PreprintPDF Available

Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves

Authors:
Preprints and early-stage research may not have been peer reviewed yet.

Abstract

Fredholm-type backstepping transformation, introduced by Coron and L\"u, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form Dxα|D_x|^\alpha for α(1,3/2]\alpha \in (1,3/2]. We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the α=3/2\alpha=3/2 threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying α>1\alpha>1, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for α>3/2\alpha>3/2. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order α=3/2\alpha=3/2.
arXiv:2202.08321v1 [math.AP] 16 Feb 2022
FREDHOLM BACKSTEPPING FOR CRITICAL OPERATORS AND
APPLICATION TO RAPID STABILIZATION FOR THE LINEARIZED
WATER WAVES
LUDOVICK GAGNON, AMAURY HAYAT, SHENGQUAN XIANG, AND CHRISTOPHE ZHANG
Abstract. Fredholm-type backstepping transformation, introduced by Coron and u, has be-
come a powerful tool for rapid stabilization with fast development over the last decade. Its
strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate
controllability. But limitations with the current approach exist for operators of the form |Dx|α
for α(1,3/2]. We present here a new compactness/duality method which hinges on Fred-
holm’s alternative to overcome the α= 3/2 threshold. More precisely, the compactness/duality
method allows to prove the existence of a Riesz basis for the backstepping transformation for
skew-adjoint operator verifying α > 1, a key step in the construction of the Fredholm backstep-
ping transformation, where the usual methods only work for α > 3/2. The illustration of this
new method is shown on the rapid stabilization of the linearized capillary-gravity water wave
equation exhibiting an operator of critical order α= 3/2.
Keywords: water waves, compactness/duality method, Fredholm transformation, backstep-
ping, rapid stabilization.
2010 MSC: 35S50, 76B15, 93B05.
Contents
1. Introduction 2
2. Main results 9
3. Strategy and outline 11
4. Compactness/duality method for Riesz basis 14
5. Construction of the feedback-isomorphism pair 22
6. Well-posedness and stability of the closed-loop system 28
7. Conclusion 34
Appendix A. Riesz basis in Hilbert spaces 35
Appendix B. Controllability and Proof of Proposition 2.1 36
Appendix C. Proof of Lemmas 4.34.5 37
Appendix D. Proof of Property (i) in Lemma 4.11 38
Appendix E. Proof that ker(T) = {0}39
Appendix F. Adapting the proof of Theorem 2.2 to Theorem 2.4 43
References 47
Universit´e de Lorraine, CNRS, Inria ´equipe SPHINX, F-54000 Nancy, France. E-mail:
ludovick.gagnon@inria.fr.
CERMICS, ´
Ecole des Ponts ParisTech, 6 - 8, Avenue Blaise Pascal, Cit´e Descartes—Champs sur Marne, 77455
Marne la Valee, France. E-mail: amaury.hayat@enpc.fr.
atiment des Math´ematiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland. E-mail:
shengquan.xiang@epfl.ch.
Universit´e de Lorraine, CNRS, Inria ´equipe SPHINX, F-54000 Nancy, France. E-mail:
Christophe.zhang@inria.fr.
1
2 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
1. Introduction
Since its introduction by Coron and u for the rapid stabilization of the Korteweg-de Vries
equation [16] and Kuramoto-Sivashinsky equation [17], the Fredholm transformation has been
applied successfully in the past decade for the rapid stabilization of a large class of equations. It
consists in finding an operator-isomorphism pair (K, T ) that maps a system of the form
tu=Au +BK u
(1.1)
to the rapidly exponentially stable system
tv= (Aλ)v
(1.2)
where Ais the generator of a strong semigroup Bis an unbounded operator and λis a posi-
tive number arbitrarily large. Compared with the original Volterra transformation introduced
by Krsti´c and Balogh [5], the Fredholm transformation possesses the advantage of presenting
a systematic approach to prove the rapid stabilization from spectral properties of the spatial
operator Aand from suitable controllability assumptions. However, the classical approach for
the Fredholm transformation fails to deal with operators of the form |Dx|αfor α(1,3/2].
Indeed, one key step in proving the existence of the Fredholm transformation Tis to prove that
the family T ϕnis a Riesz basis of the state space, where
{(ϕn, λn)}nare the eigenmodes of the spatial operator A.
The usual way to tackle this problem is to prove that the family T ϕnis quadratically close to the
eigenfunction basis ϕn(see Definition A.1 (3) for a precise statement), but the lack of growth of
the high frequency eigenvalues λnlike nαprevents the use of such criteria, leaving the Fredholm
alternative for operators behaving as |Dx|αfor α(1,3/2] an open question since then.
1.1. The compactness/duality method.
In this paper we present a new method to answer this question. This method is based on a
new compactness/duality approach to prove that the family T ϕnis indeed a Riesz basis in sharp
spaces for the whole range α > 1. The challenging part in proving that the family T ϕnis a
Riesz basis is the coercivity estimate (see the left-hand side estimate of (A.1)). We proceed by
a contradiction argument to prove this inequality. Using the expression of T ϕn, we are able to
prove that Tcan be decomposed in an invertible part and a compact part. Then, the desired
uniform inequality can be deduced from the ω-independent property. A further inspection on
the duality between the ω-independence of T ϕnin Hrand the density of Tϕnin (Hr)Hr
finally leads to the required property.
A second important step of our method is to deal with the so-called T B =Buniqueness
condition, introduced in [11] for finite-dimensional systems, used implicitly in [16,17] and intro-
duced explicitly in [12] for the first time in PDEs to deal with the nonlocal term arising from
the distributed controls. In the original approach, proposed by [16] and used since then, this
condition is solved thanks to the quadratically close property mentioned above. With this new
method, we are able to sidestep this limit thanks to a fine decomposition of the T B =Bcondi-
tion, allowing to define the transformation Talong with the feedback law K.
Beyond the α= 3/2 threshold, this new method leads to sharp Riesz basis properties for a
large class of skew-adjoint operators including Fourier multiplier based operators as long as the
high frequency scales as nαfor α > 1.
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 3
We apply this new strategy to prove the rapid stabilization of the linearized capillary-gravity
water waves equation, exhibiting a spatial operator behaving like |Dx|3/2. This is an example
corresponding to the critical case α= 3/2 which remained out of reach until now.
1.2. The capillary-gravity water waves equation.
We introduce the linearized capillary-gravity water waves equation (following [3,4,28,29])
relevant for modelling the motion and stability of perfect fluids where the surface tension and
capillarity cannot be neglected; for instance for small characteristic scales or when waves are
breaking and at certain waves frequencies ([29,40]). Thus, consider the 2-D capillary-gravity
water waves for an homogeneous, inviscid, incompressible, irrotational fluid over a flat bottom
on which an external pressure is applied. The volume of the fluid is described by
Ω(t) = {(x, y)T×R| hyη(x, t)},
where y=his the bottom of the fluid, y=η(x, t) is the deformation from the rest y= 0 of
the free surface and T=R/2πZ. The evolution of the velocity field Uof the fluid and of the
free surface are governed by 2-D free surface Euler equation,
tU+ (U.)U=−∇pge2,(x, y)Ω(t),
div U= 0,rot U= 0,(x, y)Ω(t),
U.n = 0,(x, y)T× {−h},
satisfying the boundary conditions on the free surface y=η(t, x),
(tη=p1 + |∇η|2U.n, (x, y)T× {η(t, x)},
p=patm +Pext σκ(η),(x, y)T× {η(t, x)},
where pis the pressure, gthe gravitational constant, n:= 1
1+|∇η|2(−∇η, 1)tthe outward normal
vector to the surface η,e2= (0,1)tthe unit vector, σ > 0 is the surface tension coefficient and
κ(η) = x xη
p1 + |xη|2!=2
xη
(1 + (xη)2)3/2,
is the mean curvature of the surface. The first part is the Euler equation on Udescribing
incompressible and irrotational fluids with an impermeable bottom respectively. The second
part is the boundary conditions on the free surface: the kinematic equation on the surface for
ηasserting that particles on the surface remains on the surface along time, and the pressure at
the surface, including the surface tensison and the localized external pressure Pext(t, x, η(t, x)).
The incompressible and irrotational assumption implies that the velocity field is represented by
a velocity potential Φ : R+×R2Rsuch that U=x,y Φ. The 2-D free surface Euler equation
implies that the velocity potential satisfies
tΦ + 1
2|∇φ|2+gy =(ppatm),(x, y )Ω(t),
∆Φ = 0,(x, y)Ω(t),
nΦ=0,(x, y)(0,2π)× {−h},
tη=p1 + |∇η|2nΦ,(x, y)(0,2π)× {η(t, x)}.
It was first noted by Zakharov [40] that the preceding equation on the velocity potential Φ is
an Hamiltonian system, where ψ:= Φ|y=ηand ηare (generalized) canonical variables. Moreover,
4 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
Φ is completely determined through the Laplace equation and the knowledge of ψ:= Φ|y=ηand
η. This leads to study the Dirichlet-to-Neumann map
G[η, h] : ψ7→ p1 + |∇η|2nΦ|y=η=yΦ(x, η, t)xη(x, t)xΦ(x, η, t).
We refer for instance to [3,4,28,29] and the references therein for the properties of the Dirichlet-
to-Neumann map as well as its application to the well-posedness of the Cauchy problem of the
gravity and capillary-gravity water waves.
Using the Dirichlet-to-Neumann map, one may reformulate the capillary-gravity water waves
as,
(1.3)
tηG[η, h]ψ= 0,
tψ+ +1
2|∇ψ|2(G[η, h]ψ+η.ψ)2
2(1 + |∇η|2)=σκ(η)Pext.
The Dirichlet-to-Neumann operator is nonlinear with respect to the surface elevation. We there-
fore consider the linearization around (η, ψ) = (0,0), yielding (fixing σ= 1),
(1.4) (tηG[0, h]ψ= 0,
tψ+ 2
xη=Pext,
where G[0, h] = |Dx|tanh(h|Dx|), defined as a Fourier multiplier on periodic functions. Set L
the operator
(1.5) L:= i(g2
x)G[0, h]1/2,
and let u=ψ+LG[0, h]1η, we end up with,
tu=Lu+Pext.
To be more precise, we consider the external pressure (the control) to be of the distributed
control form Pext =B1(x)w1(t)+ B2(x)w2(t). Notice that Lhas double eigenvalues (see Section
1.8), which means according to [19] two distributed controls are required to control/stabilize the
system instead of one. Hence, for ease of notations we consider
(1.6) tu=Lu+Bw(t),
where Bis a two dimensional control operator B:wC2w1B1+w2B2.
1.3. Statement of the main results. We consider in this paper the rapid stabilisation of (1.6),
that is to seek, for any λ > 0, a two-dimensional control feedback law w(t) = Ku(t, ·) such that
the solution of (1.6) satisfies
ku(t)k.eλtku0k,t(0,+).
To state our main results, we first introduce condition (2.1) for the exact controllability of (1.6)
given below in Section 2(see Proposition 2.1 and its proof in Appendix B). Using the backstepping
method with a Fredholm transformation, we are able to prove (see Theorem 2.2 and Corollary
2.3 for a precise statement)
THEOREM 1.1. Let B(H3/4)2satisfying Assumption 1concerning controllability. Then,
for any λ > 0, there exists a bounded linear operator K L(H3/4;C2)and an operator Tbeing
an isomorphism from Hr(T)to itself for any r(1,1) and maps the closed-loop system
(1.7) tu=Lu+BK(u),(t, x)R+×T,
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 5
to the system
tv=Lvλv, (t, x)R+×T,
where Lis the linearized water-wave operator given in (1.5). Consequently, the closed-loop
system (1.7)is exponentially stable in Hrfor any r(1,1).
The proof of Theorem 1.1 relies on our compactness/duality method, allowing to prove the
existence of a Fredholm operator for the backstepping method in the critical case α= 3/2. Our
method in fact extends to the more general cases (see Theorem 2.4 for a precise statement).
THEOREM 1.2 (General skew-adjoint operators).Let α > 1. Let B(H3/4)2satisfying
Assumption 1concerning controllability. Let h(s)a real valued-function satisfying
|n1n2|nα1
1.|h(n1)h(n2)|for any (n1, n2)N.
sα.|h(s)|.sαfor any s[1,+).
Then, for any λ > 0, there exists a bounded linear operator K L(H3/4;C2)and an operator T
such that Tis an isomorphism from Hr(T)to itself for any r(1/2α, α 1/2) and maps the
system
(1.8) tu=i h(|Dx|)u+BK(u),(t, x)R+×T,
to the system
tv=i h(|Dx|)vλv, (t, x)R+×T.
Consequently, the closed-loop system (1.8)is exponentially stable in Hrfor r(1/2α, α 1/2)
with decay rate λ.
In the next subsection, we introduce formally the Fredholm backstepping method for PDEs
as well as the main steps of our proof.
1.4. The Fredholm-type backstepping method. Before stating our main results, we intro-
duce formally the backstepping method for PDEs. The backstepping method to consists to prove
the existence of a feedback law w=Ku and an invertible operator Tmapping the solution uof
the equation to be stabilized,
(1.9) (ut(t) = Au(t) + Bw(t),
u(0) = u0,
onto the solution zof an exponentially stable equation (thanks to the natural dissipation of A
and the strong damping effect of λI ),
(1.10) (vt(t) = (AλI)v(t),
v(0) = v0,
where u(t) and v(t) = T u(t) belong to a Hilbert space H,Ais the generator of a semigroup
over the state space, Bis an unbounded operator satisfying some admissibility condition (see
for instance [36] for a definition) and the control w(t) = Ku(t) is of feedback form to achieve
stabilisation.
The main challenge is to find an operator-isomorphism pair (T , K) such that this mapping can
be achieved. This problem is equivalent to find (T , K) solving,
(1.11) T(A+BK) = (AλI)T ,
shown by taking formally the time derivative of z=T u and using (1.9) and (1.10).
6 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
The backstepping method was first introduced in finite dimension as a chain of integrators
for feedback laws [27], but was then extended to PDEs by Krsti´c and Balogh [5] for Volterra
transformations of the second kind,
z(t, x) = T u =u(t, x)Zx
0
k(x, y)u(t, y)dy.
The abstract equation (1.11) then transfers to the existence of a solution to a non-standard
PDE on the kernel k. In addition to leaving the classical framework of Cauchy theory, the
kernels resulting from the Volterra transformation present boundary conditions on the diagonal
k(x, x) that proves to be very difficult to handle. Despite these difficulties, several methods
have been developed to solve the PDE on the kernel of the Volterra transformation (successive
approximations [27], explicit representations [27] or method of characteristics [15]) leading to a
rich literature, the invertibility of the Volterra transformation being guaranteed.
More recently, the Fredholm-type transformation as follows
z(t, ·) = T u(t, ·) = (Id +Tcomp)u(t, ·),
was introduced by Coron and u for the rapid stabilization of the Korteweg-de Vries equation
[16] and Kuramoto-Sivashinsky equation [17] by means of the backstepping method. In this
transformation, Tis a Fredholm operator with an invertible and a compact part. Although much
more technical, the Fredholm transformation provides a systematic approach to the backstepping
method based on the spectral properties of the operator Aand the controllability properties to
prove the rapid stabilization for a large class of equations.
Let us elaborate on the techniques involved with the backstepping method with the Fredholm
transformation. A first crucial step is to consider the so-called uniqueness condition T B =Bto
change the abstract equation (1.11) into a system of two equations
T A +BK = (AλI )T ,(1.12)
T B =B.(1.13)
The uniqueness condition T B =Bwas first introduced in [11] to prove the existence of (T , K)
solving (1.12)-(1.13) in finite dimension, used implicitly in [16,17] and finally stated explicitly in
[12] to remove nonlocal terms involved in (1.11) for distributed controls. The proof of existence
of an invertible transformation Tover the state space and a feedback law stabilizing (1.9) is then
divided in the following steps:
Step 1: Let {λn+λ}nσ(A) = . Notice that (1.12) is equivalent to
T ϕn=Kn(A(λn+λ)I)1B,
where Kn=Kϕn. We prove that
˜
T ϕn= (A(λn+λ)I)1B,
is a Riesz basis family of H.
Step 2: Let B=Pnbnϕn. Use the Riesz basis properties to solve the T B =Buniqueness
condition in a suitable sense
T B =X
n
bnT ϕn=X
n
bnKn˜
T ϕn=B.
Step 3: Show from T B =Bthat (Kn)nis uniformly bounded. Then, using operator equality
and Kato’s perturbation theory, prove that T:HHis continuous and invertible.
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 7
Step 4: Thanks to the operator equality, the operator A+BK generates a semigroup on H(and
indeed it also generates semigroups in Hβwith a certain range of β). Conclude on the
rapid stabilization using the operator equality.
Aside of the seemingly different approach of hyperbolic systems [13,14,15,41,42], the proof Step
1 and 2 relied heavily in the literature on the quadratically close criterion. Roughly speaking it
amounts to show after some computations that
X
nNX
pN\{n}
1
|λnλp+λ|2<+
which holds if the eigenvalues λnof the operator Ascales as nαwith α > 3/2 but fails as soon
as α3/2.
We introduce in this article the compactness/duality approach to overcome the limitations of
the quadratically close criterion coupled with ω-independence/density properties. Indeed this
criterion corresponds to the rather strong Hilbert-Schmidt criterion for compactness. In fact the
compactness part can be proved in a more general way, and we establish ω-independence thanks
to a new duality observation, thus overcoming the apparent limit of α > 3/2 (see Remark 4.9).
This allows us to prove Step 1 for operators with eigenvalues scaling as nα(for instance |Dx|α)
with α(1,3/2].
Let us also stress that the T B =Buniqueness condition is more difficult to handle than it
seems since Bis an unbounded operator. Indeed, if Bwere to be bounded, then from
X
n
bnKn˜
T ϕn=B,
one would be tempted to deduce that the sequence bnKnbelongs to 2. Moreover, the control-
lability assumption implies that the sequence of |bn|is bounded from below (it is impossible
from the assumption that Bis bounded but let us assume it is for the sake of the argument)
and therefore one would conclude that Kn2. But then, with the expression of T, it is not
difficult to prove that in this case that the transformation Twould be compact and therefore
not invertible.
The proof of the decomposition of T B =Bfor Bunbounded and admissible still follow the
same idea, with the slight modification that T B is seen as a singular and bounded part. Then,
one adjusts the behaviour of Knby hand by letting Knc+knwhere cis a constant. If the
Riesz basis is quadratically close to the eigenfunctions, then one obtains,
X
n
bnkn˜
T ϕn=X
n
bn(ϕn˜
T ϕn)
and the right-hand side is bounded in Husing the quadratically close argument and the bound-
edness of the sequence bnin (roughly speaking provided by the admissibility). Without the
quadratically close property this direct argument fails. However, we prove here that even if the
Riesz basis is not quadratically close to the eigenfunctions, we are still able to reach the same
final conclusion by a close inspection of the left-hand side.
1.5. Related works on the backstepping with a Fredholm transform. There are essen-
tially two type of systems in the literature for which the rapid stabilisation was achieved through
the backstepping method with a Fredholm transformation : either the operator Ais of first order
(α= 1) or of second order or higher (α2). We have so far excluded from our discussion the
case α= 1 as it seems to be a very specific case with techniques on its own. Indeed, the rapid
stabilisation for hyperbolic systems was established in [14,15] through direct methods or by
identifying the isomorphism applied to the eigenbasis leading to the Riesz basis [13,41,42]. The
8 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
other results found in the literature were concerned with operators such that α2, and in these
case the Riesz basis properties was proved through the quadratically close criterion, thanks to
the sufficient growth of the eigenvalues. Following the steps described in the previous section, the
rapid stabilisation was obtained for the linearized bilinear Schr¨odinger equation [12], the KdV
equation [16], the Kuramoto-Sivashinksi equation [17], a degenerate parabolic operator [20] and
finally the heat equation for which the backstepping is proved in sharp spaces [19]. The variety
of the PDEs for which this methodology can be applied tends to show that there exists an ab-
stract theory for operators of order α > 1. Theorem 1.2 demonstrates this fact for skew-adjoint
operators for α > 1. This abstract setting could allow to lift some difficult questions raised when
trying to apply the backstepping with the Volterra transformation. One such difficult is seen for
instance for degenerate parabolic equations ([20]), where the Fredholm transformation lead to
the study of well-known spectral properties of the Sturm-Liouville equation, whereas the PDE on
the kernel of the Volterra transformation amounts to describe the propagation of bicharacteristics
from a boundary satisfying a degenerate equation, a notoriously difficult problem.
Finally, we shall remark that there are many other useful stabilization techniques in the
literature that may also apply to similar systems, for instance the damping stabilization of
waves [1,7,26], the multiplier methods [25], Riccati theory [6,9,30], Gramian method [37],
equivalence between observability and stabilizability [35], quantitative finite time stabilization
[38,39], various Lyapunov approaches [18,22,23,24] and among others.
1.6. Related works on controllability and stabilization of the water waves system.
A considerable amount of literature exists on the control of fluids. However, few works address
the controllability and stabilization of the water waves. Recently, this subject has drawn more
and more attention. The controllability properties of these systems was first investigated in [31]
using the moment method, and the controllability assumption was sharpened to quasi-linear
systems in [3], where the control is localized in the domain. A recent work [43] investigated the
water waves in 3-D, highlighting the need for the geometric control condition for the controllabil-
ity (the 2-D case satisfying automatically the geometric control condition as the control problem
reduces to a 1-D equation).
Concerning stabilization, despite fruitful stabilisation results were obtained in the literature for
fluids, only few result seems to exist regarding the water waves. We may refer to the asymptotic
stabilization and exponential stabilisation results of water waves systems [1,2] that are based
on external “damping” forces and “observability” of the closed-loop systems. Alternatively, sta-
bilizability properties of linearized water waves systems with controls acting on the boundary
have been recently studied in [32,33]. Our contribution, as a direct consequence of the Fredholm
backstepping transformation we obtain a rapid stabilisation result, that is exponential stability
with arbitrarily decay rate.
1.7. Outline of the paper. The paper is divided in the following way. First, the main results
are stated precisely in Section 2as well as their possible extensions, and the strategy of the
proof is presented in Section 3. Section 4begins with the statement of technical estimates used
throughout the article as well as the proof of the Riesz basis (Step 1 of Section 1.4) using the
compactness/duality argument. In Section 5we prove that the uniqueness condition T B =B
holds in H1/2ε, ε (0,1/2) (Step 2) and define properly the feedback law K, the isomorphism
T(Step 3) as well as the operator equality (1.12). The well-posedness and rapid stabilisation
of the closed-loop system (2.2) is proved (Step 4) in Section 6and in turn prove that the sharp
spaces for which the backstepping transform is establish coincide with the sharp space for the
well-posedness of the closed-loop system. Finally, Appendix A recalls the basic definitions for
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 9
Riesz basis, Appendix B proves the exact controllability of (1.6), Appendix C is devoted to some
basic estimates concerning the linearized water waves opeartor, Appendix D and E are dedicated
to technical proofs on the Fredholm transform for the backstepping method already existing in
the literature and Appendix F extends the proof of Theorem 2.2 to Theorem 2.4 for the general
cases.
1.8. Notations and spaces. We begin by recalling that the eigenfunctions of the operator L
on the torus coincide with the classical Fourier series {einx }nZin T. Notice that for any nN
Le±inx =i(g+n2)ntanh(hn)1/2e±inx,
thus
Lsin(nx) = λnsin(nx) and Lcos(nx) = λncos(nx),(1.14)
with λn:= in(g+n2) tanh(hn)1/2.(1.15)
Since every nonzero eigenvalue has multiplicity two, any given function can be separated by odd
part and even part which corresponds to the orthonormal basis
(1.16) ϕ1
n= (π1sin(nx)), ϕ2
n= (π1cos(nx)),for nN, ϕ2
0=2π2.
In this logic, we decompose the space Hr(T;C) as Hr
1Hr
2, for any rR, where
Hr
1:= {aHr(T;C)|a=X
nN
anϕ1
n}, Hr
2:= {aHr(T;C)|a=X
nN
anϕ2
n}.
Since we are working on T, the inner product ,·iHm
iis well-defined and given by, for any
f=Pnfnϕi
nand g=Pngnϕi
nbelong to Hm
i,
hf, giHm
1=X
nN
(nmfn)(nmgn),hf, giHm
2=f0g0+X
nN
(nmfn)(nmgn).
We finally recall some properties of G[0, h] and L: for s > 3/2, we have G[0, h] : Hs(T;K)
Hs1(T;K) and for sR, we have L:Hs(T;K)Hs3/2(T;K), for K=Ror C.
2. Main results
We first introduce the assumption leading to the exact controllability of 1.6.
Assumption 1. Let the operator B= (B1, B2)be such that B1H3/4
1and B2H3/4
2.
Then, assume the following condition,
(2.1) b06= 0 and c1<|bi
n|< c2,for i {1,2}, n N.
Thanks to the condition (2.1), we have the following controllability result for (1.6).
PROPOSITION 2.1. Let T > 0and assume that (2.1)holds. For any (u0, uf)(L2)2there
exists a control vL2(0, T )such that the unique solution of (1.6)with initial state u0satisfies
u(T) = uf.
This is proved using the moments method and using Haraux’s refined version of the Ingham’s
inequality. As the proof is similar to the one given by Reid [31] for a similar water waves system,
we postpone it to the Appendix B.
Our main result is the following
10 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
THEOREM 2.2. Let B= (B1, B2)H3/4
1×H3/4
2such that (2.1)holds. Then, for any λ >
0, there exists an explicit bounded linear operator K L(H3/4
1×H3/4
2;C2)and an isomorphism
Tfrom Hr(T)to itself for any r(1,1) that maps the system
(2.2) tu=Lu+BK(u),(t, x)R+×T,
to the system
(2.3) tv=Lvλv, (t, x)R+×T,
where Lis the linearized water-wave operator given in (1.5).
A direct consequence of this theorem is the existence of an explicit control law for the rapid
exponential stabilization of the system (1.6).
COROLLARY 2.3 (Exponential stability).For any λ > 0, there exists an explicit feedback
functional K L(H3/4
1×H3/4
2;C2)such that for any r(1,1), for any initial state u(t)|t=0 =
u0Hr, the closed-loop system (2.2)has a unique solution uC0([0,+); Hr(T;C)). In
addition, this unique solution decays exponentially with rate λ
ku(t, ·)kHr.eλtku0kHr,t(0,+).
We shall remark here that the bound r= 1 is sharp in the sense that for r1 the unbounded
operator A+BK does not anymore generate a strongly continuous semigroup in Hr. We also
underline that, while the isomorphism Tdepends on the regularity of the state space Hr, the
feedback law is, surprisingly, independent of r. This independence was already noticed in [19].
As stated in the introduction, the system (1.6) is all the more interesting as it represents the
critical case α= 3/2 where the usual method fails. To prove Theorem 2.2, we have to overcome
this difficulty by introducing a new method. This allows us to free ourselves from the bound
α= 3/2 and, in fact, the method presented in this paper is more general: it can be extended
at no cost for a large class of systems satisfying α > 1. More precisely, we have the following
theorem
THEOREM 2.4 (General skew-adjoint operators).Let α > 1. Let B= (B1, B2)H3/4
1×
H3/4
2such that (2.1)holds. Let h(s)a real valued function satisfying
|n1n2|nα1
1.|h(n1)h(n2)|for any (n1, n2)N.
sα.|h(s)|.sαfor any s[1,+).
For any λ > 0, there exists an explicit bounded linear operator K L(H3/4
1×H3/4
2;C2)and an
isomorphism Tfrom Hr(T)to itself for r(1/2α, α 1/2) that maps the system
(2.4) tu=i h(|Dx|)u+BK(u),(t, x)R+×T,
to the system
tv=i h(|Dx|)vλv, (t, x)R+×T.
Consequently, the closed-loop system (2.4)is exponentially stable in Hrfor r(1/2α, α 1/2)
with decay rate λ.
A way to adapt the proof of Theorem 2.2 to this case is given in Appendix F. The only
significant difference is in the derivation of the regularity of the stabilizing feedback (see step
(4) in Section 3below) for which one need a finer decomposition and an iteration to reach the
desired regularity.
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 11
2.1. Extension of the main results. The results of this article extends readily to more general
cases or to other boundary conditions with minor modifications of the proof. We describe below
such extensions.
2.1.1. Rapid stabilisation in Hs.The fundamental assumption guiding the appropriate Sobolev
spaces in which the rapid stabilisation holds is the controllability Assumption 1since the co-
efficients of the feedback law K, which enters in the decomposition of the isomorphism T, are
obtained through the T B =Buniqueness condition. The controllability Assumption 1such that
bi
n1 implies the controllability of the linearized system in L2and leads to an isomorphism
Tfrom Hrfor any r(1,1). If we replace this assumption by the condition that guarantees
the controllability in Hsspace for some sR, namely bi
nns, then we can adapt slightly
the proof of existence of the Riesz basis and construct the isomorphism Tin Hs+r(T) to itself
for any r(1,1) and a bounded linear operator K:Hs+3/4R2such that the closed-loop
system is exponentially stable in Hs+rfor any r(1,1).
2.1.2. Conservation of mass. In this paper we have not taken into account the “conservation of
mass” condition ZT
u(x)dx =hu(t), ϕ2
0i= 0,
which concerns the even space Hr
2. In fact by choosing Bsuch that hB2, ϕ2
0i= 0, the backstepping
method can be applied. Indeed, by following the same steps as in our proof, one can build an
isomorphism that maps the even part of the water-waves system, with mass conservation, to the
target system
tv=Lvλv,
with mass conservation, which is of course still exponentially stable. The reason for this is that
the projection on ϕ2
0commutes with the isomorphism Tand the closed-loop A+BK obtained
in the general case.
Thus, the same proof also leads to rapid stabilization in cases where there is a “conservation
of mass” condition.
2.1.3. Water waves in bounded domain. We have investigated the linearized water waves system
in a periodic domain. In fact we can also study the same system in a bounded domain with
Neumann boundary conditions, the controllability of which was obtained by Reid [31]. In this
framework, since all eigenvalues are simple, we are able to establish controllability, and rapid
stabilization by backstepping, using only one control term.
3. Strategy and outline
In this Section we briefly comment on the strategy to prove Theorem 2.2 which is the task of
the next two section, while the proof of Corollary concerning the well-posedness and stability of
the closed-loop system will be discussed later on in Section 6. Given the decomposition along
odd and even functions, showing Theorem 2.2 amounts to proving the following proposition.
PROPOSITION 3.1. Let i {1,2}. Let BiH3/4
isatisfying (2.1). For any λ > 0, there
exists a bounded linear operator Ki L(H3/4
i;C)and an isomorphism Tifrom L2to itself (which
is also an isomorphism from Hr
ito itself for any r(1,1)) which maps the system
(3.1) tu=Lu+BiKi(u), u Hr
i
12 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
to the system
(3.2) tv=Lvλv, v Hr
i,
where Lis the linearized water-wave operator given in (1.5).
If Proposition 3.1 holds, then the operator-isomorphism pair (K, T ) of Theorem 2.2 is simply
T=T1T2and K= (K1, K2). In the following we will focus on the odd space i= 1, the even
space i= 2 can be treated in the same way.
Notations: for reader’s convenience we will also drop for now on the index 1and denote again
Hr,K,T,ϕninstead of Hr
1,K1,T1,ϕ1
n.
Before proving Proposition 3.1, let us make some formal observations to get an intuition of
the problem. To map systems (3.1) onto (3.2), what we would like to do is to obtain (formally)
the following operator equality
(3.3) T(A+BK) = (Aλ)T ,
where A:= L. As noted in [12,17,19], a good approach to this aim is to add a condition on
T B, and to require instead the two following operator equalities
T A +BK = (Aλ)T ,
T B =B,
(3.4)
in a certain sense to be specified. Note that (3.4) implies formally (3.3) and requiring (3.4) instead
of (3.3) allows to deal with operator equations that are linear with (T, K ). It also usually ensures
the uniqueness of solution (see for instance [19]). In finite dimension, (3.3) corresponds to an
equivalent formulation of the pole-shifting theorem [13, Section 2]. Applying the first operator
equality on the orthonormal basis (ϕn)nNgives
(3.5) λn(T ϕn) + BK(ϕn) = (Aλ)(T ϕn),
where we used the fact that ϕnis an eigenvector of A. Observe that (3.5) is a (nonlocal)
differential equation on (T ϕn). Projecting now on a vector ϕpand recalling that bp=hB, ϕpi,
this becomes
(λn+λ)h(T ϕn), ϕpi+hB, ϕpiK(ϕn) = hA(T ϕn), ϕpi,
=h(T ϕn), Aϕpi=λph(T ϕn), ϕpi.
(3.6)
This gives the following formal expression
(3.7) T ϕn=X
pNh(T ϕn), ϕpiϕp= (K(ϕn)) X
pN
bpϕp
λnλp+λ.
These formal calculations lead us to introduce the following notations that will be used all
along the proof. We define
The families
(3.8) qn:= X
pN
ϕp
λnλp+λ, Kn=K(ϕn), n N.
The operator
(3.9) S:nrϕn7→ nrqn.
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 13
Note that Sis completely defined as an operator on Hrsince (nrϕn)nNis an orthonor-
mal basis of Hrfor any rR.
The operator
(3.10) τ:ϕn7→ bnϕn.
Note that since bnare uniformly bounded by above and below, this is an isomophism
from L2to L2, and in fact also from Hrto Hrfor any rR.
The operator Tdefined on Hr
(3.11) T:nrϕn7→ (Kn)nrτqn.
Note that this expression of Tcorresponds exactly to the expression (3.7) obtained from
the formal calculations.
In the following we will show that, for a good choice of (Kn)nN, the operator Tthus defined is an
isomorphism from Hrto itself for r(1,1), and Tand Ksatisfy (3.4) in a sense to be specified.
We are going to show successively the following steps:
(1) Show that Sis a Fredholm operator from HrHrfor any r(1,1).
(2) Show that (qn)nNis a Riesz basis for L2using a duality argument and the fact that S
is Fredholm.
(3) Further show that (nrqn)nNis a Riesz basis for Hrfor any r(1,1) by showing
it is ω-independent using a duality argument between the density of (nrqn)nNin Hr
and the ω-independence of (nrqn)nNin Hr.
(4) Provide an explicit candidate of (Kn)nNwhich satisfies T B =Bin H3/4sense. Show
that (|Kn|)nNis bounded from above and that bnKn=(λ+kn) for any nN, where
(knnε)nNlfor any ε[0,1/2).
(5) Show that Tis bounded from Hrin itself for r(1,1) and the rst operator equality
(3.4) holds in L(H3/4;H3/4).
(6) Show that Tis a Fredholm operator from H3/4to H3/4.
(7) Show that Tis an isomorphism from H3/4to H3/4using a Fredholm argument and
spectral theory in H3/4.
(8) Show that Tis an isomorphism from L2to L2and in fact an isomorphism from Hrto
itself for any r(1,1).
Let us briefly discuss step 6 to 8 as, at first sight, it seems odd to prove the invertibility in
H3/4and not in the classical L2space for instance. The main motivation is to avoid working
in the space D(A+BK) := {fL2: (A+BK )fL2}before proving the invertibility of T.
Indeed, in the preesent setting, the space D(A+BK) does not have nice properties shared by
the Sobolev spaces such as the density of Cfunctions. This comes from the fact that Bis not
regular enough, and therefore one is not able to conclude that ϕnD(A+BK) for any nN.
Hence, it is easier to first prove the invertibility in weaker but classical Sobolev spaces (step 6
14 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
and 7) before deducing the invertibility in the required spaces (step 8). In turn, the invertibility
of Tin Hrallows to construct an equivalent norm to Hr, which allows to prove that D(A+BK)
is an Hilbert space, a non-trivial task to prove without the invertibility of T. We underline that
if our setting is close to the linearized bilinear Schr¨odinger equation, the fact that the control is
real-valued in [12] allows to decouple the real and imaginary part of the solution to deal directly
with the space D(A+BK), which is not the case here.
We start by introducing some technical Lemmas in Subsection 4.1. Then we prove Proposition
3.1, following the outline above : we prove steps (1)-(3) in Section 4and steps (4)-(8) in Section
5. Finally, we prove the well-posedness of the closed-loop system obtained and Corollary 2.3 in
Section 6.
4. Compactness/duality method for Riesz basis
Following the outline described in Section 3, this section is devoted to the proofs of Steps
(1)-(3). These steps form a first important part of the proof of our main theorem: they revolve
around Riesz basis properties for some important families of functions derived from the back-
stepping operator equalities (3.4). As we have mentioned in the introduction, we introduce here
a new method based on compactness and duality, namely, we prove in a general way that the
transformations involved in our backstepping method are Fredholm operators.
4.1. Some basic estimates. In this section we introduce some technical Lemmas that will be
used in the following. For readers’ convenience we put part of proofs in Appendix C.
The first lemma is a direct consequence of the existence of c, C > 0 such that
cn3/2 |λn| Cn3/2,nN.
LEMMA 4.1. Let sR. Let ρfrom the resolvent set of the operator L. We know that
L:Hs+3/4Hs3/4is continuous,
(Lρ)1:Hs3/4Hs+3/4is continuous.
We turn to,
LEMMA 4.2. For any s < 1/2, there exists C > 0such that
X
nN\{p}
ns
|λnλp|C(p1/2+slog(p) + p3/2),pN.
Let us now show Lemma 4.2
Proof of Lemma 4.2.Let s < 1/2, we have
X
nN\{p}
ns
|λnλp|=I1+I2+I3,
where
I1=X
nN, np/2
ns
|λnλp|,
I2=X
nN\{p}, p/2<n<2p
ns
|λnλp|,
I3=
+
X
n=2p
ns
|λnλp|.
RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES SYSTEM 15
We will show that all these three terms can be bounded by C(p1/2+slog(p) + p3/2) where C
is a constant independent of p. For this, we introduce the following basic estimates (the proofs
may be found in Appendix C).
LEMMA 4.3. There exists c > 0such that for any (n, m)N
|λnλm| c|nm|3/2,
|λnλm| c|nm|n1/2.
LEMMA 4.4. For any s6=1, there exists C > 0such that for any pN
p
X
n=1
nsC(1 + p1+s).
LEMMA 4.5. For any sRand ε1>0, there exists C > 0and ε > 0such that ε(0, ε1)
and p
X
n=1
nslog(n)C(1 + p1+s+ε).
Let us look at I1, since there exists C > 0 independent of pand nsatisfying np/2, such
that |λpλn| C1p3/2, using Lemma 4.3
(4.1) X
nN,np/2
ns
|λnλp|.p3/2X
nN,np/2
ns.p3/2+p1/2+slog(p),
where in the rightmost inequality we used Lemma 4.4 if s6=1, applies and p1+sp1+slog(p)
for plarge enough, and if s=1 then we simply used that p
P
n=0
1/n=O(log(p)).
Then we turn to I2, using Lemma 4.3 we have
X
nN\{p},p/2<n<2p
ns
|λnλp|.psX
nN\{p},p/2<n<2p
1
|λnλp|.psX
nN\{p},p/2<n<2p
1
|np|p1/2.
Notice that X
nN\{p},p/2<n<2p
1
|np|X
kp/2
1
k+
p
X
k=1
1
k.log(p),
hence
(4.2) X
nN\{p},p/2<n<2p
ns
|λnλp|.p1/2+slog(p),
which gives the bound on I2.
Finally we look at I3, since n > 2p, there exists a constant C > 0 independent of psuch that
|λnλp| C1n3/2from Lemma 4.3, thus
(4.3) I3=
+
X
n=2p
ns
|λnλp|.
+
X
n=2p
ns3/2.Z+
2p
xs3/2dx .p1/2+s
where we used that s < 1/2 thus s3/2<1. Combining (4.1), (4.2) and (4.3) we deduce that
I1+I2+I3.p1/2+slog(p) + p3/2.
This ends the proof of Lemma 4.2.
16 L. GAGNON, A. HAYAT, S. XIANG, AND C. ZHANG
4.2. Step (1): a general Fredholm operator. In this Section we show the following Propo-
sition
PROPOSITION 4.6.