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**Remarks on the internal** **exponential stabilization to a** **nonstationary solution for 1D**

**Burgers equations**

**A. Kröner, S.S. Rodrigues**

**RICAM-Report 2014-02**

TO A NONSTATIONARY SOLUTION FOR 1D BURGERS EQUATIONS

AXEL KR ¨ONER AND S ´ERGIO S. RODRIGUES

Abstract. The feedback stabilization of the Burgers system to a nonstationary solu- tion using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint in 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.

MSC2010: 93B52, 93C20, 93D15, 93C50

Keywords: Burgers equations, exponential stabilization, feedback control, finite elements

Contents

1. Introduction 2

Notation 3

2. Preliminaries 3

2.1. Reduction to local null stabilization 4

2.2. Weak and strong solutions 4

2.3. Existence of a stabilizing control 5

2.4. Setting of the problem 6

3. On the dimension of the controller 7

3.1. The particular case of no support constraint 7

3.2. The general case 8

3.3. The gap due to the support constraint 10

4. Discretization 10

4.1. Discretization in space 10

4.2. Discretization in time 13

4.3. Computation of the discretized feedback rule 13

4.4. Solving the discretized Oseen–Burgers system 16

4.5. Solving the discretized Burgers system 16

5. Numerical examples: the linear Oseen–Burgers system 17

5.1. Testing with a family of reference trajectories 17

5.2. Piecewise constant controls 19

5.3. Increasing number of needed controls 22

6. Numerical examples: the Burgers system 23

6.1. Local nature of the results and nonlinear nature of the equation 23

6.2. Real versus fictitious external force behavior 25

6.3. On the discretization error 25

7. Final remarks 25

References 27

A. Kr¨oner acknowledges the hospitality and support of Commands team at ENSTA, ParisTech (Palaiseau, France), where during a research visit part of the work was carried out.

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

1

A. Kr¨oner and S. S. Rodrigues

1. Introduction

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

(1) ∂_{t}u+u∂_{x}u−ν∆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 ˆu ∈ W^{st} of (1) with ζ = 0, in a
suitable Banach space W^{st}. Then, following the procedure presented in [BRS11], we can
prove that there exists an integer M = M(|ˆu|W^{st}), a function η = η(t, x), defined for
t > 0, x ∈ Ω, such that the solution u = u(t, x) of problem (1), with ζ = χPMη, and
supplemented with the initial condition

(2) u(0, x) =u_{0}(x)

is defined on [0, +∞) and satisfies the relation |u(t) − u(t)|ˆ ^{2}_{L}2(Ω,R) ≤ Ce^{−λt}|u(0) −
u(0)|ˆ ^{2}_{L}2(Ω,R), provided the norm |u(0) − u(0)|ˆ _{L}^{2}_{(Ω,}_{R}_{)} is small enough. Here 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, the Burgers system to the reference trajectory ˆu.

Notice that the support of the controlζ is necessarily contained in that ofχ, 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}_{u}_{ˆ} (u(t)−u(t)), for a suitable family of linearˆ
continuous operators Q^{t, λ}_{u}_{ˆ} :L^{2}(Ω, R)→L^{2}(Ω, R), t≥0 (cf. [BRS11, section 3.2]).

We can see that the dimension M of the range of the controller depends only on the
norm|u|ˆW^{st} of ˆubut, up to now no precise estimate is known. In the case ˆuis 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 ˆu; see,
for example, [RT10, BT11, Bar12, BLT06, BT04] (cf. [BRS11, 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 dimensionM of the range of the internal stabilizing controller, in the case of a reference time-dependent trajectory ˆu. Notice that this case is not less important for applications because often we are confronted with external forces h that depend on time.

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

(3) M ≥^{L}/π(2ν^{−1}e)^{1}^{/}^{2}(ν^{−1}|ˆu|^{2}_{W}wk+λ)^{1}^{/}^{2},

where W^{wk} ⊃ W^{st} is a suitable Banach space. In the case our control is supported in a
small subset ω= supp(χ), we can also derive that it is “enough” to take

(4) M ≥C_{1}e^{C}^{2}(^{1+(ν}^{−1}^{λ)}^{1}^{/2}^{+(ν}^{−1}^{λ)}^{2}^{/3}^{+ν}^{−1}^{|ˆ}^{u|}Wwk+ν^{−2}|ˆu|^{2}

Wwk)

where C_{1} and C_{2} are constants depending on χ and Ω. 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 (3) cannot be (at
least not straightforwardly) used under the constraint on the support of the control.

So one question arises: can we improve (4)? To derive (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 > 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 possi- bility of getting, also in the general case, an estimate analogous to (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 well-known results and set up our problem. In section 3 we present the first estimates for a lower bound for the suitable dimensionM 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 with the general case. In section 4 we present the discretization of our problem and in sections 5 and 6 we present the results of some simulations we have performed. Finally, in section 7 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 Rr := (r, +∞), for r ∈ R, and N0 := N\ {0}. We denote by
Ω⊂Ra bounded interval. Given a vector functionu: (t, x)7→u(t, x)∈R, defined in an
open subset of R×Ω, its partial time derivative ^{∂u}/∂t will be denoted by ∂tu. The partial
spatial derivative ^{∂u}/∂xwill be denoted by ∂xu.

Given a Banach spaceX and an open subset O⊂R^{n}, let us denote byL^{p}(O, X), with
eitherp∈[1, +∞) orp=∞, the Bochner space of measurable functionsf :O →X, and
such that|f|^{p}_{X} is integrable overO, forp∈[1, +∞), and such that ess sup_{x∈O}|f(x)|_{X} <

+∞, forp=∞. In the caseX =Rwe recover the usual Lebesgue spaces. ByW^{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_{0}^{s}(O, R) = closure of {f ∈
C^{∞}(O, R)|suppf ⊂O}inH^{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^{0} its dual,
and by h·,·i_{X}^{0}_{,X} the duality between X^{0} and X. The dual space is endowed with the
usual dual norm: |f|_{X}^{0} := sup{hf, xi_{X}^{0}_{,X} | x∈X and |x|_{X} = 1}. In the case thatX is a
Hilbert space we denote the inner product by (·,·)X.

Given an open interval I ⊆ R and two Banach spaces X, 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 caseX =Y we writeH^{1}(I, X) :=W(I, X, X). Again, ifXandY
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 ¯I ⊂ R is a closed bounded interval, C( ¯I, X) stands for the space of continuous
functions f : ¯I →X with the norm |f|_{C( ¯}_{I,X}_{)}= max_{t∈}I¯|f(t)|_{X}.

C_{[a}_{1}_{,...,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.

2. Preliminaries

A. Kr¨oner and S. S. Rodrigues

2.1. Reduction to local null stabilization. We will denote V := H_{0}^{1}(Ω,R), H :=

L^{2}(Ω,R), D(∆) := V ∩H^{2}(Ω, R), and V^{0} := H^{ư1}(Ω, R). The space H is supposed to
be endowed with the usualL^{2}(Ω, R)-scalar product; the spaceV with the scalar product
(u, v)_{V} := (∇u, ∇v)_{H}. The spaceH is taken a pivot space, andV^{0} is the dual ofV. The
inclusions V ⊂H ⊂V^{0} are dense, continuous and compact. The space D(∆) is endowed
with the scalar product (u, v)_{D(∆)}:= (∆u, ∆v)_{H}.

Let us denote

(5) W^{wk} :=L^{∞}(R0, L^{∞}(Ω, R))

Fix a function h and suppose that ˆu ∈ W^{wk} solves the Burgers system (1), with ζ = 0
and initial condition ˆu0 := ˆu(0).

Let us be given a Lipschitz continuous function χ∈W^{1,}^{∞}(Ω, R) with nonempty sup-
port andλ >0. Then, given another functionu_{0} such that|u_{0}ưu(0)|ˆ _{H} is small enough,
our goal is to find an integer M ∈ N0 and a control η ∈ L^{2}(R0, H) such that the so-
lution of the problem (1)–(2), with ζ =χE^{O}0 P_{M}^{O}η is defined for all t > 0 and converges
exponentially to ˆu, that is, for some positive constant C >0 independent of u_{0}ưuˆ_{0},
(6) |u(t)ưu(t)|ˆ ^{2}_{H} ≤Ce^{ưλt}|u_{0}ưuˆ_{0}|^{2}_{H} fort ≥0.

HereP_{M}^{O} stands for the orthogonal projection in Honto the subspaceL^{2}_{M}(O, R) spanned
by the first M eigenfunctionss_{n} of the Dirichlet Laplacian inO, that is,

L^{2}_{M}(O, R) := span{s_{n} |n ∈N0, n≤M}

where O is an open interval such that supp(χ) ⊆ O ⊆ Ω, and E^{O}0 : L^{2}(O, R) → H
is the extension by zero outside O, that is 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 ∈ N0} and the corresponding system of vector fields {α_{n} | n ∈ N0} are given
explicitly by

(7) s_{n}(x) :=p

2/lsin(^{nπx}/l), α_{n} = (^{π}/l)^{2}n^{2}, ư∆s_{n} =α_{n}s_{n}, x∈ O,
where l stands for the length of O.

Let us note that, seeking for the controlηand considering the corresponding solutionu, we find that v =uưu, will solveˆ

(8) ∂_{t}vưν∆v+v∂_{x}v+∂_{x}(ˆuv) +ζ = 0, v|_{Γ} = 0, v(0) =v_{0},

with ζ =χE^{O}0P_{M}^{O}η and v_{0} =u(0)ưu(0). It is now clear that to achieve (6) it suffices toˆ
consider the problem of local exponential stabilization to zero for solutions of (8), where

“local” means that the property is to hold “provided |v_{0}|_{H} is small enough”.

2.2. Weak and strong solutions. The existence and uniqueness of strong solutions for system (8) can be proved by classical arguments, where weak and strong solutions are understood in the classical sense as in [Lio69, Tem95, Tem01].

Let us introduce, for given Banach spaces X and Y, the linear spaces
L^{2}_{loc}(R0, X) :={f | f|_{(0, T)} ∈L^{2}((0, T), X) for all T >0},
Wloc(R^{0}, X, Y) :={f | f|_{(0, T}_{)}∈W((0, T), X, Y) for all T >0}.

and the space

(9) W^{st} :=W^{wk}∩L^{2}_{u.loc}(R^{0}, H^{1}(Ω, R)),

where L^{2}_{u.loc}(R^{0}, H^{1}(Ω, R))⊂L^{2}_{loc}(R^{0}, H^{1}(Ω, R)) is the Morrey-like space
L^{2}_{u.loc}(R^{0}, H^{1}(Ω, R)) :=

f

sup

i∈N

f|_{(i, i+1)}

L^{2}((i, i+1), H^{1}(Ω, R))

<+∞

.

Theorem 2.1. Given uˆ∈ W^{wk},ζ ∈L^{2}((0, T), V^{0}), andv_{0} ∈H, then there exists a weak
solution v ∈ W((0, T), V, V^{0}) for system (8), in (0, T)×Ω. Moreover v is unique and
depends continuously on the given data (v_{0}, η):

(10) |v|^{2}_{W}_{((0, T}_{), V, V}0)≤C[^{T ,}^{|ˆ}^{u|}Wwk]

|v_{0}|^{2}_{H} +|ζ|^{2}_{L}2((0, T), V^{0})

.

Theorem 2.2. Given uˆ∈ W^{st}, ζ ∈L^{2}((0, T), H), andv_{0} ∈V, then there exists a strong
solution v ∈ W((0, T), D(∆), H) for system (8), in (0, T)×Ω. Moreover v is unique
and depends continuously on the given data (v_{0}, η):

(11) |v|^{2}_{W}_{((0, T}_{),D(∆), H)}≤C[^{T,}^{|ˆ}^{u|}Wst]

|v_{0}|^{2}_{V} +|ζ|^{2}_{L}2((0, T), H)

.

Notice that the proof of the existence and uniqueness of a weak solution can be done following the argument in [Tem01, chapter 3, section 3.2] by using the estimate

|∂_{x}(wv)|^{2}_{V}0 ≤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}.
For the existence of a strong solution we can use, in addition, the estimate

|w∂_{x}v|^{2}_{H} ≤ |w|^{2}_{L}∞(Ω,R)|∂_{x}v|^{2}_{L}2(Ω,R)≤C_{3}|w|^{2}_{V}|v|^{2}_{V}.

Definition 2.1. We say thatv ∈W_{loc}(R0, V, V^{0}) is a global weak solution for system (8),
in R0 ×Ω, if v|_{(0, T}_{)} ∈ W((0, T), V, V^{0}) is a weak solution, for the same system, in
(0, T)×Ω, for all T >0.

Definition 2.2. We say that v ∈ W_{loc}(R^{0}, D(∆), H) is a global strong solution for
system (8), in R0×Ω, ifv|_{(0, T}_{)}∈W((0, T), D(∆), H) is a strong solution, for the same
system, in (0, T)×Ω, for allT > 0.

Corollary 2.3. Given uˆ∈ W^{wk}, ζ ∈ L^{2}_{loc}(R0, V^{0}), and v_{0} ∈H, then there exists a weak
solution v ∈ Wloc(R^{0}, V, V^{0}) for system (8), in R^{0} ×Ω, which is unique and there holds
estimate (10).

Corollary 2.4. Given uˆ∈ W^{st}, ζ ∈L^{2}_{loc}(R0, H), andv_{0} ∈V, then there exists a strong
solution v ∈ W_{loc}(R0, D(∆), H) for system (8), in R0 ×Ω, which is unique and there
holds estimate (11).

Finally notice that system (1)–(2), is a particular case of (8) (with ˆu = 0), hence Theorems 2.1 and 2.2, and Corollaries 2.3 and 2.4, also hold for (1)–(2) (with h+ζ in the role ofζ).

2.3. Existence of a stabilizing control. We claim that the existence of a suitable
integer M = C[|ˆu|_{W}st] and a suitable control η ∈ L^{2}(R0, H), such that ζ = χE^{O}0P_{M}^{O}η
stabilizes system (8) to zero, can be derived following the procedure presented in [BRS11].

Indeed the procedure in [BRS11] uses two key ingredients: the smoothing property of the Oseen-Stokes system and an observability inequality for the adjoint Oseen-Stokes system.

In our setting we have the Oseen–Burgers system

(12) ∂_{t}vưν∆v+∂_{x}(ˆuv) +ζ = 0, v|_{Γ}= 0, v(0) =v_{0},
and its “time-backward” adjoint

(13) ư∂_{t}qưν∆qưu∂ˆ _{x}q+f = 0, q|_{Γ} = 0, q(T) =q_{1}

A. Kr¨oner and S. S. Rodrigues

for q_{1} ∈ H and f ∈ L^{2}((0, T), V^{0}). The system (12) is a parabolic system and hence
has the smoothing property if ˆu∈ W^{st} (cf. [BRS11, Lemma 2.1]); on the other hand the
desired internal observability inequalities for (13) can be found, for example in [Ima95,
DZZ08, Yam09].

Remark 2.1. Theorems 2.1 and 2.2, and Corollaries 2.3 and 2.4, also hold for system (12) in the role of system (8).

Remark 2.2. In [BRS11, section A.3] it is considered only the case when we takeO = Ω.

However it is straightforward to check that we can repeat the idea taking O satisfying supp(χ)⊆ O ⊆Ω. This idea to take a more general subset O is borrowed from [Rod13];

below in Remark 3.1 we will explain why considering a more generalOmay be interesting.

Also, in [BRS11] the functionχis taken inC^{1}(Ω, R), and this regularity is used to derive
the truncated observability in [BRS11, section A.3]; the same arguments can be followed
if χ is a continuous function inW^{1,∞}(Ω, R).

We can then conclude that we have the following results (cf. [BRS11, Theorem 3.1 and Proof of Theorem 3.6]).

Theorem 2.5. For given uˆ ∈ W^{st}, v_{0} ∈ H and λ > 0, there is an integer M =
C[^{λ,}^{|ˆ}^{u|}Wst] ≥1and a controlη^{u,λ}^{ˆ} (v_{0})∈L^{2}(R0, H)such that the solutionv of system (12),
with ζ =χE^{O}0P_{M}^{O}η, satisfies the inequality

(14) |v(t)|^{2}_{H} ≤C[^{λ,}^{|ˆ}^{u|}Wst]e^{−λt}|v_{0}|^{2}_{H}, t≥0,

where M and the positive constant C[|ˆu|_{W}st,λ] in (14) do not depend on v0. Moreover, the
mapping v0 7→η^{ˆ}^{u,λ}(v0) is linear and satisfies the inequality

e^{(˜}^{λ/2)t}η^{u,λ}^{ˆ} (v0)

2

L^{2}(R^{0},H) ≤C[^{λ,}^{(λ−}^{˜}^{λ)}^{−1}^{,}^{|ˆ}^{u|}Wst]|v0|^{2}_{H}, for 0≤λ < λ.˜
Furthermore the control can be taken in feedback form

ζ = e^{−λt}χE^{O}0P_{M}^{O}((χQ^{t, λ}_{u}_{ˆ} v)|_{O})
(15)

for a suitable operator Q^{t, λ}_{u}_{ˆ} : H →H, t∈R^{0}.

Theorem 2.6. LetM =C[^{λ,}^{|ˆ}^{u|}Wst] be the integer constructed in Theorem 2.5. Then there
are positive constants Θand depending only on|ˆu|_{W}^{st} andλ such that for |v_{0}|_{H} ≤ the
solution v of system (8), with ζ as in (15), is well defined for all t ≥0 and satisfies the
inequality

(16) |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 (12). Then, Theorem 2.6 says that the same feedback rule also locally stabilizes to zero the bilinear system (8).

2.4. Setting of the problem. The main goal of this work is to provide some first
estimates for the integer M in Theorem 2.5. We already know that M = C[^{λ,}^{|ˆ}^{u|}Wst]
depends onλand |ˆu|W^{st}. Following [BRS11] the bound for the integerM in Theorem 2.5
is related to a suitable observability inequality, for the adjoint system (13), of the form

|q(a)|^{2}_{H} ≤C|χq|^{2}_{L}2((a, a+T)), L^{2}(Ω,R)

for a, T ≥ 0, where C is known to depend on T, on Ω, and on the (support of the) function χ, then M will (in principle) also depend on these objects. Furthermore, since

the viscosity coefficientν plays a crucial role on the stability of the system, we expectM to depend also on it.

The dependence ofM on all these objects is the main focus of this work, in particular
the dependence on the triple (λ, |ˆu|W^{st}, ν).

3. On the dimension of the controller

Here we derive some first estimates concerning a lower bound for the integer M in Theorems 2.5 and 2.6.

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

Theorem 3.1. If χ= 1_{Ω}, in Theorem 2.5 it is sufficient to take
(17) M ≥(^{L}/π)(^{3e}/2ν)^{1}^{/}^{2}(ν^{−1}|ˆu|^{2}_{W}wk+λ)^{1}^{/}^{2},
where e is the Napier’s constant.

Proof. Letw solve

(18) w_{t} =ν∆w−∂_{x}(ˆuw) + (^{λ}/2)w, w|_{Γ} = 0, w(0) =v_{0}.
By standard arguments we can find

d

dt|w|^{2}_{H} ≤ −2ν|∇w|^{2}_{H} + 2|ˆu|_{L}^{∞}_{(Ω,}_{R}_{)}|w|_{H}|∇w|_{H} +λ|w|^{2}_{H}

≤(^{1}/2ν)|ˆu|^{2}_{L}∞(Ω,R)|w|^{2}_{H} +λ|w|^{2}_{H}
from which we can derive that

(19) |w|^{2}_{L}∞((0, T), H) ≤ e(^{(}^{1}^{/}^{2ν}^{)|ˆ}^{u|}^{2}^{W}^{wk}^{+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∆δ−∂_{x}(ˆuδ) + (^{λ}/2)δ+ (∂_{t}ϕ)w, δ|_{Γ} = 0, δ(0) =v_{0}

with δ(T) = 0. Let now M ∈ N0 be a positive integer and consider the solution δ_{M} for
the system

∂_{t}δ_{M} =ν∆δ_{M} −∂_{x}(ˆuδ_{M}) + (^{λ}/2)δ_{M} + (∂_{t}ϕ)P_{M}^{Ω}w, δ_{M}|_{Γ}= 0, δ_{M}(0) =v_{0}.
The difference d:=δ−δ_{M} solves

∂_{t}d=ν∆d−∂_{x}(ˆud) + (^{λ}/2)d+ (∂_{t}ϕ)(1−P_{M}^{Ω})w, d|_{Γ}= 0, d(0) = 0,
from which we can also derive

|d|^{2}_{L}∞((0, T), H) ≤e(^{(}^{3}^{/}^{2ν}^{)|ˆ}^{u|}^{2}^{W}^{wk}^{+λ})^{T}

|d(0)|^{2}_{H} + (^{3}/4ν)|(∂_{t}ϕ)(1−P_{M}^{Ω})w|^{2}_{L}2((0, T), V^{0})

(20)

≤T^{−2}e(^{(}^{3}^{/}^{2ν}^{)|ˆ}^{u|}^{2}^{W}^{wk}^{+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 (19), we can arrive at

|d|^{2}_{L}∞((0, T), H) ≤T^{−1}e^{2}(^{ν}^{−1}^{|ˆ}^{u|}^{2}_{W}wk+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 overT > 0, we can see that the minimizerT∗

is defined by T_{∗}^{−1} = 2(ν^{−1}|ˆu|^{2}_{W}wk+λ); then, setting T =T∗ we have that

|d|^{2}_{L}∞((0, T∗), H) ≤2(ν^{−1}|ˆu|^{2}_{W}wk +λ)e^{1}(^{3}/4ν)α_{M}^{−1}|v_{0}|^{2}_{H}.

A. Kr¨oner and S. S. Rodrigues

Now, from α_{M} = (^{M π}/L)^{2} (cf. (7), with O = Ω), setting M satisfying (17), and recalling
that δ_{M}(0) =v_{0} and δ_{M}(T_{∗}) =−d(T_{∗}), we find thatα_{M} ≥(ν^{−1}|ˆu|^{2}_{W}wk+λ)^{3e}/2ν and

|δ_{M}(T∗)|^{2}_{H} ≤ |δ_{M}(0)|^{2}_{H}.

Furthermore, from (19) and (20) we have that |δ_{M}|^{2}_{L}∞((0, T∗), H) = |δ−d|^{2}_{L}∞((0, T∗), H) ≤
C|δ_{M}(0)|^{2}_{H}.

Now, notice that we can consider system (18) in (T∗, +∞)×Ω with w(T∗) =δ_{M}(T∗),
and repeat the arguments. Recursively, we conclude that in each intervalJ_{∗}^{i} := (iT∗,(i+
1)T∗), i ∈ N0, we have |δ_{M}((i+ 1)T∗)|^{2}_{H} ≤ |δ_{M}(iT∗)|^{2}_{H} and |δ_{M}|^{2}_{L}∞(J∗^{i}, H) ≤ C|δ_{M}(iT∗)|^{2}_{H}
(with C independent ofi). Hence, we can conclude that |δ_{M}|^{2}_{L}∞(R0, H) ≤C|v_{0}|^{2}_{H}.

Next we notice thatv := e^{−(}^{λ}^{/}^{2}^{)t}δ_{M} solves (12), inR0×Ω, with the concatenated control
ζ = χP_{M}^{Ω}(e^{−(}^{λ}^{/}^{2}^{)t}(−T_{∗}^{−1})w) = −T_{∗}^{−1}e^{−(}^{λ}^{/}^{2}^{)t}χP_{M}^{Ω}w, where w|_{J}i

∗ solves (18), in J_{∗}^{i} ×Ω,
with w(iT_{∗}) = δ_{M}(iT_{∗}); from (19) and from the boundedness of {|δ_{M}(iT_{∗})|_{H} | i ∈ N},
we can conclude that the family {|w|_{L}^{2}_{(J}_{∗}^{i}_{, H)} | i ∈ N} is bounded; so we have that
e^{(}^{ˆ}^{λ}^{/}^{2}^{)t}ζ ∈L^{2}(R0, H) for all ˆλ < λ. Finally we observe that |v(t)|^{2}_{H} ≤e^{−λt}|δ_{M}|^{2}_{L}∞(R0, H) ≤

Ce^{−λt}|v_{0}|^{2}_{H}.

3.2. The general case. Letw solve the system

(21) ∂_{t}w=ν∆w−∂_{x}(ˆuw) + (^{λ}/2)w+χ˜η, 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
(22) ∂_{τ}w˘ = ∆ ˘w−∂_{x}(˘uw) + (˘ ^{λ}^{˘}/2) ˘w+χη,˘ w|˘ _{Γ} = 0, w(0) =˘ v_{0},

with (˘u, ˘λ, η) =˘ ν^{−1}(ˆu, λ, η). Next, consider the adjoint system˜
(23) −∂_{τ}q= ∆q+ ˘u∂_{x}q+ (^{˘}^{λ}/2)q, q|_{Γ} = 0, q(T) =q_{T}

with qT ∈H (here with no external force; cf. system (13)). From, for example, [DZZ08,
Theorem 2.1] and [DFCGBZ02, Theorem 2.3] (e.g., reversing time in system (23)), 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 (23) satisfies

(24) |q(0)|^{2}_{H} ≤e^{C}^{ω,Ω}(^{1+}_{T}^{1}^{+T}^{˘}^{λ+˘}^{λ}^{2}^{/3}^{+(1+T}^{)|˘}^{u|}^{2}_{W}wk)|q|^{2}_{L}2((0, T), L^{2}(ω,R).

Proposition 3.2. For every v0 ∈ H, we can find a control η˘= ¯η(v0) ∈ L^{2}((0, T), H),
driving system (22)tow(T˘ ) = 0at timet=T >0. Moreover, the mappingη¯:v0 7→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^{C}^{χ,Ω}(^{1+}T^{1}+T˘λ+˘λ^{2}^{/3}+(1+T)|˘u|^{2}

Wwk)|v_{0}|^{2}_{H}.
(25)

Sketch of the proof. The proof can be done following the arguments in [BRS11]. First, from (24) we can derive an observability of the form

(26) |q(0)|^{2}_{H} ≤e^{C}^{χ,Ω}(^{1+}T^{1}+Tλ+˘˘ λ^{2}^{/3}+(1+T)|˘u|^{2}

Wwk)|χq|^{2}_{L}2((0, T), H)

for the solution q of system (23) (cf. [BRS11, eq. (A.8)]). Then we can prove the null controllability considering the following minimization problem (cf. [BRS11, Problem 3.3])

J_{}( ˘w,η) =˘ |˘η|^{2}_{L}2 +1

|w(T˘ )|^{2}_{H} → min; with ( ˘w,η) solving (22).˘
To prove the linearity we can, next, consider the minimization problem

J∞( ˘w,η) =˘ |˘η|^{2}_{L}2 → min; with ( ˘w,η) solving (22) and ˘˘ w(T) = 0

(cf. [BRS11, Problem 3.4]).

Considering the null controllability of linear parabolic equations we also refer to [Ima95, section 2].

Theorem 3.3. In Theorem 2.5 it is sufficient to take

(27) M ≥C_{χ,}^{0} _{Ω}e^{3}^{/}^{2}^{(1+C}^{χ,Ω}^{)}(^{1+(ν}^{−1}^{λ)}^{1}^{/2}^{+(ν}^{−1}^{λ)}^{2}^{/3}^{+ν}^{−1}^{|ˆ}^{u|}Wwk+ν^{−2}|ˆu|^{2}

Wwk)|v_{0}|^{2}_{H}

where C_{χ,}^{0} _{Ω} =^{l}/π(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 (25).

Proof. Let ˘w solve (22) with ˘η= ¯η(v_{0}), and let ˘w_{M} be the solution of

∂_{τ}w˘_{M} = ∆ ˘w_{M} −∂_{x}(˘uw˘_{M}) + (^{˘}^{λ}/2) ˘w_{M} +χE^{O}0 P_{M}^{O}(¯η(v_{0})|_{O}), w|˘ _{Γ} = 0, w˘_{M}(0) =v_{0}.
Then, the differenced := ˘w−w˘_{M} solves

∂τd= ∆d−∂x(˘ud) + (^{λ}^{˘}/2)d+χE^{O}0(1−P_{M}^{O})(¯η(v0)|_{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|∇d|^{2}_{H} + 2|˘u|_{L}^{∞}_{(Ω,}_{R}_{)}|d|_{H}|∇d|_{H} + ˘λ|d|^{2}_{H}
(28)

+ 2hχE^{O}0(1−P_{M}^{O})(¯η(v_{0})|_{O}), di_{V}^{0}_{, V}.
For the last term we find

hχE^{O}0(1−P_{M}^{O})(¯η(v_{0})|_{O}), di_{V}^{0}_{, V} = (χE^{O}0 (1−P_{M}^{O})(¯η(v_{0})|_{O}), d)_{H}

≤ |¯η(v0)|_{O}|_{L}^{2}(O,R)|(1−P_{M}^{O})(χd)|_{L}^{2}(O,R)

and from|(1−P_{M}^{O})(χd)|^{2}_{L}2(O,R) ≤α^{−1}_{M}|(1−P_{M}^{O})(χd)|^{2}_{V} ≤2α^{−1}_{M}|χ|^{2}_{W}1,∞(Ω,R)|d|^{2}_{H}1

0(Ω,R), and

|∇d|^{2}_{L}2(Ω,R) ≥(^{α}^{Ω}^{1}/1+α^{Ω}_{1})|d|^{2}_{H}1

0(Ω,R), whereα^{Ω}_{1} =^{π}^{2}/L^{2}, we find

hχE^{O}0(1−P_{M}^{O})(¯η(v_{0})|_{O}), di_{V}^{0}_{, V} ≤α^{−}_{M}^{1}^{/}^{2}D_{χ,}_{Ω}|∇d|_{L}^{2}_{(Ω,R)}|¯η(v_{0})|_{O}|_{L}^{2}_{(O,R)},

with Dχ,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)}. Thus from (28),
d

dτ|d|^{2}_{H} ≤ |˘u|^{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 (25), we obtain

(29) |d|^{2}_{L}∞((0, T), H) ≤α^{−1}_{M}D_{χ,}^{2} _{Ω}e^{C}^{χ,Ω}(^{1+˘}^{λ}^{2}^{/3}^{+|˘}^{u|}^{2}_{W}wk)e^{C}^{χ,Ω}^{1} (T^{1}+2(˘λ+|˘u|^{2}

Wwk)T)|v_{0}|^{2}_{H}
with C_{χ,Ω}^{1} = max{1, C_{χ,Ω}}. Now the function E(T) = e^{C}^{χ,Ω}^{1} (T^{1}+2(˘λ+|˘u|^{2}

Wwk)T) takes its
minimum whenT =T∗, with T∗ defined by ^{1}/T_{∗}^{2} = 2(˘λ+|˘u|^{2}_{W}wk). Then, choosingT =T∗,
and recalling that ˘w_{M}(T) =−d(T) and ˘w_{M}(0) =v_{0}, we arrive at

|w˘_{M}(T∗)|^{2}_{H} ≤α^{−1}_{M}D^{2}_{χ,}_{Ω}e^{C}^{χ,Ω}(^{1+˘}^{λ}^{2}^{/3}^{+|˘}^{u|}^{2}_{W}wk)e^{2}^{3}^{/2}^{C}^{χ,Ω}^{1} (^{λ+|˘}^{˘} ^{u|}^{2}_{W}wk)^{1}^{/2}|w˘_{M}(0)|^{2}_{H}.

Therefore, choosing M ∈N0 satisfying (27), and recalling that α_{M} = (^{M π}/l)^{2}, we have
(30) α_{M} ≥D_{χ,}^{2} _{Ω}e^{3(C}^{χ,Ω}^{+1)}(^{1+˘}^{λ}^{2}^{/3}^{+|˘}^{u|}^{2}_{W}wk+˘λ^{1}^{/2}+|˘u|_{W}_{wk})

and |w˘_{M}(T_{∗})|^{2}_{H} ≤ |w˘_{M}(0)|^{2}_{H}. Moreover we can deduce from (25) and (29) that

|w˘_{M}|^{2}_{L}∞((0, T), H) =|w|˘ ^{2}_{L}∞((0, T), H)+|d|^{2}_{L}∞((0, T), H)≤C|w˘_{M}(0)|^{2}_{H}
for a suitable constant C independent of ˘w_{M}(0) =v_{0}.

Recursively, repeating the argument in the time interval (iT∗,+∞) with ˘w(iT∗) =

˘

w_{M}(iT∗) in (22), we can conclude that the solution ˘w_{M} will remain bounded for all time
τ ≥0. That is,|w˘_{M}|^{2}_{L}∞(R0, H) ≤C|v_{0}|^{2}_{H}.

A. Kr¨oner and S. S. Rodrigues

Next, we notice thatv(t) := e^{−(}^{λ}^{/}^{2}^{)t}w˘_{M}(νt) solves (12), inR0×Ω, with the concatenated
control ζ = χE^{O}0P_{M}^{O}(νe^{−(}^{λ}^{/}^{2}^{)t}η(v¯ _{[i]})(νt)|_{O}), where ¯η(v_{[i]}), i ∈ N, is the control given in
Proposition 3.2, when we consider system (22), in J_{∗}^{i}×Ω, with J_{∗}^{i} := (iT∗, (i+ 1)T∗),i∈
N0, and ˘w(iT∗) = ˘w_{M}(iT∗), in particular ¯η(v_{[i]})(νt) is defined fort∈(iν^{−1}T∗, (i+1)ν^{−1}T∗).

We can also conclude from (25) and from the boundedness of {|w˘_{M}(iT∗)|_{H} | i∈N} that
the family {|¯η(v_{[i]})|_{L}^{2}_{(J}_{∗}^{i}_{, H)} | i ∈ N} is bounded; so e^{(}^{λ}^{ˆ}^{/}^{2}^{)t}ζ ∈ L^{2}(R0, H) for all ˆλ < λ.

Finally we observe that |v(t)|^{2}_{H} ≤e^{−λt}|w˘M(νt)|^{2}_{L}∞(R0, H) ≤Ce^{−λt}|v0|^{2}_{H}.
Remark 3.1. Notice that when we shrink the support of χ the constant C_{χ,}_{Ω} in (25) is
expected to increase. This is why taking the length l of O in (27) can compensate a
little the increasing ofC_{χ,}_{Ω} in order to get a smaller bound for the number M of needed
controls.

3.3. The gap due to the support constraint. Comparing the estimates (17) and (27)
we see that there is a big gap; the former is proportional to ν^{−2}|ˆu|^{2}_{W}wk+ν^{−1}λ^{1}/2

and the
latter depends exponentially in both ν^{−1}|ˆu|_{W}^{wk} and ν^{−(}^{1}^{/}^{2}^{)}λ^{1}^{/}^{2}. For application purposes
the latter is much less convenient, so one question arises naturally: can we improve (27)?

It seems that the idea used to derive (17) cannot (at least straightforwardly) be applied under the support constraint for the controls. On the other side to derive (27) we start from an exact null controllability result and carry the cost of the respective control. This means that to improve (27) we will probably need a different idea.

In section 5, in order to understand if it is possible to improve (27), say that we also have an estimate like (17) under the support constraint, we present results of some numerical simulations comparing the number of controls M = Mneed, that we need to stabilize the system (12) to zero, to the reference value

(31) M_{ref} = (^{L}/π)ν^{−(}^{1}^{/}^{2}^{)}(ν^{−1}|ˆu|^{2}_{W}wk+λ)^{1}^{/}^{2} ∈R0;

(cf. (17)). Notice that in the case ˆu = 0, and under no support constraint, we can see
that the unstable modes of system (18) are those defined by the inequalityνα_{i} < λ, that
is,i <(^{L}/π)ν^{−(}^{1}^{/}^{2}^{)}λ^{1}^{/}^{2} =M_{ref}. Thus, in this case, it is enough (and necessary) to take the
M = bM_{ref}c controls in n

p2/Lsin(^{iπx}/L)|i∈ {1, 2, . . . , M}o

(taking χ and the family of controls considered in section 3.1). Here byc ∈N stands for the biggest integer that is strictly smaller than y >0.

4. Discretization

To perform the simulations in order to check the stabilization of systems (1) and (12), to a reference trajectory ˆu and to 0 respectively, we must discretize those systems with the feedback control ζ as in (15).

4.1. Discretization in space. We use a finite-element based approach. We introduce a regular mesh

(32) Ω_{D} := (^{L}/Nx,^{2L}/Nx, . . . , ^{(N}^{x}^{−1)L}/Nx)

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

1−i+^{x}/h, ifx∈[(i−1)h, ih] ;
1 +i−^{x}/h, ifx∈[ih,(i+ 1)h] ;
0, ifx /∈[(i−1)h,(i+ 1)h].

t t t t - t

t

0 (i−1)h ih (i+ 1)h L ^{x}

φi(x)6

1

A A A A A

Figure 1. Graph of φ_{i}.

Next any function u ∈ V can be approximated by the values it takes on Ω_{D}. More
precisely, we approximate uby the function ˜u, defined as

˜ u:=

Nx−1

X

i=1

u(ih)φ_{i}.

We define also the evaluation vectoru:= [u(ih)]^{>} := [u(1h), u(2h), . . . , u((Nx−1)h)]^{>} ∈
M_{(N}_{x}−1)×1, whereA^{>} stands for the transpose matrix ofA.

Remark 4.1. Notice that ue := PNx−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 u ∈
L(V →V^{0}). We defineL_{D} by the formula

(33) v^{>}L_{D}u=hL˜u, ˜vi_{V}^{0}_{, V} for all u, v ∈V.

Of key importance are the identity and the Laplace operator. For the identity operator Iu = u, we find that ID = [(φi, φj)H] =: M is the so-called Mass matrix, while for the Laplace operator we find ∆D =−[(∂xφi, ∂xφj)H] =: −S, whereS is the so-called Stiffness matrix. Explicitly we have the tridiagonal matrices

M:= h 6

4 1 0 0 . . . 0

1 4 1 0 . . . 0

0 1 4 1 . .. ... ... . .. . .. . .. . .. 0

0 . . . 0 1 4 1

0 . . . 0 0 1 4

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 ˆu and discretize the operator v 7→ B(ˆu)v :=

∂_{x}(ˆuv), v ∈ V. We start by noticing that, for an arbitrary w ∈ V, (∂_{x}(ˆuv), w)_{H} =

−(ˆuv, ∂_{x}w)_{H}, then we consider the approximationfuvˆ =PNx−1

j=1 uˆ_{j}v_{j}φ_{j} of ˆuv, and we find

−(fuv, ∂ˆ xw)˜ H =PNx−1

j=1 −uˆjvjwi(φj, ∂xφi)H, and

(∂x(ˆuv), w)H ≈w^{>}BD_{u}_{ˆ}v,
with B the bidiagonal matrix

B:= 1 2

0 1 0 . . . 0

−1 0 1 . . . 0 ... . .. ... ... ...

0 . . . −1 0 1 0 . . . 0 −1 0

,