Optimal shape of an underwater moving bottom generating surface waves ruled by a forced KdV equation Jérémy Dalphin∗

Ricardo Barros†‡

Abstract

It is well known since Wu & Wu (1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves, propagating ahead of the disturbance in procession. This rather intriguing phenomenon has been studied extensively by relying on reduced long-wave models, such as the generalized Boussinesq equations, or the forced Korteweg-de Vries (fKdV) equation, but also on the fully non-linear Euler (or even the Navier-Stokes) equations. One possible new application of this phenomenon could very well be surng competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the fKdV equation to investigate the shape of the moving body capable of generating the highest rst upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms b ∈ L2 (R, [0, +∞[), with uniformly bounded L2 -norms and supports embedded in a given xed compact set. We establish analytically the existence of a maximizer saturating the L2 -constraint, derive the gradient of the functional, and then implement numerically an optimization algorithm yielding the desired optimal shape.

Keywords:

shape optimization, existence theory, optimal control, surface wave generation,

numerical simulation, forced Korteweg-de Vries equation, nite-dierence methods.

AMS classication:

1

primary 49K20, secondary 35Q53, 49M29, 65M06, 76B15, 49J45, 49J50.

Introduction

The generation of water-waves is a complex phenomenon and the articial reproduction of such processes has many interesting applications for the engineering industry. In this paper, we consider a specic mechanism, initially discovered by Wu & Wu [20] from numerical simulations, and which has been recently used to develop a now-patent prototype of wavemaker [15].

The operating

principle consists in translating a moving bottom underwater to produce periodically a convenient wave upstream the forcing disturbance [19].

One possible new application of this phenomenon

could very well be the surng competitions, where in a controlled environment, such as a pool, waves can be generated by using a translating bottom (see [15]). Moreover, the translating bottom is assumed to move steadily in shallow water with a transcritical velocity. We are interested in a reduced long-wave phenomenon, but weakly dispersive and non-linear eects should also be considered since experiments display the successive and continuous propagation of solitary waves ahead the moving disturbance. As shown by Wu in [19], under such conditions, the free surface can be eectively described by the forced Korteweg-de Vries (fKdV) equation. The realm of validity of this model has been extensively studied in the literature, and the model has been shown to be in good agreement, for a wide range of parameters, with laboratory experiments and numerical solutions (see [4, 11, 19, 22] and references therein). In this article, we investigate how the shape of an underwater moving bottom can aect the amplitude of the rst upstream-progressing solitary wave. The admissible bottoms will be required to belong to the set

B

dened as:

( B :=

Z

2

b ∈ L (R, R) | supp b ⊆ [−K, K], b > 0 and kbkL2 (R,R) :=

2

b (x) dx

)

1/2 6M

,

(1)

R ∗ Centro de Modelamiento Matemático (CMM), Facultad de Ciencas Físicas y Matemáticas (FCFM), UMR 2071 CNRS-Universidad de Chile, Beauchef 851, Santiago, Chile. † Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK. ‡ Mathematics Applications Consortium for Science and Industry, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland.

1

where

K

and

M

are xed positive constants. We then propose an energy functional

F

for which

the following optimization problem is considered:

sup F (b) .

(2)

b∈B The functional

F : b ∈ B 7→ F (b) ∈ R F (b) :=

is dened by

kub k2L2 (0,T ;L2 (R,R))

Z

T

Z

0 where

T >0

is xed and

ub (x, t)

u2b (x, t) dx dt,

=

(3)

R

denotes the free-surface elevation ruled by a fKdV equation with

zero initial data:

 db ∂ub ∂ 3 ub ∂ub   =− + ub + ∈ H −1 (R, R) , ∂t ∂x ∂x3 dx   ub (x, 0) = 0,

x ∈ R, t > 0, (4)

x ∈ R.

As usual, the partial derivatives in (4) are understood in a distributional sense. For convenience, we assume the dimensionless form of all variables and functions. in the body frame of reference and, following common practice,

x

Moreover, the problem is set and

t

denote space and time

coordinates, respectively. The well-posedness of the fKdV equation with homogeneous initial data has been the object of numerous studies. The time global well-posedness in by Bona & Zhang [2].

with

s > 0,

has been established

The extension of this result to lower regularity forcing functions is not

straightforward since, contrary to the case when

s < 0.

H s (R, R),

s > 0, L2 -conservation

laws are absent when

In the particular case when the ow is critical, reected here by the fact that

b

does not

depend on time in (4), the fKdV equation is endowed with a scaling property (Lemma B.1) that can be used to obtain

a priori

estimates as in the KdV case (see [5], [18]).

These, combined

with local well-posedness results known in the literature [2, 10], allowed Tsugawa [18] to prove the existence of a unique global solution to the initial-value problem (4) for low-regular bottoms (b

: H σ 7→ ub ∈ Ct0 (Hxσ+2 ), σ > − 21 ).

Remark that much stronger regularity is often required for

other wave models. The choice of admissible bottoms (1) deserves some explanation, but, as it will be made clear, it turns out to be a rather natural one.

Indeed, non-negative bottoms with compact support

must be considered for manufacturing purposes. Moreover, the constraint imposed on the support is revealed essential to obtain the continuity of the functional (3) with respect to the

L2 -weak

topology (Proposition 3.5). Furthermore, from the physics viewpoint, to remain within the regime of validity of the mathematical model, the forcing function imposes a

L2 -norm

b must be small enough, which naturally

constraint. From the mathematics viewpoint, however, one could be tempted

to discard such constraint. We show in Proposition B.2 that by doing so, (2) becomes an ill-posed problem in the sense that the supremum is not achieved, even with smooth forcing functions. Finally, it is essential that all these requirements guarantee that (3) is a well-dened application from

L2 (R, R)

into

R.

It is shown in Proposition 3.1 that this is indeed the case.

A related problem was recently addressed by Nersisyan

et al.

[12]. In their work, the bottom

velocity is not necessarily constant, thus both the piston trajectory and shape of the underwater wave maker are optimized, subject to practical constraints. The underlying mathematical model of their study is the generalized Benjamin-Bona-Mahony (BBM) equation, which similarly to the fKdV equation can be deduced from the generalized Boussinesq equations [1, 19]. In our setting, the bottom velocity is constant and the shape of the admissible bottoms (1) becomes a distributed

L2 -control

function inside the innite domain

R

of the fKdV equation (4).

In that form, the

optimization of the bottom shape is modelled by the optimal control problem (2). We refer to the references given in [12] for other applications of control theory to the BBM and KdV equations. Our approach is motivated by the fact that so much is known about the solution properties of the fKdV model and its ability to predict the generation of a succession of solitary waves propagating upstream as a response to a forcing disturbance, moving steady with a transcritical velocity, which can eectively be used to formulate the optimization problem (2). One further reason to retain the fKdV model in the present study is the mathematical challenges it presents, view that: (i) theoretically, the existence of a maximizer to (2) for low-regular bottoms is not a straightforward problem, as it is the case for smoother ones (

2

cf. [12, Theorem 2]);

(ii) numerically, the fKdV equation is known to be quite unstable, being its regularized version often preferred for computations [19]. In this paper, we will investigate (i)(ii), using techniques from calculus of variations. One of our main contributions is the proof of existence of an optimal bottom saturating the

i.e.

L2 -constraint,

the following result holds:

Theorem 1.1. Let

T > 0, M > 0, and K > 0. We consider the set B and the functional dened in (1) and (3), respectively. Then, the optimization problem (2) is well posed in the following sense: F : b ∈ B 7→ F (b)

∃bopt ∈ B,

Moreover, any maximizer bopt of

(2)


F bopt = max F (b) . b∈B

satises kbopt kL2 (R,R) = M .

Another important contribution is the description of an ecient numerical algorithm providing an optimal shape, which could be useful for practical applications.

The uniqueness of a global

maximizer to (2) is not yet known, however numerics seem to suggest that (see Fig. 5). The paper is organized as follows. In Ÿ 2 we present the mathematical model and summarize briey some previous known results in the literature. discussed in Ÿ 3.

The optimization problem is thoroughly

We then present in Ÿ 4 the numerical scheme used for solving (4), as well as

its adjoint formulation that is used to evaluate the gradient, whose expression is incorporated in a Usawa-type optimization algorithm. The obtained numerical results are then presented and discussed in Ÿ 5.

2

Mathematical model

Consider a layer of water initially uniform in depth. a disturbance moving steadily with speed

U.

Suppose we introduce at the bottom oor

Then, depending on the magnitude of

U,

dierent

responses of this uid-mechanical system can be obtained. One, particularly intriguing, arises when the ow is transcritical. As discovered by Wu & Wu [20], near resonance, a long-wave phenomenon takes place and solitary waves, propagating ahead of the disturbance, are periodically generated. Although originally discovered numerically based on the generalized Boussinesq equations, it is desirable to have a simpler model to carry out analytical and numerical investigations of the phenomenon.

One such model, known as forced Korteweg-de Vries equation, was proposed by

Wu [19] and it consists on a one-directional, weakly non-linear and weakly dispersive long wave model that is able to retain much of the physics of the problem. The model governs the free-surface elevation

ζ(x, t)

and reads in dimensional variables as follows:

2 ∂ζ −2 c0 ∂t



U −1 c0



3 ∂ζ h2 ∂ 3 ζ db ∂ζ + ζ + 0 3+ = 0. ∂x h0 ∂x 3 ∂x dx

(5)

U is assumed positive and the equation holds for right-going waves. Here t (> 0) x = X − U t is the horizontal space variable expressed in the body frame of reference, z = −h0 + b(x) indicates a topography √ b(x) moving over the bottom oor at depth h0 , and c0 is gh0 , with g the gravitational acceleration. linear long wave speed dened as c0 := For our purposes, is the time,

As shown by [4, 11, 22], the model has a surprisingly wide range of validity, and a remarkable agreement with experiments and numerics for the fully non-linear Euler equations is achieved for

0.9 < U/c0 < 1.1

and

kbkL∞ (R,R) /h0 < 0.15.

Moreover, if the translating bottom is suciently

regular and the motion is assumed to start impulsively from rest, then it can be shown that the initial value problem for (5) is well-posed:

Lemma 2.1. Let

T > 0, U > 0, and h0 > 0 be given xed constants, and b a suciently regular bottom, such that b ∈ H ∞ (R, R) := ∩s>0 H s (R, R). Then, there exists a unique solution ζ ∈ C ∞ (0, T ; H ∞ (R, R)) of (5) with zero initial data. The proof consists in applying [2, Theorem 1.1] and using the fKdV equation (5) to gain,

recursively, regularity in time.

In Ÿ 3.1, this well-posedness result will be extended for lower

regular forcing functions. Near resonance, the generic behaviour of the free-surface elevation predicted by the fKdV equation with zero initial data can be depicted as in Figure 1. According to Wu [19], ve dierent

3

h0 far upstream and downstream on ] − ∞, x0 ] t [x2 , +∞[; a cnoidal-like1 wavetrain downstream on [x0 , x1 ]; an almost uniform state of depth h1 (< h0 ) behind the disturbance on [x1 , −K]; periodic succession of upstream advancing solitary waves on [K, x2 ]. regions can be distinguished: some uniform states of depth

0.4 Free−surface elevation at the final time Profile of the solitary wave generated Bottom topography of a cosine shape

0.3

Vertical spatial coordinate

0.2 0.1 0 −0.1 −0.2 −0.3

h0

h1

h0

−0.4 −0.5

Figure 1:

x0 −40

x1 −K 0 K −30 −20 −10 Horizontal spatial coordinate in the bottom frame

10 x2

Illustration of the generic behaviour of the solutions to the fKdV equation (5) near

resonance. The translating body at the oor bottom has a cosine shape and moves at transcritical speed,

i.e. U/c0 ≈ 1.

For comparison, the prole of a solitary-wave solution

ζKdV (x, t)

(dashed

line) in (7) is superposed to the rst upstream-progressing wave. For simplicity, here and hereafter, we shall limit ourselves to the critical case of

U = c0 .

In this

situation, the fKdV equation reduces to

2 ∂ζ 3 ∂ζ h2 ∂ 3 ζ db + ζ + 0 3+ = 0, c0 ∂t h0 ∂x 3 ∂x dx and can be cast, by a simple change of variables, into the

canonical

(6)

form given in (4), better suited

to the theoretical study of the optimization problem (2):

Lemma 2.2. Set x˜ :=

c0 45 x and t˜ := 2h 3 t, and 0 3 and ˜b(˜x) := h10 3 5 b(x). Then u(˜x, t˜) is a solution of ζ(x, t) satises (6) with zero initial data. 1 35 h0 3

consider the functions u(˜x, t˜) := h10 3 5 ζ(x, t) (4) with forcing function ˜ b(˜ x) if and only if 4

Remark that the classical KdV equation can be recovered from (6) when the forcing disturbance vanishes,

i.e. b = 0, in which case we have the well-known family of solitary-wave solutions "s ζKdV (x, t) = a sech

2

3a 4h30

 # ac0 x − x0 − t . 2h0

(7)

Fig. 1 illustrates how well the runaway solitons of the fKdV equation can be captured by the classical soliton prole (7) for the KdV equation. If the motion starts impulsively from rest, we observe that it takes a certain time

Tg

until the rst upstream-progressing solitary wave is fully

formed and breaks away from the disturbance. As time evolves, multiple (almost identical) copies

a is the amplitude of such upstreamTs can be estimated accordingly to Wu [19]

of the leading wave will be produced in a periodic way. If progressing solitary waves, then the period of generation by

64h0 Ts = c0



h0 3a

 32 .

(8)

1 Here, the term cnoidal simply refers to the prole of the periodic travelling-wave solutions to the KdV equation (see [7]). 4

The formula reveals a somewhat counterintuitive feature of the system: the higher the amplitude

Ts .

of the upstream waves, the shorter the period of generation what related and numerical evidence seem to suggest that

e.g.

(negative) forcing functions (see

Tg

The times

Tg

Ts are someTs for positive

and

is less (greater) than

[19]).

The presence of a non-constant forcing term in (6) has the eect of destroying the invariance

1 2 2 R ζ dx is not a constant and integrating the resulting equation over the real

with respect to spatial translations, and so the excess energy integral of motion.

ζ

Indeed, by multiplying (6) by

R

line, one concludes that

d dt where

Dw

is given by

Dw (t) :=

R

Z

ζ 2 dx = c0 Dw ,

R

b(x) ∂x ζ(x, t) dx R

and can be physically interpreted as the drag,

or the resistance due to unsteady wave making. As a consequence, we obtain

kζ (•, t) k2L2 (R,R) = c0

∀t > 0,

t

Z

Dw (s) ds.

(9)

0

t 7→ kζ(•, t)k2L2 (R,R) can be reasonably approximated by t 7→ c0 D w t, with D w the average value of Dw over the wave period Ts . By mass 3 and energy considerations, such value can also be estimated as D w = a /(4h0 ) (see [19]). Hence, provided the elapsed time T exceeds the time Tg necessary to allow the generation of the rst Furthermore, it can be shown numerically that the function

upstream solitary wave, the following approximation holds:

Z

T

Z

ζ 2 (x, t) dx dt ≈ c0 Dw

∀T > Tg , 0

3

R

c0 T 2 a3 T2 = . 2 8h0

(10)

Optimization problem

For the purposes of our work, the ow is assumed to be critical and properties of the corresponding fKdV solutions are exploited to formulate the optimization problem. For convenience, the problem is stated in terms of non-dimensional variables as in Ÿ 1.

In particular, the parameters in (1)

must be chosen appropriately to remain within the range of validity of the fKdV model: we set

K := Kdim /h0 = O(1)

and

M = Mdim /h0 = O(10−1 ).

Moreover, for obvious manufacturing

reasons, only non-negative forcing functions with compact support will be admitted. An ecient wave maker is one capable of generating a wave of high amplitude.

Since the

dynamics is well described by the fKdV model, we can rely on the description given in Fig. 1 to assert that from the moment the wave maker starts moving steadily at critical speed, an upstreamprogressing solitary wave will be generated after a certain time

Tg .

This wave will suddenly unlatch

from the moving bottom and will propagate throughout large distances without altering its form. It depends on one single parameter (its amplitude relative to the bottom (

cf.

a)

and moves with an excess speed

c = ac0 /2h0

(7) in dimensional variables). From a practical point of view, once the

leading wave is fully formed, the wave maker could then be stopped, without that same wave being aected. In the surng context, this elevation wave would be the wave of interest, and the one that we wish to maximize. In other words, we would like to solve

sup

a(b).

b∈Cc∞ (R,[0,+∞[) supp b⊆[−K,K] Moreover, we can use (10) to justify that maximizing functional

b 7→ F (b)

Z

T

Z

sup b∈Cc∞ (R,[0,+∞[) supp b⊆[−K,K] where

ub

b 7→ a(b)

0

u2b (x, t) dx dt,

T > 0

is discussed below.

(11)

R

is the solution corresponding to the forcing function

a nal time

is equivalent to maximize the

given in (3). Alternatively, we can then address the problem

b

in (4), and where the choice of

The formulation (11), which is a smooth unconstrained

version of (2), also allows us to carry out the numerical procedure developed in Ÿ 4.1, since the employed gradient method relies on the directional derivative of the functional (3). Furthermore,

5

the

L2 -setting

is considered for the set (1) of admissible bottoms, ensuring the numerical stability

of the optimization algorithm.

In particular, an

imposed, otherwise the supremum is not achieved,

L2 -norm

cf.

We now comment on the choice of a nal time asymptotic behaviour of (2) as

T → +∞.

constraint on the forcing functions is

Appendix

T.

B.

One could be tempted to examine the

Indeed, for problems involving conservation laws and

propagation phenomena, the introduction of a nite time

T

(and a backward adjoint system)

can potentially create undesirable articial and numerical phenomena. Consequently, it would be more convenient to replace a time-dependant optimal control problem by the time-independent one associated with its asymptotic behaviour. The issue is important and still open, apart from some recent work of Trélat & Zuazua [17], where in some cases, they estimate the error made on such approximation. However, in the considered setting, we have some good reasons not do so. Heuristically, from (10), regardless the bottom considered, the integral over time of the energy should go to innity as

T → +∞.

In addition, the dynamics of our system is completely determined

in advance. As described in Ÿ2, once the time

Tg

is reached and a rst upstream-progressing solitary

wave is produced, copies of such wave are generated every interval of time of length

Ts , given by (8).

All of these propagate ahead of the translating bottom and remain permanent in form, regardless whether or not the bottom keeps its procession. motivates the picking of a nite time in a nite pool, so that the nal time

This periodic pattern in time of the system

T . Lastly, in real applications, the phenomenon takes place T must also be nite to ensure that some waves, potentially

reected from the pool ends, do not aect our system. To choose of bottoms

b

T,

k

we proceed as follows.

b0

Consider an educated initial guess

converging to an optimal one

b

opt

and a sequence

thanks to an optimization procedure ensuring

that the amplitudes of the corresponding runaway waves are not decreasing. In particular, we

a(b0 ) 6 a(bk ) and by (8), we get Ts (b0 ) > Ts (bk ). Recalling that Ts > Tg , we deduce that Ts (b ) > Tg (bk ) for any k ∈ N, and similarly Ts (b0 ) > Tg (bopt ). Note that Ts (b0 ) is explicit from 0 (8) and a measure of a(b ) given by any simulation. Hence, any choice of T greater than Ts (b0 )

have

0

is appropriate because it guarantees that the rst upstream-progressing solitary wave depicted in Fig. 1 is always fully formed at time

T

throughout the iterative scheme.

To prove Theorem 1.1, a number of steps will be introduced in this section. Following closely Tsugawa [18], we start by establishing in Ÿ 3.1 the time global well-posedness of (4) and proving that the functional explicit

a priori

F

proposed in (3) is well-dened. We then proceed by providing in Ÿ 3.2 some

estimates that allow us to prove the existence of an optimal bottom in Ÿ 3.3.

1 2 -Hölder continuity and Fréchet dierentiability, giving us access to the gradient of (3) via the adjoint formulation of (4). Finally, in Ÿ 3.4, other properties of (3) are highlighted such as its

3.1

Global well-posedness of the fKdV equation

We start by recalling the following result of Tsugawa [18, Theorem 1.2]: the initial value problem for the equation

s ∈] −

3 4, σ

+ 3]

∂t u + u∂x u + ∂xxx u = f (x) ∈ H σ (R, R)

with

σ > − 23

and

u(•, 0) ∈ H s (R, R),

is globally well-posed in time. For our particular case, the following result holds:

Proposition 3.1. Let

and b ∈ L2 (R, R). Then, the initial-value problem (4) is well posed in the sense that it has a unique global solution ub ∈ C 0 (0, T ; H 2 (R, R)). Moreover, given any other pair (˜b, u˜b ) formed by a forcing function ˜b ∈ L2 (R, R) and its corresponding solution u˜b ∈ C 0 (0, T ; H 2 (R, R)), we have: T > 0

sup ku˜b (•, t) − ub (•, t)k2L2 (R,R) 6 4CT eCT k˜b − bkL2 (R,R) ,

t∈[0,T ]

where C := max(kub kC 0 (0,T ;H 2 (R,R)) , ku˜b kC 0 (0,T ;H 2 (R,R)) ). In particular, the functional F (b) in is a well-dened application from L2 (R, R) into R. Proof.

T > 0 is xed db σ = −1, f = − dx , and Assume

b ∈ L2 (R, R) is given. initial data u0 ≡ 0, there

and

(3)

As a particular case of [18, Theorem 1.2],

ub ∈ C 0 (0, T ; H 2 (R, R)) to the initial-value problem (4). Consider any other bottom ˜ b ∈ L2 (R, R), with corresponding 0 2 ˜ solution u˜ b ∈ C (0, T ; H (R, R)), then set δb = b − b and δu = u˜ b − ub . Clearly, one has:  " # 2  ∂ (δu) ∂ 2 (δu) d (δb)  ∂ (δu) + + ub δu + =− ∈ H −1 (R, R), x ∈ R, t ∈ [0, T ], (12) ∂t ∂x 2 ∂x2 dx   δu(x, 0) = 0, x ∈ R. with

6

exists a solution

Although the partial derivatives in (12) have to be handled with care, since they are understood in distributional sense, we can still apply the integration-by-parts formula stated in [13, Lemma 7.3]

H 1 (R, R) ⊂ L2 (R, R) ⊂ H −1 (R, R) and the fact that we have δu ∈ {w ∈ L (0, T ; H (R, R)), ∂t w ∈ L2 (0, T ; H −1 (R, R))}. For any t ∈ [0, T ], we thus obtain: Z t 2 kδu (•, t) kL2 (R,R) = kδu(•, 0)kL2 (R,R) + 2 h∂t (δu) | δuiH −1 (R,R),H 1 (R,R) (•, s) ds.

by considering the Gelfand triple

2

1

0

2

2 2 h∂t (δu) | δuiH −1 (R,R),H 1 (R,R) = h (δu) 2 +ub δu+∂xx (δu)+δb | ∂x (δu)iL (R,R),L (R,R) δu(•, 0) = 0, from which it follows for any t ∈ [0, T ]: Z tZ Z tZ ∂(δu) 2 ∂ub δb kδu (•, t) k2L2 (R,R) = 2 dx ds − (δu) dx ds. ∂x ∂x 0 R 0 R

Using (12), we get and

We may then write, by introducing the constant

C := max(kub kC 0 (0,T ;H 2 (R,R)) , ku˜b kC 0 (0,T ;H 2 (R,R)) ),

which is nite:

∀t ∈ [0, T ], kδu (•, t) k2L2 (R,R) 6 4CT kδbkL2 (R,R) + C

Z

t

kδu (•, s) k2L2 (R,R) ds.

0

t ∈ [0, T ] 7→ kδu(•, t)kL2 (R,R) ∈ R

Since

is a continuous function [13, Lemma 7.3], we can apply

Grönwall's Lemma, which yields

ku˜b (•, t) − ub (•, t)k2L2 (R,R) 6 4CT eCt k˜b − bkL2 (R,R) .

∀t ∈ [0, T ],

In particular, the uniqueness of solution to the initial-value problem (4) follows and the functional

F : b 7→ F (b) 3.2

given by (3) is a well dened application from

Some explicit

Proposition 3.2. Let T

a priori

L2 (R, R)

into

R.

estimates

and b ∈ H ∞ (R, R) := ∩s>0 H s (R, R). Then, there exist polynomials P0 , P1 , P2 in two variables with (non-negative) constant coecients, such that following estimates hold for the unique solution ub ∈ C ∞ (0, T ; H ∞ (R, R)) to initial-value problem (4):
(i) sup kub (•, t) kL2 (R,R) 6 P0 T, kbkL2 (R,R) , >0

t∈[0,T ]

sup k∂x ub (•, t) kL2 (R,R) 6 P1 T, kbkL2 (R,R)

(ii)

t∈[0,T ]

sup k∂xx ub (•, t) kL2 (R,R) 6 e

(iii)




,

  T 1+ 13 kbk2L2 (R,R)

P2 T, kbkL2 (R,R)

t∈[0,T ]

Proof.




.

b ∈ H ∞ (R, R) := ∩s>0 H s (R, R). Lemmas 2.1 and 2.2 imply the existence ∞ ∞ 2 of a unique smooth solution ub ∈ C (0, T ; H (R, R)) of (4). Since b ∈ L (R, R), we may apply db [18, Proposition 3.1] with nal time T + 1(> 1), σ = −1, f = − dx , and homogeneous initial data u0 ≡ 0, to establish the inequality:   3 sup kub (•, t)kL2 (R,R) 6 sup kub (•, t)kL2 (R,R) 6 C 1 + (1 + T ) k∂x bk3H −1 (R,R) , Let

T >0

and

t∈[0,T ]

t∈[0,T +1]

C , which does not P0 (x, y) := C(1 + (1 + x)3 y 3 ), using here

T , b, or ub . This proves assertion (i) k∂x bkH −1 (R,R) 6 kbkL2 (R,R) . We now

for some positive constant

depend on

with

the fact that

exploit the Hamiltonian structure of equation (4) (see [3]). Although energy is not conserved, as already pointed out, an extra conserved quantity is available for the fKdV equation, which is in fact a Hamiltonian for the system. Let

H

be such Hamiltonian. Then,

Z " ∀t ∈ [0, T ],

H(t) := R

∂ub ∂x

2

# 1 3 − ub − 2b ub dx = 0. 3

(13)

kgk2L∞ (R,R) 6 2kgkL2 (R,R) k∂x gkL2 (R,R) 6 kgk2H 1 (R,R) √ √ √ 1 2 2 valid for any g ∈ H (R, R), with the well-known inequalities 2xy 6 x +y and x + y 6 x + y, valid for any x, y > 0, one can deduce from (13): r   7 2 k∂x ub (•, t) kL (R,R) 6 kub (•, t) kL2 (R,R) + kub (•, t) k2L2 (R,R) + kbkL2 (R,R) . 5 Combining the Cauchy-Schwarz inequality and

7

P1 (x, y) := 2[y + P0 (x, y) + P02 (x, y)]. Finally, the same method is used to determine P2 . For this purpose, we rst show for any t ∈ [0, T ]:  Z  Z d 5 2 5 2 2 2 (∂xx ub ) + 2b ∂xx ub − ub (∂x ub ) + b u2b + u4b dx = b I ∂x ub dx, (14) dt R 3 3 36 3 R

This proves the estimate

with

(ii)

I := ∂xx ub + 12 u2b + b.

of our Proposition 3.2 with

t.

Notice that both sides of (14) depend only on time

Denote by

G(t)

the left-hand side of the equation. Straightforward calculations lead to:

Z G(t)

10 5 4 5 2 2∂xt ub [−∂xxx ub − bx ] − ub ∂x ub ∂xt ub − ∂t ub (∂x ub ) + b ub ∂t ub + u3b ∂t ub {z } | 3 3 3 9 R

=

=∂t ub +ub ∂x ub

Z = R

  4 u2b 1 1 2 ub ∂t ub ∂xx ub + + b − ∂t ub (∂x ub ) − u3b ∂t ub 3 |{z} 2 3 |{z} 9 |{z} {z } =−Ix | =−Ix =−Ix

Z = R

2 ∂x ub I b. 3

:=I

t ∈ [0, T ], following the same strategy above yields ! kbk2L2 (R,R) Z t
62 1+ k∂xx ub (•, s)k2L2 (R,R) ds + P01 T, kbkL2 (R,R) , 3 0

Integrating equality (14) on

k∂xx ub (•, t)k2L2 (R,R)

[0, t]

for any

where we have set

P01 (x, y)

:=

   
P0 (x, y)2 P1 (x, y)2 2 2 2 P0 (x, y) + P1 (x, y) + 2x y 1 + 3 4 
y
5P1 (x, y) + 2 2y 2 + P0 (x, y)2 + 2P1 (x, y)2 + 2P0 (x, y)2 + P1 (x, y)2 6 3 
5P0 (x, y)2 2 2 P0 (x, y) + P1 (x, y) + . 36

Consequently, we apply Grönwall's Lemma to the last inequality above, from which the assertion

(iii)

follows by setting

P2 (x, y) := 1 + P01 (x, y),

and concluding the proof.

We emphasize that (13)(14) hold because, under the assumption that

b

and

ub

have enough

regularity, we can dierentiate under the integral sign and commute the order of the mixed partial derivatives. Since we cannot proceed similarly in the

L2 -case,

we use an approximating argument

instead, keeping in mind the quantitative estimate of Proposition 3.1.

Corollary 3.3. Let T

and b ∈ L2 (R, R). Then, the unique solution ub ∈ C 0 (0, T ; H 2 (R, R)) given in Proposition 3.1 satises the inequality: >0







kub kC 0 (0,T ;H 2 (R,R)) 6 P0 T, kbkL2 (R,R) + P1 T, kbkL2 (R,R) + e

  T 1+ 13 kbk2L2 (R,R)


P2 T, kbkL2 (R,R) ,

with P0 , P1 , and P2 the same as dened in Proposition 3.2. Proof. Let T > 0 and b ∈ L2 (R, R). First, according to Proposition 3.1, we can consider the unique solution

ub ∈ C 0 (0, T ; H 2 (R, R))

of (4). Moreover, by density, a sequence

with compact support is strongly converging to that the sequence for any

n ∈ N.

(ubn )n∈N

b

in

L2 (R, R).

(bn )n∈N

of smooth maps

In addition, Proposition 3.2 ensures

of associated smooth maps also satises (4) and the

a priori

estimates

We deduce from the quantitative estimate of Proposition 3.1 the strong convergence

(ubn )n∈N to ub in C 0 (0, T ; L2 (R, R)). In particular, we can correctly let n → +∞ in the inequality
(i) of Proposition 3.2 applied to (ubn , bn ) in order to get kub kC 0 (0,T ;L2 (R,R)) 6 P0 T, kbkL2 (R,R) . Then, let t ∈ [0, T ] xed. Since (bn )n∈N is bounded, the sequence (∂x ubn (•, t))n∈N is uniformly 1 1 bounded in H (R, R). Consequently, there exists a subsequence that weakly converges in H (R, R) to a certain map, which has to be ∂x ub (•, t) by considering the convergence in distributional sense. of

We emphasize the fact that here the subsequence depends on the time variable so it is denoted by

(∂x ubn(t) )n∈N .

Considering the lower-semicontinuity of the norm with respect to the weak

convergence, we obtain for any

t ∈ [0, T ]:

k∂x ub (•, t)kH 1 (R,R) 6 lim inf k∂x ubn(t) (•, t)kH 1 (R,R) 6 P1 T, kbkL2 (R,R) n∈N





T (1+ 1 kbk2 2 ) 3 L (R,R) . + P2 T, kbkL2 (R,R) e

Hence, the expected inequality of Corollary 3.3 holds with

8

(b, ub ),

concluding the proof.

3.3

Existence of an optimal bottom

To prove the existence of an optimal bottom, as in Theorem 1.1, a suitable topology must be introduced on the set of admissible bottoms ensuring:

•

the compactness of any maximizing sequence associated with the supremum in (2);

•

the closedness of the set of admissible bottoms (1);

•

the (upper-semi)continuity of the energy functional (3).

L2 (R, R) into R, thus adopting the L -weak topology would seem rather natural. In addition, the compactness of B follows from the 2 fact that positivity, uniform upper bound and support are preserved by the L -weak convergence.

From Proposition 3.1, the functional (3) is a well-dened map from

2

Lemma 3.4. Let

K > 0 and M > 0. Then, the set of admissible bottoms compact for the weak topology of L2 (R, R).

(1)

is sequentially

However, the continuity of the functional (3) is not straightforward. Indeed, we are now dealing with the weak convergence of bottoms and the results of Ÿ 3.2 such as the quantitative estimate of Proposition 3.1 are useless since they involve the strong topology of

L2 (R, R).

Also, usual

compactness arguments such as the Aubin-Lions-Simon Lemma [14, Section 8 Corollary 4] cannot be directly applied here because the embedding

H 2 (R, R) ⊂ H 1 (R, R) is not compact.

Nevertheless,

we can recover some continuity by using the fact the supports of the admissible bottoms (1) are all contained in a xed compact set.

Proposition 3.5. Let T

> 0, K > 0

and b ∈ L2 (R, R) with support included in [−K, K]. Consider any sequence (bn )n∈N of square-integrable maps with supports all included in [−K, K] converging weakly to b in L2 (R, R). Then, the sequence (ubn )n∈N converges strongly to ub in C 0 (0, T ; L2 (R, R)), where ub and ubn are the unique maps of C 0 (0, T ; H 2 (R, R)) associated to b and bn , respectively, as in Proposition 3.1 for any n ∈ N. Proof. The proof is very similar to the one of Proposition 3.1. Let T > 0, K > 0 and b ∈ L2 (R, R) [−K, K]. From Proposition 3.1, the initial-value problem (4) has a unique ub ∈ C 0 (0, T ; H 2 (R, R)). Consider now any sequence (bn )n∈N of square-integrable maps 2 with supports all included in [−K, K] that is weakly converging to b in L (R, R). In particular, such a sequence is bounded so Proposition 3.1 and Corollary 3.3 ensures that the sequence (ubn )n∈N of associated maps satises (4) and the a priori estimate for any n ∈ N. We deduce that ub and (ubn )n∈N are uniformly bounded in C 0 (0, T ; H 2 (R, R)). First, consider the compact embeddings H 2 (] − K, K[, R) ⊂ H 1 (] − K, K[, R) ⊂ H −1 (] − K, K[, R). We can apply the Aubin-Lions-Simon with support included in

solution

Lemma [14, Section 8 Corollary 4] to obtain that the following embedding is also compact:



W := w ∈ L∞ 0, T ; H 2 (]−K, K[ , R) , ∂t w ∈ L∞ 0, T ; H −1 (]−K, K[ , R)
,→ C 0 0, T ; H 1 (]−K, K[ , R) . kubn kL∞ (0,T ;H 2 (]−K,K[,R)) +k∂t ubn kL∞ (0,T ;H −1 (]−K,K[,R)) W . We deduce that there exists a subsequence (ub 0 )n∈N that is strongly converging to uK n let n ∈ N. We introduce the quantities δb = bn0 − b and

From the foregoing and (4), we known that is uniformly bounded

i.e. (ubn )n∈N

is uniformly bounded in

uK ∈ C 0 (0, T ; H 1 (] − K, K[, R)) and 0 1 in C (0, T ; H (] − K, K[, R)). Then, δu = ubn0 − ub . One can check that

they satisfy the initial-value problem (12).

We emphasize

again the fact that the partial derivatives in (12) have to be handled with care since these are understood in a distributional sense. However, we can still apply the integration-by-parts formula

H 1 (R, R) ⊂ L2 (R, R) ⊂ H −1 (R, R) and the fact that δu ∈ {w ∈ L (0, T ; H (R, R)), ∂t w ∈ L (0, T ; H −1 (R, R))}. Following the same calculations than we did in the proof of Proposition 3.1, we get for any t ∈ [0, T ]: Z tZ Z tZ ∂(δu) 2 ∂ub 2 dx ds − (δu) dx ds. kδu (•, t) kL2 (R,R) = 2 δb ∂x ∂x 0 R 0 R

given in [13, Lemma 7.3] by considering the Gelfand triple

2

1

2

Finally, we use the Cauchy-Schwarz inequality and the fact that all the supports of the considered bottoms are included in

[−K, K].

kubn0 (•, t) − ub (•, t) k2L2 (R,R)

t ∈ [0, T ]:   T K ∂ubn0 ∂ub 2 (bn0 − b) (x) − (x, s) dx ds ∂x 0 −K Z ∂x t + kub kC 0 (0,T ;H 2 (R,R)) kubn0 (•, s) − ub (•, s) k2L2 (R,R) ds.

We thus get for any

Z

6

Z

0

9

Since

t ∈ [0, T ] 7→ kδu(•, t)kL2 (R,R) ∈ R

is a continuous function [13, Lemma 7.3], we can apply

Grönwall's Lemma and we obtain:

kubn0 − ub k2C 0 (0,T ;L2 (R,R)) 6 C

Z

T

Z

K

 (bn0 − b) (x)

0

−K

 ∂ubn0 ∂ub − (x, s) dx ds, ∂x ∂x

C := 2eT kub kC 0 (0,T ;H 2 (R,R)) . Hence, it remains to prove that the right-member of the above inequality converges to zero as n → +∞. For this purpose, we introduce the integrand RK Rn : t 7→ −K δb(x) ∂x (δu)(x, t) dx. Since bn0 converges weakly to b in L2 (] − K, K[, R) and ubn0 0 1 strongly to uK in C (0, T ; H (] − K, K[, R)), we get for any t ∈ [0, T ] that Rn (t) converges to zero. Moreover, the a priori estimate of Corollary 3.3 ensures that Rn (t) is uniformly bounded. Hence, RT Lebesgue's Dominated Convergence Theorem applies and |Rn (t)| dt → 0 as n → +∞. One 0 0 2 concludes from the last inequality that (ub 0 )n∈N strongly converges to ub in C (0, T ; L (R, R)). n where we have set

We also have proved the uniqueness of the limit for any other converging subsequence. Recalling that

(ubn )n∈N

is uniformly bounded, we deduce that the whole sequence converges to

Proof of Theorem 1.1.

ub .

Combining Lemma 3.4 and Proposition 3.5, it is possible to extract from

(bn0 )n∈N that is weakly converging in L2 (R, R) to a 2 certain b ∈ B , and such that ub 0 → ub∗ in C (0, T ; L (R, R)). Since we have n   ∗ |F (bn0 ) − F (b )| 6 T kubn0 − ub∗ kC 0 (0,T ;L2 (R,R)) sup kubk0 kC 0 (0,T ;L2 (R,R)) + kub∗ kC 0 (0,T ;L2 (R,R)) ,

any maximizing sequence of (2) a subsequence

∗

0

k∈N

(15) we obtain

F (b∗ ) = supb∈B F (b)

with

b∗ ∈ B

so the supremum is a maximum and problem (2) has

a global maximizer. To conclude the proof, it remains to show that such a maximizer saturates

L2 -constraint, which is proved as in Proposition B.2. Indeed, if it not the case, then choose 2 of Lemma B.1 is admissible. One can check that θ ∈]1, (M/kbopt kL2 (R,R) ) 7 ] and the bottom bopt θ opt opt opt 3 F (bθ ) > F (b ) and the optimality of b yields to ubopt = 0 on [T, θ T ]. Considering now (4), opt opt we obtain ∂x b = 0 so ubopt = 0 also on [0, T ] and F (b ) = 0, which is a contradiction. the

3.4

Other useful properties

In this section, two properties associated with the optimization problem (2) are highlighted. The rst one guarantees that the energy functional

F : B → R

while the second one concerns the Fréchet dierentiability of

1 2 -Hölder continuous, 1 with respect to H -perturbations.

given in (3) is

F

In particular, we obtain an explicit expression for the gradient of

F

by introducing the adjoint

formulation of the initial-value problem (4).

3.4.1 The 21 -Hölder continuity of the functional K > 0, we have established the (sequential) continuity of the non-linear N : b ∈ L2 (] − K, K[, R) 7→ ub ∈ C 0 (0, T ; L2 (R, R)) for the L2 -weak topology. Here, we rst 2 0 1 2 establish that N : L (R, R) 7→ C (0, T ; H (R, R)) is continuous for the L -strong topology. Then, 2 by restricting N and F to any ball of L (R, R), we obtain their Hölder continuity.

In Ÿ 3.3, for any given xed map

Proposition 3.6. Let T

> 0 and b ∈ L2 (R, R). Consider any sequence (bn )n∈N of square-integrable maps strongly converging to b in L2 (R, R). Then, the sequence (ubn )n∈N of their associated solutions given in Proposition 3.1 strongly converges to ub in C 0 (0, T ; H 1 (R, R)), where ub is the unique solution of Proposition 3.1 associated with b.

Proof.

T > 0 and b ∈ L2 (R, R). From Proposition 3.1, we can consider the unique solution ub ∈ C (0, T ; H 2 (R, R)) satisfying (4). First, we treat the smooth case. Let (bn )n∈N be a sequence ∞ 2 of maps in H (R, R) that is strongly converging to b for the L -norm. In particular, this sequence 2 is uniformly bounded in L (R, R). Applying Proposition 3.2, there exists a sequence (ubn )n∈N Let 0

of associated smooth maps satisfying (4) and the

a priori

estimates, from which we deduce that

C 0 (0, T ; H 2 (R, R)) by a constant denoted C > 0. Then, applying Proposition 3.1 with b and ˜ b = bn for any n ∈ N, we obtain that (ubn )n∈N strongly converges to ub in C 0 (0, T ; L2 (R, R)). We now prove that in fact the convergence occurs in C 0 (0, T ; H 1 (R, R)). Let (m, n) ∈ N × N. We set δu := ubn − ubm and δb := bn − bm then establish that (ubk )k∈N is a uniform Cauchy sequence by relating ∂x (δu) to δb and δu. Since both (bn , ubn ) and (bm , ubm )

(ubn )n∈N

is uniformly bounded in

10

ub by ubm ). ∂t (ubm ) = −∂x I and ∂t (δu) = −∂x J , 1 1 2 2 where we set I := ∂xx ubm + (ubm ) + bm and J := ∂xx (δu) + (δu) + δb + ubm δu. We have: 2 2 #  2 Z " Z Z 3 2 d (δu) (δu) 1 ∂ (δu) 1 2 + ubm − + δb δu dx = 2 ∂t (δu) J + ∂t (bm ) (δu) dt R 6 2 2 ∂x 2 } | {z } | R {z R = − ∂x (J 2 ) = 0 = − I δu ∂x (δu) ! Z u2bm ∂ (δu) ∂ 2 ubm + = − δu + bm dx. ∂x ∂x2 2 R satisfy (4), we get that

(δu, δb)

is a smooth solution to (12) (where we have replaced

We use the conservative structure of (4) and (12) by writing

Proceeding as below (12) (but here the functions are regular), we obtain:

5C kubn (•, t) − ubm (•, t)k2L2 (R,R) 3  + 4Ckbn − bm kL2 (R,R)  C2 + sup kbk kL2 (R,R) kubn (•, t) − ubm (•, t)kL2 (R,R) . + 2T C C + 2 k∈N

∀t ∈ [0, T ], k∂x ubn (•, t) − ∂x ubm (•, t) k2L2 (R,R) 6

(16)

t ∈ [0, T ] 7→ ∂x (ubn )(•, t) ∈ L2 (R, R) is a uniform Cauchy sequence. 0 2 converging to a certain map in C (0, T ; L (R, R)), which has to be ∂x ub by

We deduce from (16) that It is thus strongly

considering the convergence in the sense of distributions. Finally, we treat the non-regular case

(bn )n∈N be any sequence of maps in L2 (R, R) that is strongly k converging to b. By density, for any n ∈ N, there exists a sequence (bn )k∈N of smooth maps with 2 compact support that is strongly converging to bn in L (R, R). From the foregoing, we deduce that there exists kn ∈ N such that ku kn − ubn kC 0 (0,T ;H 1 (R,R)) < ε. Moreover, one can check that bn (bknn )n∈N strongly converges to b in L2 (R, R). Again, from the foregoing, there exists N ∈ N such that for any integer n > N , we have ku kn − ub kC 0 (0,T ;H 1 (R,R)) < ε. We deduce that: bn by approximations. Let

ε>0

and

∀n > N, kubn − ub kC 0 (0,T ;H 1 (R,R)) 6 kubn − ubknn kC 0 (0,T ;H 1 (R,R)) + kubknn − ub kC 0 (0,T ;H 1 (R,R)) < 2ε. To conclude the proof of Proposition 3.6,

(ubn )n∈N

converges to

ub

in

C 0 (0, T ; H 1 (R, R)).

Corollary 3.7. Let M

> 0 and T > 0. We set BM := {b ∈ L2 (R, R), kbkL2 (R,R) 6 M }. Then, there exists a constant C(T, M ) > 0 depending only on T and M such that: q     ˜ 0 2 max ku − u k , |F (b) − F ( b)| 6 C(T, M ) kb − ˜bkL2 (R,R)  ˜ b  b C (0,T ;L (R,R)) ∀(b, ˜b) ∈ BM ×BM , q   4  ku − u k 0 kb − ˜bkL2 (R,R) . ˜ C (0,T ;H 1 (R,R)) 6 C(T, M ) b b

In particular, the energy functional F : B → R given in Proof.

(3)

is 12 -Hölder continuous.

BM as in the statement. First, combining the a priori estimate of (b, ˜b) ∈ BM × BM , we deduce that the constant C > 0 appearing in the quantitative estimate of Proposition 3.1 can be bounded by one that only depends on T 1 0 2 and M . Hence, the non-linear map N : b ∈ BM 7→ ub ∈ C (0, T ; L (R, R)) is -Hölder continuous. 2 Then, similarly arguments applied to (15) with b and bn0 = ˜ b also yields to the same result for the map F : BM 7→ R. Finally, there exists two sequences (bn )n∈N and (˜ bn )n∈N of smooth maps with compact support respectively converging to b and ˜ b strongly in L2 (R, R). Proposition 3.6 ensures that the associated smooth maps (ubn )n∈N and (u˜ bn )n∈N respectively converges to ub and 0 1 u˜b in C (0, T ; H (R, R)). We can now proceed as in the proof of Proposition 3.6 so (16) holds with bn and bm = ˜ bn . By letting n → +∞ in this inequality, we deduce from the foregoing that N : b ∈ BM 7→ ub ∈ C 0 (0, T ; H 1 (R, R)) is 41 -Hölder continuous, concluding the proof. Let

M > 0, T > 0,

and

Corollary 3.3 with the fact that

3.4.2 The Fréchet dierentiability of the functional First, we prove the existence of a unique solution to the adjoint formulation of the initial-value problem (4), which is then used to explicitly evaluate the Fréchet derivative of the functional (3) with respect to

H 1 -perturbations,

giving us access to the gradient of (3).

11

Proposition 3.8. Let T

and b ∈ L2 (R, R). The unique solution ub ∈ C 0 (0, T ; H 2 (R, R)) of Proposition 3.1 satisfying (4) is considered. Then, the following nal-value problem (understood as a distributional equality) is well-posed: >0

 ∂v ∂v ∂3v   + 2ub = 0 ∈ H −1 (R, R) + ub + ∂t ∂x ∂x3   v(x, T ) = 0

x ∈ R, t ∈ [0, T ] (17)

x ∈ R,

in the sense that it has a unique global solution vb ∈ C 0 (0, T ; H 2 (R, R)). Proof.

Formerly speaking, if

vb

is a solution of (17), then the equation satised by

∂x vb

is already

studied in [2]. However, we need to specify a bit the existence result because it is stated in terms

Ys,β (see [2, Ÿ2] for details). First, we apply [18, Proposition 2.1] σ = −1, f = −∂x b, u0 ≡ 0, s = σ + 3 = 2 and β = ε + 21 , where ε > 0 is chosen small

of the so-called Bourgain space with

enough. This local existence result combined with standard global arguments [2, Proposition 5.1]

ub of (4) is the [0, T ]-restriction of a map Ub ∈ Y2,ε+1/2 . (ξ, τ ) := (−x, T − t) to transform (17) into an initial-value U (ξ, τ ) := Ub (−x, T − t) and we still have U ∈ Y2,ε+1/2 ⊂ Y1,ε+1/2 but we also

establishes that the unique solution

Then, we introduce new variables problem. We set

get ∂ξ U ∈ Y1,ε+1/2 [2, above Theorem 5.5]. We are now in position to apply [2, Theorem 2.6] with s = 1, β = ε + 21 , and f = 2∂ξ U . We deduce that there exists a unique solution W ∈ Y1,ε+1/2 satisfying W (•, 0) = 0 and ∂τ W + ∂ξ (U W ) + ∂ξξξ W = 2∂ξ U on R × [0, T ]. Finally, it remains to get back to (17). For this purpose, we consider the [0, T ]-restrictions u ∈ C 0 (0, T ; H 2 (R, R)) and w ∈ C 0 (0, T ; H 1 (R, R)) of the maps U and W [2, Lemma 2.3]. Using the 1,1 equations they satisfy, we obtain u(2−w) ∈ W (0, T ; H −2 (R, R)). From the standard linear semi0 1 group theory [2, Section 1 ŸIII], there exists a unique function v ∈ C (0, T ; H (R, R)) satisfying v(•, 0) = 0 and the Airy equation ∂τ v + ∂ξξξ v = u(2 − w) on R × [0, T ]. Looking at the equation 0 2 satised by w − ∂x v , we deduce that ∂x v = w . In particular, we obtain v ∈ C (0, T ; H (R, R) and getting back to the original variables (x, t) := (−ξ, T − τ ), one can check that the map vb (x, t) := v(ξ, τ ) is a global solution of (17) in C 0 (0, T ; H 2 (R, R)). 0 2 At last, we prove such a solution is unique. Consider two maps v1 and v2 of C (0, T ; H (R, R)) solving (17) and introduce the quantity δv := v1 −v2 . From the linearity of (17), one can check that δv satises δv(•, T ) = 0 and ∂t (δv) + ub ∂x (δv) + ∂xxx (δv) = 0. This last equality is understood in distributional sense but we can still apply the integration-by-parts formula [13, Lemma 7.3]

H 1 (R, R) ⊂ L2 (R, R) ⊂ H −1 (R, R) since we clearly have that δv ∈ {w ∈ L (0, T ; H (R, R)), ∂t w ∈ L2 (0, T ; H −1 (R, R))}. Proceeding as below (12), we get:

with respect to the Gelfand triple

2

1

∀t ∈ [0, T ], kδv(•, t)k2L2 (R,R) 6 k∂x ub kC 0 (0,T ;H 1 (R,R))

Z

T

kδv(•, s)k2L2 (R,R)) ds.

t

t ∈ [0, T ] 7→ kδv(•, t)kL2 (R,R) ∈ R [13, Lemma 7.3] and δv ≡ 0 on [0, T ] × R i.e. v1 = v2 . To conclude the proof, there exists a vb ∈ C 0 (0, T ; H 2 (R, R)) satisfying (17).

It follows from the continuity of the map Grönwall's Lemma that unique global solution

Proposition 3.9. Let T Then, for

> 0 and F : L2 (R, R) → R be well-dened by (3) (cf. Proposition any b ∈ L (R, R) and any h ∈ H 1 (R, R), the following expansion holds: ! Z Z T   ∂vb F (b + h) = F (b) + h(x) (x, t) dt dx + O k∂x hk2L2 (R,R) , ∂x R 0 2

3.1).

where vb ∈ C 0 (0, T ; H 2 (R, R)) is the unique global solution of (17) introduced in Proposition 3.8. In particular, the associated map Fb : h ∈ H 1 (R, R) 7→ F (b + h) ∈ R is Fréchet dierentiable at the origin for any bottom b ∈ L2 (R, R) and the gradient of F at b is given by Z ∂b F : x ∈ R 7−→ ∂b F (x) := 0

T

∂vb (x, t) dt, ∂x

(18)

which is a well-dened function of H 1 (R, R). Henceworth, the map b ∈ L2 (R, R) 7→ ∂b F ∈ H 1 (R, R) is referred to as the shape gradient of F .

12

Proof.

Let

T > 0, b ∈ L2 (R, R),

and

h ∈ H 1 (R, R).

First, we establish a quantitative estimate

h has a stronger regularity. ub and ub+h in C 0 (0, T ; H 2 (R, R)) again the quantities δu := ub+h − ub

similar to the one given in Proposition 3.1, using here the fact that From Proposition 3.1, there exists two associated global solutions

(b, ub ) and (b + h, ub+h ) satisfy (4). Introducing δb := (b + h) − b = h, one can check that (δb, δu) satises

such that and

(12), which has to be understood

in a distributional sense. Still, we can apply the integration-by-parts formula [13, Lemma 7.3] by

H 1 (R, R) ⊂ L2 (R, R) ⊂ H −1 (R, R) combined with the fact that δu ∈ {w ∈ L (0, T ; H (R, R)), ∂t w ∈ L2 (0, T ; H −1 (R, R))}. Proceeding as below (12), we get: Z tZ Z tZ ∂(δb) 2 ∂ub dx ds − 2 δu dx ds. ∀t ∈ [0, T ], kδu(•, t)k2L2 (R,R) = − (δu) ∂x ∂x 0 R 0 R considering the Gelfand triple

2

1

Note that the last relation holds only because we have assumed

δb = h ∈ H 1 (R, R).

Using the

t ∈ [0, T ]: Z
t kδu(•, s)k2L2 (R,R) ds + T k∂x hk2L2 (R,R) . 6 1 + k∂x ub kC 0 (0,T ;H 1 (R,R))

Cauchy-Schwarz inequality, we obtain for any

kδu(•, t)k2L2 (R,R)

0 We can now apply Grönwall's Lemma to the map

t ∈ [0, T ] 7→ kδu(•, t)kL2 (R,R) ∈ R,

which is

continuous [13, Lemma 7.3], and it comes:

√ T kub+h − ub kC 0 (0,T ;L2 (R,R)) 6 k∂x hkL2 (R,R) T e 2 (1+k∂x ub kC 0 (0,T ;H 1 (R,R)) ) . Then, Proposition 3.8 ensures that (17) has a unique global solution

0

(19)

2

vb ∈ C (0, T ; H (R, R)).

Hence, we can correctly compute again the integration-by-parts formula of [13, Lemma 7.3] by

H 1 (R, R) ⊂ L2 (R, R) ⊂ H −1 (R, R) combined with the fact that (δu, vb ) ∈ {w ∈ L (0, T ; H (R, R)), ∂t w ∈ L2 (0, T ; H −1 (R, R))}2 . Since vb (•, T ) = δu(•, 0) = 0, considering the Gelfand triple

2

1

we have:

Z

T

Z h∂t (δu) | vb iH −1 (R,R),H 1 (R,R) (•, t) dt +

0 = 0

T

0

h∂t vb | δuiH −1 (R,R),H 1 (R,R) (•, t) dt

Proceeding as below (12), one may obtain from the previous relation:

Z

T

Z

Z

2 0 Recalling that

T

2

Z

ub δu dx dt =

δb = h

0

R

R

(δu) ∂vb dx dt + 2 ∂x

Z

T

Z δb

0

R

∂vb dx dt. ∂x

and introducing the map (3), we deduce from the last relation:

Z RF (h) := F (b + h) − F (b) −

Z

T

h(x) R

0

!   Z TZ 1 ∂vb ∂vb (x, t) dt dx = dx dt. (δu)2 1 + ∂x 2 ∂x 0 R

vb ∈ C 0 (0, T ; H 2 (R, R)), we establish   1 2 |RF (h)| 6 T kδukC 0 (0,T ;L2 (R,R)) 1 + k∂x vb kC 0 (0,T ;H 1 (R,R)) , 2

Consequently, using the fact that

RF (h) = O(k∂x hk2L2 (R,R) ). R Since h ∈ H (R, R) 7→ h(x)[ [0,T ] ∂x vb (x, t) dt] dx ∈ R is a continuous linear form, the uniqueness R 1 of the dierential ensures that the functional Fb : h ∈ H (R, R) 7→ F (b + h) ∈ R is Fréchet 2 dierentiable at the origin i.e. F is Fréchet dierentiable at any bottom b ∈ L (R, R) and the and inserting the estimate (19) into the last one above, we eectively get

1

R

shape gradient is well dened by (18). To conclude the proof of Proposition 3.9, one can check that in fact, we have

∂b F ∈ H 1 (R, R)

since

vb ∈ C 0 (0, T ; H 2 (R, R)).

To conclude Ÿ 3, note that higher regularity is required on the perturbations

h

in order to get

the directional dierentiability of (3). Otherwise, we just have:

2

2

∀(b, h) ∈ L (R, R) × L (R, R),

Z F (b + h) = F (b) +

h (x) ∂b F (x) dx + O(khkL2 (R,R) ), R

which can be proved by combining the quantitative estimate of Proposition 3.1 with the arguments given in the proof of Proposition 3.9. In shape optimization, this is a known fact: the existence of a shape gradient usually require more regularity than the one directly deduced from the problem. However, a careful study of the optimality conditions might imply that the maximizer is more regular than expected. Improving the regularity of a maximizer is usually a very dicult question. Numerically, as shown in Figure 6, the simulations indicate a discontinuity on the right side of the optimal bottom so there are good reasons to think the optimal bottom is not

13

H 1 -regular.

4

Numerical approach to the problem

In this section, we present the numerical procedure used to solve problem (2). In particular, we aim to develop a numerical scheme allowing a fast and precise resolution of the fKdV equation because it will be incorporated in the loop of the optimization algorithm.

Here and hereafter,

the dimensional form (6) for the fKdV equation will be used. All simulations in this work were performed on a standard laptop with Matlab with xed values

4.1

h0 = 1 m

and

g = 9.81 m.s−2 .

Numerical scheme for the fKdV equation

A naive discretization of equation (6) using nite dierences is expected to perform rather poorly. Contributing to this is the fact that, most likely, some physical properties are not conserved, the third-order derivative introduces numerical dispersion, and the forcing term breaks some possible symmetries. For the KdV equation, which is just a particular case of (6), some well-known ecient methods can be found in the literature (see

e.g.

[21, 8, 16]). All these three methods are, however,

subject to a drastic stability condition of the form

∆t = O(∆x3 ).

To overcome this inconvenience,

Furihata proposes in [9] an implicit nite-dierence scheme that is unconditionally stable. This numerical scheme is known to be very stable because it also preserves the physical properties of the equation, namely the mass and the Hamiltonian. Such method can be easily adapted to discretize (6), as follows. Noticing that (6) can be cast into the form

(2/c0 )∂t u = ∂x (δu G), where δu G = −(3/2h0 )u2 − (h20 /3)∂xx u − b,   2  1 3 h20 ∂u ∂u ,u = − u + − b u. G ∂x 2h0 6 ∂x

we have

∆x > 0 and a small time step ∆t > 0, the domain is R × [0, T ] ≈ (xi , tn )(i,n)∈Z×J0,N K with xi = i∆x, tn = n∆t and N ∆t = T . n n In the numerical scheme, Ui approximates ui := u(xi , tn ). We then introduce the two discrete + − operators δi (•) = [(•)i+1 − (•)i ]/∆x and δi (•) = [(•)i − (•)i−1 ]/∆x. According to the general

First, considering a small space step uniformly discretized:

procedure described by Furihata [9, Section 5.1], and adopting the same notation, we thus have

Gd (U )i := −(1/2h0 )Ui3 + (h20 /12)[δi+ Ui )2 + (δi− Ui )2 ] − Bi Ui , and so:  δGd U 2 + Ui Vi + Vi2 h2   := − i − 0 δi+ δi− (Ui + Vi ) − Bi    δ(U, V )i 2h0 6     1 δGd 2 Uin+1 − Uin δGd    = − . c0 ∆t 2∆x δ(U n+1 , U n )i+1 δ(U n+1 , U n )i−1 The remarkable feature of the above discretization is that the solutions set of non-linear equations satisfy for any

X

Gd (U n )i ∆x =

Z

n ∈ J0, N K

[G (∂x u, u)] (x, tn ) dx = 0 R

i∈Z

(see [9]):

and

X

Uni ∆x =

(Uin )(i,n)∈Z×J0,N K

of this

u (x, tn ) dx = 0.

(20)

Z R

i∈Z

In other words, the scheme preserves, at a discrete level, the mass and the Hamiltonian structure of equation (6). With the view of incorporating a fast numerical scheme into the loop of the optimization algorithm, we propose to simplify the numerical method by linearizing it according to the approximation

(un+1 )2 + (uni )2 = 2un+1 uni + O(∆t2 ), i i

valid for any

i∈Z

and

u(xi , •) ∈ C 2 (R, R).

Our

discretization of (6) amounts to solving the following linear system:

2 Uin+1 − Uin h2 = − 0 c0 ∆t 6

n+1 n+1 n+1 n+1 n n n n Ui+2 − 2Ui+1 + 2Ui−1 − Ui−2 Ui+2 − 2Ui+1 + 2Ui−1 − Ui−2 + 2∆x3 2∆x3

−

!

n+1 n+1 n n Ui+1 − Ui−1 Ui−1 3 Ui+1 Bi+1 − Bi−1 − . 2h0 2∆x 2∆x (21)

This will be the numerical scheme retained in this work, given that it oers a good compromise between a fast and accurate resolution and has some good numerical properties, In particular, it is consistent and unconditionally stable, and preserves mass.

14

cf.

Appendix A.

To test the performance of our numerical scheme, we set

b = 0, in which case the KdV equation

is recovered, and compare it against the ecient methods given in [8, 16, 21].

As we know, in

the particular case when the forcing term is absent, we have a family of solitary-wave solutions

ζKdV

dened by (7). This traveling-wave solution corresponds to the solution of the KdV equation

with initial condition

ζKdV (•, 0),

i.e., the initial prole remains unaltered as it propagates, which

can be used as a benchmark test for the dierent schemes. The equation is solved numerically on

[−L, L] × [0, T ]

with initial condition

ζKdV (•, 0),

by imposing periodic boundary conditions. The

numerical errors obtained with respect to the exact solution and computational time required by each one of the methods are depicted in Figure 2. As it can be seen from the gure, as long as the stability condition

∆t = O(∆x3 )

is required,

our scheme is slower, although as precise, as its competitors. However, Proposition A.2 ensures the non-dissipative character of our method, which allows us to relax this stability constraint. By doing so, say by considering

∆t = O(∆x), we are able to drastically reduce the computational time

while keeping the same order of precision, as shown in the last entry of the chart legend in Fig. 2.

−3

L2−norm of the error |u(.,t)−uth(.,t)|L2(−L,L)

1.5

x 10

Zabusky and Kruskal: 6 min Fornberg and Whitham: 8 min Trefethen: 12 min Algorithm used: 19 min Algorightm used (with ∆t=0.1): 2 s 1

0.5

0

0

20

40 60 Time variable t

80

100

Figure 2: Performance comparison between dierent numerical schemes for the KdV equation. The problem is solved on

[−L, L]×[0, T ] with periodic boundary conditions, and the numerical solutions

are compared against the analytical solitary-wave solution. The computational time required by each one of the methods is specied in the chart legend. Parameters are set as: a = 0.2, x0 = 0, ∆x = 0.1, L = 15, ∆t = 0.00025, and T = 100. The last line of the chart legend has been obtained by changing the value of ∆t (= 0.1) while keeping the same values of the other parameters.

4.1.1 Numerical boundary conditions In contrast to what was seen above, in the absence of a forcing term, we will solve equation (6) on a nite domain, say from

−L

to

L,

without imposing periodic boundary conditions. Since the

ow is assumed to be critical, a positive constant

L

is chosen large enough to contain all the ve

I ∈ N such that I∆x = L and we Uin+1 = Uin = 0 for any i ∈ / J−I, IK. In particular, the discretization (21) becomes a linear n n+1 system of 2I + 1 equations, which reads A U = B n in matrix form. Instead of increasing the execution time with the choice of a large L that would depend on T , a smooth lter f is applied on distinct regions depicted in Fig. 1.

We may then introduce

have

the approximated map at each time step. The introduction of a lter has the aim of suppressing the waves at the left-end of the domain

[−L, −L + ∆L],

15

with

∆L ∈]0, L[

being the interval size on

which the lter acts, while preserving the solution behavior outside this region. More precisely:

(

Uin+1 = fh(xi )[(An)−1 B n ]i i f (x) = 12 1 + cos π ∆L−(L+x) 1[−L,−L+∆L] (x) + 1]−L+∆L,L] (x). ∆L U n+1 to zero on [−L, −L + ∆L] and it does not [−L + ∆L, L] as shown in Figure 3. More importantly, it

This procedure ensures a smooth decreasing of aect the approximation of

u(•, tn+1 )

on

allows a small numerical domain, which greatly reduces the computational time of our simulations. Finally, concerning the right-end of the domain, knowing that waves entering this region are solitary waves, we can use their explicit expression (7) and their generation period (8) in order to tune the nal time

T

and prevent them from reaching the boundary (see Ÿ 3 for further details on the choice

of a nal time

4.2

T ).

Description of the optimization algorithm

First, we explain in Ÿ 4.2.1 how the shape gradient (18) of the functional (3) is computed. Then, the

L2 -constrained

optimization problem (2) is replaced by an innite sequence of unconstrained ones

thanks to the introduction of a Lagrange multiplier in Ÿ 4.2.2. Finally, the Usawa-type procedure is presented and discussed in Ÿ 4.2.3, relying on the combination of two projected gradient methods.

4.2.1 Computation of the shape gradient of the functional The techniques used to obtain (17) in Proposition 3.8 can be easily adapted to obtain the adjoint

ζ = u, associated to the optimization problem (2). Formally, the calcula(2/c0 )∂t v + (3/h0 )u ∂x v + (h20 /3)∂xxx v + 2u = 0 on R × [0, T ] and v(•, T ) = 0. Then,

formulation of (6) with tions yield

the discretization is performed in the same way as (21) was obtained from (6), with the result:

2 Vin+1 − Vin c0 ∆t

=

h2 − 0 6

n+1 n+1 n+1 n+1 n n n n Vi+2 − 2Vi+1 + 2Vi−1 − Vi−2 Vi+2 − 2Vi+1 + 2Vi−1 − Vi−2 + 2∆x3 2∆x3

−

n+1 n+1 n n − Vi−1 3Uin Vi+1 3Uin+1 Vi+1 − Vi−1 − − 2h0 2∆x 2h0 2∆x

!


Uin+1 + Uin . (22)

An+1 V n = B n+1 . Using the trick of Ÿ 4.1.1, a n smooth decreasing of V to zero on [−L, −L + ∆L]

As time is reversed, the system gets the matrix form

f˜ is applied at each time step to ensure a [L − ∆L, L], where ∆L ∈]0, L[ is the interval size on which the lter acts. We thus have: ( Vni = f˜(xi )[(An+1 )−1 B n+1 ]i h  i f˜(x) := f (x) − 1[L−∆L,L] (x) + 1 1 + cos π ∆L+(x−L) 1[L−∆L,L] (x).

lter and

2

∆L

f˜ alters severely the numerical solution of (22) on the whole [−L, L], whereas the lter f only interfered on the interval [−L, −L + ∆L]

As illustrated in Figure 3, the lter computational domain

where it acts. Fortunately, the eects of the lter are negligible on the computation of the shape gradient (18), which is the pertinent quantity for the numerical calculations. We can also observe

∂b Ffilter are almost proportional, i.e. ∂b Ffilter = ∂b F and ∂b Ffilter can be used indierently in the optimization algorithm since the new bottom is evaluated by b+γ∂b Ffilter = b+γ(1+γfilter )∂b F , where γ > 0 is small. Finally, note that the adjoint equation is a linearization of (6). In particular, that the corresponding shape gradients

(1 + γfilter )∂b F ,

where

γfilter

∂b F

and

is small. Consequently,

similar results as those presented in (20) and Propositions A.1A.2 can be obtained for (22). To conclude, at each step of the optimization algorithm, (21) and (22) must be solved to compute (18), whose integral is approximated according to Simpson's rule.

4.2.2 Introduction of a Lagrange multiplier L : (b, λ) ∈ L2 (R, R)×[0, +∞[7→ L(b, λ) ∈ R be the Lagrangian associated with (2) and dened L(b, λ) := F (b) + λ(M 2 − kbk2L2 (R,R) ). Consider the following optimization problem:

Let by

G(λ) :=

sup b∈L2 (R,[0,+∞[) supp b⊆[−K,K]

16

L(b, λ).

(23)

4 L2−error |v(x,.)−vfilter(x,.)|L2(0,T)

L2−error |u(x,.)−ufilter(x,.)|L2(0,T)

0.12 2

L −error of the fKdV solution 0.1 0.08 0.06 0.04 0.02 0 −50

0 Space variable x in the bottom frame

2

L −error of the adjoint 3

2

1

0 −50

50

350

0 Space variable x in the bottom frame

50

1.03 Ratio dbF(x)/dbFfilter(x)

Shape gradient dbF(x)

Linearity between the gradients 300 250 200 150 100 −1

Without filter With a filter −0.5 0 0.5 Space variable x in the bottom frame

1.025 1.02 1.015 1.01 1.005 1 −1

1

−0.5 0 0.5 1 Space variable x in the bottom frame

Figure 3: Illustration of the eects on the solution due to the lters. The numerical simulation is implemented with the following parameters:

∆x = 0.05, L = 50, ∆L = 20, ∆t = 0.1,

The method of Usawa consists in relaxing the

L2 -constraint

by solving

inf λ>0 G(λ)

and

T = 30.

instead of (2).

In general, the two problems are not equivalent since we only have:

inf G(λ) > sup F (b).

λ>0

(24)

b∈B

We were not able to prove analytically that equality holds in (24). Adapting some arguments of

λ > 0 and a non-negative square-integrable map bλ L(bλ , λ) = inf λ>0 G(λ). If one could show the uniqueness of such map bλ , then we could that kbλ kL2 (R,R) = M , in which case (24) would be an equality and the global maximizer of

Ÿ 3, we could however show the existence of such that prove

(2) unique. Despite the lack of a theoretical proof, every single numerical simulation we performed

inf λ>0 G(λ) yielded an optimal bottom bopt saturating the L2 -constraint to a very high opt kL2 (R,R) = M up to 8 digits). This strongly suggests the validity of our of precision (kb

solving order

claim and justies, at least numerically and

a posteriori, the use of Usawa's method to our problem.

We thus combine: (i) a gradient method for the primal problem (23), in which the shape gradient of the Lagrangian is needed and given by

∂b L(b, λ) := ∂b F − 2λb; inf λ>0 G(λ), using P+ (•) := max(0, •).

(ii) a projected gradient method for the dual problem of the projector on

[0, +∞[

which is given by

the explicit expression

It is worth pointing out that (23) still contains the constraint of non-negativity of the bottom, which is also required to have support contained in

[−K, K].

Therefore, we have to replace in (i)

a gradient descent by a projected gradient method. The algorithm thus reads:


 bnew = P+ bold + γ∂b L bold , λold 1[−K,K] where

γ, κ > 0

h  i then λnew = P+ λold − κ M 2 − kbnew k2L2 (R,R) ,

are parameters to tune, and where

17

1[−K,K]

equals one on

[−K, K]

otherwise zero.

4.2.3 Behaviour of the Usawa-type algorithm The general behaviour expected from the algorithm is depicted in Figure 4. The initial bottom cosine-shaped prole satisfying the

L2 -constraint.

In particular,

λ=0

and

L(b, λ) = F (b).

b is a

Hence,

the initial deformation is only ruled by the functional. Then, it is expected from the algorithm to increase the bottom height because the wave elevation is very sensitive to it. However, after some iterations, the bottom will not satisfy the

L2 -constraint

and thus

λ > 0.

The

L2 -constraint

begins

to act in the deformation process in order to bring back the bottom into the admissible set. Consequently, oscillations of the Lagrangian are expected around a saddle point corresponding to an equilibrium between the will of the functional and the constraints. The convergence of the algorithm mainly depends on how quickly the which is thus ruled by the parameter

κ

κ.

L2 -constraint intervenes in the optimization process,

As shown in Fig. 4, the convergence occurs if the parameter

is able to reduce progressively the oscillations observed on the evolution of all the characteristic

parameters, such as the functional, the Lagrange multiplier and the amplitude of the solitary wave. An educated choice of

κ has been dicult to set up.

Indeed, too small

κ mean a delay in the process

of penalization of the functional whereas high values make the constraints immediately signicant. In both cases, high oscillations will be observed. A typical value is Then, we comment the choice of

γ.

κ = 30 000.

The setting up of (23) via the gradient method (i) relies on

the optimal local direction for steepest descent. Hence, the parameter expansion around zero are involved. A typical value is

γ = 10−4 .

γ

must be small since Taylor

Moreover, if

γ

is too important,

the new bottom will always leave the admissible set or violate the physical limit of the fKdV model. Finally, a stop criteria

ι > 0 is chosen small.

It corresponds to the tolerance allowed on the precision

kbnew − bold kL2 (R,R) < ι. 6 γkL(bold , λold )kL2 (−K,K) and

of variables. The algorithm stops when the constraints are satised and A typical value is

−3

ι = γ10

new . Indeed, we have kb

numerics suggest that equality hold if

γ

− bold k

L2 (R,R)

is chosen small enough.

50

0

20

40

60 80 100 120 140 Number of iterations in the optimization algorithm

Lagrange multiplier

0 Wave elevation at final time Norm of the gradient |dbL|L2(−K,K)

Functional F(b) Lagrangian L(b,λ)

100

800 Evolution of the optimality condition 600 400 200 0

1 Initial elevation Optimal elevation 0.5

0

−0.5 −50

0 Space variable in the bottom frame

50

160

180

200

4000 Evolution of the Lagrange multiplier 3000 2000 1000 0 0 50 100 150 200 Number of iterations in the optimization algorithm

0 50 100 150 200 Number of iterations in the optimization algorithm

Bottom topography

Functional

150

0.2 0.15

Initial bottom Optimal bottom

0.1 0.05 0 −1

−0.5 0 0.5 Space variable in the bottom frame

1

Figure 4: Illustration of the algorithm convergence. The computational time is 20 minutes for the

√ M = 0.03, supp b ⊆ [−1, 1], ∆x = 0.05, L = 50, ∆t = 0.1, T = 30, κ = 30 000, γ = 0.00005, and ι = γ10−3 . parameters:

18

5

Results and concluding remarks

Figure 4 illustrates how, given an initial cosine-shaped prole satisfying the

L2 -constraint,

our

method converges to an optimal bottom responsible for generating a leading solitary wave with almost a two-fold increase of amplitude with respect to the one generated by the initial bottom. Both initial bottom and optimal bottom have the same support restriction, but while the initial prole is continuous (when viewed as a function over the real line), the optimal one is not.

A

discontinuity can be observed at both ends of the support. The oscillating behaviour of the algorithm, and especially its diculty to converge, is well present in this gure. This indicates that the soliton height is highly sensitive to the amplitude of the bottom topography and corroborates the ndings in [22]. We were interested in determining how this optimal shape depended on dierent prescribed initial conditions.

Figure 5 displays various initial shapes that are used to compute an optimal

bottom. Our numerical results point to the fact that the maximizer is not much sensitive to the particular starting shape. This is quantied on the lower panel of Figure 5, in which we present the distance between the optimal bottom

bref opt

resulting from an initial cosine shape and the optimal

bottoms obtained from the other initial shapes. Discrepancies were found to be of order presented in this gure is the value of the rst-order optimality condition, the Lagrangian, for which obtained values were of order

10−4 .

i.e.

10−7 .

Also

the shape gradient of

These observations strongly suggest

the uniqueness of a critical point to (2) and the convergence of various initial bottoms to an unique optimal shape. We deduce from Theorem 1.1 that the common shape given on the upper panel of Figure 5 is

the

maximizer of (2).

Bottom topography b(x)

0.2

0.15

Optimal Cosine Cube Left−triangle Right−triangle Bowl Semi−ellipses

0.1

0.05

0 −1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 Space variable x in the bottom frame

−7

3 2.5 2 1.5 1 0.5 0 −1

−0.5 0 0.5 Space variable x in the bottom frame

0.8

1

−4

x 10

Shape gradient of the optimal shape

Distance |bopt(x)−bref (x)| opt

3.5

0.6

1

4

x 10

2 0 −2 −4 −6 −1

−0.5 0 0.5 Space variable x in the bottom frame

1

Figure 5: The optimization algorithm is performed for various initial bottoms. Here, each optimal

√ M = 0.03, supp b ⊆ [−1, 1], ∆x = 0.05, L = 50, ∆L = 20, ∆t = 0.1, T = 30, κ = 30000, γ = 0.00025, and ι = γ10−3 . shape has been computed in 4 minutes for 90 iterations with the following parameters:

Then, we study the inuence of the constraint

supp b ⊆ [K− , K+ ]

on the discontinuities of the

optimal bottom that may manifest at the ends of the support considered. On the left panel of Figure

19

6, we set

K+ = 1

K− ∈ {−3.5, −2.5, −1.5, −0.5}. K− ∈ {−3, −2, −1, 0}. In all cases,

and display the optimal proles computed for

On the right panel, we set

K+ = 1.5

and consider the cases

the right-end discontinuity is present (although not at the same height). end discontinuity seems to strongly depend on the support restriction. resemblant the shapes are in the cases order

K+ = 1

and

K+ = 1.5;

In contrast, the left-

Also, it is striking how

in fact, almost indiscernible (of

10−4 ). 0.2

0.2 Initial bottom supp(b)⊆[−0.5,1] supp(b)⊆[−1.5,1] supp(b)⊆[−2.5,1] supp(b)⊆[−3.5,1]

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 −4

Initial bottom supp(b)⊆[0,1.5] supp(b)⊆[−1,1.5] supp(b)⊆[−2,1.5] supp(b)⊆[−3,1.5]

0.18 Optimal bottom topography b(x)

Optimal bottom topography b(x)

0.18

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

−3 −2 −1 0 1 Space variable x in the bottom frame

0 −4

2

−3 −2 −1 0 1 Space variable x in the bottom frame

2

Figure 6: Inuence of the support restriction. For the same initial bottom, we determine how the optimal shape of the bottom varies with dierent supports. Each optimal shape has been computed in 4 minutes for 90 iterations with the same parameters as in Figure 5. It is worth pointing out the shortcomings of an inviscid model as the one adopted here. However the gains in amplitude were rather important in the case displayed in Figure 4, this only seems to be the case because the initial bottom is far from saturating the

L2 -constraint.

As a matter of

fact, when in Figure 5 the shape of the optimal bottom is compared to the highest admissible cube (rectangle to be more precise), the dierence in amplitudes between the two leading wave is not that signicant. Even if in reality that could be the case, it is not dicult to imagine that drag forces due to a cube would be much more important than those due to the optimal shape computed here (which reminds a section of an airfoil), hence with a far greater energy consumption associated to it. For a more realistic description of the problem, weak dissipative eects should be included in the model allowing in particular to characterize the adherence of the water on the bottom to better study the dependency between its shape and the amplitude of the generated wave. Despite those reductions, we still recover results which look very like the kind of shapes people use in practice when manufacturing experimental prototypes for generating surng waves in a pool thanks to a translating bottom underwater [15]. To conclude, we justify numerically the choice of the

L2 -setting

to study (2).

Clearly, from

b ∈ kbkH 1 (R,R) 6 M . However, in this case, our numerical approach −1 perturbation ∂b L(b, λ) = ∂b F − 2λ(b − ∂xx b) ∈ H (R, R). The computation

a theoretical point of view, it would have been much easier to consider admissible bottoms

H01 (] − K, K[, [0, +∞[) provides an irregular

satisfying

immediately diverges, leading to errors and oscillations. The well-posedness of (4) is only proved at least for bottoms in

1

H− 2 ,

although it is actually not known whether this exponent is optimal [18,

Remark 1.2]. Even for the KdV equation, the space

3

H− 4

is critical for well-posedness [5]. Hence,

there is few hope to get a stable program. In comparison, the great advantage of the

L2 -setting

is that the shape gradient of our Lagrangian remains a square integrable map so our numerical algorithm is stable with respect to the class of admissible bottoms.

Acknowledgements This work started in March 2010 under the original idea of E. Zuazua, who guided the author J. D. during an internship at the Basque Center for Applied Mathematics (Spain). J. D. gratefully

20

acknowledges E. Zuazua for his support, dynamism, and guidance during this period, which now, looking back, feels like good old times. The work was then considerably improved while J. D. was nishing his Master degree at the Institut Elie Cartan de Lorraine (France), for which the author would like to acknowledge nancial support and thank A. Henrot for the encouragement to pursue this study. R. B. acknowledges the support of Science Foundation Ireland under grant 12/IA/1683.

Appendix A

Stability analysis of the numerical scheme

Proposition A.1. The discretization

(21) takes the form L∆x,∆t u = 0 and it approximates the equation (6) written as ∂t u + Lu = 0. If we assume ∆t = O(∆x), then (21) is consistent and rstorder accurate: ∀u ∈ C 4 (R × [0, +∞[ , R), ∂t u + Lu = 0 =⇒ ∂t u + Lu = L∆x,∆t u + O(∆x).

Proof.

1 + − 1 s± x [(•)(x, t)] := (•)(x±∆x, t) to dene δx := 2∆x (sx −sx ), − 3 1 2 4 := − 2 + sx ) and δx := δx δx . Let u ∈ C (R × [0, +∞[, R) be such that ∂t u + Lu = 0. ∆x2 ∆x3 ± 4 A Taylor expansion yields sx u = u ± ∆x ∂x u + 2 ∂xx u ± 6 ∂xxx u + O(∆x ), from which we 1 2 2 2 deduce ∂x u = δx u + O(∆x ) and ∂xx u = δx u + O(∆x ). These estimations are then combined to 3 obtain ∂xxx u = δx u + O(∆x). Therefore, we have an approximation of the linear terms of L:

δx2

We introduce the shift operators

1 + ∆x2 (sx

Lu =

3c0 c0 h20 3 c0 u ∂x u + δx u + δx1 b + O(∆x). 2h0 6 2

(25)

+ + 1 s+ t [(•)(x, t)] := (•)(x, t + ∆t) and δt := ∆t (st − 1), the same type + + + of arguments yields to ∂t u = δt u + O(∆t) and ∂t u = st ∂t u + O(∆t) = −st Lu + O(∆t). We thus + + + + 1 1 get: ∂t u + Lu = δt u + Lu + O(∆t) = δt u + (Lu − ∂t u) + O(∆t) = δt u + (1 + st )Lu + O(∆t). 2 2 Then, we assume that ∆t = O(∆x) and we treat the non-linear term of Lu as follows. We have: + + + 1 st +1 2 2 2 1 2 1 (1 + s+ t )(u ∂x u) = δx [ 2 u ] + O(∆x ) = δx [u st u + O(∆t )] + O(∆x ) = δx (u st u) + O(∆x). Hence, we can deduce the expected estimation ∂t u + Lu = L∆x,∆t u + O(∆x) with: Introducing the time operators

L∆x,∆t u := δt+ u +


c0 h20
3
c0 1 3c0 1 δx u s+ 1 + s+ δx u + δx b. t u + t 4h0 12 2

Proposition A.2. Consider the discretization

(21) of equation (6) with forcing term b = 0. Let ∆t , µ = c60 h20 and s = ∆x . Then, the Von Neumann stability analysis provides an amplication factor g : [−π, π] → C of the following form:

β =

3c0 0 2h0 kζkC ([−L,L]×[0,T ],R)



1 − iA (ξ) g(ξ) := 1 + iA (ξ)

where

 β µ A (ξ) := s (sin ξ) + (cos ξ − 1) . 2 ∆x2

In particular, we have |g| = 1 ensuring the non-dissipative feature of the method: the scheme (21) 2A is unconditionally stable. Moreover, the numerical dispersion Ψ = arg(g) = −arctan( 1−A 2 ) is sµ 3 compared to the analytical one whose expression is given by Ψref (ξ) := −sβξ + ∆x2 ξ . We obtain Ψ(ξ) = Ψref (ξ) + EΨ (ξ) + O(ξ 7 ) where: EΨ (ξ) =

Proof.

sβ 6

 1+

s2 β 2 2



ξ3 −



 s5 β 5 s3 β 3 sβ s3 β 2 µ sµ + + + + ξ5. 80 24 120 4∆x2 4∆x2

We refer to [6, (38)(51)] for details on the proof of this result.

disagreement between our expression of

EΨ

We stress, however, a

and the one provided in [6, (50)]. This seems to result

Ψ by using Taylor series. More precisely, it 1 2 −2A 4 7 arctan( 1−A 2 ) = −2A[1 − 3 A − 3A ] + O(A ), as the correct expression −2A 1 2 1 4 arctan( 1−A2 ) = −2A[1 − 3 A + 5 A ] + O(A7 ).

from a mistake made in [6, (48)], when expanding

is

wrongly stated that

is

given by

Appendix B

The necessity of a

L2 -constraint

We recall the scaling property for (4) introduced by Tsugawa [18]:

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Lemma B.1. Let b(x) be a forcing function with enough regularity, as required in Lemma 2.1, and u(x, t) be the unique smooth solution of the initial value problem (4). For any θ ∈ R, dene the maps uθ : (x, t) 7→ θ2 u(θx, θ3 t) and bθ : x 7→ θ4 b(θx). Then, uθ is precisely the solution of (4) with forcing function bθ . We can then state the following result:

Proposition B.2. Let K > 0 and T > 0. Then, the optimization problem (11) has no global maximizer. Proof. Assume, by contradiction, that there exists a maximizer b to (11). Then, from Lemma 2.1, we can consider its associated smooth solution

ub .

Introducing the bottoms

(bθ )θ>1

of Lemma B.1,

one can check they are admissible for problem (11). Moreover, we deduce from Lemma B.1 that

R θ3 T R 2 F (bθ ) = 0 u (x, t) dx dt. Using the optimality of b, we obtain F (bθ ) = F (b) for any θ > 1 so R b ub = 0 on [T, θ3 T ]. From (6), we get ∂x b = 0 thus ub = 0 also on [0, T ]. Thus F (b) = 0, which contradicts the optimality of b.

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