Axel Kr Öner
email: axel.kroener@inria.fr

S Érgio

S Rodrigues
email: sergio.rodrigues@oeaw.ac.at.

REMARKS ON THE INTERNAL EXPONENTIAL STABILIZATION TO A NONSTATIONARY SOLUTION FOR 1D BURGERS EQUATIONS

Keywords: Burgers equations, exponential stabilization, feedback control, finite elements AMS subject classifications. 93B52, 93C20, 93D15, 93C50

The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the control a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examples are presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible.

1. Introduction. Let L > 0 be a positive real number. We consider the controlled Burgers equations in the interval Ω = (0, L) ⊂ R:

(1.1) ∂ t u + u∂ x u -ν∂ xx u + h + ζ = 0, u| Γ = 0.
Here u stands for the unknown velocity of the fluid, ν > 0 is the viscosity, h is a fixed function, Γ = ∂Ω stands for the boundary {0, L} of Ω, and ζ is a control taking values in the space of square-integrable functions in Ω, whose support, in x, is contained in a given open subset ω ⊂ Ω.

Let us be given a positive constant λ > 0, a continuous Lipschitz function χ ∈ W 1,∞ (Ω, R) with nonempty support, and a solution û ∈ W of (1.1) with ζ = 0, in a suitable Banach space W. Then, following the procedure presented in [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF], we can prove that there exists an integer M , a function η = η(t, x), defined for t > 0, x ∈ Ω, such that the solution u = u(t, x) of problem (1.1), with ζ = χP M η, and supplemented with the initial condition (1.2) u(0, x) = u 0 (x) is defined on [0, +∞) and satisfies the relation |u(t) -û(t)| 2 L 2 (Ω, R) ≤ Ce -λt |u(0)û(0)| 2 L 2 (Ω, R) , provided |u(0) -û(0)| L 2 (Ω, R) < , for small enough . Here M , C, and can be taken depending only on (|û| W , λ), and P M is the orthogonal projection in L 2 (Ω, R) onto the subspace

L 2 M (Ω, R) := span{sin( iπx L ) | i ∈ N, 1 ≤ i ≤ M }.
That is, the internal control ζ = χP M η stabilizes exponentially, with rate λ 2 , the Burgers system to the reference trajectory û.

Notice that the support of the control ζ is necessarily contained in that of χ, and that the control is finite-dimensional. Furthermore, we also know that the control can be taken in feedback form, ζ(t) = e -λt χP M χQ t, λ û (u(t) -û(t)), for a suitable family of linear continuous operators Q t, λ û : L 2 (Ω, R) → L 2 (Ω, R), t ≥ 0 (cf. [7, section 3.2]). We can see that the dimension M of the range of the controller depends on the norm |û| W of û but, up to now no precise estimate is known. In the case û is independent of time it is possible to give, for the case of the Navier-Stokes equations, a rather sharp description of its dimension M , though the range of the controller depends on û; see, for example, [START_REF] Badra | Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system[END_REF][START_REF] Barbu | Stabilization of Navier-Stokes equations by oblique boundary feedback controllers[END_REF][START_REF] Barbu | Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers[END_REF][START_REF] Barbu | Internal stabilization of Navier-Stokes equations with finitedimensional controllers[END_REF][START_REF] Raymond | Boundary feedback stabilization of the two-dimensional Navier-Stokes equations with finite-dimensional controllers[END_REF] (cf. [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]Remark 3.11(c)]). The procedure uses the spectral properties of the Oseen-Stokes system and cannot be (at least not straightforwardly) used in the time-dependent case.

The aim of this paper is to establish some first results concerning the dimension M of the range of the internal stabilizing controller, in the case of a reference timedependent trajectory û. Notice that this case is not less important for applications because often we are confronted with external forces h that depend on time.

In [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF], the proof of existence of an M -dimensional stabilizing control uses a contradition argument (cf. [7, proofs of Lemma A.4 and Proposition A.3]) which makes it difficult to find an estimate for M . Here we prove the existence of a stabilizing control by a more constructive procedure.

In the case we impose no restriction on the support of the control, more precisely, if we take χ(x) = 1 for all x ∈ Ω, then we obtain that is it enough to take

(1.3) M ≥ L π ( 3e 2 ) 1 2 (ν -2 |û| 2 W + ν -1 λ) 1 2
, where e is the Napier's constant. In the case our control is supported in a small subset ω = supp(χ), we can also derive that it is "enough" to take

(1.4) M ≥ C 1 e C2 1+(ν -1 λ) 1 2 +(ν -1 λ) 2 3 +ν -1 |û| W +ν -2 |û| 2 W
where C 1 and C 2 are constants depending on χ and Ω. Estimates (1.3) and (1.4) are the main results of this paper. We easily see that the estimate in the case of the support constraint in much less reasonable, if we think about an application. The reason of the gap is that the idea used to derive (1.3) cannot be (at least not straightforwardly) used for general χ(x). So one question arises: can we improve (1.4)? To derive (1.4) we depart from an exact null controllability result, carrying the cost associated with the respective control. For stabilization, with a given (finite) positive rate λ 2 > 0, we do not need to reach zero; that is why we believe the estimate can be improved, if we can avoid using the exact controllability result.

We have performed some numerical simulations whose results suggest that the possibility of getting, also in the general case, an estimate analogous to (1.3) is plausible. We focus on the 1D Burgers equations because the simulations are much simpler to perform in this setting. However, we believe that the difficulties to find an estimate for M will be analogous for the 2D and 3D Burgers and Navier-Stokes systems, and for a suitable class of parabolic systems.

The rest of the paper is organized as follows. In section 2 we recall some wellknown results and set up our problem, in particular we recall that the problem can be reduced to the stabilization to zero of the Oseen-Burgers system. In section 3, for the linearized Oseen-Burgers system, we present the first estimates for a lower bound for the suitable dimension M of the controller; section 3.1 deals with the particular case where we impose no restriction on the support of the control and section 3.2 deals with the general case. In section 4 we consider the full nonlinear Oseen-Burgers system. The discretization of our problem is presented in section 5, and in sections 6 and 7

we present the results of some simulations we have performed. Finally, in section 8 we give a few more comments on the results.

Notation. We write R and N for the sets of real numbers and nonnegative integers, respectively, and we define R r := (r, +∞), for r ∈ R, and N 0 := N \ {0}. We denote by Ω ⊂ R a bounded interval. Given a vector function u : (t, x) → u(t, x) ∈ R, defined in an open subset of R × Ω, its partial time derivative ∂u ∂t will be denoted by ∂ t u. The partial spatial derivative ∂u ∂x will be denoted by ∂ x u, and ∂ ∂x ∂ ∂x by ∂ xx . Given a Banach space X and an open subset O ⊂ R n , let us denote by L p (O, X), with either p ∈ [1, +∞) or p = ∞, the Bochner space of measurable functions f : O → X, and such that |f | p X is integrable over O, for p ∈ [1, +∞), and such that ess sup x∈O |f (x)| X < +∞, for p = ∞. In the case X = R we recover the usual Lebesgue spaces. By W s,p (O, R), for s ∈ R, denote the usual Sobolev space of order s. In the case p = 2, as usual, we denote H s (O, R) := W s,2 (O, R). Recall that H 0 (O, R) = L 2 (O, R). For each s > 0, we recall also that H -s (O, R) stands for the dual space of

H s 0 (O, R) = closure of {f ∈ C ∞ (O, R) | supp f ⊂ O} in H s (O, R). Notice that H -s (O, R) is a space of distributions.
For a normed space X, we denote by |•| X the corresponding norm, by X its dual, and by •, • X ,X the duality between X and X. The dual space is endowed with the usual dual norm: |f | X := sup{ f, x X ,X | x ∈ X and |x| X = 1}. In the case that X is a Hilbert space we denote the inner product by (•, •) X .

Given an open interval I ⊆ R and two Banach spaces X and Y , we write W (I, X, Y ) := {f ∈ L 2 (I, X) | ∂ t f ∈ L 2 (I, Y )}, where the derivative ∂ t f is taken in the sense of distributions. This space is endowed with the natural norm

|f | W (I, X, Y ) := |f | 2 L 2 (I, X) + |∂ t f | 2 L 2 (I, Y ) 1 2 .
In the case X = Y we write H 1 (I, X) := W (I, X, X). Again, if X and Y are endowed with a scalar product, then also W (I, X, Y ) is. The space of continuous linear mappings from X into Y will be denoted by L(X → Y ).

If Ī ⊂ R is a closed bounded interval, C( Ī, X) stands for the space of continuous functions

f : Ī → X with the norm |f | C( Ī,X) = max t∈ Ī |f (t)| X .
C [a1,...,a k ] denotes a nonnegative function of nonnegative variables a j that increases in each of its arguments.

C, C i , i = 1, 2, . . . , stand for unessential positive constants.

Preliminaries.

2.1. Reduction to local null stabilization. We will denote

V := H 1 0 (Ω, R), H := L 2 (Ω, R), D(∂ xx ) := V ∩ H 2 (Ω, R), and V := H -1 (Ω, R).
The space H is supposed to be endowed with the usual L 2 (Ω, R)-scalar product; the space V with the scalar product (u, v) V := (∂ x u, ∂ x v) H . The space H is taken a pivot space, and V is the dual of V . The inclusions V ⊂ H ⊂ V are dense, continuous and compact. The space D(∂ xx ) is endowed with the scalar product (u, v) D(∂xx

) := (∂ xx u, ∂ xx v) H . Let us denote (2.1) W := L ∞ (R 0 , L ∞ (Ω, R))
and, for given Banach spaces X and Y ,

L 2 loc (R 0 , X) := {f | f | (0, T ) ∈ L 2 ((0, T ), X) for all T > 0}, W loc (R 0 , X, Y ) := {f | f | (0, T ) ∈ W ((0, T ), X, Y ) for all T > 0}.
Fix a function h ∈ L 2 loc (R 0 , V ) and suppose that û ∈ W ∩ W loc (R 0 , V, V ) solves the Burgers system (1.1), with ζ = 0 and initial condition û0 := û(0) ∈ H.

Let us be given a Lipschitz continuous function χ ∈ W 1, ∞ (Ω, R) with nonempty support and λ > 0. Then, given another function u 0 such that |u 0 -û(0)| H is small enough. Let also O = (l 1 , l 2 ) be an open interval, such that supp(χ) ⊆ O ⊆ Ω.

Our goal is to find an integer M ∈ N 0 and a control η ∈ L 2 (R 0 , H) such that the solution of the problem (1.1)

-(1.2), with ζ = χE O 0 P O M (η| O )
is defined for all t > 0 and converges exponentially to û, that is, for some positive constant C > 0 independent of u 0 -û0 ,

(2.2) |u(t) -û(t)| 2 H ≤ C e -λt |u 0 -û0 | 2 H for t ≥ 0.
Here P O M stands for the orthogonal projection in L 2 (O, R) onto the subspace spanned by the first M eigenfunctions s n of the Dirichlet Laplacian in O, that is, onto

L 2 M (O, R) := span{s n | n ∈ N 0 , n ≤ M } where E O 0 : L 2 (O, R) → H is the extension by zero outside O, defined by E O 0 f (x) := f (x) if x ∈ O 0 if x ∈ Ω \ O
. Recall that it is well-known that the complete system of (normalized) Dirichlet eigenfunctions

{s n | n ∈ N 0 } ⊂ D(∂ xx ) and the corresponding system of eigenvalues {α n | n ∈ N 0 } are given explicitly by (2.3) s n (x) := ( 2 l ) 1 2 sin( nπ(x-l1) l ), α n = ( π l ) 2 n 2 , -∂ xx s n = α n s n , x ∈ O,
where l = l 2 -l 1 stands for the length of O.

Let us notice that, seeking for the control η and considering the corresponding solution u, we find that v = u -û, will solve the Oseen-Burgers system (2.4)

∂ t v -ν∂ xx v + v∂ x v + ∂ x (ûv) + ζ = 0, v| Γ = 0, v(0) = v 0 , with ζ = χE O 0 P O M (η| O ) and v 0 = u(0) -û(0).
It is now clear that to achieve (2.2) it suffices to consider the problem of local exponential stabilization to zero for solutions of (2.4), where "local" means that the property is to hold "provided |v 0 | H is small enough".

Weak solutions.

The existence and uniqueness of weak solutions for system (2.4) can be proved by classical arguments, where weak solutions are understood in the classical sense as in [37, chapter 1, sections 6.1 and 6.4], [42, sections 2.4 and 3.2], [43, chapter 3, section 3].

Theorem 2.1. Given û ∈ W, ζ ∈ L 2 ((0, T ), V ), and v 0 ∈ H, then there exists a weak solution v ∈ W ((0, T ), V, V ) for system (2.4), in (0, T ) × Ω. Moreover v is unique and depends continuously on the given data (v 0 , η):

(2.5) |v| 2 W ((0, T ), V, V ) ≤ C [T, |û| W ] |v 0 | 2 H + |ζ| 2 L 2 ((0, T ), V ) .
Notice that the proof of the existence and uniqueness of a weak solution can be done following the argument in [43, chapter 3, section 3.2] by using the estimate

|∂ x (wv)| 2 V ≤ C|w| 2 L ∞ (Ω, R) |v| 2 L 2 (Ω, R) ≤ C 1 |w| 2 H 1 (Ω, R) |v| 2 L 2 (Ω, R) ≤ C 2 |w| 2 V |v| 2 H . Definition 2.2. We say that v ∈ W loc (R 0 , V, V ) is a global weak solution for system (2.4), in R 0 × Ω, if v| (0, T ) ∈ W ((0, T ), V, V
) is a weak solution, for the same system, in (0, T ) × Ω, for all T > 0.

Corollary 2.3. Given û ∈ W, ζ ∈ L 2
loc (R 0 , V ), and v 0 ∈ H, then there exists a weak solution v ∈ W loc (R 0 , V, V ) for system (2.4), in R 0 × Ω, which is unique and there holds estimate (2.5).

Finally notice that system (1.1)-(1.2), is a particular case of (2.4) (with û = 0), hence Theorem 2.1 and Corollary 2.

∂ t v -ν∂ xx v + ∂ x (ûv) + ζ = 0, v| Γ = 0, v(0) = v 0 ,
to zero, with a desired exponential rate λ 2 > 0. We also provide some first estimates, concerning a lower bound for the integer M , depending on the triple (λ, |û| W , ν). Later, the results will follow for system (2.4), provided |v 0 | H is small enough, by a fixed point argument.

Remark 3.1. Theorem 2.1 and Corollary 2.3, also hold for system (3.1) in the role of system (2.4).

It is well known that controllability properties for system 3.1 are closely related to observability properties for the "time-backward" adjoint Oseen-Burgers

(3.2) -∂ t q -ν∂ xx q -û∂ x q + f = 0, q| Γ = 0, q(T ) = q 1
for q 1 ∈ H and f ∈ L 2 ((0, T ), V ); below, in section 3.2, we will use some suitable observability inequalities for this adjoint system.

3.1. The particular case χ = 1 Ω . We consider the case O = Ω and χ = 1 Ω , with 1 Ω (x) := 1 for all x ∈ Ω. In particular, there is no constraint in the support of the controller.

Theorem 3.2. For given û ∈ W and λ > 0, set

(3.3) M ≥ L π ( 3e 2 ) 1 2 ( 1 ν 2 |û| 2 W + 1 ν λ) 1 2 ,
where e is the Napier's constant. Then for any given v 0 ∈ H, there is a control

η λ,û,ν (v 0 ) ∈ L 2 (R 0 , H) such that the corresponding solution v of system (3.1), with ζ = χE O 0 P O M (η λ,û,ν | O ), satisfies the inequality (3.4) |v(t)| 2 H ≤ (1 + e 1 2 )e -λt |v 0 | 2 H , t ≥ 0.
Moreover, the mapping v 0 → η λ,û,ν (v 0 ) is well defined, is linear, and satisfies

e ( λ/2)t η λ,û,ν (v 0 ) 2 L 2 (R0,H) ≤ 4e 1 2 λ- λ ( 1 ν |û| 2 W + λ)|v 0 | 2 H , for 0 ≤ λ < λ. Proof. Let w solve (3.5) ∂ t w = ν∂ xx w -∂ x (ûw) + λ 2 w, w| Γ = 0, w(0) = v 0 . By standard arguments we can find d dt |w| 2 H ≤ -2ν|∂ x w| 2 H + 2|û| L ∞ (Ω, R) |w| H |∂ x w| H + λ|w| 2 H ≤ 1 2ν |û| 2 L ∞ (Ω, R) |w| 2 H + λ|w| 2
H from which we can derive that

(3.6) |w| 2 L ∞ ((0, T ), H) ≤ e ( 1 2ν |û| 2 W +λ)T |v 0 | 2 H . Now let ϕ(t) := 1 -t T ∈ C 1 ([0, T ], R)
, and set δ := ϕw. Notice that δ solves

∂ t δ = ν∂ xx δ -∂ x (ûδ) + λ 2 δ + (∂ t ϕ)w, δ| Γ = 0, δ(0) = v 0
with δ(T ) = 0. Let now M ∈ N 0 be a positive integer and consider the solution δ M for the system

∂ t δ M = ν∂ xx δ M -∂ x (ûδ M ) + λ 2 δ M + (∂ t ϕ)P Ω M w, δ M | Γ = 0, δ M (0) = v 0 . The difference d := δ -δ M solves ∂ t d = ν∂ xx d -∂ x (ûd) + λ 2 d + (∂ t ϕ)(1 -P Ω M )w, d| Γ = 0, d(0) = 0,
from which we can also derive

|d| 2 L ∞ ((0, T ), H) ≤ e 3 2ν |û| 2 W + λ T |d(0)| 2 H + 3 4ν (∂ t ϕ)(1 -P Ω M )w 2 L 2 ((0, T ), V ) ≤ T -2 e 3 2ν |û| 2 W + λ T 3 4ν α -1 M |w| 2 L 2 ((0, T ), H)
and, from |w| 2 L 2 ((0, T ), H) ≤ T |w| 2 L ∞ ((0, T ), H) and (3.6), we can arrive at

|d| 2 L ∞ ((0, T ), H) ≤ T -1 e 2(ν -1 |û| 2 W +λ)T 3 4ν α -1 M |v 0 | 2 H .
Since we are interested in the stabilization of the system, we can see T as a parameter at our disposal. Minimizing the right hand side over T > 0, we can see that the minimizer T * is defined by T -1 * := 2(ν -1 |û| 2 W + λ); then, setting T = T * we have that

(3.7) |d| 2 L ∞ ((0, T * ), H) ≤ 2(ν -1 |û| 2 W + λ)e 1 3 4ν α -1 M |v 0 | 2 H . Now, from α M = ( M π L ) 2 (cf. (2.
3), with O = Ω), setting M satisfying (3.3), and recalling that δ M (0) = v 0 and δ M (T * ) = -d(T * ), we find

(3.8) α M ≥ (ν -1 |û| 2 W + λ) 3e 1 2ν and |δ M (T * )| 2 H ≤ |δ M (0)| 2 H . Further, from (3.6) and (3.7), we find |δ M | 2 L ∞ ((0, T * ), H) = |δ -d| 2 L ∞ ((0, T * ), H) ≤ C δ M |δ M (0)| 2 H , with C δ M := e ( 1 2ν |û| 2 W +λ)T * + 2(ν -1 |û| 2 W + λ)e 1 3 4ν α -1 M ≤ e 1 2 + 2(ν -1 |û| 2 W + λ)e 1 3 4ν α -1 M ≤ e 1 2 + 1 =: Υ δ .
Now, notice that we can consider system (3.5) in (T * , +∞) × Ω with w(T * ) = δ M (T * ), and repeat the arguments. Recursively, we conclude that in each interval

J i * := (iT * , (i + 1)T * ), i ∈ N 0 , we have |δ M ((i + 1)T * )| 2 H ≤ |δ M (iT * )| 2 H and |δ M | 2 L ∞ (J i |v 0 | 2 H ≤ 1 λ- λ (2(ν -1 |û| 2 W + λ)) 2 e 1 2 |v 0 | 2 H .
That is, e λ 2 t η λ,û,ν

2 L 2 (R0, H) ≤ 4e 1 2 λ- λ (ν -1 |û| 2 W + λ) 2 |v 0 | 2 H .
3.2. The general case. Let w solve the system (3.9)

∂ t w = ν∂ xx w -∂ x (ûw) + λ 2 w + χη, w| Γ = 0, w(0) = v 0 .
To simplify the exposition we rescale time as t = τ ν . Then w(τ ) := w( τ ν ) solves

(3.10) ∂ τ w = ∂ xx w -∂ x (ȗ w) + λ 2 w + χη, w| Γ = 0, w(0) = v 0 ,
with (ȗ, λ, η) = ν -1 (û, λ, η). Next, consider the adjoint system (3.11) -∂ τ q = ∂ xx q + ȗ∂ x q + λ 2 q, q| Γ = 0, q(T ) = q T with q T ∈ H (here with no external force; cf. system (3.2)). From, for example, [18, Theorem 2.1] and [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF]Theorem 2.3] (e.g., reversing time in system (3.11)), we have that given an open set ω ⊆ Ω, there exists a constant C ω,Ω > 0, depending on ω and Ω, such that for any time T > 0, the weak solution q for (3.11) satisfies

(3.12) |q(0)| 2 H ≤ e Cω,Ω 1+ 1 T +T λ+ λ 2 3 +(1+T )|ȗ| 2 W |q| 2 L 2 ((0, T ), L 2 (ω, R) .
Proposition 3.3. For every v 0 ∈ H, we can find a control η = η(v 0 ) ∈ L 2 ((0, T ), H), driving system (3.10) to w(T ) = 0 at time t = T > 0. Moreover, the mapping η : v 0 → η(v 0 ) is linear and continuous: η ∈ L(H → L 2 ((0, T ), H)), and there is a constant C χ,Ω such that

|η(v 0 )| 2 L 2 ((0, T ), H) ≤ e Cχ,Ω 1+ 1 T +T λ+ λ 2 3 +(1+T )|ȗ| 2 W |v 0 | 2 H . (3.13)
Sketch of the proof. The proof can be done following the arguments in [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]. First, from (3.12) we can derive an observability of the form

(3.14) |q(0)| 2 H ≤ e Cχ,Ω 1+ 1 T +T λ+ λ 2 3 +(1+T )|ȗ| 2 W |χq| 2 L 2 ((0, T ), H)
for the solution q of system (3.11) (cf. [7, eq. (A.8)]). Then we can prove the null controllability considering the following minimization problem (cf. [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]Problem 3.3])

J ( w, η) = |η| 2 L 2 + 1 | w(T )| 2
H → min; with ( w, η) solving (3.10).

Next we can consider the minimization problem J ∞ ( w, η) = |η| 2 L 2 → min; with ( w, η) solving (3.10) and w(T ) = 0 whose unique minimizer (w, η)(v 0 ) depends linearly on v 0 (cf. [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]Problem 3.4]).

Considering the null controllability of linear parabolic equations we also refer to [23, section 2] and [46, section 5.2.2], and references therein.

Theorem 3.4. For given û ∈ W and λ > 0, set

(3.15) M ≥ l π C 0 χ, Ω e 3 2 (1+Cχ,Ω) 1+( λ ν ) 1 2 +( λ ν ) 2 3 + 1 ν |û| W + 1 ν 2 |û| 2 W ,
where

C 0 χ, Ω = (2+2( L π ) 2 ) 1 2 |χ| W 1, ∞ (Ω, R)
, l is the length of O, and C χ,Ω is the constant from (3.13). Then for any given v 0 ∈ H, there is a control

η λ,û,ν (v 0 ) ∈ L 2 (R 0 , H) such that, taking ζ = χE O 0 P O M (η λ,û,ν | O )
, the corresponding solution v of system (3.1) satisfies, for t ≥ 0, the inequality

(3.16) |v(t)| 2 H ≤ K χ,Ω e -λt |v 0 | 2 H , with K χ,Ω := 1 + e ( λ ν + 1 ν 2 |u| 2 W ) 1 2 + (C 0 χ, Ω ) 2 α1 e 3(Cχ,Ω+1) 1+( λ ν ) 1 2 +( λ ν ) 2 3 + 1 ν |û| W + 1 ν 2 |û| 2 W .
Moreover, the mapping v 0 → η λ,û,ν (v 0 ) is well defined, is linear, and satisfies for 0 ≤ λ < λ the inequality

e λ 2 t η û,λ (v 0 ) 2 L 2 (R0,H) ≤ νe Cχ,Ω 1+3( λ ν ) 1 2 +( λ ν ) 2 3 +3 1 ν |û| W + 1 ν 2 |û| 2 W 1 -e ( λ-λ)(2(νλ+|û| 2 W )) -1 2 |v 0 | 2 H .
Proof. Let w solve (3.10), for t ∈ (0, T ), with the control η = η(v 0 ) given by Proposition 3.3, and let wM be the solution of

∂ τ wM = ∂ xx wM -∂ x (ȗ wM ) + λ 2 wM + χE O 0 P O M (η(v 0 )| O ), wM | Γ = 0, wM (0) = v 0 .
Then, the difference d := w -wM solves

∂ τ d = ∂ xx d -∂ x (ȗd) + λ 2 d + χE O 0 (1 -P O M )(η(v 0 )| O ), d| Γ = 0, d(0) = 0,
and taking the scalar product with d, in H, we can arrive at

d dτ |d| 2 H ≤ -2 |∂ x d| 2 H + 2 |ȗ| L ∞ (Ω, R) |d| H |∂ x d| H + λ |d| 2 H (3.17) + 2 χE O 0 (1 -P O M )(η(v 0 )| O ), d V , V .
For the last term we find

χE O 0 (1 -P O M )(η(v 0 )| O ), d V , V = (χE O 0 (1 -P O M )(η(v 0 )| O ), d) H ≤ |η(v 0 )| O | L 2 (O, R) (1 -P O M )(χd| O ) L 2 (O, R) ,
and from

(1

-P O M )(χd| O ) 2 L 2 (O, R) ≤ α -1 M (1 -P O M )(χd| O ) 2 V ≤ 2α -1 M |χ| 2 W 1, ∞ (Ω, R) |d| 2 H 1 0 (Ω, R)
(which makes sense for a.e. t ∈ (0, T ), since from 2 χE O 0 (1

-P O M )(η(v 0 )| O ), d V , V ≤ 2 χE O 0 (1 -P O M )(η(v 0 )| O ) H |d| H and (3.17), by standard arguments it follows that d ∈ L ∞ ((0, T ), H) ∩ L 2 ((0, T ), V )) and |∂ x d| 2 L 2 (Ω, R) ≥ α Ω 1 1+α Ω 1 |d| 2 H 1 0 (Ω, R) , where α Ω 1 = π 2
L 2 , we find

χE O 0 (1 -P O M )(η(v 0 )| O ), d V , V ≤ α -1 2 M D χ, Ω |∂ x d| L 2 (Ω, R) |η(v 0 )| O | L 2 (O, R) , with D χ, Ω = 2 1+α Ω 1 α Ω 1 1 2 |χ| W 1, ∞ (Ω, R) = (2+2( L π ) 2 ) 1 2 |χ| W 1, ∞ (Ω, R) . Then, from (3.17), d dτ |d| 2 H ≤ |ȗ| 2 L ∞ (Ω, R) |d| 2 H + λ |d| 2 H + α -1 M D 2 χ, Ω |η(v 0 )| O | 2 L 2 (O, R)
and, using (3.13), we obtain

(3.18) |d| 2 L ∞ ((0, T ), H) ≤ α -1 M D 2 χ, Ω e Cχ,Ω 1+ λ 2 3 +|ȗ| 2 W e C 1 χ,Ω ( 1 T +2( λ+|ȗ| 2 W )T ) |v 0 | 2 H with C 1 χ,Ω = max{1, C χ,Ω }. Now the function E(T ) = e C 1 χ,Ω ( 1 T +2( λ+|ȗ| 2
W )T ) takes its minimum when T = T * , with T * defined by 1

T 2 * = 2( λ+|ȗ| 2 W ).
Then, choosing T = T * , and recalling that wM (T ) = -d(T ) and wM (0) = v 0 , we arrive at

| wM (T * )| 2 H ≤ α -1 M D 2 χ, Ω e Cχ,Ω 1+ λ 2 3 +|ȗ| 2 W e 2 3 2 C 1 χ,Ω ( λ+|ȗ| 2 W ) 1 2 | wM (0)| 2 H .
Thus, choosing M ∈ N 0 satisfying (3.15) and recalling that α M = ( M π l ) 2 , we have

(3.19) α M ≥ D 2 χ, Ω e 3(Cχ,Ω+1) 1+ λ 2 3 +|ȗ| 2 W + λ 1 2 +|ȗ| W and | wM (T * )| 2 H ≤ | wM (0)| 2
H . Moreover we can deduce from (3.13) and (3.18) that

| wM | 2 L ∞ ((0, T * ), H) = | w -d| 2 L ∞ ((0, T * ), H) ≤ C d M | wM (0)| 2 H , with C d M := e ( λ+|ȗ| 2 W )T * + (α -1 1 + α -1 M )D 2 χ, Ω e Cχ,Ω 1+ λ 2 3 +|ȗ| 2 W e C 1 χ,Ω ( 1 T * +2( λ+|ȗ| 2 W )T * ) ≤ e ( λ+|ȗ| 2 W ) 1 2 + α -1 1 D 2 χ, Ω e 3(Cχ,Ω+1) 1+ λ 2 3 +|ȗ| 2 W + λ 1 2 +|ȗ| W + 1 =: Υ d .
Recursively, repeating the argument in the time interval (iT * , +∞) with w(iT * ) = wM (iT * ) in (3.10), we can conclude that the solution wM will remain bounded for all time

τ ≥ 0. That is, | wM | 2 L ∞ (R0, H) ≤ Υ d |v 0 | 2 H . Next, we notice that v(t) := e -λ 2 t wM (νt) solves (3.1), in R 0 × Ω, with the con- catenated control ζ = χE O 0 P O M (νe -λ 2 t η( wM (iT * ))(νt)| O )
, where η( wM (iT * )), i ∈ N, is the control given in Proposition 3.3, when we consider system (3.10), in J i * × Ω, with J i * := (iT * , (i + 1)T * ), i ∈ N 0 , and w(iT * ) = wM (iT * ), in particular η( wM (iT * ))(νt) is defined for t ∈ (iν -1 T * , (i + 1)ν -1 T * ). We can also conclude from (3.13) and from the boundedness of {| wM (iT

* )| H | i ∈ N} that the family {|η( wM (iT * ))| L 2 (J i * , H) | i ∈ N} is bounded; so e λ 2 t ζ ∈ L 2 (R 0 , H) for all λ < λ. Finally we observe that |v(t)| 2 H ≤ e -λt | wM (νt)| 2 H ≤ Υ d e -λt |v 0 | 2 H , and for η λ,û,ν (t) := νe -λ 2 t η( wM (iT * ))(νt), t ∈ J i * , it follows e λ 2 t η λ,û,ν 2 L 2 (R0, H) = lim j→+∞ j i=0 (i+1)T * ν iT * ν e ( λ-λ)s ν 2 |η( wM (iT * ))(νs)| 2 H ds ≤ lim j→+∞ ν j i=0 e ( λ-λ) iT * ν (i+1)T * iT * |η( wM (iT * ))(τ )| 2 H dτ ≤ ν 1-e ( λ-λ) T * ν e Cχ,Ω 1+ 1 T * +T * λ+ λ 2 3 +(1+T * )|ȗ| 2 W |v 0 | 2 H
from which, using the equality

T * = (2( λ + |ȗ| 2 W )) -1 2
, we can arrive to the estimate

e λ 2 t η λ,û,ν 2 L 2 (R0, H) ≤ νe C χ,Ω 1+3( λ ν ) 1 2 +( λ ν ) 2 3 +3 1 ν | û| W + 1 ν 2 | û| 2 W 1-e ( λ-λ)(2(νλ+| û| 2 W )) -1 2 |v 0 | 2 H .
Remark 3.5. Notice that when we shrink the support of χ the constant C χ, Ω in (3.13) is expected to increase; we cannot expect the right hand side of (3.15) to go to 0 as the length l of O does. we see that there is a big gap; the former being proportional to

1 ν 2 |û| 2 W + 1 ν λ 1 2
, while the latter depends exponentially on both 1 ν |û| W and ( λ ν )

1 2 . For application purposes the latter is much less convenient, so one question arises naturally: can we improve (3.15)?

It seems that the idea used to derive (3.3) cannot (at least straightforwardly) be applied in the general case. On the other side to derive (3.15) we start from an exact null controllability result and carry the cost of the respective control. This means that to improve (3.15) we will probably need a different idea.

In section 6, in order to understand if it is possible to improve (3.15), say that we also have an estimate like (3.3) in the general case, we present results of some numerical simulations comparing the number of controls M = M need , that we need to stabilize the system (3.1) to zero, with the following reference real numbers

(3.20) M ref := L π (ν -2 |û| 2 W + ν -1 λ) 1 2 ; M exp := L π e M ref .
The value M ref is motivated by (3.3), and the value M exp by (3.15); notice that l π e M ref is a lower bound for the right hand side of (3.15); we take L π instead of l π in front of e M ref in order to avoid giving the wrong idea that (3.15) goes to 0 with l (cf. Remark 3.5).

Notice that in the case û = 0, we can see that the unstable modes of system (3.5) are those defined by the inequality

να i < λ 2 , that is, i < L π ν -1 2 ( λ 2 ) 1 2 = 2 -1 2 M ref < M ref .
Thus, in this case and with χ = 1 Ω , it is enough (and necessary) to take the

M = 2 -1 2 M ref controls in ( 2 L ) 1 2 sin( iπx L ) | i ∈ {1, 2, .
. . , M } (taking the family of controls considered in section 3.1). Here y ∈ N stands for the biggest integer that is strictly smaller than y > 0.

3.4. Feedback control and Riccati equation. By the dynamic programing principle, for example following the arguments in [7, section 3.2], considering the family of minimization problems (3.21, s)

Minimize J s (v, η) := Rs e λt (ν |v(t)| 2 V + |η(t)| 2 H ) dt on the space X := (v, η) e λ 2 t (v, η) ∈ W (R s , V, V ) × L 2 (R s , H) and, for t ∈ R s , (v, ζ) solves (3.1) with v(s) = w ∈ H and ζ = χE O 0 P O M (η| O )
where s runs over [0, +∞) and R s = (s, +∞), we can derive the following result.

Theorem 3.6. The controls ζ given in Theorems 3.2 and 3.4 can be taken in feedback form Observe also that

ζ = e -λt χE O 0 P O M ((χQ t, λ û v)| O ) (3.22) for a suitable family of operators Q t, λ û : H → H, t ≥ 0, with Q t, λ û L(H→H) ≤ C [λ, û, 1 ν ] e λt . Furthermore, the family {Q t, λ û | t ≥ 0} is continuous in the weak operator topology and Q := Q t, λ û satisfies the differential Riccati equation (3.23) Q -Q(-ν∂ xx + B(û)) -(-ν∂ xx + B(û)) * Q -QB O M B O M * Q -e λt ν∂ xx = 0 where B(û)v := ∂ x (ûv), and B O M : H → H and its adjoint B O M * : H → H given by (3.24) B O M η := e -λ 2 t χE O 0 P O M (η| O ), B O ( Qw 1 , w 2 ) H = (QAw 1 , w 2 ) H + (A * Qw 1 , w 2 ) H + (QB O M B O M * Qw 1 , w 2 ) H + (e λt ν∂ xx w 1 , w 2 ) H holds for all (w 1 , w 2 ) ∈ D(∂ xx ) × D(∂ xx ), with A = A(t) := -ν∂ xx + B(û(t)).
(Q s, λ û w, w) H = J s (v s * , η s * ) > 0 for all w = 0, that is, Q s, λ û is definite positive. Remark 3.8. Notice that (∂ xx ) * = ∂ xx and B(û) * = -û∂ x .
Notice also that from Theorems 3.2 and 3.4 (taking, e.g., (2λ, λ) in place of (λ, λ)), we have that the space X in problem (3.21) is nonempty.

Remark 3.9. For any T > 0 and w ∈ H, the function q := Qv 0 * solves the system

(3.2) with f = -e λt ν∂ xx v 0 * and q(T ) = Q T, λ û v 0 * (T ), where (v 0 * , η 0 * ) = (v 0 * , η 0 * )(w) is the minimizer of problem (3.21, 0).
We already know that Q satisfies (3.23), we can also show that it is unique in the class of operators e λt C with

C :=    R ∈ L ∞ (R 0 , L(H →H)) R(t) is self-adjoint positive definite for all t ≥ 0, the family { R(t) | t ≥ 0} is continuous in the weak operator topology    . Lemma 3.10. If R satisfies (3.23) and R := e -λt R ∈ C, then the feedback control ζ = e -λt χE O 0 P O M ((χRv)| O ) = B O M B O M * Rv stabilizes exponentially system (3.1) to zero with rate λ 2 . Proof. We find that d dt (Rv, v) H = ( Ṙv, v) H + (R∂ t v, v) H + (Rv, ∂ t v) H = ((RA + A * R + RB O M B O M * R + e λt ν∂ xx )v, v) H + (-R(A + B O M B O M * R)v, v) H + (Rv, -(A + B O M B O M * R)v) H = -e λt (ν|v| 2 V + |e -λ 2 t B O M * Rv| 2 H ). Notice that χE O 0 P O M ((χe -λ 2 t B O M * Rv)| O ) = B O M B O M * Rv. Thus we have that (Rv, v) H
is decreasing and, after integration, that

J s (v, e -λ 2 t B O M * Rv) = (R(s)v(s), v(s)) H - lim T →+∞ (R(T )v(T ), v(T )) H ≤ (R(s)v(s), v(s)) H ≤ | R| L ∞ (R0, L(H→H)) e λs |v(s)| 2 H . This inequality, together with ∂ t (e λ 2 t v) = λ 2 e λ 2 t v + e λ 2 t (ν∂ xx v -∂ x (ûv) + B O M B O M * Rv) and Rs |e λ 2 t B O M B O M * Rv| 2 H dt ≤ C 0 Rs |B O M * Rv| 2 H dt ≤ C 0 J s (v, e -λ 2 t B O M * Rv), imply that ∂ t (e λ 2 t v) is in L 2 (R s , V ), with |∂ t (e λ 2 t v)| L 2 (Rs, V ) ≤ Ce λs |v(s)| H . Hence it follows that |e λ 2 t v| C([s, +∞), H) ≤ C 1 e λ 2 s |v(s)| H , for suitable positive constants C 0 , C and C 1 . That is, the feedback control ζ = B O M B O M * Rv stabilizes system (3.1) to zero with rate λ 2 , |v(t)| H ≤ C 1 e -λ 2 (t-s) |v(s)| H . The uniqueness of Q will follow from the uniqueness of Q 1 := e -λt Q ∈ C satisfying (3.25) Q1 -Q 1 A -A * Q 1 -Q 1 BB * Q 1 -ν∂ xx + λQ 1 = 0 with B = B(t) := e λ 2 t B O M .
From the exponential stability, with rate λ 2 , of system (3.1) with ζ given by (3.22), it follows that

(3.26) ∂ t z -ν∂ xx z + ∂ x (ûz) -λ 2 z + BB * Q 1 z = 0, z| Γ = 0, z(0) = z 0 , is stable, that is, there is a constant C > 0 independent of z 0 such that |z(t)| H ≤ C |z 0 | H , for all t ∈ R 0 .
Actually we can prove that it is uniformly exponentially stable, that is, there are α > 0 and K > 0 such that

(3.27) |z(t)| H ≤ Ke -α(t-t0) |z(t 0 )| H for all 0 ≤ t 0 ≤ t.
Indeed, notice that we can consider the system (3.1) in the interval of time R t0 = (t 0 , +∞), instead of R 0 and we obtain the analogous to Theorems 3.2 and 3.4, replacing the initial time t = 0 by t = t 0 . This means that if we denote by S(t, t 0 )w the solution of system (3.26) for t ∈ R t0 with initial condition z(t 0 ) = z t0 we will have that |S(t, t 0 )w|

2 L 2 (R0, H) ≤ C |z t0 | 2 H
, where C is given in Theorems 3.2 and 3.4 and can be taken independent of t 0 . The uniform exponential stability follows then by [14, Theorem 1], see also [47, Remark 3.11. The operator (or family of operators) S(t, t 0 ) is sometimes called "evolutionary process" as in [14, section 1], "Green operator" as in [36, chapter IV, section 3], or "evolution operator" as in [13, section 2].

Theorem 3.12. The solution of (3.25) is unique in C.

Proof. We follow ideas from [START_REF] Curtain | The infinite-dimensional Riccati equation for systems defined by evolution operators[END_REF][START_REF] Da Prato | Uniform asymptotic stability of evolutionary processes in a Banach space[END_REF][START_REF] Ungureanu | Nonlinear differential equations of Riccati type on ordered Banach spaces[END_REF], see also [35,

chapter 1]. Let Q 2 ∈ C solve (3.

25). Then with

A 1 = A + BB * Q 1 -λ 2 I and A 2 = A + BB * Q 2 -λ 2 I
, where I is the identity operator, the difference

D := Q 2 -Q 1 solves Ḋ = DA 1 + A * 1 D + DBB * D, (3.28a) Ḋ = DA 2 + A * 2 D -DBB * D. (3.28b)
Let w ∈ H and let S i (t, s)w stand for the solution of system (3.26) in the interval of time t ∈ R s , with z(s) = w and with Q i in the place of Q 1 , i ∈ {1, 2}. Then we have that ∂ t S i (t, s)w = -A i (t)S i (t, s)w, and we find

∂ t S i (t, s) = -A i (t)S i (t, s) and ∂ t S i (t, s) * = -S i (t, s) * A i (t) * . For t ∈ (s, T ) we also have 0 = ∂ t S i (T, s)w = ∂ t (S i (T, t)S i (t, s)w), which gives us 0 = (∂ t S i (T, t))S i (t, s) + S i (T, t)∂ t S i (t, s), that is, ∂ t S i (T, t) = S i (T, t)A i (t) and ∂ t S i (T, t) * = A i (t) * S i (T,
Ġ1 = S 1 (t, s) * (DBB * D)(t)S 1 (t, s) and Ġ2 = -S 2 (t, s) * (DBB * D)(t)S 2 (t, s).
Then we obtain (G

1 (t)w, w) H -(G 1 (s)w, w) H = t s |(B * D)(r)S 1 (r, s)w| 2 H dr and (G 2 (t)w, w) H -(G 2 (s)w, w) H = - t s |(B * D)(r)S 2 (r, s)w| 2 H dr. Then, from G 1 (s) = D(s) = G 2 (s), we arrive to (3.29) (G 2 (t)w, w) H ≤ (D(s)w, w) H ≤ (G 1 (t)w, w) H .
Notice that from Lemma 3.10 and (3.27) we have that

|S i (t, t 0 )w| H ≤ K i e -αi(t-t0) |w| H for all 0 ≤ t 0 ≤ t,
for suitable positive constants K i and α i . Thus from (3.29), it follows

(3.30) -DK 2 2 e -2α2(t-s) |w| 2 H ≤ (D(s)w, w) H ≤ DK 2 1 e -2α1(t-s) |w| 2 H .
with D := |D| L ∞ (R0, L(H→H)) . Letting t go to +∞ we obtain (D(s)w, w) H = 0; hence since w can be taken arbitrary and D(s) is self-adjoint, it follows that 0 = (D(s)(w 1 + w 2 ), w 1 + w 2 ) H = 2(D(s)w 1 , w 2 ) H for any (w 1 , w 2 ) ∈ H × H; necessarily D(s) = 0 and D = 0 because s can be taken arbitrary. We know (cf. Remark 3.7) that Q 1 (t) = e -λt Q(t) is self-adjoint and positive definite for all t ≥ 0. From (3.25) we can also conclude that if, at some T > 0, we impose a final condition Q 1 (T ) = Q T 1 with Q T 1 self-adjoint and positive definite, then Q 1 (t) remains self-adjoint and positive definite for all t ∈ [0, T ]. Indeed from

(3.31) Q1 = Q 1 A 1 + A * 1 Q 1 -Q 1 BB * Q 1 + ν∂ xx
we can see that Q 1 can be written as

Q 1 (t) = S 1 (T, t) * Q T 1 S 1 (T, t) + T t S 1 (s, t) * (Q 1 BB * Q 1 -ν∂ xx )(s)S 1 (s, t) ds (3.32)
and, for u = 0, we have

(Q 1 (t)u, u) H = (Q T 1 S 1 (T, t)u, S 1 (T, t)u) H (3.33) + T t |B * Q 1 S 1 (s, t)u| 2 H + ν |∂ x S 1 (s, t)u| 2 H ds > 0. Further, if Q 2 also solves (3.25) with Q 2 (T ) = Q T 1 , then necessarily D(s) := Q 2 (s) -Q 1 (s) = 0 for all s ∈ (0, T ), because from (3.29) if D(T ) = 0 we can derive that 0 = (G 2 (T )w, w) H ≤ (D(s)u, u) H ≤ (G 1 (T )w, w) H = 0. Remark 3.13. Denoting F 1 := Q 1 BB * Q 1 -ν∂ xx and differentiating (3.32) we find Q1 = ∂ t Q 1 = A 1 (t) * S 1 (T, t) * Q T 1 S 1 (T, t) + S 1 (T, t) * Q T 1 S 1 (T, t)A 1 (t) -F 1 (t) + T t A 1 (t) * S 1 (s, t) * F 1 (s)S 1 (s, t) + S 1 (s, t) * F 1 (s)S 1 (s, t)A 1 (t) ds = Q 1 (t)A 1 (t) + A 1 (t) * Q 1 (t) -F 1 (t),
that is, we recover (3.31). Remark 3.14. From (3.32) we see that

Q 1 (t) is obtained from Q T 1 = Q 1 (T ).
Thus, the Riccati equations (3.31), (3.23), and (3.25) must be solved backwards in time.

4. The nonlinear system. The next result is a corollary of Theorem 3.6. It will follow by a fixed point argument.

Theorem 4.1. Let M be the integer in Theorem 3.6 (i.e., as in either (1.3) or (1.4)). Then there are positive constants Θ and = (Θ) depending only on λ, |û| W , and ν such that for |v 0 | H ≤ the solution v of system (2.4), with ζ as in (3.22), is well defined for all t ≥ 0 and satisfies the inequality

(4.1) |v(t)| 2 H ≤ Θe -λt |v 0 | 2 H for t ≥ 0.
Notice that the feedback rule is found to globally stabilize to zero the linear Oseen-Burgers system (3.1). Then, Theorem 4.1 says that the same feedback rule also locally stabilizes to zero the bilinear system (2.4).

The proof of Theorem 4.1 will be done following the arguments in [7, section 4]. The nonlinear system (2.4) with ζ as in (3.22)

reads (4.2) ∂ t v -ν∂ xx v + v∂ x v + ∂ x (ûv) + K t, λ v = 0, v| Γ = 0, v(0) = v 0 with K t, λ v := e -λt χE O 0 P O M ((χQ t, λ û v)| O ). Given λ > 0, we denote by Z λ the space of functions z ∈ C([0, +∞), H) ∩ L 2 (R 0 , V ) such that |z| Z λ := e λ 2 • z(•) 2 L ∞ (R0, H) + e λ 2 • z(•) 2 L 2 (R0, V ) 1 2 < ∞.
Let us fix a constant Θ > 0 and a function v 0 ∈ H, and introduce the following subset of Z λ :

Z λ Θ := {z ∈ Z λ | z(0) = v 0 , |z| 2 Z λ ≤ Θ|v 0 | 2 H }.
We define a mapping Ξ :

Z λ → C([0, +∞), H) ∩ L 2 loc (R 0 , V ) that takes a function a ∈ Z λ to the solution b of the problem (4.3) ∂ t b -ν∂ xx b + ∂ x (ûb) + K t b + a∂ x a = 0, b| Γ = 0, b(0) = v 0 . Recall that K •, λ L ∞ (R0, L(H→H)) ≤ C 1 and |a∂ x a| V ≤ C 2 |a| H |a| V . Lemma 4.2.
Let M be the integer in Theorem 3.6. Then, there exists Θ = Θ(λ, |û| W , ν) > 0 such that the following property holds: for any γ ∈ (0, 1) one can find a constant = Θ, γ > 0 such that for any v 0 ∈ H with |v 0 | H ≤ the mapping Ξ takes the set Z λ Θ into itself and satisfies the inequality

(4.4) |Ξ(a 1 ) -Ξ(a 2 )| Z λ ≤ γ|a 1 -a 2 | Z λ for all a 1 , a 2 ∈ Z λ Θ .
Proof.

Step 1. For suitable Θ and = Θ , Ξ maps Z λ Θ into itself. By the Duhamel formula, we can write b as

|b(t)| 2 H ≤ 2|S(t, 0)b(0)| 2 H + 2 t 0 |S(t, s)(a∂ x a)(s)| H ds 2 ≤ 2C 3 e -(λ+β)t |b(0)| 2 H + 2C 3 e -(
ν s+1 s |b(τ )| 2 V dτ ≤ |b(s)| 2 H + C 6 |û| 2 W + K •, λ 2 L ∞ (R0, L(H→H)) |b| 2 L 2 ((s, s+1), H) + C 7 s+1 s |a∂ x a(τ )|
+ C 6 |û| 2 W + K •, λ 2 L ∞ (R0, L(H→H)) e λ(s+1) |b| 2 L ∞ ((s, s+1), H) ≤ e -βs C 8 |b(0)| 2 H + |a| 4 Z λ + C 9 s+1 s e λτ |a(τ )| 2 H |a(τ )| 2 V dτ.
Thus, since e λτ ≤ e -λs e 2λτ ≤ e -βs e 2λτ for τ ∈ (s, s + 1), summing up we obtain

R0 e λτ |b(τ )| 2 V dτ ≤ C 10 |b(0)| 2 H + |a| 4 Z λ +∞ j=1
e -βj . Then from (4.6) we arrive to

|b| Z λ ≤ C 11 |b(0)| 2 H + |a| 4 Z λ , which implies that for a ∈ Z λ Θ we have |Ξ(a)| 2 Z λ ≤ C 11 1 + Θ 2 |v 0 | 2 H |v 0 | 2 H
Setting Θ = 2C 11 and choosing Θ > 0 so small that Θ Θ ≤ 1, we see that if

|v 0 | H ≤ Θ , then Ξ maps the set Z λ Θ into itself.
Step 3. Given γ ∈ (0, 1), Ξ is a γ-contraction for smaller = Θ, γ . Let us take two functions a 1 , a 2 ∈ Z λ Θ and set a := a 1 -a 2 and b := Ξ(a 1 ) -Ξ(a 2 ). Then the function b satisfies (4.3) with b(0) = 0 and a 1 ∂ x a 1 -a 2 ∂ x a 2 in the place of a∂ x a. From

|a 1 ∂ x a 1 -a 2 ∂ x a 2 | H = |a 1 ∂ x a + a∂ x a 2 | H ≤ C 4 (|a 1 | V + |a 2 | V ) |a| V ; |a 1 ∂ x a 1 -a 2 ∂ x a 2 | V ≤ C 12 (|a 1 | H + |a 2 | H ) |a| V ;
and proceeding as above we can arrive to

|Ξ(a 1 ) -Ξ(a 2 )| 2 Z λ ≤ C 13 |a 1 | 2 Z λ + |a 2 | 2 Z λ |a| 2 Z λ ≤ 2C 13 Θ |v 0 | 2 H |a 1 -a 2 | 2 Z λ .
Choosing ˜ Θ, γ > 0 so small that 2ΘC 13 ˜ 2 Θ, γ ≤ γ 2 , we see that if |v 0 | H ≤ ˜ Θ, γ , then (4.4) holds. Therefore the lemma holds with Θ, γ = min{ Θ , ˜ Θ, γ }.

Proof of Theorem 4.1. If |v 0 | H ≤ Θ, γ , the contraction mapping principle implies that there is a unique fixed point v ∈ Z λ Θ for Ξ. It follows from the definition of Ξ and Z λ Θ that v is a solution of problem (4.2) and satisfies (4.1). We claim that v is the unique solution of (4.2) in the space C([0, +∞), H) ∩ L 2 (R 0 , V ). Indeed, if w is another solution, then the difference z = v -w satisfies

z t -ν∂ xx z + z∂ x z + ∂ x (wz) + ∂ x (ûz) + K t, λ z = 0, z(0) = 0.
Multiplying this equation by z, in H, and following a standard procedure, we arrive

to d dt |z| 2 H + ν |z| 2 V ≤ C 14 |w| 2 V + |û| 2 W |z| 2
H which implies z(t) = 0, for all t ≥ 0. Remark 4.3. Though it would be possible to derive more precise estimates on the Θ and in Theorem 4.1, it would lead to a more cumbersome exposition, and these estimates are not the main focus of this work.

Discretization.

To perform the simulations in order to check the stabilization of systems (1.1) and (3.1), to a reference trajectory û and to zero respectively, we must discretize those systems with the feedback control ζ as in (3.22).

Discretization in space.

We use a finite-element based approach. We introduce an uniform mesh

   1 -i + x h , if x ∈ [(i -1)h, ih] ; 1 + i -x h , if x ∈ [ih, (i + 1)h] ; 0 , if x / ∈ [(i -1)h, (i + 1)h] .
Next any function u ∈ V can be approximated by the values it takes on Ω D . More precisely, we approximate u by the function ũ, defined as

ũ := Nx-1 i=1 u(ih)φ i .
We define the evaluation vector u := [u(ih)] := [u(1h), u(2h), . . . , u((N x -1)h)] ∈ M (Nx-1)×1 , where A stands for the transpose matrix of A.

Remark 5.1. Notice that u := Nx-1 i=1 u i φ i , is a piecewise (affine) linear function that takes the same values as u at the points of the mesh Ω D . Also notice that, since we are dealing with homogeneous Dirichlet boundary conditions, only the values at interior points are unknown for the solution of our system

The next step is the weak discretization matrix L D of a given linear operator L ∈ L(V → V ). We define L D by the formula

(5.2) v L D u = Lũ, ṽ V , V for all u, v ∈ V.
Of key importance are the identity and Laplace operators. For the identity operator Iu = u, we find that I D = [(φ i , φ j ) H ] =: M is the so-called mass matrix, while for the Laplace operator we find (∂ xx ) D = -[(∂ x φ i , ∂ x φ j ) H ] =: -S, where S is the so-called stiffness matrix. Explicitly we have the tridiagonal matrices

M := h 6        4 1 0 0 . . . 0
       and S := 1 h        2 -1 0 0 . . . 0 -1 2 -1 0 . . . 0 0 -1 2 -1 . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 0 -1 2 -1 0 . . . 0 0 -1 2       
.

Next, we recall the reference solution û and discretize the operator v → B(û)v = ∂ x (ûv), v ∈ V . We start by noticing that, for an arbitrary w ∈ V , (∂ x (ûv), w) H = -(ûv, ∂ x w) H , then we consider the approximation ûv = Nx-1 j=1 ûj v j φ j of ûv, and we find -( ûv, ∂ x w) H =

Nx-1 i=1

Nx-1 j=1 -û j v j w i (φ j , ∂ x φ i ) H , and

(∂ x (ûv), w) H ≈ w BD ûv, with B := 1 2        0 1 0 0 . . . 0 -1 0 1 0 . . . 0 0 -1 0 1 . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 0 -1 0 1 0 . . . 0 0 -1 0        and D û :=      û1 0 0 . . . 0 0 û2 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 ûNx-2 0 0 . . . 0 0 ûNx-1      .
Notice also that, rewriting v∂ x v as 1 2 B(v)v, we can discretize v∂ x v as 1 2 BD v v. Remark 5.2. Notice that above we consider the operator v → ∂ x (ûv) as a composition ∂ x • m û, where m û denotes the pointwise multiplication by û, and then we just take the product of the discretized factors. Of course, we can also discretize directly and, after some computations we can find that B(û) D is a tridiagonal matrix B û:

B û ii = - k∈{i-1, i+1} 1≤k≤Nx -1 ûk Ω (φ k φ i ∂ x φ i )dx = 1 6 û2, if i = 1 ûi+1 -ûi-1, if i ∈ {2, . . . , Nx -2} -û Nx -2 , if i = Nx -1 ; B û ij = - k∈{i, j} 1≤k≤Nx -1 ûk Ω (φ k φ j ∂ x φ i )dx = 1 6 2ûi+1 -ûi, if j = i + 1; -2ûi-1 + ûi, if j = i -1; 0, if |i -j| R ≥ 2 .
We see that the composition based procedure leads to a simpler result. We have also performed some simulations with the direct discretization (for the nonlinear system) and, though we have noticed no substantial difference, we must say that the direct discretization could lead to better results under suitable data.

To discretize the operators in the feedback control rule in (3.22), we start by rewriting it, recalling (3.24), as

Fv := B O M B O M * Q t, λ û v, (5.3)
and we notice that what we essentially need is an approximation Fv of Fv, when we only know the approximation v of v.

We will construct Fv in a few steps. For the multiplication operator v → χv we can of course take D χ v = χv as an approximation of χv. For the orthogonal projection P O M we start by noticing that

P O M (v| O ) = M n=1 (v| O , s n ) L 2 (O, R) s n = M n=1 (v, E O 0 s n ) H s n ,
then we can take the approximation

P O M (v| O ) ≈ M n=1 E O 0 s n Mv s n ,
from which we set the discrete approximation

P M v ≈ E O 0 P O M (v| O ) with P M := S M M, and
S M := M n=1 E O 0 s n E O 0 s n .
Finally, the linear operator Q t, λ û is, at this moment, unknown and (an approximation) has to be found. Note that denoting by

Q D = (Q t, λ û ) D the discretization of Q t, λ û , we may take M -1 Q D v ≈ Q t, λ
û v and discretize the feedback rule (5.3) as follows: first we take the approximation B O M * ξ ≈ e -λ 2 t P M D χ ξ, then from (5.2) and (Fv, w)

H = B O M * Q t, λ û v, B O M * w H , for (v, w) ∈ H × H, we find (Fv, w) H ≈ B O M * Q t, λ û v, B O M * w H = B O M * w MB O M * Q t, λ û v ≈ e -λt (P M D χ w) M P M D χ M -1 Q D v = w ⊥ MM -1 e -λt (P M D χ ) M P M D χ M -1 Q D v = w ⊥ Me -λt RR Q D v where (5.4) R := M c P M D χ M -1
and M c , satisfying M c M c = M, is the Cholesky factor of M (notice that M is positive definite). Thus we take Fv = Fv with (5.5)

F := e -λt RR Q D . Remark 5.3. Denoting Q = Q t, λ û , the approximation M -1 Q D v ≈
Qv can be understood in the following formal sense: we have that w

⊥ Q D v ≈ (Qv, w) H ≈ w ⊥ MQv, thus we can write w ⊥ MM -1 Q D v ≈ w ⊥ MQv, that is, (M -1 Q D v, w) H ≈ (Qv, w) H ,
where in the last expression the vector ρ

= (ρ 1 , ρ 2 , . . . , ρ Nx-1 ) = M -1 Q D v is to be seen as an element in Y = span{φ i | i ∈ {1, 2, . . . , N x -1}} ⊂ L 2 (Ω, R), that is, ρ is to be understood as Nx-1 i=1 ρ i φ i . Further, notice that in this way the opera- tor M -1 Q D is symmetric in Y, because if (v, w) ∈ Y × Y we have (v, w) = ( v, w) and (M -1 Q D v, w) H = w ⊥ MM -1 Q D v = w ⊥ Q D v = v ⊥ Q D w = v ⊥ MM -1 Q D w = (M -1 Q D w, v) H .

Discretization in time.

For discretization in time of system (3.1), considered in a time interval [0, T ], where T is a positive real number, we introduce an uniform mesh (5.6) [0, T ] D := 0, T Nt , 2T Nt , . . . , (Nt-1)T

Nt

, T consisting of the points in [0, T ] that are proportional to the time step k := T Nt , with N t ∈ N 0 . Then, any function u ∈ H 1 ((0, T ), V ) is approximated by the values it takes in [0, T ] D × Ω D , that is, we essentially approximate u = u(t, x) by a matrix [u] ∈ M (Nx-1)×(Nt+1) whose jth column is the vector u(jk, •). That is, [u] ij = u(jk, ih), for i ∈ {1, 2, . . . , N x -1} and j ∈ {0, 1, 2, . . . , N t } 5.3. Computation of the discretized feedback rule. We recall the operator Q = Q s, λ û satisfying, for t > 0, the differential Riccati equation (3.23).

Discretization of the differential Riccati equation. To construct the approximation

Q D the operator Q, we can look for Q D solving ∂ t Q D -Q D X -X Q D -e -λt Q D RR Q D + e λt νS = 0, t > 0,
with R as in (5.4) and

X = X(t) = M -1 νS + BD û(t) . (5.7)
Equivalently, we can look for P = e -λt Q D solving (5.8)

∂ t P -P X -X P -P RR P + νS + λP = 0, t > 0.

Remark 5.4. Notice that from the relation Xv ≈

-ν∂ xx v + B(û)v, we have that (Q(-ν∂ xx + B(û))v, w) H ≈ w Q D Xv. Similarly ((-ν∂ xx v + B(û)) * Qv, w) H ≈ w X Q D v, and (QFv, w) H ≈ w Q D Fv.

Initialization of the differential Riccati equation.

Since we need to solve (3.23) backwards in time (cf. Remark 3.14), we will also solve system (5.8) backwards in time; thus the question is: how to initialize the system? Roughly speaking, it seems that we would need to know P (+∞), and even if we know this (limit) value it is not clear how we could use it.

Recall that, our main goal is to approach the desired solution û(t) as time t increases but, in a real application we also want to have an effective controller that, for example, guarantees us that after some time t = T > 0 we are indeed closer than we were at initial time t = 0, say, e.g., |v( T )| 2 H ≤ 1 2 |v(0)| 2 H . Also, in applications it is reasonable to think of a problem set for a possibly very long time range t ∈ [0, T ], but never for an infinite time range.

Thus we suppose we are interested in the evolution for time t ∈ [0, T ], then we may suppose that for time t > T our solution is stationary, that is, we may study the same problem but, now we suppose that û(t) = û(T ), for all t ≥ T . Notice, however, that this does not reduce the full problem to the stationary case, because in the interesting time range t ∈ (0, T ) the reference trajectory û(t) remains unchanged. Now we can find P T solving the algebraic Riccati equation (5.9) P (-X(T ) + λ 2 I) + (-X(T ) + λ 2 I) P -P RR P + νS = 0, and we can see that, P T will solve the autonomous system (5.8) for t ≥ T (under the supposition û(t) = û(T ) for t ≥ T ), see also [47, sections 1.4 and 4.4].

Then, it remains to solve (5.8) for t ∈ [0, T ] with the final condition P (T ) = P T .

Solving the Riccati systems.

• General procedure. To solve the algebraic Riccati system (5.9) we use the software available from [START_REF] Benner | Morlab package software[END_REF]; in this way we find P T .

To solve (backwards in time) the differential system (5.8), for t ∈ [0, T ], with the initial condition P (T ) = P T , we proceed as follows. Recall the mesh [0, T ] D of the interval [0, T ], defined in (5.6). We have P Nt := P (N t k) = P (T ) = P T ; next, recursively, we construct P j := P (jk) from P j+1 , for j ∈ {0, 1, . . . , N t -1}, as we start by rewriting (5.8) as

-∂ t P = P (-X + λ 2 I) + (-X + λ 2 I) P -P RR P + νS =: R F (P )
and, we use the Crank-Nicolson inspired scheme

-2 k (P j+1 -P j ) = R F (P j ) + R F (P j+1 ),
from which we obtain R F (P j ) -2 k P j + R F (P j+1 ) + 2 k P j+1 = 0, that is, (5.10)

P j (-X + λ 2 I -1 k I) + (-X + λ 2 I -1 k I) P j -P j RR P j + Z j+1 = 0
with Z j+1 = R F (P j+1 ) + 2 k P j+1 + νS. Hence P j solves again an algebraic Riccati equation and we can still use the software in [START_REF] Benner | Morlab package software[END_REF].

• Initial guess. The software in [START_REF] Benner | Morlab package software[END_REF] (see also [START_REF]A matlab repository for model reduction based on spectral projection[END_REF]) uses a Newton method to solve an algebraic Riccati equation like (5.9). We have to provide an initial starting guess Y 0 such that -X(T ) + λ 2 I -RR Y T 0 Y 0 is stable. This is of course a nontrivial task (see, e.g., the discussion after (1.4) in [START_REF] Kesavan | On a degenerate Riccati equation[END_REF]) and we look for the initial guess in three steps:

(i) We set M = +∞ and χ = 1 Ω . That is, we impose no constraints neither on the dimension nor on the support of the controller. In this case we can see that R = M c M -1 and RR = M -1 . Then from (5.7) we can expect that

-X(T ) + λ 2 I -M -1 Y T 0 Y 0 = -M -1 νS + BD û(T ) + λ 2 I -M -1 Y T 0 Y 0 will be stable for Y 0 = βM c , with √ 2β ≥ β 0 := (ν -1 |û(T )| 2 L ∞ (Ω, R) + λ) 1 2
. Notice that, proceeding as in the beginning of section 3.1 we see that a weak solution w for w t = ν∂ xx w -∂ x (û(T )w) + λ 2 w -β 2 w, will satisfy the estimate

d dt |w| 2 H ≤ -ν|∂ x w| 2 H + ν -1 |û(T )| 2 L ∞ (Ω, R) |w| 2 H + λ|w| 2 H -2β 2 |w| 2 H .
That is, the lower bound β 0 works for the continuous system. However, when taking β strictly bigger than β 0 we may expect that the stability is preserved for the discretized system, if N x and N t are big enough. Hence we set β = (ν T . (ii) We set M = +∞ and the true χ. That is, now we include the constraints on the support of the controller. In this case R = M c D χ M -1 , see (5.4). In some cases it may happen that P T is not a "good" initial guess. In some cases (as we have observed in some simulations) the step from (+∞, 1 Ω ) to (+∞, χ) seems to be too big, in other words P , and denote the solution by P

-1 |û(T )| 2 L ∞ (Ω, R) + λ)
[1+lρ] T . After N H steps we have found a solution P [START_REF] Badra | Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system[END_REF] T for

P (-X(T ) + λ 2 I) + (-X(T ) + λ 2 I) P -P H 1 H 1 P + νS = 0 with H 1 = M c D χ M -1 . (iii)
We set the true M and the true χ. That is, finally we include also the constraints on the dimension of the controller. In this case R is given by (5.4). Analogously to step (ii) we consider the homotopy

H τ = (1 -τ )D χ + τ P M D χ , set H τ := (M c H τ M -1
) and, starting with

P [2]
T we find, recursively after N H steps, a solution P [START_REF] Banks | The linear regulator problem for parabolic systems[END_REF] T for

P (-X(T ) + λ 2 I) + (-X(T ) + λ 2 I) P -P H 1 H 1 P + νS = 0 with H 1 = M c P M D χ M -1 = R. That is P [3]
T solves (5.9). Of course, the number of homotopy steps N H may be taken different in steps (ii) and (iii). To get the convergence of the Newton method used to solve the algebraic Riccati equations at each homotopy step we may need, depending on the situations, to increase the number of homotopy steps N H .

Notice, however that in step (iii) increasing N H can be sufficient for convergence at each homotopy step only if M is big enough. Indeed we can see that the algebraic Riccati equation will have a solution up to the homotopy step before the last, because from the observability inequality (3.14) we can also derive

|q(0)| 2 H ≤ (1 -τ ) -2 C |(1 -τ )χq| 2 L 2 ((0, T ), H) , and from |(1 -τ )χq| 2 L 2 ((0, T ), H) = (1 -τ )(1 -P O M )(χq| O ) 2 L 2 ((0, T ), L 2 (O, R)) + (1 -τ )P O M (χq| O ) 2 L 2 ((0, T ), L 2 (O, R)) ≤ (1 -τ )(1 -P O M )(χq| O ) 2 L 2 ((0, T ), L 2 (O, R)) + P O M (χq| O ) 2 L 2 ((0, T ), L 2 (O, R)) = (1 -τ )(χq| O ) + (1 -(1 -τ ))P O M (χq| O ) 2 L 2 ((0, T ), L 2 (O, R)) , we arrive at |q(0)| 2 H ≤ (1-τ ) -2 C (1 -τ )χq + τ E O 0 P O M (χq| O ) 2 L 2 ((0, T ), H) .
Then, from this observability inequality it will follow that there exists a stabilizing control (for system (3.1)) of the form

ζ(t) = F τ η := (1 -τ )χη(t) + τ χE O 0 P O M (η(t)| O ), for τ < 1.
Reasoning as in section 3.4, by the dynamic programing principle it will follow that the control can be taken in feedback form

ζ(t) = F τ F * τ Q t τ v(t)
, where e λt Q t τ solves the Ricatti equation (3.25) with F τ in the role of B, which corresponds on the discrete level to the case we take R = (M c [(1 -τ )D χ + τ P M D χ ]M -1 ) in (5.8).

For the last homotopy step, that is, for τ = 1 the observability will hold if M is big enough (following the arguments in [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]). Also, from Theorem 3.4 a stabilizing control exists if M is big enough, so we cannot guarantee the existence of a stabilizing control of the form

ζ(t) = χE O 0 P O M (η(t)| O )
for arbitrary (small) M , and then we cannot guarantee the existence of a solution for the algebraic Riccati equation (3.25).

In step (ii), increasing N H should be sufficient to get the convergence at each homotopy step, because reasoning as above we can conclude that there exists a stabilizing control of the form

ζ(t) = (1 -τ )η(t) + τ χη(t), for all τ ∈ [0, 1].
Therefore, if convergence is not reached at a homotopy step in (ii) or at a homotopy step before the last in (iii) we probably need either more homotopy steps or to refine our mesh; if convergence is not reached only at the last homotopy step in (iii), then probably the number of controls is not enough.

In the simulations we present here, we have taken no more than N H = 20 in the second step and no more than N h = 10 in the third step. Notice however that increasing the number of homotopy steps does not mean that the computational time will be much bigger because the Newton method may converge faster at each homotopy step.

Finally, in the process of solving the differential Riccati equation, to find P j solving (5.10) we provide the natural initial guess P j+1 . Again, we cannot guarantee that the solution will always exist. If this process fails at some j-step, we can try to refine the mesh (in particular, by increasing the number N t of time steps in (5.6)); if that does not work it probably means that the number of controls M is not sufficient.

5.4.

Solving the discretized Oseen-Burgers system. Once we have constructed P we can simulate the evolution of the system (3.1); we look for v(t) := v(t, •) that solves the system (5.11)

∂ t v + νM -1 Sv + M -1 BD ûv + R RP (v) = 0, v(0) = v 0 ,
and expect v to go exponentially to 0 as time increases, with an a priori prescribed rate λ 2 > 0 as time goes to infinity (cf. (3.4) and (3.16)); recall that P depends on λ (cf. (5.8)). Notice that from (5.5), (5.4), and P = e -λt Q D , it follows Fv = R T RP v.

Again, we will approximate v(t) ≈ [v(jk)], j ∈ {0, 1, . . . , N t }, and we apply a Crank-Nicolson inspired algorithm to solve system (5.11). For simplicity we denote F j := R RP j , for j ∈ {0, 1, . . . , N t }. Set v 0 := v(0k) = v 0 ; then the idea is to construct, recursively, v j+1 := v((j + 1)k) from v j := v(jk) by the scheme

2 k (v j+1 -v j ) = -νM -1 S(v j + v j+1 ) -M -1 BD ûj v j + BD ûj+1 v j+1 -F j v j + F j+1 v j+1 ,
with ûj := û(jk), j ∈ {0, 1, . . . , N t }. Then, working a little the above scheme, we can obtain

v j+1 = A -1 ⊕ A v j -k 2 A -1 ⊕ BD ûj v j + BD ûj+1 v j+1 (5.12) -k 2 A -1 ⊕ M F j v j + F j+1 v j+1 , .
with A ⊕ := M + k 2 νS and A := Mk 2 νS. Notice that the unknown v j+1 is still present on the right hand side of (5.12). In the argument of the feedback operator we will replace v j+1 by a preliminary guess v j+1 G , and approximate BD ûj+1 v j+1 by

BD ûj v j + k(BD ûj v j -BD ûj-1 v j-1
) (where we define v -1 := v 0 = v 0 ). In this way we arrive at the scheme

v j+1 = A -1 ⊕ A v j -kA -1 ⊕ (1 + k 2 )BD ûj v j -k 2 BD ûj-1 v j-1 (5.13) -k 2 A -1 ⊕ M F j v j + F j+1 v j+1 G .
We set v j+1 G as the "uncontrolled" output

v j+1 G := A -1 ⊕ A v j -kA -1 ⊕ (1 + k 2 )BD ûj v j -k 2 BD ûj-1 v j-1 . 5.5.
Solving the discretized Burgers system. Concerning the evolution of the system (1.1)-(1.2). We look for u(t) := u(t, •) that solves the system (5.14)

∂ t u + νM -1 Su + 1 2 M -1 BD u u + h + R RP (u -û) = 0, u(0) = u 0 ,
and expect u to go exponentially to û, with an a priori prescribed rate λ 2 > 0, as time increases (with R as in (5.4)). However this would be meaningful if û were a solution for the uncontrolled discrete system, which is not true. The solution of the uncontrolled discrete system (5.15)

∂ t ûS + νM -1 Sû S + 1 2 M -1 BD ûS ûS + h = 0, ûS (0) = û0
will be an approximation ûS of û. There is no reason to expect that e λ 2 t (u(t) -û(t)) will remain bounded for t ∈ R 0 .

Nevertheless there is a way to check the rate of exponential stabilization λ. We will just have to compute the discrete (fictitious) external force h f , that makes û a solution of the discrete system, that is, (5.16)

∂ t û + νM -1 Sû + 1 2 M -1 BD û û + h f = 0, û(0) = û0
. Before that we present the scheme we apply. Suppose for the moment, that we know h f . Then, we follow the idea in section 5.4 and arrive at the scheme

u j+1 = A -1 ⊕ A u j -k 2 A -1 ⊕ (1 + k 2 )BD u j u j -k 2 BD u j-1 u j-1 (5.17) -k 2 A -1 ⊕ M h j f + h j+1 f + F j (u j - ûj ) + F j+1 (u j+1 G - ûj+1 ) ,
with the preliminary "uncontrolled" guess u j+1 G given by

u j+1 G := A -1 ⊕ A u j -k 2 A -1 ⊕ (1 + k 2 )BD u j u j -k 2 BD u j-1 u j-1 -k 2 A -1 ⊕ M h j f + h j+1 f .
It remains to explain how we construct the force h f . Actually, from our scheme we can deduce that we only need to know the terms k

2 A -1 ⊕ M h j f + h j+1 f
, for j ∈ {0, 1, . . . , N t -1}, that we can easily compute as

k 2 A -1 ⊕ M h j f + h j+1 f = -û j+1 + A -1 ⊕ A ûj -k 2 A -1 ⊕ (1 + k 2 )BD ûj ûj -k 2 BD ûj-1 ûj-1 ,
(where we define û -1 := û0 ).

6.

Numerical examples: the linear Oseen-Burgers system. We present some results of the numerical simulations we have performed concerning the stabilization of system (3.1) to zero. Below, v u stands for the solution of the uncontrolled (discretized) system (i.e., ζ = 0), and v (or v λ ) stands for the solution of the (discretized) system under the action of a (discretized) feedback controller

ζ = χη, with η = e -λt E O 0 P O M ((χQ t, λ û v)| O ) as in (3.22). If nothing is said in contrary O = (inf{Ω ∩ supp(χ)}, sup{Ω ∩ supp(χ)}).
We follow a "trying and checking" procedure, we fix M and check the results of the simulations.

6.1. Testing with a family of reference trajectories. We set

ν = 1 10 , λ = 2, Ω = (0, π), O = ( 3 2 , 5 2 ), and (6.1)
χ(x) = E O 0 sin((x -3 2 )π)| O .
That is, χ = E O 0 s 1 (cf. section 2.1). Next we set the family of reference trajectories

(6.2) û = û(i,j) = C nr (sin(-t) sin(ix) -cos(3t) sin(jx)) ,
where the constant C nr is chosen so that |û| W = 1. In this case we have that M ref = √ 120 ≈ 10.95 and M exp ≈ 57208.12; so our question is if the number M of needed controls stay "close" to M ref or to M exp (cf. section 3.3). We will test with the smaller number M = 4, and v 0 (x) = sin(2x). The function χ and the four controls are plotted in figure 1.

O 0 η | η ∈ span{s i | i ∈ {1, 2, 3, 4}}}.
In figure 2 we can check that the feedback control is able to stabilize the system with the desired rate. Then, we change the initial condition to v 0 (x) = sin(x)sin(6x), and test for some other reference trajectories (with higher frequencies) in the family (6.2); in figure 3 we see that the feedback control is still able to stabilize the system with the desired rate; of course, the squared norm |v| 2 H is to be understood as the discrete approximation v Mv (cf. section 5.1).

Remark 6.1. There is no particular reason to test with M far below M ref ; trivially, if M controls are enough to stabilize the system, then taking more controls we can also stabilize the system. Initial data in L 2 (Ω, R)\H 1 (Ω, R). We set λ = 4, ν = 1 10 and χ as in (6.1). But now we set v 0 (x) = x 1 2 and the reference trajectory û = C nr 1 [0, π 2 ] (sin(-t) sin(2x) -

0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 time t λt + log(|v(t)| 2 H / |v0| 2 H ) (i, j) = (1, 1) (i, j) = (1, 2) (i, j) = (1, 3) (i, j) = (3, 1) (i, j) = (3, 2) (i, j) = (3, 3) λ = 2 M = 4
(a) With feedback control.

λt + log(|vu(t)| 2 H / |v0| 2 H ) (i, j) = (1, 1) (i, j) = (1, 2) (i, j) = (1, 3) (i, j) = (3, 1) (i, j) = (3, 2) (i, j) = (3, 3) λ = 2 (b) Without control.
Fig. 2. Convergence rate is achieved with the feedback control.

0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 time t λt + log(|v(t)| 2 H / |v0| 2 H ) (i, j) = (1, 8) (i, j) = (8, 2) (i, j) = (8, 7) (i, j) = (8, 8) λ = 2 M = 4
(a) With feedback control. cos(3t) sin(2x)), where

(6.3) 1 [a, b] (x) := 1 , if x ∈ [a, b] 0 , if x ∈ Ω \ [a, b] , a, b ∈ R
and C nr is taken so that |û| W = 1. We can see in figure 4 that two controls stabilize the system (3.1) to zero with the desired rate. In figure 5 we see the controls corresponding to the cases we take either two or three controls. Notice that in this case the initial condition is in H \ V and the support of the control is disjoint from that of û.

6.2. Increasing number of needed controls. We set û = 0, Ω = (0, π). In this example we show that for any given n ∈ N 0 we can construct χ supported in a subset ω ⊂ ω ⊂ Ω, λ > 0, and an initial condition v 0 , such that the first 2n controls cannot stabilize the system (3.1) to zero with the rate λ. However, by increasing the number of controls we can obtain the desired stabilization. Notice that here we look for χ = 1 Ω (cf. last paragraph in section 3. (cf. definition of the functions s n in section 2.1). Now, notice that since i ≤ 2n, we have that 0 < 2n + 1 ± i < 2(2n + 1), thus we can conclude that (v 0 , χP O 2n η) H = 0. Therefore since the eigenspace span{v 0 } is preserved by the Laplacian we can conclude that the control cannot change the dynamics on this space. Thus, we conclude that the rate of convergence is at most 2ν(2n + 5) 2 . Now we set ν = 1 10 ; from above we know that for n ∈ {1, 2} the rate of convergence λ = 20 > 81 5 is not achieved with the first 2n controls. Simulations below show that, in these examples, it is enough to add one more control to achieve the rate. In particular, we have

M = 2n + 1 ≤ 5 < ( λ ν ) 1 2 = 10 √ 2 = M ref < M exp = e 10 √ 2
. In Figures 6 and7 we see the results of the simulations for the cases n = 1 and n = 2; we can check the stabilization rate to zero of the heat system (i.e., system (3.1) with û = 0). 6.3. Instability of the system. Increasing |û| W and decreasing ν brings more instability to the system, which leads to the necessity to take a bigger number M of controls. To illustrate the instability of the (uncontrolled) system (3.1), and the response of the controller, we can just take a stationary reference trajectory. The main advantage is that we do not need to solve the differential Riccati equation that is the more expensive numerical step. Notice, however that (as far as we know) an estimate depending on the norm |û| W is not known also in this case; for the Oseen-Stokes system estimates are known, but depending on û (cf. the discussion and given references in section 1). We will set (6.4) û(t, x) = -1 sin(5x), and v 0 (x) = sin(πx)

where is a constant that we will use to change the norm |û| W of û.

6.3.1. Changing the norm of the reference trajectory. Here we take λ = 4, M = 4, and ν = 1 10 . In figure 8 we see that the uncontrolled system becomes more instable as decreases, that is, as |û| W increases. We can also see that the four controls work up to = 0.067, but not for = 0.0665. This could mean that either the number of controls in not enough anymore or that our discretization is not fine enough (notice that for smaller the magnitudes |∂ x û(t, x)| R become bigger, we will come back to this issue hereafter in section 7.

Changing the viscosity.

Here we take λ = 4, M = 4 and û(t, x) = 2 sin(5x). In figure 9 we see that the uncontrolled system becomes more instable as ν decreases. We can also see that the four controls work up to ν = 0.001, but not for ν = 0.0005. Again, either the number of controls in not enough anymore or our discretization is not fine enough (notice that for smaller ν the magnitudes |∂ x û(t, x)| R become bigger when compared to ν). Notice that in all the cases we have M exp ≥ e 6.3.3. Changing the desired decreasing rate. Here we take M = 4, û = sin(5x), and ν = 1 10 . In figure 10 we see that with the four controls we can get at least rate of convergence λ = 17. We also see that |v

(t)| 2 H ≤ e C λ e -λt |v 0 | 2 H
, where C λ is the maximum of the corresponding curves. Since in all cases we have λ > C λ we see that the controller is effective at time t = 1, that is, the norm has been squeezed at time t = 1. On the other hand, for example for λ = 15, up to time t = 0. 12(a) could explain the cusp in the control plot in figure 11; we guess that one control is not enough, because the cost function t → (Q(t)v(t), v(t)) H must be a strictly decreasing function s ≤ t (cf. section 3.4). In figure 12(b) we can see that two controls can stabilize the system, and that the cost decreases as M increases.

(Q(s)v(s), v(s)) H = (Q(t)v(t), v(t)) H + t s e λτ (|∂ x v(τ )| 2 H + |η(τ )| 2
((Q(t)v(t), v(t))H ) time t M = 1 λ = 4 ǫ = 1
(a) One control is not stabilizing the system. In figure 13 we can see that the feedback control is able to stabilize, with the desired rate, the nonlinear system (1.1)-(1.2) to the trajectory û, provided that d 0 is small enough. We can see that, for |δ| R > 1, the stabilization rate is not guaranteed; while for |δ| ≤ 1 it holds. For example, for |δ| R ≤ 1 we can see that the local maxima of the plotted curves seem either to converge to a real number or to decrease, while for |δ| R > 1 those local maxima seem to go to infinity. Notice that the radius 1 here is suitable for this example, for other settings the stabilization may hold only for smaller |δ| R .

In figure 14 we can see that the uncontrolled systems does not go exponentially to û (at least not with the rate λ = 2); here we have plotted the curves corresponding to some of those values of δ in figure 13 (for the other the behavior is similar).

We can also see the nonlinear nature of the equations, because changing the sign of the initial condition leads to different curves.

Remark 7.1. The results correspond to simulations in which we have taken a fictitious external force h f (i.e., an approximation of h) that makes û a solution of the discrete system (cf. section 5.5).

λt + log(|v(t)| 2 H / |v0| 2 H ) δ = -0.1 δ = 0.1 δ = -0.5 δ = 0.5 δ = -1 δ = 1 λ = 2 M = 4 0 1 2 3 4 5 -1 0 1 2 3 time t λt + log(|v(t)| 2 H / |v0| 2 H ) δ = -1.5 δ = 1.5 δ = -2 δ = 2 λ = 2 M = 4
Fig. 13. Convergence rate to û holds locally.

Real versus fictitious external force behavior.

Here we are in the same setting as in section 7.1. But, now we fix δ = 1, and consider û in the longer time interval t ∈ [0, 10]. We compare the numerical results in the case when we take the real external force h = -∂ t û -û∂ x û + ν∂ xx û with those in the case when we take the fictitious external force h f (cf. Remark 7.1). We denote d = ū -ū and d r = ūr -ū, where ūr solves (5.14) (that is, with the real external force h), and ū solves (5.14) with hf in the place of h. In figure 15, we confirm the rate of convergence of ū to ū in the entire time interval, while for d r the rate is confirmed until time t = 6; after time t = 6 we see that d r remains bounded, this just means that the magnitude of ūr -ū has reached that of the discretization error of our solver, and consequently we cannot expect the magnitude of ūr -ū to decrease more.

+ log(|d(t)| 2 H / |d0| 2 H ) λt + log(|dr(t)| 2 H / |d0| 2 H ) log(|dr(t)| 2 H / |d0| 2 H ) λ = 2 M = 4
Fig. 15. Ficticious versus real external force.

7.3. On the discretization error. Here we are in the same setting as in section 7.1 with δ = 1. We observe that, following the scheme (5.17), with the exact external force h (in the place of h f ), the discrete solution ûS for (5.15) will be close to û. Moreover, ûS converges to û as (k, h) → (0, 0).

The main goal of the presented simulations in the previous sections is to show that an estimate like (3.3) should hold, in general, instead of (3.15), and it is not our intention to compare our algorithm/discretization to solve the Burgers and Oseen-Burgers systems with existing ones. That is why we have not performed a rigorous numerical analysis concerning the convergence of the scheme. Though, we would like to say that from numerical experiments we have performed we expect linear convergence. We present some results that indeed suggest that the error ûS -û is proportional to h + k.

Let û = û(i, j) be as in (6.2), with (i, j) = [START_REF] Barbu | Internal stabilization of Navier-Stokes equations with finitedimensional controllers[END_REF][START_REF] Barbu | Internal stabilization of Navier-Stokes equations with finitedimensional controllers[END_REF]. In figure 17 the error ûS -û is proportional to h + k; notice that as we squeeze (h, k) by the factor 1 2 , the plots are squeezed by a factor (not bigger than) 1 2 . Also, in the examples in figure 18, corresponding to û = û(i, j) as in (6.2) with (i, j) = (1, 4) and (i, j) = (3, 2), the error is proportional to h + k.

We must however recall that it is well-known that in general, when solving Burgers equation with small viscosity by finite elements, spurious oscillations may appear in the numerical solution if the convection part is dominant. They can be reduced by a reduction of the mesh size or by stabilization of the numerical scheme (e.g., see [START_REF] Braack | Stabilized finite element methods for the generalized Oseen problem[END_REF][START_REF] Donea | Finite Element Methods for Flow Problems[END_REF][START_REF] Johnson | Finite element methods for linear hyperbolic problems[END_REF]). In our simulations the reference trajectory is smooth with moderately bounded gradient. Thus we can choose moderately small mesh parameters, such that no spurious oscillations appear. When we apply our scheme to a different example where the gradient ∂ x u of the exact solution reaches big magnitudes, then we will observe spurious oscillations on a coarse mesh; the oscillations vanish by decreasing the mesh parameters, see figure 19. In 19(a) we have taken the same space step and time step and viscosity as in [1, figure 3(a)]. We see we get the same behavior near (t, x) = (1, 1), where the gradient of the exact solution has a big magnitude |∂ x u(1, 1)| R , see also [16, section 5.6.5].

For references on numerical methods for feedback control and stabilization of the Burgers equation we refer the reader to [START_REF] Banks | The linear regulator problem for parabolic systems[END_REF][START_REF] Hinze | Analysis of instantaneous control for the Burgers equation[END_REF][START_REF] Kunisch | Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition[END_REF][START_REF]Galerkin proper orthogonal decomposition methods for a general equation in fluid mechanics[END_REF][START_REF] Kunisch | HJB-POD based feedback design for the optimal control of evolution problems[END_REF][START_REF] Kunisch | POD-based feedback control of the Burgers equation by solving the evolutionary HJB equation[END_REF][START_REF] Thevenet | Nonlinear feedback stabilization of a twodimensional burgers equation[END_REF]. 8. Final remarks. We have presented some estimates on the number of internal controls M we need to exponentially stabilize the Burgers system to a given reference trajectory û = û(t, x). In the case we take χ = 1 Ω , in particular there is no constraint on the support of the control, we can derive a better estimate comparing with the general case (cf. sections 3.1 and 3.2), and we have presented the results of some numerical simulations that suggest that an estimate like that obtained in the case χ = 1 Ω might hold also in the general case. Throughout the paper we consider controls given in the form M i=1 η i (t)χE O 0 s i (x). Usually, the controls at our disposal depend on each specific application, of course we can always consider another family of controls Ψ = {ψ i | i ∈ N 0 }, and perform the simulations for controls like M i=1 κ i (t)ψ i (x), and perhaps also derive the corresponding estimates on M following the procedure in sections 3.1 and 3.2.

We have focused on the viscous 1D Burgers system. However, we are convinced that the challenge to find an estimate for M , trough a condition like (3.8) (preferable to a condition like (3.19)), will present analogous difficulties for the cases of 2D and 3D Burgers and Navier-Stokes systems, and also for a wide class of parabolic systems.

Our results do not apply to the case of nonviscous Burgers equation (i.e., to the case we take ν = 0 in (1.1)), that is a completely different problem. We do not even know if a finite number M of controls is enough to stabilize the system (in a general situation the number M of needed controls will go to +∞ as ν goes to 0).

The existence of a finite-dimensional feedback control supported on a small subset of the boundary and stabilizing the system to reference nonstationary solution, is work still going on (see [START_REF] Rodrigues | Boundary observability inequalities for the 3D Oseen-Stokes system and applications[END_REF][START_REF] Rodrigues | Local exact boundary controllability of 3D Navier-Stokes equations[END_REF] for some work on this direction). Also in this case, it will be interesting to have an estimate on the dimension of the controller.

The value ν = 1 10 we use in most of the simulations is (perhaps) too big for many applications. Of course we can take smaller ν but, in that case we may need to take also a finer mesh in order to guarantee that the stabilization observed for the discretized system in the numerical simulations, will also hold for the continuous system. Notice that, when the numerical solution for system (1.1) goes to û as time increases, we can extrapolate that also the evaluations u(t, ih), i ∈ {1, 2, . . . , N x -1}, of the continuous solution at the spatial mesh points will go to û(t, ih) as time increases. Recall that if |u(t) -û(t)| H goes to 0 as t increases, then also |u(t) -û(t)| V does (provided that û ∈ {v ∈ W | sup τ ≥0 |∂ x û| L 2 ((τ, τ +1), L 2 (Ω, R)) < +∞}, due to the smoothing property of the system (3.1), see [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]Lemma 2.1]). However, the fact that |u(t, ih) -û(t, ih)| R goes to 0, for all i ∈ {1, 2, . . . , N x -1}, as time increases, is in general not enough to conclude that u goes to û. Indeed, from [25, Theorem 4.2] (for the case of the Navier-Stokes system in a two-dimensional Torus) we can derive that to conclude that u goes to û, the space step h should be taken proportional to

ν 2 1-2 log(ν)
(for small ν); and supposing that a similar estimate holds for the 1D Burgers system, it would follow that the number N x of space points (determining nodes) should be proportional to 1-2 log(ν)

ν 2
. Notice that the computational effort and computational time will increase with N x . We refer also to [START_REF]Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations[END_REF] and [19, chapter III, section 2], and references therein, concerning the estimates on the number of determining nodes.

The mathematical theory concerning stabilization to time-dependent trajectories (cf. [START_REF] Barbu | Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations[END_REF]) is not so developed as for stabilization to a stationary state (cf. [2, 4-6, 8, 38, 39]). However, since these problems arise in applications, methods to solve these problems numerically have already been developed (see, e.g., [START_REF] Fursikov | Feedback stabilization for the Navier-Stokes equations: Theory and calculations[END_REF][START_REF] Kornev | The method of asymptotic stabilization to a given trajectory based on a correction of the initial data[END_REF][START_REF]A problem of asymptotic stabilization by the right-hand side[END_REF] and references therein), notice that in this setting "trajectory" will often mean a suitable evolutionary discrete process u 0 ∈ Z, u i+1 = S(u i ) ∈ Z, i ∈ N, where Z is a Hilbert space. Other approaches can be found, for example, in [START_REF] Krstic | Nonlinear control of the viscous Burgers equation: Trajectory generation, tracking, and observer design[END_REF] (in particular, see section 4 concerning trajectory tracking) and in [START_REF] Gunzburger | Perspectives in Flow Control and Optimization, Advances in Design and Control[END_REF] (in particular, see section 7.1 concerning linear feedback control of Navier-Stokes flows).

  3 also hold for (1.1)-(1.2) (with h + ζ in the role of ζ).

3 .

 3 The Oseen-Burgers system. The dimension of the controller. Here we look for a control in the form ζ = χE O 0 P O M (η| O ), with η ∈ L 2 (R 0 , H), that stabilizes exponentially the linearized Oseen-Burgers system(3.1)

3. 3 .

 3 The gap between (3.3) and (3.15). Comparing estimates (3.3) and (3.15),

  See, for example, [15, section 5.4], [35, chapter 1, Theorem 1.4.6.4 and Corollary 1.5.3.3].

  chapter 3, Theorem 3.1].

(4. 5 ) 0 S

 50 b(t) = S(t, 0)b(0) + t (t, s)(a∂ x a)(s) ds where S(t, s)w denotes the solution of the system (4.3), for time t ≥ s with initial condition b(s) = w, and a = 0. Then, we derive

2 V 2 H + e λ C 7 s+1 s e λτ |a∂ x a(τ )| 2 V

 222 dτ ≤ e λ(s+1) |b(s)| dτ

( 5

 5 .1) Ω D := L Nx , 2L Nx , . . . , (Nx-1)L Nx consisting of the interior points of Ω that are multiples of the space step h = L Nx , with 2 ≤ N x ∈ N. As basis functions we take the classical hat-functions φ i ∈ V defined for x ∈ Ω and each i ∈ {1, 2, . . . , N x -1} by φ i (x) :=

1 2

 1 and solve (5.9), i.e.P (-X(T ) +λ 2 I) + (-X(T ) + λ 2 I) P -P M -1 P + νS = 0,providing the initial guess Y 0 = βM c . Let us denote the solution by P[START_REF] Atwell | Stabilized finite element methods and feedback control for Burgers equation[END_REF]

[ 1 ]

 1 T is too far from the solution corresponding to R = M c D χ M -1 . Having this in mind we connect the operators I and D χ by the homotopy H τ = (1 -τ )I + τ D χ , τ ∈ [0, 1] and set H τ := (M c H τ M -1 ) . Now let us fix N H ∈ N 0 and set the homotopy step ρ = 1 N H and solve P (-X(T ) + λ 2 I) + (-X(T ) + λ 2 I) P -P H ρ H ρ P + νS = 0, providing the initial guess Y 0 = P [1] T . Let us denote the solution by P [1+ρ] T . Recursively we solve, for l ∈ 2, . . . , N H , P (-X(T ) + λ 2 I) + (-X(T ) + λ I) P -P H lρ H lρ P + νS = 0, providing the initial guess Y 0 = P [1+(l-1)ρ] T

  The first four sinus.

Fig. 1 .

 1 Fig. 1. Basis for the control space {χE O 0 η | η ∈ span{s i | i ∈ {1, 2, 3, 4}}}.

  + log(|vu(t)| 2 H / |v0| 2 H ) (i, j) = (1, 8) (i, j) = (8, 2) (i, j) = (8, 7) (i, j) = (8, 8) λ = 2 (b) Without control.

Fig. 3 .

 3 Fig. 3. Convergence rate is achieved with the feedback control.

3 2 O

 32 3).Let n ∈ N 0 . Set v 0 (x) = sin((2n + 5)x), . We claim that the controls χP O 2n η cannot stabilize the equation with rate λ > 2ν(2n + 5) 2 . Indeed, we can write(v 0 , χP O 2n η) H = 2n i=1 η i O v 3 0 s i dO = 2n i=1 η i ( l 2 ) s 3 2n+1 s i dO and also O s 3 2n+1 s i dO = 1 4 O (1 -c 2(2n+1) )(c 2n+1-i -(a) The trajectory û. (|v(t)| 2 H / (|v0| 2 H ) M = 2 4t + log(|v(t)| 2 H / (|v0| 2 H ) M = 3 log(|vu(t)| 2 H / (|v0| 2 H ) (b) Convergence rates of vu and v.

Fig. 4 .

 4 Fig. 4. Two controls stabilize the system to zero with desired rate.

Fig. 5 . 1 2

 51 Fig. 5. The controls.

= 3 (

 3 b) Convergence rates of vu and v.

Fig. 6 .

 6 Fig. 6. Case n = 1. The first three controls can stabilize the heat system.

  + log(|v(t)| 2 H / |v0| 2 H ) λt + log(|vu(t)| 2 H / |v0| 2 H ) λ = 20 M = 5(b) Convergence rates of vu and v.

Fig. 7 .

 7 Fig. 7. Case n = 2. The first five controls can stabilize the heat system.

√ 44 ≈

 44 3). Notice that in all the cases we have M ref ≥ √ 44 ≈ 6.63 and M exp ≥ e 759.95. In particular, M ( =0.067) ref ≈ 149.49 is already big compared to M .

Fig. 8 .

 8 Fig. 8. Instability increases as decreases.

√ 8 ≈ 1 2

 81 16.92, M (ν=1) ref = √ 8 ≈ 2.83 < 4 and M (ν≤0.5) ref ≥ √ 22 ≈ 4.69 > 4. In particular, notice that for ν ∈ {0.5, 1} we have M ≈ M ref and the corresponding plots in figure 9(a) remain below 0.423 ≈ log(1 + e ) which suits (3.4) better than (3.16).

Fig. 9 .

 9 Fig. 9. Instability increases as ν decreases.

Fig. 10 .

 10 Fig. 10. Behavior as the desired exponential rate changes.

6 λ = 4 ǫ = 1 (

 61 b) The control.

Fig. 11 .

 11 Fig. 11. Behavior as M increases.

  log

6 λ = 4 ǫ = 1 (

 61 Q(t)v(t), v(t))H ) time t M = 2 M = b) Two controls stabilize the system.

Fig. 12 .

 12 Fig. 12. The cost decreases as the number of controls increase.

5 δ = - 1 δ = - 2 Fig. 14 .

 51214 Fig. 14. Uncontrolled case.

  (a) The reference trajectory û. (b) The discretization error.

Fig. 16 .

 16 Fig. 16. The difference between the discrete ûS and exact û solutions. (h, k) = ( π 121 , 5 500 ).

8 Fig. 17

 817 Fig. 17. The discretization error. û = û(8, 8) (cf. (6.2)), (h 0 , k 0 ) = ( 1 121 , 1 100 ).

.

  Fig. 17. The discretization error. û = û(8, 8) (cf. (6.2)), (h 0 , k 0 ) = ( 1 121 , 1 100 ).

  ) -û(t)|H (h, k) = (h0, k0) (h, k) = (h0, k0)/2 (h, k) = (h0, k0)/4 (h, k) = (h0, k0)/8(a) û = û(1,4) .

  ) -û(t)|H (h, k) = (h0, k0) (h, k) = (h0, k0)/2 (h, k) = (h0, k0)/4 (h, k) = (h0, k0)/8 (b) û = û(3, 2) .

Fig. 18 .

 18 Fig. 18. The discretization error. (h 0 , k 0 ) = ( 1 121 , 1 100 ).

  (a) Mesh size (h, k) = ( 1 80 , 1 100 ). (b) Mesh size (h, k) = ( 1 320 , 1 200 ).

Fig. 19 .

 19 Fig. 19. Spurious oscillations occur on coarser meshes.

  2 we can guarantee that |v(t)| 2 H ≤ e 9 e -15t |v 0 | 2 H , in particular |v(0.2)| 2 H ≤ e 6 |v 0 | 2 H , that is we cannot guarantee that the norm has been squeezed, but at time t = 0.8 we find

	|v(0.8)| 2 H ≤ e 11 e -15 8 10 |v 0 | 2 H = e -1 |v 0 | 2 H , and we see that the norm is already squeezed.
	6.3.4. Changing the number of controls. Here we take û = sin(5x), and
	ν = 1 10 , and λ = 4. In figure 11 we see that |v(t)| 2 H ≤ e C v M e -λt |v 0 | 2 H , and |η(t)| 2 H ≤ e C η M e -λt |v 0 | 2 H , with C v M and C η M decreasing as M increases. Figure

Acknowledgments. The authors would like to thank Karl Kunisch for fruitful discussions on the subject, and the anonymous referees for their remarks that lead to an improvement of the paper.

§ S. R. acknowledges partial support from the Austrian Science Fund (FWF): P 26034-N25