Rapid Exponential Stabilization of a Boussinesq System of KdV-KdV Type

Abstract

This paper studies the exponential stabilization of a Boussinesq system describing the two-way propagation of small amplitude gravity waves on the surface of an ideal fluid, the so-called Boussinesq system of the Korteweg-de Vries type. We use a Gramian-based method introduced by Urquiza to design our feedback control. By means of spectral analysis and Fourier expansion, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.

Full text

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Communications in Contemporary Mathematics
©World Scientific Publishing Company
Rapid Exponential Stabilization of a Boussinesq System
of KdV–KdV Type
Roberto de A. Capistrano–Filho
Departamento de Matem´atica
Universidade Federal de Pernambuco
Avenida Professor Luiz Freire S/N
Recife, Pernambuco 50740-545, Brazil.
roberto.capistranofilho@ufpe.br
Eduardo Cerpa∗
Instituto de Ingenier´ıa Matem´atica y Computacional
Facultad de Matem´aticas
Pontificia Universidad Cat´olica de Chile
Avda. Vicu˜na Mackenna 4860
Macul, Santiago, Chile
eduardo.cerpa@mat.uc.cl
Fernando A. Gallego
Departamento de Matem´atica
Universidad Nacional de Colombia
Cra 27 No. 64-60, 170003
Manizales, Colombia
fagallegor@unal.edu.co
This paper studies the exponential stabilization of a Boussinesq system describing the
two-way propagation of small amplitude gravity waves on the surface of an ideal fluid,
the so-called Boussinesq system of the Korteweg–de Vries type. We use a Gramian-based
method introduced by Urquiza to design our feedback control. By means of spectral
analysis and Fourier expansion, we show that the solutions of the linearized system
decay uniformly to zero when the feedback control is applied. The decay rate can be
chosen as large as we want. The main novelty of our work is that we can exponentially
stabilize this system of two coupled equations using only one scalar input.
Keywords: KdV–KdV system; Gramian-based method; Stabilization; Feedback control
Mathematics Subject Classification 2020: 93D15, 35Q53, 93B05
∗Corresponding author.
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2Capistrano–Filho, Cerpa & Gallego
1. Introduction
1.1. Setting of the problem
Boussinesq introduced in [3] several nonlinear partial differential equations to ex-
plain certain physical observations concerning the water waves, e.g. the emergence
and stability of solitons. Unfortunately, several systems derived by Boussinesq
proved to be ill-posed, so that there was a need to propose other systems with
better mathematical properties. In that direction, the four parameter family of
Boussinesq systems
ηt+vx+ (ηv)x+avxxx −bηxxt = 0,
vt+ηx+vvx+cηxxx −dvxxt = 0,(1.1)
was introduced by Bona et al. in [1] to describe the motion of small amplitude
long waves on the surface of an ideal fluid under the gravity force and in situations
where the motion is sensibly two-dimensional. In (1.1), ηis the elevation of the fluid
surface from the equilibrium position and vis the horizontal velocity in the flow.
The parameters a,b,c,dare required to fulfill the relations
a+b=1
2θ2−1
3,c+d=1
21−θ2≥0,(1.2)
where θ∈[0,1] and thus a+b+c+d=1
3. As it has been proved in [1], the initial
value problem for the linear system associated with (1.1) is well posed on Rif and
only if the parameters a, b, c, d fall in one of the following cases
(C1) b, d ≥0, a ≤0, c ≤0;
(C2) b, d ≥0, a =c > 0.
The well-posedness of the system (1.1) on the line (x∈R) was investigated in
[2]. Considering (C2) with b=d= 0, then necessarily a=c= 1/6. Using the
scaling x→x/√6, t→t/√6 gives a coupled system of two Korteweg–de Vries
(KdV) equations equivalent to (1.1) for which a=c= 1, namely







ηt+wx+wxxx + (ηw)x= 0,in (0, L)×(0,+∞),
wt+ηx+ηxxx +wwx= 0,in (0, L)×(0,+∞),
η(x, 0) = η0(x), w(x, 0) = w0(x),in (0, L),
(1.3)
which is the so-called Boussinesq system of KdV–KdV type.
The goal of this paper is to investigate the boundary stabilization for the linear
Boussinesq system of KdV–KdV type







ηt+wx+wxxx = 0,in (0, L)×(0,+∞),
wt+ηx+ηxxx = 0,in (0, L)×(0,+∞),
η(x, 0) = η0(x), w(x, 0) = w0(x),in (0, L),
(1.4)

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Rapid Stabilization for the KdV–KdV system 3
with boundary conditions
(η(0, t) = 0, η(L, t)=0, ηx(0, t) = f(t),in (0,+∞),
w(0, t)=0, w(L, t)=0, wx(L, t)=0,in (0,+∞),(1.5)
where f(t) is the boundary control. We are mainly concerned with the following
problem.
Stabilization Problem: Can one find a linear feedback control law
f(t) = F[(η(·, t), w(·, t)], t ∈(0,∞),
such that the closed-loop system (1.4)with boundary condition (1.5)is exponentially
stable?
1.2. Previous results
Abstract methods have been developed to obtain the rapid stabilization of linear
partial differential equations. Among them, we cite the works [11,15,16] based on the
Gramian approach. In this paper we are interested in applying this method to design
the feedback control law for the system (1.4)-(1.5). The method presented here was
successfully applied by Cerpa and Cr´epeau in [4] to study the rapid stabilization of
the KdV equation. Although this method is typically used in a single equation, this
approach has not yet widespread applied to coupled systems.
Stability properties of systems (1.3) or (1.4) on a bounded domain have been
studied by several authors. The pioneering work, for the system under consideration
in this work, is due to Rosier and Pazoto in [13]. They showed the asymptotic
behavior for the solutions of the system (1.4) satisfying the boundary conditions











w(0, t) = wxx(0, t)=0,on (0, T ),
wx(0, t) = α0ηx(0, t),on (0, T ),
w(L, t) = α2η(L, t),on (0, T ),
wx(L, t) = −α1ηx(L, t),on (0, T ),
wxx(L, t) = −α2ηxx (L, t),on (0, T ).
(1.6)
In (1.6), α0,α1and α2denote some nonnegative real constants. Under the above
boundary conditions, they observed that the derivative of the energy associated to
the system (1.3), satisfies
dE
dt =−α2|η(L, t)|2−α1|ηx(L, t)|2−α0|ηx(0, t)|2,
where
E(t) = 1
2ZL
0
(η2+w2)dx.
This indicates that the boundary conditions play the role of a damping mechanism,
at least for the linearized system. In [13] the authors provide the following result.
Theorem A (Pazoto and Rosier [13]) Assume that α0≥0, α1>0, and that
α2= 1 Then there exist two constants C0, µ0>0such that for any (η0, w0)∈

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4Capistrano–Filho, Cerpa & Gallego
L2(0, L)×L2(0, L), the solution of (1.4)with boundary condition (1.6)satisfies
k(η(t), w(t))kL2(0,L)×L2(0,L)≤C0e−µ0tk(η0, w0)kL2(0,L)×L2(0,L),∀t≥0.
Recently, Capistrano–Filho and Gallego [5] investigated the system (1.4) with
two controls in the boundary conditions
(η(0, t)=0, η(L, t)=0, ηx(0, t) = f(t),in (0,+∞),
w(0, t) = 0, w(L, t)=0, wx(L, t) = g(t),in (0,+∞),(1.7)
and deal with the local rapid exponential stabilization by using the backstepping
method. They designed boundary feedback controls
f(t) = F1(η(·, t), ω(·, t)) and g(t) = F2(η(·, t), ω(·, t)),
that lead to the stabilization of the system. The authors proved that the solution of
the closed-loop system decays exponentially to zero in the L2(0, L)–norm and the
decay rate can be tuned to be as large as desired.
Theorem B (Capistrano–Filho and Gallego [5]) Let L∈(0,+∞)\N where
N:= 2π
√3pk2+kl +l2;k, l ∈N∗.(1.8)
For every λ > 0, there exist a continuous linear feedback control law
F:= (F1, F2) : L2(0, L)×L2(0, L)→R×R,
and positive constant C > 0. Then, for every (η0, w0)∈L2(0, L)×L2(0, L),
the solution (η, w)of (1.4)with boundary conditions (1.7)belongs to space
C([0, T ]; (L2(0, L)×L2(0, L))) and satisfies
k(η(t), w(t))kL2(0,L)×L2(0,L)≤Ce−λ
2tk(η0, w0)kL2(0,L)×L2(0,L),∀t≥0.
It is important to emphasize that our goal in this paper is to stabilize the system
(1.4) using only one feedback control, improving thus the result in [5]. Concerning
controllability, the paper [6] studied different configurations for the position of the
control, in particular they proved the following.
Theorem C (Capistrano–Filho et al. [6]) Let T > 0and L∈(0,+∞)\N. For all
states (η0, w0),(η1, w1)∈L2(0, L)2one can find a control f∈L2(0, T )such that
the solution
(η, w)∈C[0, T ],L2(0, L)2∩L20, T, H1(0, L)2
of (1.4)-(1.5)satisfies η(T, x) = η1(x)and w(T, x) = w1(x),x∈(0, L).
As in the case of the KdV equation [14, Lemma 3.5], when L∈ N, the linear
system (1.4)-(1.5) is not controllable. To prove Theorem C, the authors used the
classical duality approach based upon the Hilbert Uniqueness Method (H.U.M.)
due to Lions [9], which reduces the exact controllability of the system to some

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Rapid Stabilization for the KdV–KdV system 5
observability inequality to be proved for the adjoint system. Then, to establish the
required observability inequality, was used the compactness-uniqueness argument
due to Lions [10] and some multipliers, which reduces the analysis to study a spectral
problem. The spectral problem is finally solved by using a method introduced by
Rosier in [14], based on the use of complex analysis, namely, the Paley-Wiener
theorem. As we will prove later, in this paper we obtain a controllability result
as in Theorem C but in different state space. This is required by our stabilization
method.
1.3. Main result and outline of the work
In order to present the main result of the article, let us now define the spaces that
we will work on. To do that, we first need a spectral analysis (see Section 3for
details) for the operator A:D(A)⊂L2(0, L)2→L2(0, L)2given by
A(η, w) = (−w0−w000 ,−η0−η000)
and
D(A) = (η, w)∈[H3(0, L)∩H1
0(0, L)]2:η0(0) = w0(L)=0.
As can be seen in [6], the operator Ahas a compact resolvent and it can be diag-
onalized in an orthonormal basis, i.e., the spectrum σ(A) of Aconsists only of eigen-
values and the eigenfunctions form an orthonormal basis of X0. Thus, due to the re-
sults presented in [6], there exists an orthonormal basis {(θ+
n, u+
n)n∈Z∪(θ−
n, u−
n)n∈Z}
in [L2
C(0, L)]2, endowed with the natural scalar product
((θ, u),(ϕ, ω)) = ZL
0θ(x)ϕ(x) + u(x)ω(x)dx,
composed of eigenfunctions of Asatisfying
A(θ+
n, u+
n) = iλn(θ+
n, u+
n)
and
A(θ−
n, u−
n)=(−iλn)(θ−
n, u−
n),
where ±λnare the eigenvalues. Consider then this orthonormal basis in [L2
C(0, L)]2
(θ+
n, u+
n)n∈Z∪(θ−
n, u−
n)n∈Z,
and let Z=span{(θ+
n, u+
n)∪(θ−
n, u−
n)}. For any s∈R, consider the norm




X
n∈Zcn,+(θ+
n, u+
n) + cn,−(θ−
n, u−
n)



s
:= X
n∈Z
(1 + |λn|)2
3s(|cn,+|2+|cn,−|2)!
1
2
which is used in the following definition.

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Definition 1.1. Let s∈R. The spaces Hswill be defined as the completion of Z
with respect of the norm k·ks. In each space Hs, one has the orthonormal basis
{(1 + |λn|)−s
3θ+
n,(1 + |λn|)−s
3u+
n}n∈Z∪ {(1 + |λn|)−s
3θ−
n,(1 + |λn|)−s
3u−
n}n∈Z.
Our result deals with the stabilization problem already mentioned showing the
following theorem in the H1−level.
Theorem 1.1. Let L∈(0,+∞)\N and ω > 0. Then, there exist a continuous
linear map
Fω:H1−→ R
and a positive constant C, such that for every (η0, w0)∈H1, the solution (η, w)of
the closed-loop system (1.4)-(1.5), with f(t) = Fw(η(t), w (t)) satisfies
k(η(t), w(t))kH1≤Ce−2ωtk(η0, w0)kH1,∀t≥0.
Theorem 1.1 is shown using the result proved by Urquiza in [15, Theorem 2.1]. It
is important to emphasize that our control acts only on one equation and through
a boundary condition, instead of four or two controls as in Theorem A and B,
respectively.
The content of this article is divided as follows. Section 2is devoted to presenting
the Urquiza approach, which requires four hypotheses to be satisfied called (H1),
(H2), (H3) and (H4). In Section 3we deal with some preliminary results including
the proof of (H1). We will note that (H2) is easily verified. Next, we prove the
hypotheses (H3) and (H4) in Section 4. Section 5is dedicated to the construction
of the feedback and to finish the proof of Theorem 1.1. Some final comments are
provided in Section 6.
2. Urquiza approach
In this section, we present the Urquiza method [15] to prove rapid exponential
stabilization of the following system







ηt+wx+wxxx = 0,in (0, L)×(0,+∞),
wt+ηx+ηxxx = 0,in (0, L)×(0,+∞),
η(x, 0) = η0(x), w(x, 0) = w0(x),in (0, L),
(2.1)
with boundary conditions
(η(0, t) = 0, η(L, t) = 0, ηx(0, t) = f(t),in (0,+∞),
w(0, t)=0, w(L, t)=0, wx(L, t)=0,in (0,+∞).(2.2)
We will extensively use the space operator already mentioned given by
A(η, w)=(−w0−w000,−η0−η000),(2.3)

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with domain
D(A) = (η, w)∈[H3(0, L)∩H1
0(0, L)]2:η0(0) = w0(L) = 0(2.4)
where
D(A)⊂X0:= [L2(0, L)]2.
2.1. Gramian method
Let us first write our system in the abstract framework. Set Athe operator defined
by (2.3)-(2.4) and Bgiven by
B:R−→ D(A∗)0
s7−→ Bs (2.5)
where s∈Rand Bs is a functional given by
Bs :D(A∗)−→ R
(u, v)7−→ Bs(u, v) := −svx(0).(2.6)
We will see in Prop. 3.1 that D(A) = D(A∗), which are obviously closed and dense
in X0. Thus, we have that B∗is
B∗:D(A)−→ R
(u, v)7−→ B∗(u, v) = −vx(0).(2.7)
Note that system (2.1)-(2.2) takes the abstract form
(˙y(x, t) = Ay(x, t) + Bv(t),in [D(A∗)]0,
y(x, 0) = y0(x).(2.8)
Here y0= (η0, w0)∈X0is the initial condition and the control is v(t) = −f(t).
In order to get the rapid exponential stabilization, we use the Urquiza approach
[15]. Let us explain the method on the abstract control system (2.8) with state y(t)
in a Hilbert space Yand control s(t) in a Hilbert space U. Here, the initial condition
y0∈Y,Ais a skew-adjoint operator in Ywhose domain is dense in Y, and Bis
an unbounded operator from Uto Y.
The method to prove rapid stabilization consists on building a feedback control
using the following four hypothesis for the operators Aand B:
(H1) The skew-adjoint operator Ais an infinitesimal generator of a strongly contin-
uous group in the state space Y.
(H2) The operator B:U→D(A∗)0is linear and continuous.
(H3) (Regularity property) For every T > 0 there exists CT>0 such that
ZT
0kB∗e−tA∗yk2
Udt ≤CTkyk2
Y,∀y∈D(A∗).

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(H4) (Observability property) There exist T > 0 and cT>0 such that
ZT
0kB∗e−tA∗yk2
Udt ≥cTkyk2
Y,∀y∈D(A∗).
With these hypotheses in hand, the next result holds. Its proof mainly relies on gen-
eral results about the algebraic Riccati equation associated with the linear quadratic
regulator problem (see [7]).
Theorem 2.1. (See [15, Theorem 2.1]) Consider operators Aand Bunder as-
sumptions (H1)-(H4). For any ω > 0, we have
(i) The symmetric positive operator Λωdefined by
(Λωx, z)Y=Z∞
0B∗e−τ(A+ωI)∗x, B∗e−τ(A+ωI )∗zUdτ, ∀x, z ∈Y,
is coercive and is an isomorphism on Y.
(ii) Let Fω:= −B∗Λ−1
ω. The operator A+BFωwith D(A+BFω)=Λω(D(A∗)) is
the infinitesimal generator of a strongly continuous semigroup on Y.
(iii) The closed-loop system (system (2.8)with the feedback law v=Fω(y)) is expo-
nentially stable with a decay equals to 2ω, that is,
∃C > 0,∀y∈Y , ket(A+BFω)ykY≤C e−2ωt kykY.
In order to apply this method, we have to verify the four hypotheses. This will
be done in the next sections. It is worth mentioning that the observability property
(H4) is equivalent to the controllability of system (2.8) in the appropriate spaces.
3. Preliminaries
To apply Theorem 2.1 to our linear Boussinesq control system is necessary to check
the hypotheses (H1)-(H4). First, we will prove that operator Adefined by (2.3)
satisfies (H1). Note that by the definition of the operator B, see (2.6), it is easy to
see that (H2) also follows true. Moreover, in this section we establish the asymptotic
behavior of eigenfunctions.
3.1. Hypothesis (H1)
We first comment that in order to verify hypothesis (H1) it is enough to prove that
Ais a skew-adjoint operator in H1⊂X0=L2(0, L)2. In fact, this will imply that
Ahas a semigroup property on H1. Remember that H1plays the role of Yin the
general statement of Urquiza’s method.
Prop 3.1. Ais a skew-adjoint H1and thus generates a group of isometries (etA)t∈R
in H1.
Proof. First, it is clear that D(A) is dense in H1. We have to prove that A∗=−A
in H1. Note that we have −A⊂A∗(i.e. (θ, u)∈D(A∗) and A∗(θ, u) = −A(θ , u)

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Rapid Stabilization for the KdV–KdV system 9
for all (θ, u)∈D(A)). Indeed, for any (η, v),(θ , u)∈D(A), we have the following
series representation by the orthonormal basis {(θ+
n, u+
n)n∈Z∪(θ−
n, u−
n)n∈Z}given
by Definition 1.1
(η, v) = X
n∈Zc+
n(θ+
n, u+
n) + c−
n(θ−
n, u−
n)
and
(θ, u) = X
n∈Zd+
n(θ+
n, u+
n) + d−
n(θ−
n, u−
n).
In this case, we have that
A(η, u) = X
n∈Ziλnc+
n(θ+
n, u+
n)−iλnc−
n(θ−
n, u−
n)
and
A(θ, u) = X
n∈Ziλnd+
n(θ+
n, u+
n)−iλnd−
n(θ−
n, u−
n),
respectively. Therefore, it yields that
((θ, u), A(η, v ))H1=X
n∈Z1 + |λn|2
2
3c+
n(iλd+
n) + c−
n(−iλnd−
n)
=X
n∈Z1 + |λn|2
2
3−iλnc+
nd+
n+iλnc−
nd−
n
=−(A(θ, u),(η, v ))H1.
Now, let us prove now that A∗⊂ −A. Pick any (θ, u)∈D(A∗). Then, we have
for some constant C > 0
|((θ, u), A(η, v ))X0| ≤ Ck(η, v)kX0∀(η, v)∈D(A),
i.e.
ZL
0
[θ(vx+vxxx) + u(ηx+ηxxx )]dx≤C ZL
0
[η2+v2]dx!
1
2
,(3.1)
for all (η, v)∈D(A). Picking v= 0 and η∈C∞
c(0, L), we infer from (3.1) that
ux+uxxx ∈L2(0, L), and hence that u∈H3(0, L). Similarly, we obtain that
θ∈H3(0, L). Integrating by parts in the left hand side of (3.1), we obtain that
|θ(L)vxx(L)−θ(0)vxx (0) + θx(0)vx(0) + u(L)ηxx(L)−u(0)ηxx (0) −ux(L)ηx(L)|
≤C ZL
0
[η2+v2]dx!,∀(η, v)∈D(A).
It easily follows that
θ(0) = θ(L) = θx(0) = u(0) = u(L) = ux(L)=0,
so that (θ, u)∈D(A) = D(−A). Thus D(A∗) = D(−A) and A∗=−A, which ends
the proof of this proposition.

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3.2. Behavior of the traces
As already mentioned (see [6] for details), there exists an orthonormal basis
(θ+
n, u+
n)n∈Z∪(θ−
n, u−
n)n∈Z
in [L2
C(0, L)]2, composed of eigenfunctions of Asatisfying
A(θ+
n, u+
n) = iλn(θ+
n, u+
n)
and
A(θ−
n, u−
n)=(−iλn)(θ−
n, u−
n),
where the real numbers {λn}n∈Zare the eigenvalues. Moreover, they have the fol-
lowing asymptotic form
λn=












π+ 12π(k1+n)
6L3
+O(n),as n→+∞
−7π+ 12π(k2−n)
6L3
+O(n),as n→ −∞
(3.2)
for some numbers k1, k2∈Z.Next result provides the behavior of boundary traces
associated with the orthonormal basis {(θ+
n, u+
n)n∈Z∪(θ−
n, u−
n)n∈Z}. The proof is
given in Appendix A.
Prop 3.2. There exist positive constants C±
1and C±
2, such that
lim
|n|→∞ |θ±
n,x(L))|
|n|=C±
1and lim
|n|→∞ |u±
n,x(0)|
|n|=C±
2.(3.3)
4. Proof of hypothesis (H3) and (H4)
In this section we are interested in proving the hypothesis (H3) and (H4). We start
presenting some auxiliary results that will be used to prove the regularity condition
and observability inequality, respectively.
4.1. Auxiliary lemmas
To find the regularity needed and to prove the observability inequality we use the
following classical Ingham inequality, see e.g., [8] and [12] for details.
Lemma 4.1. Let T > 0and {βn}n∈Z⊂Rbe a sequence of pairwise distinct real
numbers such that
lim
|n|→∞(βn+1 −βn)=+∞.
Then, the series
h(t) = X
n∈Z
γneiβntconverges in L2(0, T ),

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Rapid Stabilization for the KdV–KdV system 11
for any sequence {γn}n∈Zsatisfying Pn∈Zγ2
n<∞. Moreover, there exist two
strictly positive constant C1and C2such that,
C1X
n∈Z
γ2
n≤ZT
0|h(t)|2dt ≤C2X
n∈Z
γ2
n.
Now, observe that following the spectral analysis for the operator Agiven in [6,
Appendix: Proof of Theorem 3.11], we obtain that
θ±
n(x) := ∓i
√2vn(L−x) and u±
n(x) = 1
√2vn(x),(4.1)
where {vn}n∈Zare the eigenvectors of the operator Bdefined as
By=−y000(L−x)−y0(L−x),
with domain D(B) = y∈H3(0, L)∩H1
0(0, L) : y0(L)=0, which is closely related
with the operator A. So, operator Bhas the following properties that can be seen
in [6, Appendix: Proof of Theorem 3.11].
Lemma 4.2. The operator Bis self-adjoint in L2(0, L). Moreover, the following
claims hold:
(i) If L∈(0,∞)\ N , then
B−1:L2(0, L)→H3(0, L)
is well defined continuous operator. Here, Nis defined by (1.8);
(ii) There is an orthonormal basis {vn}n∈Zin L2(0, L)composed of eigenvectors of
B:vn∈D(B)and Bvn=λnvnfor all n∈Nfor some λn∈R.
Finally, the next result is a direct consequence of the spectral analysis for the
operator Aand ensures the well-posedness for the homogeneous system associated
to the system (2.1)-(2.2).
Lemma 4.3. For any (η0, w0) = Pn∈Zzn,+
0(θ+
n, u+
n) + zn,−
0(θ−
n, u−
n)∈Hs, there
exists a unique solution of
((ηt, wt) = A(η, w),
(η(0), w(0)) = (η0, w0),
belonging of C(R, Hs)and given by
(η(x, t), w(x, t)) = X
n∈Zeiλntzn,+
0(θ+
n, u+
n) + e−iλntzn,−
0(θ−
n, u−
n).
Additionally, as {λn}n∈Z⊂R, we have
k(η(t), w(t))ks=k(η0, w0)ks.

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12 Capistrano–Filho, Cerpa & Gallego
4.2. Proof of (H3)
Let us take
(η0, w0) = X
n∈Zzn,+
0(θ+
n, u+
n) + zn,−
0(θ−
n, u−
n)∈H1.
Note that by Lemma 4.3, we have
(η(x, t), w(x, t)) = X
n∈Zeiλntzn,+
0(θ+
n, u+
n) + e−iλntzn,−
0(θ−
n, u−
n),
it implies that
wx(0, t) = X
n∈Zeiλntzn,+
0u+
n,x(0) + e−iλntzn,−
0u−
n,x(0).
Thanks to (4.1), we deduce that





θ+
n,x(L) := −i
√2vn,x(0), θ−
n,x(L) := −i
√2vn,x(0),
u+
n,x(0) = 1
√2vn,x(0), u−
n,x(0) = 1
√2vn,x(0).
(4.2)
Thus, there exists a positive constant C, such that
|wx(0, t)|2≤CX
n∈Z|zn,+
0|2|u+
n,x(0)|2+|zn,−
0|2|u−
n,x(0)|2
≤CX
n∈Z|v+
n,x(0)|2|zn,+
0|2+|zn,−
0|2|.
Hence,
|wx(0, t)|2≤CX
n∈Z
|vn,x(0)|2
(1 + |λn|)2
3h(1 + |λn|)2
3|zn,+
0|2+|zn,−
0|2i.
Using the asymptotic behavior (3.2), there exists a positive constant C1such that
|vn,x(0)|2
(1 + |λn|)2
3≤C1,∀n∈Z.
Therefore,
|wx(0, t)|2≤C1X
n∈Z
(1 + |λn|)2
3|zn,+
0|2+|zn,−
0|2=C1k(η0, w0)k2
1,(4.3)
and so condition (H3) holds in H1-norm.
4.3. Proof of (H4)
Note that (3.2) implies that sequences {λn}n∈Zsatisfies the following gap condition
lim
|n|→∞(λn+1 −λn)=+∞.

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Rapid Stabilization for the KdV–KdV system 13
Then, from Lemma 4.1 and relations (4.1) and (4.3), we obtain that
C2 X
n∈Z|zn,+
0u+
n,x(0)|2+X
n∈Z|zn,−
0u−
n,x(0)|2!≤ZT
0|wx(0, t)|2dt
≤C1k(η0, w0)k2
1,
(4.4)
for some positive constants C1and C2. By using (4.2) and (4.4), we have that
ZT
0|wx(0, t)|2dt ≥C3X
n∈Z|vn,x(0)|2|zn,+
0|2+|zn,−
0|2
=C3X
n∈Z
|vn,x(0)|2
(1 + |λn|)2
3h(1 + |λn|)2
3|zn,+
0|2+|zn,−
0|2i,
(4.5)
holds for some C3>0. Observe that we can estimate the right hand side of (4.5)
in terms of any Hs–norm for s≥1. To finalize the proof of the hypothesis (H4) for
the H1–norm, we can not lose any coefficient zn,±
0. Thus, we claim the following.
Claim 1. vn,x (0) 6= 0 for all n∈Z.
Indeed, suppose by contradiction that there exists n0∈Zsuch that vn0,x(0) = 0.
This implies that
(θ+
n0,x(L), u+
n0,x(0)) = 0 and (θ−
n0,x(L), u−
n0,x(0)) = 0.(4.6)
In particular, considering u(x) = θ+
n0(x) + u+
n0(x), there exist λn0∈Csuch that
(u000 +u0+λn0u= 0,
u(0) = u(L) = u0(0) = u0(L)=0.
From [14, Lemma 3.5], it follows that L∈ N, which is a contradiction, and Claim
1 holds.
Lastly, due to the asymptotic behavior (3.2) and Claim 1, there exists a positive
constant C4>0 such that
|vn,x(0)|2
(1 + |λn|)2
3≥C4,
for all n∈N, Thus, follows by (4.6) that
ZT
0|wx(0, t)|2dt ≥C4k(η0, w0)k2
1(4.7)
Therefore, relation (H4) is satisfied. As we already mentioned, this property gives
us an additional result. The exact controllability of (2.1)-(2.2) with control space
L2(0, T ) and state space H1.

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14 Capistrano–Filho, Cerpa & Gallego
5. Rapid stabilization: Control design
In this section we design the feedback law using the Urquiza approach to show the
rapid exponential stabilization for solutions of the system (2.1)-(2.2). Recall that
this system takes an abstract form (2.8) and the operators Aand Bare given by
(2.3)-(2.4) and (2.5)-(2.6), respectively. With this in hand, we are in position to
prove our main result.
5.1. Proof of Theorem 1.1
For any (p0, q0), (r0, s0) in H1and ω > 0, consider the bilinear form defined by
aω((p0, q0),(r0, s0)) := Z∞
0
e−2ωτ qx(0, τ )sx(0, τ)dτ. (5.1)
Here (p, q) and (r, s) are solutions of

















pτ+qx+qxxx = 0,
qτ+px+pxxx = 0,
p(0, τ ) = p(L, τ) = px(0, τ ) = 0,
q(0, τ ) = q(L, τ) = qx(L, τ )=0,
p(x, 0) = p0(x), q(x, 0) = q0(x)
and

















rτ+sx+sxxx = 0,
sτ+rx+rxxx = 0,
r(0, τ ) = r(L, τ) = rx(0, τ )=0,
s(0, τ ) = s(L, τ) = sx(L, τ )=0,
r(x, 0) = r0(x), s(x, 0) = s0(x),
respectively. Finally, consider the following operator Λω:H1−→ H−1satisfying
the relation
hΛω(p0, q0),(r0, s0)iH−1,H1=aω((p0, q0),(r0, s0)),(5.2)
for all (p0, q0),(r0, s0)∈H1and ω > 0. Note that, from (2.7) we have that
hΛω(p0, q0),(r0, s0)iH−1,H1=Z∞
0
e−2ωτ qx(0, τ )sx(0, τ)dτ,
=Z∞
0
e−2ωτ B∗(p(x, τ ), q(x, τ ))B∗(r(x, τ), s(x, τ ))dτ.
Thanks to the Theorem 2.1, the operator Λωis coercive and an isomorphism. On
the other hand, set the functional
Fω:H1−→ R
(z1, z2)7−→ Fω((z1, z2)) := q0
0(0),

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Rapid Stabilization for the KdV–KdV system 15
where (p0, q0) is the solution of the following Lax-Milgram problem
aω((p0, q0),(r0, s0)) = h(z1, z2),(r0, s0)iH−1,H1,∀(r0, s0)∈H1.(5.3)
From (5.2), we deduce that (z1, z2)=Λω(p0, q0) in H−1. Moreover, observe that
Fω(z1, z2) = q0
0(0) = −B∗(p0, q0) = −B∗Λ−1
ω(z1, z2),∀(z1, z2)∈H1.
Thus, we are in the hypothesis of Theorem 2.1, which one can be applied and
guarantees the rapid exponential stabilization to the solutions of the system (2.1)-
(2.2). It means that for any ω > 0, there exists a continuous linear feedback control
f(t) = Fω(η(t), w(t))
with Fω=−B∗Λ−1
ωwhere Λωis given by (5.1)-(5.2) and a positive constant C, such
that for every initial conditions (η0, w0)∈H1, the solution (η, w) of the closed-loop
system (2.1)-(2.2), satisfies
k(η(t), w(t))kH1≤Ce−2ωtk(η0, w0)kH1,
with a decay equals to 2ω.
6. Further comments
We have applied the Gramian approach to build some boundary feedback law to
prove the rapid stabilization for a coupled KdV–KdV type system. Considering one
control acting on the Neumann boundary condition at the right-hand side of the
interval where the system evolves we are able to prove that the closed-loop system
is locally exponentially stable with a decay rate that can be chosen to be as large
as we want. Below we present some final remarks.
•Theorem A guarantees the stabilization of the KdV-KdV system with four con-
trols and Theorem B ensures the rapid stabilization with two controls. However,
Theorem 1.1 gives us a best result for the linear system, that is, we are able to
make the solutions of the linear system go to zero with only one control acting
at the boundary. It is important to point out here that the drawback is that
we are not able to treat the nonlinear case. This is due to the lack of any Kato
smoothing effect, as in the case of a single KdV [4], which leaves the rapid sta-
bilization for the full system (1.3) with boundary conditions (1.5) completely
open to study. The same lack of smoothing effect brings us to work in the state
space H1in order to have traces (and then controls) in the space L2(0, T ). This
regularity is unlikely to be sharp and we think that a better result in fractional
spaces should be expected.
•Note that we can also prove that there exists a continuous linear feedback
control
g(t) = Fω(η(t), w(t))

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16 Capistrano–Filho, Cerpa & Gallego
such that the closed-loop system (1.4) with boundary conditions
(η(0, t)=0, η(L, t)=0, ηx(0, t) = 0,in (0,+∞),
w(0, t) = 0, w(L, t)=0, wx(L, t) = g(t),in (0,+∞),
satisfies
k(η(t), w(t))kH1≤Ce−2ωtk(η0, w0)kH1,
with a decay equals to 2ω. To prove this consider the operator Bgiven by
B:R−→ D(A∗)0
s7−→ Bs := Ls
where s∈Rand Lsis a functional given by
Ls:D(A∗)−→ R
(u, v)7−→ Ls(u, v) := sux(L).
With these information in hand and the following observability inequality
ZT
0|ηx(L, t)|2dt ≥Ck(η0, w0)k2
1, C > 0,
the result follows using the same idea as done in the proof of Theorem 1.1.
Appendix A. Proof of Proposition 3.2
Following the ideas of [6, Appendix A. Proof of Theorem 3.11], we observe that vn
takes the form
vn(x) =
3
X
j=1
ajherjx−ierj(L−x)i,(A.1)
with aj=aj(n)∈C, for j= 1,2,3, where
3
X
j=1
ajerjL−i= 0,
3
X
j=1
aj1−ierjL= 0,
3
X
j=1
rjajerjL+i= 0
(A.2)
and rj,j= 1,2,3, are pairwise distinct such that
r1=r1(n)∼ −iλ1/3
n, r2=r2(n)∼ −ipλ1/3
n, r3=r3(n)∼ −ip2λ1/3
n,(A.3)
for p=ei2π
3. Note that the equations in (A.2) imply that
3
X
j=1
aj=
3
X
j=1
ajerjL= 0,

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Rapid Stabilization for the KdV–KdV system 17
that is,
a3=−a1−a2
and
a1er1L−er3L+a2er2L−er3L= 0.
Moreover, if we assume λn→ ∞, we have
r1=−iλ1/3
n+O(λ−1/3
n)∼ −iλ1/3
n
r2=−ipλ1/3
n+O(λ−1/3
n)∼ √3
2+i
2!λ1/3
n
and
r3=−ip2λ1/3
n+O(λ−1/3
n)∼ −√3
2+i
2!λ1/3
n.
Additionally, if λn→ −∞, we have
r1=−iλ1/3
n+O(λ−1/3
n)∼i|λ1/3
n|,
r2=−ipλ1/3
n+O(λ−1/3
n)∼ − √3
2+i
2!|λ1/3
n|
and
r3=−ip2λ1/3
n+O(λ−1/3
n)∼ √3
2−i
2!|λ1/3
n|.
Note that the previous relations implies
|er1L| → 1,|er2L| → +∞,|er3L| → 0,as n→ ∞ (A.4)
and
|er1L| → 1,|er2L| → 0,|er3L| → +∞,as n→ −∞. (A.5)
The convergences (A.4) and (A.5) ensures that,
λn→ ±∞,as n→ ±∞.
With these relation in hand, the following claim can be verified.
Claim 2. The behaviors (3.3)hold whenever there exist positive constant C1and
C2such that
lim
|n|→∞ |v0
n(0)|
|n|=C1and lim
|n|→∞ |v0
n(L)|
|n|=C2.(A.6)

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18 Capistrano–Filho, Cerpa & Gallego
In fact, to obtain the limit (A.6), we have to analyze the asymptotic behavior of
the aj(n) terms. First, note that kvnkL2(0,L)= 1, since {vn}n∈Zis an orthonormal
basis thanks to the Lemma 4.2, thus
1 = ZL
0
3
X
j=1
ajherjx−ierj(L−x)i
2
dx
=ZL
0

3
X
j=1
A2
j(x)+2
3
X
i,j=1,i6=j
Ai(x)Aj(x)
dx,
(A.7)
where Aj(x) = ajerjx−ierj(L−x). Then,
ZL
0
A2
j(x)dx =ZL
0
a2
je2rjx−2ierjxerj(L−x)−e2rj(L−x)dx
=a2
je2rjx
2rj−2ierjLx+e2rj(L−x)
2rjL
0
hence
ZL
0
A2
j(x)dx =−2ia2
jerjLL. (A.8)
On the other hand,
ZL
0
Ai(x)Aj(x)dx =ZL
0
aiajerix−ieri(L−x)erjx−ierj(L−x)dx
=aiaje(ri+rj)x
ri+rj−ie(ri−rj)xerjL
ri−rj
+ie−(ri−rj)xeriL
ri−rj
+e(ri+rj)(L−x)
ri+rjL
0
,
therefore
ZL
0
Ai(x)Aj(x)dx = 2iaiaj
erjL−eriL
ri−rj
,∀i6=j. (A.9)
Putting together (A.8) and (A.9) in (A.7), we get
−2iL a2
1er1L+a2
2er2L+a2
3er3L
+ 4ia1a2
er2L−er1L
r1−r2
+a1a3
er3L−er1L
r1−r3
+a2a3
er3L−er2L
r2−r3= 1.(A.10)
Thanks to the second and first relation in (A.2), respectively, we have
a2=−Γa1, a3=−a1(1 −Γ),
where Γ = er1L−er3Ler2L−er3L−1. Now, using (A.10), we obtain
−2iLa2
1er1L+ Γ2er2L+ (1 −Γ)2er3L
+ 4ia2
1−Γer2L−er1L
r1−r2−(1 −Γ)er3L−er1L
r1−r3
+ Γ(1 −Γ)er3L−er2L
r2−r3= 1.
(A.11)

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Rapid Stabilization for the KdV–KdV system 19
Moreover, note that
1−Γ = er2L−er1L
er2L−er3L
and
Γ(1 −Γ) = (er1L−er3L)(er2L−er1L)
(er2L−er3L)2.
Therefore, we have
Γ(1 −Γ)er3L−er2L
r2−r3
=−(er1L−er3L)(er2L−er1L)
(r2−r3)(er2L−er3L),
Γer2L−er1L
r1−r2
=(er1L−er3L)(er2L−er1L)
(r1−r2)(er2L−er3L)
and
(1 −Γ)er3L−er1L
r1−r3
=−(er1L−er3L)(er2L−er1L)
(r1−r3)(er2L−er3L).
Using the previous equalities in (A.11), follows that
−2iLa2
1er1L+ Γ2er2L+ (1 −Γ)2er3L
+ 4ia2
1
(er1L−er3L)(er2L−er1L)
(er2L−er3L)1
r1−r3−1
r2−r3−1
r1−r2= 1
or, equivalently,
−2ia2
1Ler1L+ Γ2er2L+ (1 −Γ)2er3L−2Γ(er2L−er1L)φ(r1, r2, r3)= 1,
where
φ(r1, r2, r3) = 1
r1−r3−1
r2−r3−1
r1−r2
.
Noting that
|a1(n)|2=1
2|L(er1L+ Γ2er2L+ (1 −Γ)2er3L)−2Γ(er2L−er1L)φ(r1, r2, r3)|,
follows that
1
2 (L|er1L|+L|Γ2er2L|+L|(1 −Γ)2er3L|+ 2|Γ||er2L−er1L||φ(r1, r2, r3)|)
≤ |a1(n)|2≤1
2|L|er1L| − L|Γ2er2L| − L|(1 −Γ)2er3L| − 2|Γ||er2L−er1L||φ(r1, r2, r3)| |.
(A.12)
Additionally, from (A.3), we have
r1−r2=−i(1 −p)λ1/3
n+O(λ−1/3
n),
r2−r3=−ip(1 −p)λ1/3
n+O(λ−1/3
n)

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20 Capistrano–Filho, Cerpa & Gallego
and
r1−r3=−i(1 −p2)λ1/3
n+O(λ−1/3
n),
which allow us to conclude that φ(r1, r2, r3)→0, as |n|→∞.
Observe that
|Γ|=
(er1L−er3L)
er2L(1 −er3Le−r2L)≤|er1L|+|er3L|
|er2L||1−er3Le−r2L|
|Γ2er2L|=
(er1L−er3L)2
er2L(1 −er3Le−r2L)2≤(|er1L|+|er3L|)2
|er2L||1−er3Le−r2L|2.
and
|(1 −Γ)2er3L|=1−er1L−er3L
er2L−er3L2
er3L≤(|er2L|+|er1L|)2
|er3L||er2Le−r3L−1|2.
Due to the asymptotic behavior of {λn}n∈N(3.2) and (A.4), we get
|Γ| → 0,|Γ2er2L| → 0 and |(1 −Γ)2er3L| → 0,as n→ ∞. (A.13)
Note that |Γer2L| → 1, which implies that
|Γ||er2L−er1L||φ(r1, r2, r3)| → 0 as n→ ∞.
On the other hand, thanks to (A.5) follows that
|Γ| → 1,|Γ||er2L−er1L| → 1,|Γ2er2L| → 0 and |(1−Γ)2er3L| → 0,as n→ −∞.
(A.14)
Thus, using (A.13), (A.14) and passing to the limit in (A.12), we deduce that
lim
|n|→∞ |a1(n)|=1
√2L.(A.15)
Let us prove (A.6). From (A.1) we have that
v0
n(x) =
3
X
j=1
ajrjherjx+ierj(L−x)i,
then
v0
n(0) =
3
X
j=1
ajrj1 + ierjL
and
v0
n(L) =
3
X
j=1
ajrjerjL+i.

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Rapid Stabilization for the KdV–KdV system 21
Thanks to (A.2), it follows that
v0
n(0) = a1r11 + ier1L+a2r21 + ier2L−(a1+a2)r31 + ier3L
=a1r11 + ier1L+a2(r2−r3) + a2r2i(er2L−er3L) + a2ier3L(r2−r3)−a1r31 + ier3L
=a1r11 + ier1L+a2(r2−r3)−a1r2i(er1L−er3L) + a2ier3L(r2−r3)−a1r31 + ier3L
=a1r11 + ier1L−r3−r2ier1L+a2(r2−r3)(1 + ier3L) + a1ier3L(r2−r3)
=a1r11 + ier1L−r3−r2ier1L−a1(er1L−er3L)
(er2L−er3L)(r2−r3)(1 + ier3L) + a1ier3L(r2−r3)
We analyze the case when n→+∞, the case when n→ −∞ can be shown analo-
gously. Noting that
er1L−er3L
er2L−er3L=Oe−√3
2λ1/3L,
r2−r3=Oλ1/3
and
1 + ier3L=O1 + e−√3
2λ1/3L,
we have that
a1(er1L−er3L)
(er2L−er3L)(r2−r3)(1 + ier3L) + a1ier3L(r2−r3) = Oe−√3
2λ1/3L.
Note that (1 −p4) = (1 −p), due the fact that p=ei2π
3. Thus,
v0
n(0) = a1(r1−r3) + ier1L(r1−r2) + Oλ1/3
ne−√3
2λ1/3
nL
=a1(−iλ1/3
n)(1 −p2) + ier1L(−iλ1/3
n)(1 −p) + O(1)
=a1(−iλ1/3
n)(1 −p2) + ier1L(−iλ1/3
n)(1 −p4) + O(1)
=a1(−iλ1/3
n)(1 −p2)1 + ier1L(1 + p2) + O(1)
=a1(−iλ1/3
n)(1 −p2)1 + ier1Lp2+O(1).
Since er1L∼e−iλ1/3
nL∼e−iπ/6∼ip2, there exists K+
1∈C\ {0}, such that
v0
n(0) ∼K+
1a1(n)λ1/3
n.
Similarly, there exists K+
2∈C\ {0}, such that
v0
n(L)∼K+
2a1(n)λ1/3
n.
Analogously, when n→ −∞, we get
v0
n(0) ∼K−
1a1(n)λ1/3
n
and
v0
n(L)∼K−
2a1(n)λ1/3
n,
for some complex constants nonzero K−
1and K−
2, respectively. Finally, (3.2) and
(A.15) ensure that (A.6) follows and, consequently, Proposition 3.2 is achieved.

October 27, 2021 16:48 WSPC/INSTRUCTION FILE capistrano-cerpa-
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22 Capistrano–Filho, Cerpa & Gallego
Acknowledgments:
R. de A. Capistrano–Filho was supported by CNPq 408181/2018-4, CAPES-PRINT
88881.311964/2018-01, CAPES-MATHAMSUD 88881.520205/2020-01, MATHAM-
SUD 21-MATH-03 and Propesqi (UFPE). E. Cerpa was supported by ANID Mil-
lennium Science Initiative Program NCN19-161 and Basal Project FB0008. F. A.
Gallego was supported by MATHAMSUD 21-MATH-03 and the 100.000 Strong in
the Americas Innovation Fund.
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