Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain

EXACT BOUNDARY CONTROLLABILITY FOR THE

KORTEWEG-DE VRIES EQUATION ON A BOUNDED

DOMAIN

LIONEL ROSIER

Abstract. The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial deriva- tives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the rst spatial derivative at the right endpoint is assumed to be con- trolled, a quite dierent analysis shows that exact controllability holds too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for suciently small initial and nal states.

1. Introduction and main results The Korteweg-de Vries (KdV) equation

y

^t+

yy

^x+

y

^xxx= 0

(1.1)

is a well known instance of a nonlinear dispersive partial dierential equa- tion, which may serve as a model for (among other things) propagation of small amplitude long water waves in a uniform channel.

Recently stabilizability and controllability results have been obtained in 8] for the forced Korteweg - de Vries equation with periodic boundary con- ditions

y

^t+

yy

^x+

y

^xxx =

f

(

tx

)

0

< x <

1

0

< t < T

(1.2)

@

^k

y

@x

^k(

t

0) =

@

^k

y

@x

^k(

t

1)

0

k

2

(1.3)

the control action

f

(

tx

) being localized:

supp

f

(

t:

)

ab

]0

1]

:

(1.4) On the other hand, there are to date few results concerning the boundary controllability of KdV equation. In 16] boundary controllability of linear

Laboratoire d'analyse numerique, Universite d'Orsay, b^at. 425, 91405 Orsay, France.

E-Mail: [email protected].

Received by the journal April 16, 1996. Revised November 22, 1996. Accepted for publication December 27, 1996.

This research was funded by NSF grant DMS 9530973.

c Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

KdV equation

y

^t+

y

^xxx = 0

0

< x <

2

0

< t < T

(1.5)

@

^k

y

@x

^k(

t

0) =

@

^k

y

@x

^k(

t

2

)

k

^2f0

2^g

(1.6)

@y @x

⁽

t

2

) ^;

@y

@x

⁽

t

0) =

h

(

t

)

0

< t < T

(1.7) is obtained as a consequence of the uniform stabilizability of this time- reversible (linear) system in both t-directions, whose proof is rather long and technical. Moreover only initial and terminal states

y

⁰ and

y

^T for which

Z

2

0

;

y

^T(

x

)^;

y

⁰(

x

)

dx

= 0 (1.8) are concerned and the control

h

(

t

) is not explicitly given. The rst goal of this paper is to directly study the (exact) boundary controllability prob- lem for linear KdV equation by means of the Hilbert Uniqueness Method (H.U.M). This approach has been successfully applied for studying con- trollability of wave and plate equations, and more recently of Schrodinger equation (see for instance 1], 5], 7], 11], 12], 13], 14]). The rst result of this paper concerning boundary controllability of linear KdV equation is as follows:

Theorem 1.1. Let

H

^p2=^f

w

²

H

²(0

2

) :

w

(0) =

w

(2

)

w

⁰(0) =

w

⁰(2

)^g and

T >

0. Then for any

y

⁰

y

^{T 2}(

H

^p2)⁰ (the dual space of

H

^p2), there exist

h

⁰

h

¹

h

^{2 2}

L

²(0

T

) such that the solution

y

²

C

(0

T

]

(

H

^p2)⁰) of the boundary initial-value KdV equation:

y

^t+

y

^xxx = 0

(

xt

)²(0

2

)(0

T

) (1.9)

y

(

t

2

)^;

y

(

t

0) =

h

⁰(

t

)

(1.10)

y

^x(

t

2

)^;

y

^x(

t

0) =

h

¹(

t

)

(1.11)

y

^xx(

t

2

)^;

y

^xx(

t

0) =

h

²(

t

)

(1.12)

y

(0

:

) =

y

⁰ (1.13)

satises

y

(

T:

) =

y

^T.

Notice that explicit controls may be given. Unfortunately, the state

y

is only known to belong to

C

(0

T

]

(

H

^p2)⁰) so it seems quite dicult to deduce from Theorem 1.1 controllability results for nonlinear KdV equation (1.1). Lastly it should be more realist to set boundary conditions on the value or the rst (spatial) derivative of the function

y

rather than on the second derivative

y

^xx, if we have in mind to test such controllability result or the validity of KdV equation as a model with an experimental device: For instance we may observe the propagation of water waves of small amplitude in a bounded channel, the level -or the slope- of the surface of the uid being controlled at the ends of the channel. Notice that, as it was suggested in 2], the extra term

y

^xshould be in this case incorporated in the equation in order to obtain an appropriate model for water waves in a uniform channel when coordinates

x y

are taken with respect to a xed frame. This inclusion is not without consequence in controllability results: condition on the lenght

L

of the domain appears. The main result concerning (exact) boundary

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

controllability for linear KdV equation (with boundary control on

y

^x at

x

=

L

) is the following:

Theorem 1.2. Let ^N =^f2

^{qk2+k l+l3 2}

kl

²

IN

^g. Then for any

T >

0 and

L

² (0

+¹)^nN, for any

y

⁰

y

^{T 2}

L

²(0

L

), there exists

h

²

L

²(0

T

) such that the mild solution

y

²

C

(0

T

]

L

²(0

L

))^\

L

²(0

TH

¹(0

L

)) of

y

^t+

y

^x+

y

^xxx = 0 (1.14)

y

(

t

0) =

y

(

tL

) = 0 (1.15)

y

^x(

tL

) =

h

(

t

) (1.16)

y

(0

x

) =

y

⁰ (1.17)

satises

y

(

T:

) =

y

^T.

This theorem is proved by means of H.U.M and multiplier method. It turns out that the study of (1.14)-(1.17) as a boundary initial-value problem is more delicate than for (1.9)-(1.13), and that - because of the extra term

y

^x in (1.14) - the (key) observability result holds true i

L

^{62 N}. On the other hand the solution

y

belongs this time to a functional space in which we may give a sense to the nonlinear term

yy

^x in (1.1).

As a consequence of previous controllability result we get the main result of the paper, that is the exact boundary controllability of nonlinear KdV equation on a bounded domain:

Theorem 1.3. Let

T >

0 and

L >

0. Then there exists

r

⁰

>

0 such that for any

y

⁰

y

^{T 2}

L

²(0

L

) with^k

y

^0kL2(0L)

< r

⁰,^k

y

^TkL2(0L)

< r

⁰, there exists

y

²

C

(0

T

]

L

²(0

L

))^\

L

²(0

TH

¹(0

L

))^\

W

¹¹(0

TH

^;2(0

L

)) solution of

y

^t = ^;(

y

^x+

yy

^x+

y

^xxx) in ^D0(0

TH

^;2(0

L

)) (1.18)

y

(

:

0) = 0 in

L

²(0

T

)

(1.19)

and such that

y

(0

:

) =

y

⁰

y

(

T:

) =

y

^T. If moreover

L

^{62 N}, then we may in addition assume that

y

(

:L

) = 0 in

L

²(0

T

) and take

y

^x(

:L

) in

L

²(0

T

) as control function.

Theorem 1.3 is proved by means of Banach contraction xed point theo- rem and Theorem 1.2.

The paper is organized as follows: In Section 2 we derive from H.U.M. a direct proof of the exact boundary controllability result by Russell-Zhang for (linear) KdV equation (Th. 1.1). In Section 3 we consider another boundary controllability problem for (linear) KdV equation, in which only the value of rst spatial derivative (at

x

=

L

) of the state function is assumed to be controlled: this boundary initial-value problem is rst shown to admit solutions (Prop. 3.7). Next an observability result is given (Prop. 3.9) and used to show (applying H.U.M) the exact boundary controllability for linear KdV equation with these boundary conditions (Th. 1.2). Finally in Section 4 we derive from Th. 1.2 the main result of this paper, that is the exact

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

(local) boundary controllability of (nonlinear) KdV equation on a bounded domain (Th. 1.3).

2. Exact boundary controllability of linear KdV equation by means of control on datah

@ k

y

@x k

(t:) i

2

0

k =0 1 2.

In this section, for reasons of simplicity, we restrict ourselves to the case where the space domain 0

L

] is 0

2

], although Theorem 1.1 holds true for arbitrary

L >

0. On the other hand all the functions of this section are assumed to take complex values. For

k

^2f2

3^gwe set

H

^pk :=

u

²

H

^k(0

2

)

d

^j

u

dx

^{j(0) =}

d

^j

u

dx

^j(2

) for 0

j k

^;1

:

(2.1) (

H

^k(0

2

) denotes classical Sobolev space on the interval (0

2

).) For

n

²

ZZ

, let

u

^(

n

_{) = 12}

Z

2

0

u

(

t

)

e

^;int

dt

(2.2) be the nth Fourier coecient of

u

²

L

²(0

2

). It is easy to see that, for all

u

²

L

²(0

2

),

k

^2f2

3^g:

u

²

H

^{pk () X}

n2ZZ

(

n

^kj

u

^(

n

)^j)²

<

¹

(2.3) and that the Sobolev norm ^k

u

^kk :=^;Pkj=0R02j

u

^(j)(

x

)^j2

dx

¹² reduces to

k

u

^kk =^;X

n2ZZ

(1 +

n

²+

:::

+

n

^2k)^j

u

^(

n

)^j212 (2.4) for

u

²

H

^pk. In what follows the (Hilbert) space

H

^pk is endowed with the norm ^k

:

^kk.

Let

A

denote the operator

Au

= ^;

u

⁰⁰⁰ on the domain ^D(

A

) =

H

^p3

L

²(0

2

). As it is mentioned in 16],

A

generates a strongly continuous unitary group ^;

S

(

t

)^t2IRon

L

²(0

2

). Let

T >

0. For

u

^T =^Pn2ZZ

c

ⁿ

e

^int2

L

²(0

2

), the (mild) solution of the uncontrolled problem:

u

^t+

u

^xxx = 0

x

²(0

2

)

t

²

IR

(2.5)

u

(

t

0) =

u

(

t

2

)

(2.6)

u

^x(

t

0) =

u

^x(

t

2

)

(2.7)

u

^xx(

t

0) =

u

^xx(

t

2

)

(2.8)

u

(

T:

) =

u

^T (2.9)

is given by

u

(

tx

) = ^X

n2ZZ

c

ⁿ

e

^i;n3(t;T)+nx

:

(2.10) (In the following,

u

will denote the solution of (2.5)-(2.9) associated with

u

^T.)

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

We now consider the following non-homogeneous problem: Given

y

^{0 2}(

H

^p),

h

⁰

h

¹

h

² in

L

²(0

T

), nd

y

such that:

y

^t+

y

^xxx = 0

x

²(0

2

)

t

²(0

T

) (2.11)

y

(

t

2

)^;

y

(

t

0) =

h

⁰(

t

)

(2.12)

y

^x(

t

2

)^;

y

^x(

t

0) =

h

¹(

t

)

(2.13)

y

^xx(

t

2

)^;

y

^xx(

t

0) =

h

²(

t

)

(2.14)

y

(0

:

) =

y

⁰

:

(2.15)

We rst prove that 1) equations (2.11)-(2.15) admit a unique solution

y

²

C

(0

T

]

(

H

^p2)⁰) in a sense to be made precise, and 2) this solution is the classical one whenever

y

^{0 2 D}(

A

), and

h

⁰

h

¹

h

² are smooth enough and vanish at 0. More precisely, we prove the following:

Proposition 2.1. 1) Assume

h

⁰

h

¹

h

^{2 2}

C

⁰²(0

T

]) :=^f

h

²

C

²(0

T

]

IC

)

h

(0) = 0^g and

y

⁰²

H

^p3. Then there exists a unique solution

y

²

C

(0

T

]

H

³(0

2

))^\

C

¹(0

T

]

L

²(0

2

)) of (2.11)-(2.15). Moreover for any

u

^{T 2}

H

^p3 and any

S

²0

T

] we have

Z

2

0

u

(

Sx

)

y

(

Sx

)

dx

= ^{Z 2}

0

u

(0

x

)

y

⁰(

x

)

dx

; Z

S

0

u

^xx(

t

0)

h

⁰(

t

)^;

u

^x(

t

0)

h

¹(

t

) +

u

(

t

0)

h

²(

t

)

dt:

(2.16) 2) For

u

^{T 2}

H

^p2,

u

²

C

(0

T

]

H

^p2) and

u

^xx(

:

0) makes sense in

L

²(0

T

).

3) Assume now only that

y

^{0 2}(

H

^p2)⁰ and

h

⁰

h

¹

h

^{2 2}

L

²(0

T

). Then there exists a unique

y

²

C

(0

T

]

(

H

^p2)⁰) such that

8

u

^{T 2}

H

^p2

⁸

S

²0

T

]

< y

(

S

)

u

(

S:

)

>

^(Hp2)0Hp2=

< y

⁰

u

(0

:

)

>

^(Hp2)0Hp2

; Z

S

0

;

u

^xx(

t

0)

h

⁰(

t

)^;

u

^x(

t

0)

h

¹(

t

) +

u

(

t

0)

h

²(

t

)

dt:

(2.17) Proof. 1) Let

⁰

¹

²²

C

¹(0

2

]) be such that

^{(k )i} (0) = 0 and

^{(k )i} (2

)

=^{; ki} for

ik

= 0

1

2. The change of function

z

(

tx

) :=

h

⁰(

t

)

⁰(

x

) +

h

¹(

t

)

¹(

x

) +

h

²(

t

)

²(

x

) +^;

S

(

t

)

y

⁰(

x

) +

y

(

tx

) (2.18) yields an equivalent problem: Find

z

such that

z

^t+

z

^xxx =

f

(

tx

) :=^X2

i=0

;

h

⁰ⁱ(

t

)

ⁱ(

x

) +

h

ⁱ(

t

)

⁰⁰⁰ⁱ (

x

) (2.19)

z

(

t

2

) =

z

(

t

0)

(2.20)

z

^x(

t

2

) =

z

^x(

t

0)

(2.21)

z

^xx(

t

2

) =

z

^xx(

t

0)

(2.22)

z

(0

:

) = 0

:

(2.23)

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

Since

f

²

C

(0

T

]

L

(0

2

)), this non-homogeneous problem admits (see 15]) a unique solution

z

²

C

(0

T

]

H

^p3) ^\

C

¹(0

T

]

L

²(0

2

)). This proves the rst assertion in 1). Let

u

^{T 2}

H

^p3. Then

u

²

C

(0

T

]

H

^p3) ^\

C

¹(0

T

]

L

²(0

2

)). Integrations by parts in

Z

S

0 Z

2

0

(

u

^t+

u

^xxx)

ydxdt

= 0 (2.24) lead to

; Z

S

0 Z

2

0

u

(

y

^t+

y

^xxx)

dxdt

+

Z

2

0

^;

uy

(

:x

)]^S0

dx

+

Z

S

0

(

u

^xx

y

^;

u

^x

y

^x+

uy

^xx)(

t:

)]²⁰

dt

= 0

:

(2.25) Taking into account (2.6)-(2.8) and (2.11)-(2.15), we get (2.16).

2) By (2.10), for

t

¹

t

^{2 2}0

T

]

u

(

t

¹

x

)^;

u

(

t

²

x

) = ^X

n2ZZ

c

ⁿ

e

^;in3T;

e

^in3t1;

e

^in3t2

e

^inx

:

(2.26) If

u

^{T 2}

H

^p2,^Pn2ZZj

n

²

c

^nj2

<

¹ and it follows from Lebesgue's theorem that

P

n2ZZ

j

n

²

c

ⁿ

e

^;in3T;

e

^in3t1;

e

^{in3t2j2 !}0 as

t

^{1 !}

t

². Then

u

²

C

(0

T

]

H

^p2).

Hence

u

(

:

0) and

u

^x(

:

0) exist in

C

(0

T

])

L

²(0

T

). The same argument shows that if

u

^{T 2}

H

^p3,

u

²

C

(0

T

]

H

^p3) and

u

^xx(

t

0) = ^X

n2ZZ

;

;

n

²

e

^;in3T

c

ⁿ

e

^in3t

:

(2.27) The sum in (2.27) makes sense in

L

²(0

T

) whenever ^Pn2ZZ;

n

^2j

c

^nj2

<

¹, that is

u

^{T 2}

H

^p2. From now on

u

^xx(

:

0) denotes, for

u

^{T 2}

H

^p2, the sum in (2.27). Notice that the linear map

u

^{T 7!}

u

^xx(

:

0) is continuous since

k X

n2ZZ

;

;

n

²

e

^;in3T

c

ⁿ

e

^{in3tk2L2(0T) ;}

T

2

^{+ 1}

X

n2ZZ

;

n

^2j

c

^nj2

(2.28) where

x

] denotes the integral part of a real number

x

.

3) Identifying

L

²(0

2

) with its dual by means of the (conjugate linear) map

y

^7!(

:y

)^{L2(02 )}, we have the following diagram:

H

^p2

L

²(0

2

) =

L

²(0

2

)⁰(

H

^p2)⁰ (2.29) where each embedding is dense and compact. Moreover

< yu >

^(Hp2)0Hp2 = (

uy

)^{L2(02 )} (=

Z

2

0

uydx

) (2.30) for

u

²

H

^p2and

y

²

L

²(0

2

). It follows that (2.17) holds true for

h

⁰

h

¹

h

²

2

C

⁰²(0

T

]) and

y

⁰

u

^{T 2}

H

^p3. Since

H

^p3 is dense in

H

^p2, using 2) we see that (2.17) also be true for

u

^{T 2}

H

^p2. For

y

^{0 2}(

H

^p2)⁰ and

h

⁰

h

¹

h

^{2 2}

L

²(0

T

), we dene a "weak" solution of (2.11)-(2.15)as a function

y

²

C

(0

T

]

(

H

^p2)⁰) such that (2.17) holds true for all

u

^{T 2}

H

^p2 and all

S

²0

T

]. Let

S

²0

T

]

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

be xed. We rst show that (2.17) denes

y

(

S

) in (

H

^p) in a unique manner.

It follows from the proof of 2) that the map

u

^{T 2}

H

^{p27!;Z S}

0

u

^xx(

t

0)

h

⁰(

t

)^;

u

^x(

t

0)

h

¹(

t

) +

u

(

t

0)

h

²(

t

)

dt

²

IC

(2.31) is a continuous linear form. On the other hand (see (2.3), (2.4) and (2.10)) the map

u

^{T 2}

H

^{p2 7!}

u

(

S:

) ²

H

^p2 is clearly an automorphism of Hilbert space. Hence for each

S

²0

T

],

y

(

S

) is uniquely dened in (

H

^p2)⁰. Moreover for

S

²0

T

],

k

y

(

S

)^k(Hp2)0

= sup

ku(S:)k

2 1

j

< y

(

S

)

u

(

S:

)

>

^j

= sup

ku

T k

2 1

< y

⁰

u

(0

:

)

>

^(Hp2)0Hp2

; Z

S

0

;

u

^xx(

t

0)

h

⁰(

t

)^;

u

^x(

t

0)

h

¹(

t

) +

u

(

t

0)

h

²(

t

)

dt

sup

ku

T k21

;

k

y

^0k(Hp2)0

:

^k

u

(0

:

)^kHp2+^k

u

^xx(

:

0)^kL2(0T)

:

^k

h

^0kL2(0T) +^k

u

^x(

:

0)^kL2(0T)

:

^k

h

^1kL2(0T)+^k

u

(

:

0)^kL2(0T)

:

^k

h

^2kL2(0T)

C

^;k

y

^0k(Hp2)0+^k

h

^0kL2(0T)+^k

h

^1kL2(0T)+^k

h

^2kL2(0T)

(2.32) where

C

is a positive constant which does not depend on

S

or on

y

⁰

h

⁰

h

¹

h

². Since

y

²

C

(0

T

]

L

²(0

2

))

C

(0

T

]

(

H

^p2)⁰) (2.33) for

y

^{0 2}

H

^p3 and (

h

⁰

h

¹

h

²)²

C

⁰²(0

T

])³, and since

H

^p3 is dense in

L

²(0

L

) and

C

⁰²(0

T

]) is dense in

L

²(0

T

), it follows from (2.32) that

y

²

C

(0

T

]

(

H

^p2)⁰) in the general case too.

For applying H.U.M. we need the following (key) observability result.

Proposition 2.2. Let

T >

0. Then there exist positive numbers

C

^1T

C

^2T such that for every

u

^{T 2}

H

^p2

C

^1Tk

u

^{Tk22 k}

u

(

:

0)^k2L2(0T)+^k

u

^x(

:

0)^k2L2(0T)+^k

u

^xx(

:

0)^k2L2(0T)

C

^2Tk

u

^Tk22

:

(2.34)

Proof. We have in

L

²(0

T

) :

u

(

t

0) = ^X

n2ZZ

c

ⁿ

e

^in3(t;T)

(2.35)

u

^x(

t

0) = ^X

n2ZZ

inc

ⁿ

e

^in3(t;T)

(2.36)

u

^xx(

t

0) = ^X

n2ZZ

;

n

²

c

ⁿ

e

^in3(t;T)

:

(2.37)

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

Hence

k

u

(

:

0)^k2L2(0T)+^k

u

^x(

:

0)^k2L2(0T)+^k

u

^xx(

:

0)^k2L2(0T)

T

2

^{+ 1}

X

n2ZZ

(1 +

n

²+

n

⁴)^j

c

^nj2

C

^2Tk

u

^Tk22

:

(2.38)

if we set

C

^2T :=^2T+1. To prove the left inequality we rst take

T

⁰²(0

T

) and

>

^2T0.

Let

N

²

IN

be such that

n

²

ZZ

^j

n

^j

N

⁾ (

n

+ 1)^3;

n

³

:

(2.39) By Ingham's inequality (see 6]) there exists

C

^T0

>

0 such that

8(

a

ⁿ)^n2ZZ2

l

²(

ZZ

) ^X

jnjN

j

a

^nj2

C

^{T0 Z T0}

0

X

jnjN

a

ⁿ

e

^in3(t;T)

2

dt:

(2.40) Let ^Zn := Span(

e

^inx) for

n

²

ZZ

and ^Z := ^n2ZZZn

H

^p2. We dene a semi-norm

p

in ^Z by

8

u

^2Z

p

(

u

) = ^;j

u

(0)^j2+^j

u

⁰(0)^j2+^j

u

⁰⁰(0)^j212 (2.41)

=

j X

n2ZZ

u

^(

n

)^j2+^jX

n2ZZ

in u

^(

n

)^j2+^jX

n2ZZ

;

n

²

u

^(

n

)^j2

!1

2(2.42) (For

u

^2Z, ^

u

(

n

) = 0 for^j

n

^jlarge enough.) Let

u

^{T 2Z\}(^jnj<NZn)^?, that is

c

ⁿ= 0 for^j

n

^j

< N

or for^j

n

^jlarge enough. Using (2.10), (2.40) we get

k

u

^Tk22 = ^X

jnjN

(1 +

n

²+

n

⁴)^j

c

^nj2

C

^{T0 Z T0}

0

p

^;

u

(

t:

)²

dt:

(2.43) Since

T > T

⁰, it follows from (2.38), (2.43) and a result by Komornik (see 7 Th. 5.2]) that there exists a constant

C

^1T

>

0 such that

8

u

^{T 2Z}

C

^1Tk

u

^{Tk22 Z T}

0

p

^;

u

(

t:

)²

dt

= ^k

u

(

:

0)^k2L2(0T)+^k

u

^x(

:

0)^k2L2(0T)

+ ^k

u

^xx(

:

0)^k2L2(0T)

:

(2.44) We get the left inequality (in (2.34)) in the general case (

u

^{T 2}

H

^p2) by a density argument.

We may now complete the proof of Theorem 1.1 by applying H.U.M.

Proof of Theorem 1.1. We rst remark that we may assume

y

⁰= 0. Indeed if

y

⁰

y

^T are given in (

H

^p2)⁰, if there exist

h

⁰

h

¹

h

^{2 2}

L

²(0

T

) such that the (weak) solution ~

y

of (2.11)-(2.14) and ~

y

(0

:

) = 0 satises ~

y

(

T:

) =

y

^{T ;}

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

S

(

T

)

y

⁰, then

y

(

t:

) :=

S

(

t

)

y

⁰+ ~

y

(

t:

) is the (weak) solution of (2.11)-(2.15) (with the same control functions) and is such that

y

(

T:

) =

y

^T. In what follows we assume

y

⁰ = 0. For

u

^{T 2}

H

^p2 we let (

u

^T) denote

y

(

T

)²(

H

^p2)⁰, where

y

is the (weak) solution of (2.11)-(2.15) and

h

⁰,

h

¹ and

h

² are chosen in the following way:

h

⁰(

t

) := ^;

u

^xx(

t

0)

(2.45)

h

¹(

t

) :=

u

^x(

t

0)

(2.46)

h

²(

t

) := ^;

u

(

t

0)

:

(2.47) (As above u stands for the solution of (2.5)-(2.9)associated with

u

^T.) Clearly :

H

^{p2 !}(

H

^p2)⁰is a (conjugate linear) continuous map. Moreover

<

(

u

^T)

u

^T

>

^(Hp2)0Hp2

=

Z

T

0

j

u

(

t

0)^j2+^j

u

^x(

t

0)^j2+^j

u

^xx(

t

0)^j2

dt

C

^1Tk

u

^Tk22 (2.48)

by Propositions 2.1 and 2.2. It follows from Lax-Milgram theorem (see 19 p.92]) that is invertible. The proof of Theorem 1.1 is complete.

Remark 2.3. (i) If

T

= 2

the observability result is obvious. Indeed for any

u

^{T 2}

H

^p2

k

u

^Tk22 =^k

u

(

:

0)^{k2L2(02 )}+^k

u

^x(

:

0)^{k2L2(02 )}+^k

u

^xx(

:

0)^{k2L2(02 )}

:

(2.49) (ii) Setting!

h

¹(

t

) = 0 instead of

u

^x(

t

0) we also get (for some constant

C

^1T

>

0)

8

u

^{T 2}

H

^p2

<

(

u

^T)

u

^T

>

^(Hp2)0Hp2

C

^!1Tk

u

^Tk22

(2.50) so is also onto.

(iii) If we assume

h

⁰(

t

) =

h

²(

t

) = 0, the same arguments as above yield the following result: for every

T >

0 and

y

⁰

y

^{T 2} (

H

^p1)⁰ there exists

h

^{1 2}

L

²(0

T

) such that the solution

y

²

C

(0

T

]

(

H

^p1)⁰) of (1.9)-(1.13) (with

h

⁰=

h

² = 0) satises

y

(

T:

) =

y

^T. Indeed it may be shown that for

u

^{T 2}

H

^p1

u

^x(

:

0) makes sense in

L

²(0

T

) and that

C

^3Tk

u

^{Tk21 k}

u

^x(

:

0)^k2L2(0T)

C

^4Tk

u

^Tk21 for some constants

C

^3T

C

^4T

>

0. When we compare these results with those by Russell-Zhang, we remark that 1) the spaces in which we get exact controllability are larger than

L

²(0

L

) (

L

²(0

L

)(

H

^p1)⁰(

H

^p2)⁰) 2) the assumption (1.8) is not used here and 3) the control may be explicitly given : for reaching

y

^{T 2} (

H

^p2)⁰ we may choose

h

⁰

h

¹

h

² given by (2.45)- (2.47), with

u

^T = ^;1(

y

^T;

S

(

T

)

y

⁰). However, even if

y

⁰

y

^{T 2}

L

²(0

2

),

y

is only known to belong to

C

(0

T

]

(

H

^p2)⁰). So this method is not appropriate for starting the study of controllability of nonlinear KdV equation. In the following section, another (more appropriate) exact controllability result for linear KdV equation with dierent boundary conditions is given.

Esaim: Cocv, March 1997, Vol.2, pp. 33{55

3. Exact boundary controllability of linear KdV equation by means of control on ^yx(tL).

From now on, the scalar space is

IR

. In this section

L

stands for some positive number. We shall prove the controllability in

L

²(0

L

) of

y

^t+

y

^x+

y

^xxx = 0 (3.1)

y

(

t

0) =

y

(

tL

) = 0 (3.2)

y

^x(

tL

) =

h

(

t

) (3.3)

y

(0

x

) =

y

⁰ (3.4)

where

h

²

L

²(0

T

) stands for the control function. More precisely we shall prove that, for any

L >

0,

T >

0, for any

y

⁰

y

^{T 2}

L

²(0

L

) there exists

h

²

L

²(0

T

) such that a mild solution

y

²

C

(0

T

]

L

²(0

L

))^\

L

²(0

TH

¹(0

L

))^\

H

¹(0

TH

^;2(0

L

)) of (3.1)-(3.4)( which veries (3.1) in

D

0(0

TH

^;2) and (3.4) in

L

²(0

L

) ), may be found such that

y

(

T:

) =

y

^T. We begin by showing the well-posedness of the (initial-value) homoge- neous problem:

y

^t+

y

^x+

y

^xxx = 0 (3.5)

y

(

t

0) =

y

(

tL

) = 0 (3.6)

y

^x(

tL

) = 0 (3.7)

y

(0

x

) =

y

⁰(

x

)

:

(3.8) Let

A

denote the operator

Aw

=^;

w

^000;

w

⁰on the (dense) domain^D(

A

)

L

²(0

L

) dened by

D(

A

) =^f

w

²

H

³(0

L

)

w

(0) =

w

(

L

) =

w

⁰(

L

) = 0^g

:

(3.9) The following result holds:

Proposition 3.1.

A

generates a strongly continuous semigroup of contrac- tions on

L

²(0

L

).

Proof. It is easy to see that

A

is closed. Let

w

^2D(

A

). Then (

wAw

)^L2(0L) =

Z

L

0

w

(

x

)^;;

w

⁰⁰⁰(

x

)^;

w

⁰(

x

)

dx

=

Z

L

0

w

⁰(

x

)

w

⁰⁰(

x

)

dx

^;

ww

⁰⁰]^{L0 ;}

w

² 2 ]^L0

=

w

⁰² 2 ]^L0

= ^;

w

⁰(0)²

2 0

:

(3.10)

Hence

A

is dissipative. It may be seen that

A

(

w

) =

w

⁰+

w

⁰⁰⁰ with do- main ^D(

A

) = ^f

w

²

H

³(0

L

)

w

(0) =

w

(

L

) =

w

⁰(0) = 0^g, so that (

wA

(

w

)) = ^{;w0(L)2 2} 0 and

A

is dissipative too. Now the result fol- lows from 15 cor. 4.4 chapter 1].

From now on we let (

S

(

t

)^t0) denote the semi-group of contractions asso- ciated with

A

, and we let

B

denote the (Banach) space

C

(0

T

]

L

²(0

L

))^\

Esaim: Cocv, March 1997, Vol.2, pp. 33{55