BOUNDARY FEEDBACK STABILIZATION OF THE TWO DIMENSIONAL NAVIER-STOKES EQUATIONS WITH FINITE

AIMS’ Journals

Volume

X, Number0X, XX200X pp.X–XX

BOUNDARY FEEDBACK STABILIZATION OF THE TWO DIMENSIONAL NAVIER-STOKES EQUATIONS WITH FINITE

DIMENSIONAL CONTROLLERS

Abstract. We study the boundary stabilization of the two-dimensional Navier- Stokes equations about an unstable stationary solution by controls of finite di- mension in feedback form. The main novelty is that the linear feedback control law is determined by solving an optimal control problem of finite dimension.

More precisely, we show that, to stabilize locally the Navier-Stokes equations, it is sufficient to look for a boundary feedback control of finite dimension, able to stabilize the projection of the linearized equation onto the unstable subspace of the linearized Navier-Stokes operator. The feedback operator is obtained by solving an algebraic Riccati equation in a space of finite dimension, that is to say a matrix Riccati equation.

Jean-Pierre Raymond

Universit´ e de Toulouse & CNRS Institut de Math´ ematiques, UMR 5219

31062 Toulouse Cedex 9, France

Laetitia Thevenet

Universit´ e de Toulouse & CNRS Institut de Math´ ematiques, UMR 5219

31062 Toulouse Cedex 9, France

1. Introduction. Control of fluid flows by feedback is a challenging problem both from the theoretical and numerical points of view, see [15, 35] and the references therein. In this paper, we are interested in determining boundary feedback con- trol laws of finite dimension able to stabilize the two dimensional Navier-Stokes equations in a neighborhood of an unstable stationary solution.

The system that we are going to consider may be written in the form

z ⁰ = Az + F (z + Lu) + B m u, z(0) = z 0 , (1.1) where (A, D(A)) is the infinitesimal generator of an analytic semigroup in a real Hilbert space Z, the control operator B m belongs to L(U, (D(A ^∗ )) ⁰ ), U is another Hilbert space, L ∈ L(U, Z). We further assume that the resolvent of A is compact in Z, and the nonlinear mapping F obeys F (0) = 0 and F ⁰ (0) = 0. We are interested in finding a control u, in feedback form and of finite dimension, that is of the form

u(t) =

K

X

i=1

z(t), ξ i

Z ζ i , (1.2)

2000 Mathematics Subject Classification. Primary: 93B52, 93C20, 93D15; Secondary: 35Q30, 76D55, 76D05, 76D07.

Key words and phrases. Dirichlet control, feedback control, stabilization, Navier-Stokes equa- tions, Oseen equations, Riccati equation, finite dimensional controllers.

Both authors are supported by the ANR-project CORMORED BLAN-08-0115-02.

1

so that the closed loop system z ⁰ = Az + F z + Σ ^K _i=1 z(t), ξ _i

Z Lζ _i

+ Σ ^K _i=1 z(t), ξ _i

Z B _m ζ _i , z(0) = z ₀ , (1.3) is locally exponentially stable for all z 0 in some ball B(0, R) in Z. We are going to see that, when equation (1.1) corresponds to the Navier-Stokes equations, the functions (ξ i ) 1≤i≤K may be chosen of the form ΠQB m ζ i , where ζ i = B _m ^∗ ε i and ε i

are eigenfunctions or generalized eigenfunctions of A ^∗ , Q is a projection operator and Π can be found by solving a Riccati equation of finite dimension. For that, we apply the following programme.

(i) We decompose the original state space Z into a finite dimensional unstable subspace Z u and an infinite dimensional stable subspace Z s . Both subspaces are invariant under e ^tA .

(ii) We project the linearized system

z ⁰ = Az + B _m u, z(0) = z ₀ , (1.4) onto the stable and the unstable subspaces. The projection is defined firstly in the complexified space Z +iZ and next in the real space Z . We prove that the projected system onto the unstable subspace is stabilizable by controls u of the form (1.2).

(iii) We determine a feedback control law, of the form (1.2), able to stabilize the finite dimensional projected unstable system by solving a Riccati equation of finite dimension.

(iv) We prove that the feedback law determined in step (iii) stabilizes locally the nonlinear system (1.1).

The idea of using finite dimensional controllers for the stabilization of linear parabolic systems goes back to R. Triggiani [33] (see also [26]). The idea of using linear feedback law for proving local stabilization of semilinear partial differential equations is not new and goes back to I. Lasiecka [25]. In that case the proof of local stabilization relies on a fixed point argument, see [24]. In the context of an internal control of the Navier-Stokes equations such an approach has been developed in [8]. The drawback of this method is that it does not give an obvious Lyapunov functional for the closed loop nonlinear system. This explains why other approaches have been investigated. They consist in solving the optimal control problem

(P) inf n

J (z, u) | (z, u) is solution to (1.4) o , with a cost functional J is of the form

J (z, u) = 1 2

Z ∞ 0

|Cz(t)| ² _Y dt + 1 2

Z ∞ 0

|u(t)| ² _U dt,

where C is a linear operator (not necessarily bounded) from Z into another Hilbert space Y . We consider the cases when (P) admits a unique solution. The value function of problem (P) is of the form

z 0 7−→ 1

2 (Πz 0 , z 0 ) Z ,

and the optimal pair (z

z₀

, u

z₀

) obeys the feedback law u

z₀

(t) = −B ^∗ _m Πz

z₀

(t). The operator Π may be bounded or unbounded in Z depending on the choice of C.

This choice is crucial both for the characterization of Π and for proving that the

linear feedback law stabilizes locally the nonlinear system (1.1). In [6, 12] for the

internal stabilization of the Navier-Stokes equations and in [9] for the boundary

stabilization of the Navier-Stokes equations, the observation operator C is chosen

so that the feedback generator A Π = A − B m B ^∗ _m Π is dissipative with respect to the product (Π·, ·) Z and the value function of problem (P ) is a Lyapunov function of the closed loop nonlinear system. This choice for C is refered as the high gain functional approach [10, 11]. In the case of an internal control [6], the operator Π is unbounded in Z and it is the unique solution to an algebraic Riccati equation which is well posed in D(A) (a similar result is obtained in [12] for controls of finite dimension). In the case of a boundary control [9], Π is still an unbounded operator in Z, but it is not characterized by a Riccati equation satisfied in Z or in D(A). (The equation is satisfied only in D(A ² _Π ) which is not known and therefore this equation is not useful.) Moreover the ’high gain functional approach’ for a boundary control has been developed only in the case of tangential controls. It is not known if such an approach is still valid for controls which are not tangential to the boundary. The drawback with tangential controls is that the stabilizability is not necessarily guarenteed. It is established only under a smallness condition on the data (see [34, Theorem 2.2, and assumption (1.7) of Theorem 1.2]).

The approach consisting in taking C equal either to the identity in Z or to a smoothing operator (the low gain functional approach) has been developed for the boundary control of the Navier-Stokes equations in [29], for the two dimensional case, and in [30], for the three dimensional case. In those cases the Riccati equation is well posed in Z . In [29], we have proved that when C = I the operator Π is bounded from Z into D(A) (which is slightly better than the results already known in the literature [27, Theorem 2.2.1 (a ₃ )]). In [30, 31] we have shown, for the first time to the authors knowledge, that the smoothing properties of Π can be improved by taking an operator C more smoothing than the identity. The existence of Lyapunov functionals for the corresponding closed loop nonlinear systems has been solved very recently by M. Badra [2, 3, 4].

However in all these approaches [12, 9, 29, 30], the pair (A, C) is completely detectable, and the feedback control laws determined in [6, 12], in the case of an internal control, and in [29], in the case of a boundary control, are obtained by solving an algebraic Riccati equation stated in a space of infinite dimension.

In the present paper we introduce a qualitative jump by showing that it is pos- sible to stabilize a nonlinear system with a linear feedback law determined with an operator C for which the pair (A, C) is no longer completely detectable. In addi- tion, the Riccati equation that we have to solve is of finite dimension (the equation is stated in R ^K×K where K is the dimension of the unstable space of the linearized Navier-Stokes operator). To the authors knowledge, this type of result is completely new in the context of local stabilization of nonlinear systems. (Of course for linear systems the idea goes back to [32].) Furthermore, following [23], we could even take C = 0. In that case the feedback law is determined by looking for the maximal solution to a degenerate Riccati equation.

For numerical calculations an approximation scheme has to be used for solving the Navier-Stokes system. To determine the Riccati equation that we have to solve in the present paper, one needs to determine a few eigenvalues and the corresponding eigenfunctions and generalized eigenfunctions of the operators A and A ^∗ . Even if these numerical calculations can be delicate, they are much easier than solving an algebraic Riccati equation of high dimension.

Let us describe more precisely our problem. Let Ω be a bounded and connected

domain in R ² with boundary Γ of class C ⁴ , ν > 0, and consider a couple (w, χ) –

a velocity field and a pressure – solution to the stationary Navier-Stokes equations

in Ω

−ν∆w + (w · ∇)w + ∇χ = f and div w = 0 in Ω, w = u ^∞ _s on Γ. (1.5) We assume that w belongs to V ³ (Ω) = {v ∈ H ³ (Ω; R ² ) | div v = 0 in Ω}, and that it is an unstable solution of the Navier-Stokes equations. We consider the control system

∂z

∂t − ν∆z + (z · ∇)z + ∇q = 0, div z = 0 in Ω × (0, ∞), z = u ^∞ _s + M u ˆ on Σ _∞ = Γ × (0, ∞), z(0) = w + y 0 in Ω,

(1.6) and the corresponding system satisfied by ˆ y = z − w

∂ˆ y

∂t − ν∆ˆ y + (ˆ y · ∇)w + (w · ∇)ˆ y + (ˆ y · ∇)ˆ y + ∇ p ˆ = 0, in Ω × (0, ∞), div ˆ y = 0 in Ω × (0, ∞),

y ˆ = M u ˆ on Σ ∞ , ˆ y(0) = y 0 in Ω.

(1.7)

The operator M ∈ L(L ² (Γ; R ² )) is used to localize the control in a part of the boundary (see section 2.1). In order to stabilize ˆ y with a prescribed exponential decay rate e ^−αt , α > 0, we set

y = e ^αt y, ˆ p = e ^αt p, ˆ u = e ^αt u. ˆ Then, (y, p) is solution to the system

∂y

∂t − ν ∆y − αy + (y · ∇)w + (w · ∇)y + e ^−αt (y · ∇)y + ∇p = 0, div y = 0 in Ω × (0, ∞),

y = M u on Σ _∞ , y(0) = y ₀ in Ω.

(1.8)

In [29], the first author has determined a linear feedback law able to stabilize the nonlinear system (1.8). As mentioned above, in [29] the Riccati equation is stated in a space of infinite dimension. In the present paper, we want to find a control u of the form

u(t, x) =

n

_c

X

i=1

v i (t)ζ i (x) ∈ L ² (0, ∞; L ² (Γ; R ² )), (1.9) able to stabilize equation (1.8) and for which v i ∈ L ² (0, ∞), i = 1, . . . , n c , are written in feedback form. (The functions {ζ i } ⁿ _i=1

^c

are not a priori known and have to be determined). Let us explain how we proceed. Following [28, 29], we first write the linearized equation associated with (1.8) in the form

P y ⁰ = AP y + BMu = AP y + (λ ₀ I − A)P D _A M u in (0, ∞), P y(0) = y 0 ,

(I − P )y = (I − P )D A M u in (0, ∞),

(1.10)

where P is the so-called Helmholtz or Leray projection operator, A is the linearized Navier-Stokes operator, D _A is a Dirichlet operator and B is a control operator (see section 2). Let us notice that the associated nonlinear system

Py ⁰ = AP y + BMu + F (Py + (I − P)D _A M u) in (0, ∞), P y(0) = y ₀ ,

is of the form (1.1) if we set z = P y, B m = BM, L = (I − P)D A M and F(y) =

e ^−αt P ((y · ∇)y).

Next, we decompose V ⁰ _n (Ω) = n

y ∈ L ² (Ω; R ² ) | div y = 0 in Ω, y·n = 0 on Γ o in the form

V ⁰ _n (Ω) = Y _α

−

⊕ Y _α ,

where Y α ⊂ D(A) is the finite dimensional unstable susbspace of A and Y _α

⁻

is the stable subspace. Similarly, we have

V ⁰ _n (Ω) = Y ^∗ _α

−

⊕ Y _α ^∗ ,

where Y _α ^∗ ⊂ D(A ^∗ ) is the finite dimensional unstable susbspace of A ^∗ and Y _α ^∗

−

is the stable subspace. In section 3, we prove that there exist a basis {e 1 , · · · , e _K } of Y _α and a basis {ε 1 , · · · , ε _K } of Y ^∗ _α such that

e i , ε j

= Z

Ω

e i (x)ε j (x)dx = δ _i ^j ,

where δ _i ^j is the Kronecker symbol. This type of result is already established for parabolic equations in [19] and in [18, 20] for linearized Navier-Stokes equations.

Let Q denote the projection onto Y α along Y _α

⁻

. With such a choice for the basis of Y α and Y ^∗ _α , we obtain a very simple expression of Q and of Q ^∗ , the adjoint of Q. We consider the system

QP y ⁰ = AQP y + QBM u, QP y(0) = Qy ₀ . (1.11) In section 4, we prove that equation (1.11) is stabilizable by a control u of the form (1.9), where {ζ 1 , · · · , ζ n

_c

} is a basis of U = vect n

M B ^∗ ε j | j = 1, · · · , K o

. In section 5, we introduce a linear quadratic control problem of which the Riccati equation is

Π = Π ^∗ ∈ L(Y _α , Y ^∗ _α ), Π ≥ 0,

ΠA α + A ^∗ _α Π − ΠQB M f M f ^∗ B ^∗ Q ^∗ Π + Q ^∗ Q = 0,

(1.12) where M f is defined in (4.6). Let us notice that it is a finite dimensional algebraic Riccati equation. Finally, in section 6 we show that the feedback law

v _i (t) = −

ΠQ BM ζ _i , P y(t)

V⁰_n

(Ω)

= − Z

Ω

ΠQ BM ζ _i Py(t) dx, with u given by (1.9) stabilizes the Navier-Stokes equation locally about w.

Even if this result seems interesting, we would like to explain what is its practical interest for numerical computations. Such an approach consisting in decoupling the linearized Navier-Stokes equations into a stable and an unstable part has been used by S. Ahuja and C. W. Rowley in [1] to design reduced order models. Here, our goal is to use this decomposition to define a finite dimensional Riccati equation. Even if the domain of stability of the feedback law determined here is small, the result is still interesting. Indeed, there is an efficient algorithm to solve large scale Riccati equations, the so-called Newton–Kleinman method (see [14], and the references therein). The drawback of the Newton–Kleinman method is that it requires an initial guess for which the corresponding closed loop system is stable. The feedback that we determine may provide such an initial guess. By this way, we can hope to enlarge the domain of stability of the feedback law. Let us finally mention that, following [2, 3, 4], it should be possible to define a Lyapunov function of the closed loop system obtained by coupling the Navier-Stokes equations with the finite dimensional feedback control that we have determined.

Some months after the submission of this paper, we were informed of [5]. One

objective in [5] is to use the tools introduced in the present paper to characterize the

minimal dimension of static and dynamic controllers able to stabilize system (1.8).

Here our purpose is mainly to characterize feedback controllers of finite dimension by a matrix Riccati equation (see section 5.3).

2. Functional framework.

2.1. Notation and assumptions. Let us introduce the following spaces H ^s (Ω; R ^N )

= H ^s (Ω), L ² (Ω; R ^N ) = L ² (Ω), the same notation conventions will be used for trace spaces and for the spaces H ₀ ^s (Ω; R ^N ). We also introduce different spaces of free divergence functions and some corresponding trace spaces

V ^s (Ω) = n

y ∈ H ^s (Ω) | div y = 0 in Ω, hy · n, 1i _H

−1/2

(Γ),H

^1/2

(Γ) = 0 o

, s ≥ 0, V ^s _n (Ω) = n

y ∈ H ^s (Ω) | div y = 0 in Ω, y · n = 0 on Γ o

for s ≥ 0, V ^s ₀ (Ω) = n

y ∈ H ^s (Ω) | div y = 0 in Ω, y = 0 on Γ o

for s > 1/2, V ^s (Γ) = n

y ∈ H ^s (Γ) | hy · n, 1i _H

−1/2

(Γ),H

^1/2

(Γ) = 0 o

for s ≥ −1/2 .

In the above setting n denotes the unit normal to Γ outward Ω. We shall use the following notation Σ _∞ = Γ × (0, ∞). We also set

V ^s,σ (Ω × (0, ∞)) = H ^σ (0, ∞; V ⁰ (Ω)) ∩ L ² (0, ∞; V ^s (Ω)) for s, σ ≥ 0, and

V ^s,σ (Σ _∞ ) = H ^σ (0, ∞; V ⁰ (Γ)) ∩ L ² (0, ∞; V ^s (Γ)) for s, σ ≥ 0.

For an open subset Γ c of Γ, we introduce a weight function m ∈ C ² (Γ) with values in [0, 1], with support in Γ c , equal to 1 in Γ 0 , where Γ 0 is an open subset in Γ c . Associated with this function m, we introduce the operator M ∈ L(V ⁰ (Γ)) defined by

M u(x) = m(x)u(x) − m R

Γ m Z

Γ

mu · n

n(x).

By this way, we can replace the condition supp(u) ⊂ Γ c by considering a boundary condition of the form

z − w = M u ˆ on Σ _∞ .

For all ψ ∈ H ^1/2+ε

⁰

(Ω), with ε ⁰ > 0, we denote by c(ψ) the constant defined by c(ψ) = 1

|Γ|

Z

Γ

ψ, (2.1)

where |Γ| is the (N − 1)-dimensional Lebesgue measure of Γ. Let us recall that P , the so-called Leray or Helmholtz projector, is the orthogonal projection in L ² (Ω) onto V ⁰ _n (Ω).

2.2. Properties of some operators. In this subsection we briefly recall the def- initions and properties of some operators already used in [29]. The proof of these results can be found in [29]. We denote by (A, D(A)) and (A ^∗ , D(A ^∗ )) the un- bounded operators in V ⁰ _n (Ω) defined by

D(A) = H ² (Ω) ∩ V ₀ ¹ (Ω), Ay = νP ∆y + αy − P ((w · ∇)y) − P((y · ∇)w), D(A ^∗ ) = H ² (Ω) ∩ V ¹ ₀ (Ω), A ^∗ y = νP ∆y + αy + P ((w · ∇)y) − P((∇w) ^T y).

Since w ∈ V ³ (Ω) and div w = 0, we can verify that there exists λ 0 > 0 in the

resolvent set of A satisfying

(λ 0 I − A)y, y

V⁰_n

(Ω) ≥ ν 2 |y| ²

_V1

0

(Ω) for all y ∈ D(A), and

(λ 0 I − A ^∗ )y, y

V⁰_n

(Ω) ≥ ν 2 |y| ²

_V1

0

(Ω) for all y ∈ D(A ^∗ ).

(2.2)

Theorem 2.1. The unbounded operator (A − λ ₀ I) (respectively (A ^∗ − λ ₀ I)) with domain D(A−λ ₀ I) = D(A) (respectively D(A ^∗ −λ ₀ I)) is the infinitesimal generator of a bounded analytic semigroup on V ⁰ _n (Ω). Moreover, we have

D((λ 0 I − A) ^θ ) = D((λ 0 I − A ^∗ ) ^θ ) = [V ⁰ _n (Ω), D(A)] θ

for all 0 ≤ θ ≤ 1.

Observe that the semigroups (e ^t(A−λ

⁰

^I) ) _t≥0 and (e ^t(A

^∗

^−λ

⁰

^I) ) _t≥0 are exponentially stable on V ⁰ _n (Ω) and that

ke ^t(A−λ

⁰

^I) k _L(V

0

n

(Ω)) ≤ Ce ^−ωt and ke ^t(A

^∗

^−λ

⁰

^I) k _L(V

0

n

(Ω)) ≤ Ce ^−ωt , for all ω < ν/2 (see [13, Chapter 1, Theorem 2.12]).

Let us introduce D _A and D _p , two Dirichlet operators associated with A, defined as follows (see [29, p. 796]). For u ∈ V ⁰ (Γ), we set D _A u = y and D _p u = q where (y, q) is the unique solution in V ^1/2 (Ω) × (H ^1/2 (Ω)/ R ) ⁰ to the equation

λ ₀ y − ν ∆y − αy + (w · ∇)y + (y · ∇)w + ∇q = 0 in Ω, div y = 0 in Ω, y = u on Γ.

Lemma 2.2. (i) The operator D A is a bounded operator from V ⁰ (Γ) into V ⁰ (Ω), moreover it satisfies

|D A u|

_Vs+1/2

(Ω) ≤ C(s)|u|

_Vs

(Γ) for all 0 ≤ s ≤ 2 .

(ii) The operator D ^∗ _A ∈ L(V ⁰ (Ω), V ⁰ (Γ)), the adjoint operator of D A ∈ L(V ⁰ (Γ), V ⁰ (Ω)), is defined by

D ^∗ _A g = −ν ∂z

∂n + πn − c(π)n, (2.3)

where (z, π) is the solution of

λ 0 z − ν∆z − αz − (w · ∇)z + (∇w) ^T z + ∇π = g div z = 0 in Ω,

z = 0 on Γ, (2.4)

and c(π) is defined by (2.1).

Lemma 2.3. The operator M ∈ L(V ⁰ (Γ)) is symmetric.

We introduce the operator B = (λ 0 I − A)P D A ∈ L(V ⁰ (Γ), (D(A ^∗ )) ⁰ ).

Proposition 2.4. The operator adjoint B ^∗ ∈ L(D(A ^∗ ), V ⁰ (Γ)) satisfies B ^∗ Φ = D _A ^∗ (λ ₀ I − A ^∗ )Φ and

B ^∗ Φ = −ν ∂Φ

∂n + ψn − c(ψ)n, for all Φ ∈ D(A ^∗ ), with

∇ψ = (I − P ) h

ν∆Φ + (w · ∇)Φ − (∇w) ^T Φ i , and c(ψ) defined by (2.1). Moreover, the following estimate holds

|B ^∗ Φ|

_Vs−3/2

(Γ) ≤ C|Φ|

_Vs

(Ω)∩V

₀¹

(Ω) ,

for all Φ ∈ V ^s (Ω) ∩ V ¹ ₀ (Ω) with s ≥ 2.

3. Projected systems. In order to introduce the generalized eigenfunctions of the operator A, we consider the complexified space

V _n ⁰ (Ω) = V ⁰ _n (Ω) ⊕ iV ⁰ _n (Ω).

The first equation in (1.10) may be extended to spaces of functions with complex values as follows

P y ⁰ = AP y + BM u, P y(0) = y ₀ , (3.1) where y ₀ , y and u are now functions with complex values.

3.1. The resolvent of the operator A. We first study the resolvent of the op- erator A.

Lemma 3.1. The resolvent of A is compact and the spectrum of A is discrete.

Proof. See [18, Lemma 3.1].

Now, we give a decomposition of the resolvent of A by using Laurent series. Let λ j belong to the spectrum of A. For λ in the neighbourhood of λ j , the resolvent of A can be expressed in a Laurent series

R(λ, A) =

+∞

X

k=−∞

(λ − λ j ) ^k R k (λ j )

with R k (λ j ) = 1 2πi

Z

|λ−λ

j

|=ε

R(λ, A)

(λ − λ j ) ^k+1 dλ, and ε > 0 small enough.

(3.2)

Lemma 3.2. The expansion (3.2) of the resolvent in a Laurent series in a neigh- bourhood of λ j contains finitely many terms with negative power of λ − λ j , that is

R(λ, A) =

+∞

X

k=−m(λ

j

)

(λ − λ j ) ^k R k (λ j ). (3.3) Proof. The proof is done in [19, Lemma 3.3].

Since the spectrum of A is a pointwise spectrum, we may always choose α > 0 such that

· · · ≤ <λ N

_α

+1 < 0 < <λ N

_α

≤ · · · ≤ <λ 1

for some N α ∈ N ^∗ . We consider the continuous contour γ 0 in the complex half-plane {λ ∈ C | <λ ≤ 0} made up of a segment of the line {<λ = 0} and the two branches of γ on rays { Argλ = ±θ} with θ > ^π ₂ . Thanks to this new contour, we obtain another expression of the semigroup given in the following lemma.

Lemma 3.3. The semigroup e ^tA may be written in the form

e ^tA = 1 2πi

Z

γ

₀

(λI − A) ⁻¹ e ^λt dλ +

N

α

X

j=1

e ^λ

^j

^t

m(λ

_j

)

X

n=1

t ⁿ⁻¹

(n − 1)! R −n (λ j ).

Proof. See [19, p. 603].

3.2. Canonical Systems. For 1 ≤ j ≤ N α , we set

E(λ j ) = Ker (A − λ j I) and `(j) = dim E(λ j ).

E(λ _j ) is the eigenspace associated to the eigenvalue λ _j and `(j) is the geometric multiplicity of λ _j . We also introduce the generalized eigenspace

G(λ _j ) = Ker ((A − λ _j I) ^m(λ

^j

⁾ ) and N (λ _j ) = dim G(λ _j ),

where m(λ j ) is the multiplicity of the pole λ j of the resolvent (see Lemma 3.2) and N (λ j ) is the algebraic multiplicity of λ j .

If λ j is an eigenvalue of A, then λ j is an eigenvalue of A ^∗ . Since A has real coefficients, λ j is an eigenvalue of A and λ j is an eigenvalue of A ^∗ . We also introduce the generalized eigenspaces of A ^∗

G ^∗ (λ j ) = Ker ((A ^∗ − λ j I) ^m(λ

^j

⁾ ).

Let us define the multiplicity of an eigenvector.

Definition 3.4. (See [18, 19]). We say that (e ^k ₁ , e ^k ₂ , · · · , e ^k _r ) forms a chain of gen- eralized eigenvectors, when the following relations hold

(λ j I − A)e ^k ₁ = 0, (λ j I − A)e ^k ₂ + e ^k ₁ = 0, · · · , (λ j I − A)e ^k _r + e ^k _r−1 = 0.

If the maximal order of the chain of generalized eigenvectors corresponding to e ^k ₁ is m then the number m is called the multiplicity of the eigenvector e ^k ₁ .

We consider special bases of generalized eigenvectors.

Definition 3.5. (See [18, 19]). A basis of G(λ j ) of the form n

e ^k _i | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o is called a canonical system if n

e ^k ₁ | k = 1, · · · , `(j) o

is a basis of E(λ j ) defined in the following way

• e ¹ ₁ is an eigenvector with maximum possible multiplicity m ^j ₁ ,

• e ^k ₁ is an eigenvector with maximum possible multiplicity m ^j _k such that e ^k ₁ is not linearly expressible in terms of e ¹ ₁ , · · · , e ^k−1 ₁ ,

and, for k = 1, · · · , `(j) and i = 2, · · · , m ^j _k , (A − λ _j I)e ^k _i = e ^k _i−1 .

Obviously, we have m(λ _j ) = max(m ^j ₁ , · · · , m ^j <sub>`(j)</sub> ) and N(λ <sub>j</sub> ) = P `(j)

k=1 m ^j _k . We remark that if λ j is an eigenvalue of the operator A with multiplicity m(λ j ), then λ j is an eigenvalue of the operator A ^∗ with the same multiplicity. That is why, we can define another canonical system associated to λ j for A ^∗

n ε ^k _i | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o ,

where n

ε ^k ₁ | k = 1, · · · , `(j) o

is a basis of Ker (A ^∗ − λ _j I) such that

• ε ¹ ₁ is an eigenvector with maximum possible multiplicity m ^j ₁ ,

• ε ^k ₁ is an eigenvector with maximum possible multiplicity m ^j _k such that ε ^k ₁ is

not linearly expressible in terms of ε ¹ ₁ , · · · , ε ^k−1 ₁ ,

and, for k = 1, · · · , `(j) and i = 2, · · · , m ^j _k , (A ^∗ − λ j I)ε ^k _i = ε ^k _i−1 . In what follows, we denote by

·, ·

the complex inner product in V _n ⁰ (Ω), that is

f, g

= Z

Ω

f g dx.

Definition 3.6. For a couple (e _i , ε _j ) ∈ (V _n ⁰ (Ω)) ² , we denote by e _i ε _j the operator defined by

(e i ε j )f = f, ε j

e i

for all f ∈ V _n ⁰ (Ω).

Theorem 3.7. For any canonical system n

ε ^k _i | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o of A ^∗ corresponding to the eigenvalue λ j , there is a uniquely determined canonical system n

e ^k _i | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o

of A for λ j such that the principal part of the resolvent can be expressed in the following way in a neighbourhood of λ _j

−1

X

p=−m(λ

j

)

(λ − λ j ) ^p R p (λ j )

=

`(j)

X

k=1



 e ^k ₁ ε ^k ₁ (λ − λ j ) ^m

^jk

+ e ^k ₁ ε ^k ₂ + e ^k ₂ ε ^k ₁ (λ − λ j ) ^m

^jk

⁻¹

+ · · · + e ^k ₁ ε ^k

m

^j_k

+ e ^k ₂ ε ^k

m

^j_k

−1 + · · · + e ^k

m

^j_k

ε ^k ₁ (λ − λ _j )



.

Proof. See [19, Theorem 3.1] or [18, 22].

3.3. The complex projected system. We consider the space Z _α = ⊕ ^N _j=1

^α

G(λ _j ).

We denote by N = P N

_α

j=1 N (λ j ) its dimension. With [21, p. 178 – 182], we first notice that the space V _n ⁰ (Ω) can be decomposed as follows

V _n ⁰ (Ω) = Z _α

⁻

⊕ Z α ,

where Z _α

⁻

is the stable space of A, that is to say Z _α

⁻

∩ D(A) is invariant under A.

Similarly, we have the decomposition

V _n ⁰ (Ω) = Z _α ^∗

−

⊕ Z _α ^∗ ,

where Z _α ^∗ = ⊕ ^N _j=1

^α

G ^∗ (λ j ) and where Z _α ^∗

−

∩ D(A ^∗ ) is invariant under A ^∗ . The space Z α will be equipped with the norm

|y| Z

_α

= |y| _V

0 n

(Ω) .

Let γ _α be a simple closed curve enclosing (λ ₁ , · · · , λ _N

_α

) but no other point of the spectrum of A, and oriented counterclockwise. The operator

P α = 1 2πi

Z

γ

α

(λI − A) ⁻¹ dλ

is the projection onto Z _α parallel to Z _α

−

(see [21, p. 178–182]).

Lemma 3.8. For all j ∈ {1, · · · , N α }, we consider

ε ^k _i (λ j ) | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o

, a canonical system of A ^∗ corresponding to the eigenvalue λ j , and

the corresponding canonical system of A, n

e ^k _i (λ j ) | k = 1, · · · , `(j), i = 1, · · · , m ^j _k o , associated with λ j and determined in Theorem 3.7. Then, we have

e ^k _i

¹

1

(λ j

₁

), ε ^k _i

²

2

(λ j

₂

)

= δ _j ^j

²

1

δ _k ^k

²

1

δ ⁱ

²

m

^j_k¹

1

+1−i

1

,

for all j ₁ , j ₂ ∈ {1, · · · , N _α }, k ₁ ∈ {1, · · · , `(j <sub>1</sub> )}, k <sub>2</sub> ∈ {1, · · · , `(j ₂ )}, i 1 ∈ n

1, · · · , m ^j _k

¹

1

o

, i 2 ∈ n

1, · · · , m ^j _k

²

2

o

(where the Kronecker symbols δ _j ^j

²

1

, δ ^k _k

²

1

and δ ⁱ

²

m

^j_k¹

1

+1−i

1

are equal to 1 if the index is equal to the exponent and 0 otherwise).

Proof. Let j be in {1, · · · , N _α }. With the definition of P _α and the definition of R ₋₁ (λ _j ) in (3.2), we have

P α =

N

α

X

j=1

1 2πi

Z

|λ−λ

j

|=ε

(λI − A) ⁻¹ dλ =

N

α

X

j=1

R −1 (λ j ).

Let us set z = e ^k _i

¹

1

(λ _j

₁

), j ₁ ∈ {1, · · · , N _α }, k ₁ ∈ {1, · · · , `(j ₁ )} and i ₁ ∈ n

1, · · · , m ^j _k

¹

1

o . We have P α (z) = z because z belongs to Z α . Due to Theorem 3.7, we obtain

R ₋₁ (λ j )(z)

= P `(j) k=1

z, ε ^k

m

^j_k

(λ j )

e ^k ₁ (λ j ) + z, ε ^k

m

^j_k−1

(λ j )

e ^k ₂ (λ j ) + · · · +

z, ε ^k ₁ (λ j ) e ^k

m

^j_k

(λ j ).

Since n

e ^k _i (λ j ) | j = 1, · · · , N α , k = 1, · · · , `(j), i = 1, · · · , m ^j _k o

is a basis of Z α , we clearly obtain the result and the proof is complete.

Remark 3.9. With Lemma 3.8, we obtain P α z =

N

α

X

j=1

`(j)

X

k=1 m

^j_k

X

i=1

z, ε ^k _m

j

k

+1−i (λ j ) e ^k _i (λ j ).

Since dim(Z _α ) < ∞, we can extend, in a continuous way, the operator P _α to (D(A ^∗ )) ⁰ as follows

P _α z =

N

α

X

j=1

`(j)

X

k=1 m

^j_k

X

i=1

D z, ε ^k

m

^j_k

+1−i (λ _j ) E

(D(A

^∗

))

⁰

,D(A

^∗

)

e ^k _i (λ _j ) for all z ∈ (D(A ^∗ )) ⁰ . Let us observe that the operator P _α belongs to L((D(A ^∗ )) ⁰ , Z _α ).

Remark 3.10. Due to Lemma 3.3, we notice that P _α e ^At =

N

α

X

j=1

e ^λ

^j

^t

m(λ

_j

)

X

n=1

t ⁿ⁻¹

(n − 1)! R _−n (λ _j ).

The system (3.1) projected onto Z _α along to Z _α

−

is

P _α y ⁰ = AP _α y + P _α BM u, P _α y(0) = P _α y ₀ . (3.4) From the definition of P _α , it follows that

P _α BM u =

N

α

X

j=1

`(j)

X

k=1 m

^j_k

X

i=1

u, M B ^∗ ε ^k

m

^j_k

+1−i (λ _j )

e ^k _i (λ _j ),

where u, M B ^∗ ε ^k

m

^j_k

+1−i (λ _j )

=

u, M B ^∗ ε ^k

m

^j_k

+1−i (λ _j )

V

⁰

(Γ)

= Z

Γ

u M B ^∗ ε ^k

m

^j_k

+1−i (λ _j ).

We can rewrite it as a system of ordinary differential equations in R ^N with N = P N

_α

j=1 N(λ j ). For that, we introduce the coordinates of P α y in the basis (e ^k _i (λ j )) _1≤j≤N

α

,1≤k≤`(j), 1≤i≤m

^j_k

P α y =

N

_α

X

j=1

`(j)

X

k=1 m

^j_k

X

i=1

y _i ^k (λ j )e ^k _i (λ j ).

For 1 ≤ j ≤ N _α , we denote by Y (λ _j ) ∈ R ^{N (λ}

^j

⁾ , the vectors Y (λ j ) =

(y _i ¹ (λ j )) _1≤i≤m

j

1

, · · · , (y ^`(j) _i (λ j )) _1≤i≤m

j

`(j)

, and the vector Y ∈ R ^N defined by

Y = (Y (λ ₁ ), · · · , Y (λ _N

_α

)) ^T . Similarly, we denote by U (λ j ) ∈ R ^N(λ

^j

⁾ , the vectors

U (λ _j ) = u, M B ^∗ ε ¹

m

^j₁

+1−i (λ _j )

1≤i≤m

^j₁

, · · · ,

u, M B ^∗ ε ^`(j)

m

^j_`(j)

+1−i (λ _j )

1≤i≤m

^j_`(j)

! ,

and the vector V(u) ∈ R ^N defined by V(u) = (U (λ 1 ), · · · , U(λ N

_α

)) ^T . Due to Lemma 3.8, we can observe that P α y is solution to system (3.4) if and only if Y (t), the vector of coordinates of P α y, is solution of the differential system

Y ⁰ = J Y + V(u), Y (0) = y ₀ , ε ^k

m

^j_k

+1−i (λ _j )

1≤j≤N

α

,1≤k≤`(j),1≤i≤m

^j_k

, (3.5) where J = diag(J 1 , · · · , J N

_α

) and, for 1 ≤ j ≤ N α , J j is constituted of Jordan blocks associated with λ j . If in (3.5) we take u of the form

u(x, t) =

K

X

m=1

v _m (t) ζ _m (x),

with ζ m ∈ vec{M B ^∗ ε ^k _i (λ j ) | 1 ≤ k ≤ `(j), 1 ≤ j ≤ N α , 1 ≤ i ≤ m ^j _k }, then system (3.5) is of the form

Y ⁰ = J Y + BV, (3.6)

with V = (v 1 , · · · , v K ) ^T , B ∈ C ^{N ×K} , N = P N

_α

j=1 N(λ j ), B =





 B(λ 1 )

.. . B(λ N

α

)





 , B(λ j ) =







B ¹ (λ j ) .. . B ^`(j) (λ _j )





 , B ^k (λ j ) =









B ^k ₁ (λ _j ) .. . B ^k

m

^j_k

(λ j )







 ,

with 1 ≤ j ≤ N _α , 1 ≤ k ≤ `(j), and for 1 ≤ i ≤ m ^j _k , B _i ^k (λ _j ) = h

(ζ ₁ , M ^∗ B ^∗ ε ^k

m

^j_k

+1−i (λ _j )) _V

0

(Γ) , · · · , (ζ _K , M ^∗ B ^∗ ε ^k

m

^j_k

+1−i (λ _j )) _V

0

(Γ)

i ∈ C ^K .

We do not study here the controllability of system (3.6). We shall prove the con-

trollability of the corresponding real system in section 4 for a particular family

(ζ ₁ , · · · , ζ _K ).

3.4. The real projected system. We recall that <V _n ⁰ (Ω) = V ⁰ _n (Ω) and we in- troduce the subspaces Y α = <Z α and Y _α

⁻

= <Z _α

⁻

. We have

V ⁰ _n (Ω) = Y α

⁻

⊕ Y α ,

Y _α ⊂ ∩D(A) is invariant under A and Y _α

−

∩ D(A) is invariant under A too. We also introduce Y _α ^∗ = <Z _α ^∗ and Y _α ^∗ = <Z _α ^∗

−

, and we have

V ⁰ _n (Ω) = Y ^∗ _α

−

⊕ Y _α ^∗ .

It is obvious that Y ^∗ _α ⊂ D(A ^∗ ) and Y ^∗ _α

−

∩ D(A ^∗ ) are both invariant under A ^∗ . We follow the proof of [19, Lemma 6.2] to construct two bi-orthogonal families that will be very useful to define the projection on Y _α parallel to Y _α

−

and the projection on Y ^∗ _α parallel to Y ^∗ _α

−

.

Since the operator A has real coefficients, λ j and λ j either both are or both are not eigenvalues of A. Moreover e ^k _i (λ _j ) is a corresponding eigenfunction associated with λ j (or a generalized eigenfunction) if and only if e ^k _i (λ j ) is an eigenfunction (or a generalized eigenfunction) associated with λ j . A similar assertion applies to eigenvalues and generalized functions of the operator A ^∗ . Due to that, we can choose a canonical system (e ^k _i (λ _j )) _i,k associated to λ _j such that

e ^k _i (λ _j ) = e ^k _i (λ _j ).

As a consequence, if λ _j is real, the chosen canonical system associated to λ _j is real too. Similarly, we choose the eigenfunctions and generalized eigenfunctions of the operator A ^∗ such that

ε ^k _i (λ j ) = ε ^k _i (λ j ).

Let us consider the sets

F 1 = {j ∈ {1, · · · , N α } | =λ j > 0} and F 2 = {j ∈ {1, · · · , N α } | =λ j = 0} . Then, we set

B _1,1 = n√

2 <e ^k _i (λ _j ), √

2 =(e ^k _i (λ _j )) | j ∈ F ₁ , k = 1, · · · , `(j), i = 1 · · · m ^j _k o and

B 1,2 = n

e ^k _i (λ j ) | j ∈ F 2 , k = 1, · · · , `(j), i = 1, · · · , m ^j _k o . We also set

B _2,1 = n√

2 <ε ^k

m

^j_k

+1−i (λ _j ), √ 2 =ε ^k

m

^j_k

+1−i (λ _j ) | j ∈ F ₁ , k = 1, · · · , `(j), i = 1, · · · , m ^j _k o and

B _2,2 = n ε ^k

m

^j_k

+1−i (λ _j ) | j ∈ F ₂ , k = 1, · · · , `(j), i = 1, · · · , m ^j _k o . Let us notice that

B 2,1 = n√

2 <ε ^k _m

j

k

+1−i (λ j ), √ 2 =ε ^k _m

j

k

+1−i (λ j ) | j ∈ F 1 , k = 1, · · · , `(j), i = 1, · · · , m ^j _k o . The families B ₁ = B _1,1 ∪ B _1,2 and B ₂ = B _2,1 ∪ B _2,2 are linearly independent. From Lemma 3.8 (see also [19, Lemma 6.2]), it follows that

√

2 <e ^k _i

₁¹

(λ j

1

), √

2 <ε ^k _i

₂²

(λ j

2

)

= √

2 =e ^k _i

₁¹

(λ j

1

), √

2 =ε ^k _i

₂²

(λ j

2

)

= δ _j ^j

₁²

δ ^k _k

²

1

δ ⁱ

²

m

^j_k¹

1

+1−i

1

,

<e ^k _i

¹

1

(λ j

₁

), =ε ^k _i

²

2

(λ j

₂

)

= 0 and

=e ^k _i

¹

1

(λ j

₁

), <ε ^k _i

²

2

(λ j

₂

)

= 0, for all j 1 , j 2 ∈ F 1 , k 1 ∈ {1, · · · , `(j 1 )}, k 2 ∈ {1, · · · , `(j 2 )}, i 1 ∈ n

1, · · · , m ^j _k

¹

1

o , i 2 ∈ n

1, · · · , m ^j _k

²

2

o ,

e ^k _i

¹

1

(λ j

₁

), ε ^k _i

²

2

(λ j

₂

)

= δ _j ^j

²

1

δ _k ^k

²

1

δ ⁱ

²

m

^j_k¹

1

+1−i

1

,

for all j 1 , j 2 ∈ F 2 , k 1 ∈ {1, · · · , `(j 1 )}, k 2 ∈ {1, · · · , `(j 2 )}, i 1 ∈ n

1, · · · , m ^j _k

¹

1

o , i 2 ∈ n

1, · · · , m ^j _k

²

2

o

. Moreover the vectors belonging to B 1,1 are orthogonal to vec- tors belonging to B 2,2 and the vectors belonging to B 1,2 are orthogonal to vectors belonging to B 2,1 . Thus we can rewrite the families B 1 and B 2 in the form

B 1 = n

e 1 , · · · , e K

o

and B 2 = n

ε 1 , · · · , ε K

o

(3.7) so that

e _i , ε _j

= δ _i ^j , (3.8)

where δ _i ^j is a Kronecker symbol and

·, ·

is the inner product in the real space V ⁰ _n (Ω).

Let us consider the operator Q defined by

Qf = <(P α f ), for all f ∈ V ⁰ _n (Ω).

From the definition of P α and from (3.8), it follows that Qf =

K

X

j=1

f , ε _j

e _j for all f ∈ V ⁰ _n (Ω).

Since Y α = <Z α and Y _α

⁻

= <Z _α

⁻

, it is clear that Q is the projection onto Y α

parallel to Y _α

⁻

. Moreover, we have Y α = vect n

e j | j = 1, · · · , K o

and Y _α ^∗ = vect n

ε j | j = 1, · · · , K o . Proposition 3.11. The adjoint operator Q ^∗ ∈ L(V ⁰ _n (Ω)) is defined by

Q ^∗ f =

K

X

j=1

f , e j

ε j for all f ∈ V _n ⁰ (Ω) and it is the projection onto Y _α ^∗ parallel to Y ^∗ _α

−

.

Proof. It is a direct consequence of the definition of the operator Q.

Proposition 3.12. We can characterize the space Y _α

⁻

as follows Y _α

⁻

= n

f ∈ V ⁰ _n (Ω) | f , g

= 0 for all g ∈ Y ^∗ _α o .

Proof. Let f belong to Y _α

⁻

. We have Qf = 0. From the definition of Q, since {e 1 , · · · , e _K } is a basis of Y _α , we obtain

f , ε _j

= 0, for all j = 1, · · · , K, and therefore

Y _α

⁻

⊂ n

f ∈ V ⁰ _n (Ω) | f , g

= 0 for all g ∈ Y _α ^∗ o

.

Let f ∈ V ⁰ _n (Ω) be such that f , g

= 0 for all g ∈ Y _α ^∗ . Obviously, we have f , ε _j

= 0, for all j = 1, · · · , K. From the definition of the operator Q, we obtain Qf = 0. Thus, we have f = (I − Q)f and the proof is complete.

Remark 3.13. Similarly, we can characterize the space Y _α ^∗

−

as follows Y _α ^∗

−

= n

Φ ∈ V _n ⁰ (Ω) | Φ, h

= 0 for all h ∈ Y α

o .

Corollary 3.14. We can identify the dual of Y α with Y _α ^∗ , and the dual of Y _α

⁻

with Y _α ^∗

−

.

Proof. The space V _n ⁰ (Ω) is identified with its dual. The space Y α is a closed subspace of V _n ⁰ (Ω). Thus

(Y α ) ⁰ = V ⁰ _n (Ω)/(Y α ) ^⊥ = V ⁰ _n (Ω)/Y ^∗ _α

−

, where (Y α ) ^⊥ = n

Φ ∈ V _n ⁰ (Ω) | Φ, h

= 0 for all h ∈ Y α

o

= Y ^∗ _α

−

. Since V ⁰ _n (Ω)

= Y ^∗ _α ⊕ Y ^∗ _α

−

, the subspace V ⁰ _n (Ω)/Y ^∗ _α

−

is isomorph to Y _α ^∗ and (Y α ) ⁰ may be iden- tified with Y ^∗ _α . The proof is similar to show that the dual of Y _α

−

can be identified with Y ^∗ _α

−

.

From their definitions, Q and Q ^∗ are linear and continuous from V ⁰ _n (Ω) to re- spectively Y α and Y ^∗ _α . Since Y α and Y ^∗ _α are spaces of finite dimension, we can extend the operator Q to (D(A ^∗ )) ⁰ and Q ^∗ to (D(A)) ⁰ in the following way.

Remark 3.15. The operators Q ∈ L((D(A ^∗ )) ⁰ , Y _α ) and Q ^∗ ∈ L((D(A)) ⁰ , Y _α ^∗ ) may be defined by

Qf =

K

X

j=1

D f , ε j

E

(D(A

^∗

))

⁰

,D(A

^∗

) e j and Q ^∗ f =

K

X

j=1

D f , e j

E

(D(A))

⁰

,D(A) ε j , for all f in (D(A ^∗ )) ⁰ = (D(A)) ⁰ .

We consider the subspace U of L ² (Γ) defined by U = vect n

M B ^∗ ε j | j = 1, · · · , K o

. (3.9)

We denote by {ζ 1 , · · · ζ n

_c

} a basis of U .

Remark 3.16. From Remark 3.15 and the definition of the space U we can deduce that M B ^∗ Q ^∗ belongs to L((D(A)) ⁰ , U ).

For notational simplicity, we still denote by A _α the restriction of A to Y _α . Let A _α

−

be the unbounded operator in Y _α

−

defined by

D(A _α

−

) = D(A) ∩ Y _α

−

, A _α

−

y = Ay for all y ∈ D(A _α

−

).

It is easy to check that the adjoint of (A _α

⁻

, D(A _α

⁻

)) is the unbounded operator (A ^∗ _α

−

, D(A ^∗ _α

−

)) in Y _α ^∗

−

defined by

D(A ^∗ _α

−

) = D(A ^∗ ) ∩ Y _α ^∗

−

, A ^∗ _α

−

y = A ^∗ y for all y ∈ D(A ^∗ _α

−

).

The space D(A) (respectively D(A ^∗ )) is equipped with the norm y 7→ |(λ ₀ I − A)y|

_V0

n

(Ω) (respectively y 7→ |(λ 0 I −A ^∗ )y|

_V0

n

(Ω) ). Let us recall that D(A) = D(A ^∗ ),

and actually the two norms are equivalent. The space D(A α

⁻

) is a closed subspace

of D(A) and it is dense in Y _α

−

. Thus, we shall equip D(A _α

−

) with the norm of

D(A). Similarly, D(A ^∗ _α

−

) is closed in D(A ^∗ ) and dense in Y ^∗ _α

−

, and D(A ^∗ _α

−

) will be equipped with the norm of D(A ^∗ ).

The operator A ^∗ _α

−

can be considered either as an unbounded operator in Y ^∗ _α

−

(or in V ⁰ _n (Ω)) with domain D(A ^∗ _α

−

) or as an isomorphism from D(A ^∗ _α

−

) to Y _α ^∗

−

. Similarly, the operator A _α

−

can be considered either as an unbounded operator in Y _α

−

(or in V ⁰ _n (Ω)) with domain D(A _α

−

) or as an isomorphism from D(A _α

−

) to Y _α

−

. Since A ^∗ _α

−

∈ Isom(D(A ^∗ _α

−

), Y ^∗ _α

−

), we have (A ^∗ _α

−

) ^∗ ∈ Isom(Y _α

−

, (D(A ^∗ _α

−

)) ⁰ ) (here, Isom(E, F ) denotes the space of isomorphisms from E onto F ). The operator (A ^∗ _α

−

) ^∗ can also be viewed as an unbounded operator in (D(A ^∗ _α

−

)) ⁰ with domain Y _α

⁻

. As in [13, Chapter 3, p. 160], it can be shown that this unbounded operator is the extension of A _α

⁻

to (D(A ^∗ _α

−

)) ⁰ . For simplicity, it will be still denoted by A _α

⁻

. Let us observe that we have the following decomposition

D(A) = Y α ⊕ D(A _α

−

) and D(A ^∗ ) = Y ^∗ _α ⊕ D(A ^∗ _α

−

). (3.10) In Proposition 3.19, we are going to see that (I − Q) belongs to L((D(A ^∗ )) ⁰ , (D(A ^∗ _α

−

)) ⁰ ). For that, we need a precise characterization of (D(A ^∗ _α

−

)) ⁰ , which is given in Proposition 3.18.

Lemma 3.17. The space (D(A ^∗ )) ⁰ can be decomposed as follows (D(A ^∗ )) ⁰ = Y α ⊕ Y _α

−

|·|

_(D(A∗))0

, where Y _α

−

|·|

_(D(A∗))0

denotes the closure of Y _α

−

in the norm (D(A ^∗ )) ⁰ . Proof. Step 1. We first prove the identity Y _α

−

|·|

_(D(A∗))0

= E _α , where E α =

f ∈ (D(A ^∗ )) ⁰ | D f , g E

(D(A

^∗

))

⁰

,D(A

^∗

)

= 0 for all g ∈ Y ^∗ _α

.

Let f belong to Y _α

⁻

^|·|

^(D(A∗))0

. There exists (f n ) n∈

N

, such that f n belongs to Y _α

⁻

for all n ∈ N , and lim n→∞ f n = f in (D(A ^∗ )) ⁰ . From Proposition 3.12, for all n ∈ N , we have

D f _n , g E

(D(A

^∗

))

⁰

,D(A

^∗

)

= f _n , g

= 0 for all g ∈ Y _α ^∗ .

We show that Y _α

⁻

^|·|

^(D(A∗))0

⊂ E α by passing to the limit in the previous identity when n tends to infinity. Let f belong to E α . There exists (f n ) n∈

N

, such that f n

belongs to V ⁰ _n (Ω) for all n ∈ N , and lim n→∞ f n = f in (D(A ^∗ )) ⁰ . For all n ∈ N , we have (I − Q)f _n ∈ Y _α

−

. Moreover, from the definition of E _α and (3.8), we can check that Qf = 0. Thus, we have (I −Q)f n −f = (I −Q)(f n −f ). It follows that (I −Q)f n

tends to f in (D(A ^∗ )) ⁰ when n tends to infinity and the equality Y _α

−

|·|

_(D(A∗))0

= E α

is proved.

Step 2. We show that (D(A ^∗ )) ⁰ = Y α ⊕ E α . From Remark 3.15, for all f ∈ (D(A ^∗ )) ⁰ , we have Qf ∈ Y α . Since (I − Q)f ∈ E α for all f ∈ (D(A ^∗ )) ⁰ , the proof is complete.

Since the space D(A _α

−

) is continuously and densely imbedded in Y _α

−

and D(A ^∗ _α

−

) is continuously and densely imbedded in Y _α ^∗

−

, by duality we have

Y ^∗ _α

−

, → (D(A _α

−

)) ⁰ and Y _α

−

, → (D(A ^∗ _α

−

)) ⁰ ,

with dense and continuous imbeddings if (D(A _α

⁻

)) ⁰ is equipped with the dual norm

of D(A α

⁻

) and if (D(A ^∗ _α

−

)) ⁰ is equipped with the dual norm of D(A ^∗ _α

−

). Since the

two norms | · | D(A) and | · | D(A

^∗

) are equivalent, we have the following proposition.

Proposition 3.18. We have the identity (D(A ^∗ _α

−

)) ⁰ = Y _α

−

|·|

_(D(A∗))0

.

Proposition 3.19. The operator (I − Q) is linear and continuous from (D(A ^∗ )) ⁰ to (D(A ^∗ _α

−

)) ⁰ .

Proof. Due to Proposition 3.18 and Lemma 3.17, we have (I − Q)f ∈ (D(A ^∗ _α

−

)) ⁰ for all f in (D(A ^∗ )) ⁰ . We notice that for all Φ ∈ D(A ^∗ _α

−

), we have Φ = (I − Q ^∗ )Φ.

Then, we have D

(I − Q)f, Φ E

(D(A

^∗

α−

))

⁰

,D(A

^∗

α−

)

= D f , Φ E

(D(A

^∗

))

⁰

,D(A

^∗

)

for all Φ ∈ D(A ^∗ _α

−

), and the proof is complete.

Let us set

y α = Qy and y _α

⁻

= (I − Q)y.

The linearized equation

y ⁰ = Ay + BM u in (0, ∞), y(0) = y ₀ , (3.11) may be split as follows

y _α ⁰ = A α y α + QBM u in (0, ∞), y α (0) = Qy 0 ,

y _α ⁰

−

= A _α

−

y _α

−

+ (I − Q)BM u in (0, ∞), y _α

−

(0) = (I − Q)y ₀ . (3.12) 4. Stabilizability of the real projected system by finite dimensional con- trols. In this section, we study the controllability of the projected system

y α 0 = A α y α + QBM u = A α y α +

n

_c

X

i=1

v i (t)QBM ζ i in (0, ∞), y α (0) = Qy 0 ,

(4.1)

where {ζ 1 , · · · , ζ n

_c

} is the basis of U defined in (3.9). We can rewrite this system as a differential system in R ^K by introducing the coordinates of y α in the basis {e 1 , · · · , e K }

y _α =

K

X

j=1

y _j e _j .

From Remark 3.15, it follows that QBM ζ i =

K

X

j=1

D

BM ζ i , ε j

E

(D(A

^∗

))

⁰

,D(A

^∗

) e j =

K

X

j=1

ζ i , M B ^∗ ε j

V⁰

(Γ) e j . Thus, by setting Y = col[y ₁ , · · · , y _K ] and V = col[v ₁ , · · · , v _n

_c

], equation (4.1) can be written as follows

Y ⁰ = AY b + BV b in (0, ∞), (4.2)

where A b is of the form

A b =















 Λ 1

Λ 2 0

0 . . .

Λ N

_α

















and

B b =















ζ 1 , M B ^∗ ε 1

V⁰

(Γ) · · ·

ζ n

_c

, M B ^∗ ε 1

V⁰

(Γ)

.. . .. .

ζ 1 , M B ^∗ ε K

V⁰

(Γ) · · ·

ζ n

_c

, M B ^∗ ε K

V⁰

(Γ)













 .

In Proposition 4.2 (see below), we prove that system (4.1) (and therefore system (4.2) too) is stabilizable. If we assume that the family

{M B ^∗ ε _j | j = 1, · · · , K } (4.3) is linearly independent, then rank( B) = b K and the Kalman controllability condition is satisfied, but this condition is not necessary for the controllability of system (4.2) (see Remark 4.1 below).

Remark 4.1. As noticed above, the linearly independence of the family (4.3) is not a necessary condition to the controllability. To illustrate this, let us consider the case where the family (4.3) is linearly dependent. To simplify, let us choose K = 2, n c = 1 and

A b = λ 1 0 λ

! .

The matrix B b ∈ R ^2×1 is given by

B b =









ζ 1 , M B ^∗ ε 1

V⁰

(Γ)

ζ 1 , M B ^∗ ε 2

V⁰

(Γ)







 .

We can choose ζ ₁ = M B ^∗ ε ₂ . We have (ζ ₁ , M B ^∗ ε ₂ )

_V0

(Γ) 6= 0 and the Kalman controllability criterion is satisfied.

Let us now give a direct proof of the stabilizability of system (4.1) based on the fact that the system (3.11) is stabilizable by a control of infinite dimension.

Proposition 4.2. System (4.1) is exactly controllable.

Proof. Since system (4.1) is of finite dimension, it is sufficient to prove that it is completely stabilizable. Thus we have to prove that, for all ρ > 0, there exists a constant C > 0 and a control V = (v ₁ , · · · , v _n

_c

) ^T ∈ L ² (0, ∞; R ⁿ

^c

) such that the solution y _α to (4.1) obeys

|y α (t)|

_V0

n

(Ω) ≤ Ce ^−ρt |y 0 |

_V0 n

(Ω) .

Step 1. It has been proved in [29], that system (3.11) is completely stabilizable.

Thus for a given ρ > 0, there exists a control ˜ u such that the solution y

u

˜ of (3.11) obeys |y

u

˜ (t)|

V_n⁰

(Ω) ≤ Ce ^−ρt |y 0 |

V⁰_n

(Ω) . Since Q is a continuous operator from V _n ⁰ (Ω) to Y _α , we have

|Qy

u

˜ (t)|

V_n⁰

(Ω) ≤ Ce ^−ρt |y 0 |

V⁰_n

(Ω) . (4.4) Since Q and A α commute, we have

Qy

u

˜ (t) = e ^A

^α

^t Qy 0 + Z t

0

e ^A

^α

^(t−τ) QBM u(τ)dτ. ˜