( 2 ) ω u i ( 2 ) , φ i 1 * ( 2 ) ω u 2 ( 2 ) , φ i 2 * ( 2 ) ω u i ( 2 ) , φ i 2 * ( 2 ) ω u 1 ( 2 ) , φ i i * ( 2 ) ω u i ( 2 ) , φ i i * ( 2 ) ω ] : i × K ,

where now the duality pairing , ω involves two scalar functions. Next, for d = 2 , we shall establish the Kalman algebraic rank condition for the finite-dimensional, unstable, feedback problem (4.21), by employing not the full strength of the 2-dimensional vectors u i and φ i j * as in (4.29) involving the matrix U i in (4.23), but instead replacing U i with U i ( 2 ) . This way, only the second (scalar) components u i ( 2 ) and φ i j * ( 2 ) of the 2-dimensional vectors u i and φ i j * in (C.1) are needed. Recall that we are dealing with a pair { ω , Γ ~ } as in Figure 1. The counterpart of Theorem 4.1 is now the following theorem.

Theorem C.1

Let d = 2 . It is possible to select boundary vectors f 1 , , f K in F W 2 - 1 / q , q ( Γ ~ ) with support on Γ ~ , and scalar second components { u 1 ( 2 ) , , u i ( 2 ) } as in (C.1), such that rank [ W i , U i ( 2 ) ] = i , i , , M .

Proof

We shall appropriately modify the proof of Theorem 4.1. As in this proof, the crux is to establish that we cannot have simultaneously

(C.2) ν ψ i i * = j = 1 i - 1 α j ν ψ i j * in L q ( Γ ~ ) and φ i i * ( 2 ) = j = 1 i - 1 α j φ i j * ( 2 ) in L σ q ( ω )

with the same constant α 1 , , α i - 1 in both expressions.

Claim

Statement (C.2) is false.

By contradiction, suppose that both linear combinations in (C.2) hold true. Next, as in (4.31), define the ( d + 1 ) -vector Φ * { φ * , ψ * } by

(C.3) Φ * [ φ * ψ * ] = j = 1 i - 1 [ α j φ i j * α j ψ i j * ] - [ φ i i * ψ i i * ] = j = 1 i - 1 α j Φ i j * - Φ i i * , i = 1 , , M , q 2 ,

in W σ q ( Ω ) L σ q ( Ω ) × L q ( Ω ) , with Φ i j * { φ i j * , ψ i j * } eigenvector of A q , N * or A q * , as in (4.2). Then, in view of (C.2), we now obtain

(C.4) φ * ( 2 ) 0 in ω , ν ψ * | Γ ~ 0

(in place of (4.32)). As in the proof of Theorem 4.1, since the Φ i j * are eigenvectors of A q , N * or A q * , so is the vector Φ * { φ * , ψ * } defined in (C.3). Thus, Φ * { φ * , ψ * } satisfies the PDE version (4.33a)–(4.33d) of the eigenproblem for the dual A q * , which is now augmented with the over-determined conditions in (C.4). The fact that now φ * | Γ = 0 by (4.33d) on the entire boundary Γ permits a fortiori the argument of [89, Theorems 6, 7] to hold true. Namely, invoke φ * ( 2 ) 0 in 𝜔 in (C.4) to obtain, as in the proof of Theorem 4.1, the over-determined problem (4.34) for ψ * on 𝜔, with the conclusion that

(C.5) ψ * 0 in ω , as in (4.35) .

Next, the divergence condition (4.33c) yields, in view of (C.4), div φ * = φ x 1 * ( 1 ) + φ x 2 * ( 2 ) = φ x 1 * ( 1 ) = 0 in 𝜔. Hence φ * ( 1 ) ( x 1 , x 2 ) c ( x 2 ) in 𝜔, where c ( x 2 ) is a function constant with respect to x 1 and depending only on x 2 in 𝜔. Next, let P = { x 1 ( P ) , x 2 ( P ) } be an arbitrary point of 𝜔. Consider the line ℓ passing through the point 𝑃 and parallel to the x 1 -axis. On such a line ℓ, the value φ * ( 1 ) ( x 1 , x 2 ( P ) ) c 2 ( x 2 ( P ) ) is constant with respect to x 1 , as long as ℓ intersects 𝜔. By definition of the small set 𝜔 supported by Γ ~ , there is a non-empty open subset ω ~ ω , where the following happens: for all points 𝑃 in ω ~ , the line ℓ remains in ω ~ and hits the boundary Γ ~ , where condition (4.33d) applies for φ * ( 1 ) | Γ = 0 . Thus, φ * ( 1 ) 0 in ω ~ ω . Recalling (C.4), we finally have φ * = { φ * ( 1 ) , φ * ( 2 ) } 0 in ω ~ ω , along with ψ * 0 in ω ~ by (C.5). We can then apply [89, Theorem 5 for ω ~ ] and conclude that

φ * = { φ * ( 1 ) , φ * ( 2 ) } 0 in Ω , ψ * 0 in Ω , p const . in Ω .

The rest of the proof proceeds as in Theorem 4.1 following (4.36). Theorem C.1 is established. ∎

Case d = 3

Now we express the 3-dimensional vectors u i and φ i j * of Section 4 in terms of their components as

(C.6) u i = [ u i ( 1 ) , u i ( 2 ) , u i ( 3 ) ] , φ i j * = [ φ i j * ( 1 ) , φ i j * ( 2 ) , φ i j * ( 3 ) ] , i = 1 , , M , j = 1 , , i .

We now distinguish two sub-cases.

Sub-case { 1 , 3 } . Here, we extract only the first and third components, while we omit the second component. Accordingly, we introduce the following 2-dimensional vectors u i ( 1 , 3 ) and φ * ( 1 , 3 ) and the corresponding matrix U i ( 1 , 3 ) , (C.7)

u i ( 1 , 3 ) = [ u i ( 1 ) , u i ( 3 ) ] , φ i j * ( 1 , 3 ) = [ φ i j * ( 1 ) , φ i j * ( 3 ) ] ,
U i ( 1 , 3 ) = [ [ u 1 ( 1 ) u 1 ( 3 ) ] , [ φ i 1 * ( 1 ) φ i 1 * ( 3 ) ] ω [ u i ( 1 ) u i ( 3 ) ] , [ φ i 1 * ( 1 ) φ i 1 * ( 3 ) ] ω [ u 1 ( 1 ) u 1 ( 3 ) ] , [ φ i i * ( 1 ) φ i i * ( 3 ) ] ω [ u i ( 1 ) u i ( 3 ) ] , [ φ i i * ( 1 ) φ i i * ( 3 ) ] ω ]

Sub-case { 2 , 3 } . Here, we extract only the second and third components, while we omit the first component. Accordingly, we introduce the following 2-dimensional vectors u i ( 2 , 3 ) and φ * ( 2 , 3 ) and the corresponding matrix U i ( 2 , 3 ) , (C.8)

u i ( 2 , 3 ) = [ u i ( 2 ) , u i ( 3 ) ] , φ i j * ( 2 , 3 ) = [ φ i j * ( 2 ) , φ i j * ( 3 ) ] ,
U i ( 2 , 3 ) = [ [ u 1 ( 2 ) u 1 ( 3 ) ] , [ φ i 1 * ( 2 ) φ i 1 * ( 3 ) ] ω [ u i ( 2 ) u i ( 3 ) ] , [ φ i 1 * ( 2 ) φ i 1 * ( 3 ) ] ω [ u 1 ( 2 ) u 1 ( 3 ) ] , [ φ i i * ( 2 ) φ i i * ( 3 ) ] ω [ u i ( 2 ) u i ( 3 ) ] , [ φ i i * ( 2 ) φ i i * ( 3 ) ] ω ]
Next, for d = 3 , we shall establish the Kalman algebraic rank condition for the finite-dimensional, unstable, feedback problem (4.21) by replacing the matrix U i in (4.23) with either the matrix U i ( 1 , 3 ) in (C.7) (sub-case { 1 , 3 } ) or with the matrix U i 2 , 3 in (C.8) (sub-case { 2 , 3 } ). This way, the third (scalar) components u i ( 3 ) and φ i j * ( 3 ) of the 3-dimensional vectors u i and φ i j * in (C.6) are needed in both sub-cases. The counterpart of Theorem 4.1 (or Theorem C.1) is now the following theorem.

Theorem C.2

Let d = 3 . It is possible to select boundary vectors f 1 , , f K in F W 2 - 1 / q , q ( Γ ~ ) with support on Γ ~ , and

  1. either 2-dimensional vectors u i ( 1 , 3 ) = { u i ( 1 ) , u i ( 3 ) } as in (C.7)

  2. or 2-dimensional vectors u i ( 2 , 3 ) = { u i ( 2 ) , u i ( 3 ) } as in (C.8)

such that
rank [ W i , U i ( 1 , 3 ) ] = i , i , , M , in case (i) ,
rank [ W i , U i ( 2 , 3 ) ] = i , i , , M , in case (ii) ,
respectively.

Proof

We shall only give a proof in the sub-case { 1 , 3 } , as the proof of the sub-case { 2 , 3 } is the same mutatis mutandis. As in the proof of Theorem 4.1 (or Theorem C.1), the crux is to establish that we cannot have simultaneously

(C.9) ν ψ i i * = j = 1 i - 1 α j ν ψ i j * in L q ( Γ ~ ) and φ i i * ( 1 , 3 ) = j = 1 i - 1 α j φ i j * ( 1 , 3 ) in L σ q ( ω )

with the same constants α , , α i - 1 in both expressions. ∎

Claim

Statement (C.9) is false.

By contradiction, suppose that both linear combinations in (C.9) hold true. Next, define the ( d + 1 ) -vector Φ * { φ * , ψ * } as in (4.31) (or (C.3)). Then, in view of (C.9), we now obtain

(C.10) φ * ( 1 , 3 ) { φ * ( 1 ) , φ * ( 3 ) } 0 in ω , ν ψ * | Γ ~ 0

(in place of (4.32) or (C.4)). As in Theorem 4.1 (or Theorem C.1), the Φ * { φ * , ψ * } is an eigenvector of A q * and thus satisfies the corresponding PDE version – that is, problem (4.33a)–(4.33d), augmented this time with the over-determined conditions in (C.10). Next, invoke φ * ( 3 ) 0 in 𝜔 from (C.10) to obtain, as in the proof of Theorem 4.1, the over-determined problem (4.34) for ψ * in 𝜔, with the conclusion that

(C.11) ψ *