Correction to “Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity”

Correction to \PARTIAL EXACT CONTROLLABILITY

ANDEXPONENTIAL STABILITY IN

HIGHER-DIMENSIONAL LINEAR THERMOELASTICITY"

WEIJIU LIU

In 1], as a consequence of (3.16), we were assuming that

k

v

⁰(

t

)^k20

e

^{1;w T}

E

(

vT

)

in lines 5 and 6 on page 41. But, the correct inequality is

k

v

⁰(

t

)^k20 2

e

^{1;w (T;t)}

E

(

vT

)

:

Consequently, the proof of Theorem 2.3 is not correct. We give here a corrected proof that needs however an additional smallness condition on

.

We acknowledge Aissa Guesmia for pointing out this mistake to us.

We now give a corrected and weaker version of Theorem 2.3 with a com- plete proof. We adopt the notation and numbering of formulas, Theorems and Denitions from 1].

T^HEOREM 2.3. Let ;¹ and ;² be given by (1.4) and (1.5), respectively, satisfying (2.12). Assume that the function

a

(

x

) satises (2.19) or (2.20).

Suppose that

< !

₂

e:

Let

T

⁰ be large enough so that

2

e

^{2(1;! T0)}

<

1^; 4

²

²

²

e

²

!

²

(2

:

33) and

T

T

⁰, where

!

is the constant in Theorem 2.1. Then for any(

u

⁰

u

¹)²

W

, there exists a boundary control function (

xt

) with

p

m

²⁽

L

²(²))ⁿ

such that the solution of (2.30) satises (2.31). Moreover, there exists a positive constant

c

, independent of (

u

⁰

u

¹), such that

k p

m

^k(L2(2))n

c

^k(

u

⁰

u

¹)^kW

:

(2

:

34)

Dept of Applied Mechanics and Engineering Sciences, Univ. of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA. E-mail: [email protected]

Received April 29, 1998. Accepted for publication May 27, 1998.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by TEX.

Proof of Theorem 2.3. The main idea of the proof is rst to construct a linear operator by using the Lame system and the system of thermoelasticity, and then to show that

I

^; is an isomorphism by using the uniform stabilization of these systems.

Given (

v

⁰

v

¹)²

W

, we consider the Lame system

8

>

>

>

<

>

>

>

:

v

^00;

v

^;(

+

)^rdiv

v

= 0 in

Q

v

= 0 on ¹

@v @

^{+ (}

+

)div(

v

)

+

am

v

^;

m

v

⁰ = 0 on ²

v

(

T

) =

v

⁰

v

⁰(

T

) =

v

¹ in

(3

:

15)

which has a unique solution with

(

v

(

t

)

v

⁰(

t

))²

C

(0

T

]

W

)

:

Moreover, by Theorem 2.2 (note that we are assuming that

a

(

x

) satises (2.19) or (2.20), and in this case, Theorem 2.2 on the stabilization of the Lame system holds), there exists a positive constant

!

such that

E

(

vt

)

E

(

vT

)

e

^{1;! (T;t)}

⁸

t

²0

T

] (3

:

16) where

E

(

vt

_{) = 12}^Z

h

j

v

⁰(

xt

)^j2+

^jr

v

(

xt

)^j2+ (

+

)^jdiv

v

(

xt

)^j2i

dx

+ 12

Z

;

2

am

^j

v

(

t

)^j2

d

;

:

Using the solution

v

of (3.15), we then consider

8

<

:

^0;

=

div(

v

⁰) in

Q

= 0 on

(0) = 0 in

⁽³

:

17)

and

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

w

^00;

w

^;(

+

)^rdiv

w

+

^r

=^;

^r

in

Q

^0;

+

div

w

⁰= 0 in

Q

= 0 on

w

= 0 on ¹

@w @

^{+ (}

+

)div(

w

)

+

am

w

+

m

w

⁰ = 0 on ²

w

(0) =

v

(0)

w

⁰(0) =

v

⁰(0) (0) = 0 in

:

(3

:

18)

Since

div(

v

⁰)²(

L

²(0

T H

^;1()))ⁿ

it follows that (3.17) has a unique solution

with

²

C

(0

T

]

L

²())^\

L

²(0

T H

⁰¹())

:

In addition, multiplying (3.17) by

and integrating over

Q

, we obtain 12^k

(

T

)^k20+

Z

T

0

kr

(

t

)^k20

dt

=

^{Z T}

0 Z

div(

v

⁰)

dxdt

^{Z T}

0

kdiv(

v

⁰(

t

))^k;1kr

(

t

)^k0

dt

(use (2

:

32))

12

Z

T

0

kr

(

t

)^k20

dt

+

²

² 2

Z

T

0

k

v

⁰(

t

)^k20

dt

(use (3

:

16))

12

Z

T

0

kr

(

t

)^k20

dt

+

²

²

E

(

vT

)

Z

T

0

e

^{1;! (T;t)}

dt

12

Z

T

0

kr

(

t

)^k20

dt

+

²

²

e

! E

⁽

vT

)

which implies

k

(

T

)^k20+

Z

T

0

kr

(

t

)^k20

dt

²

²

²

e

! E

⁽

vT

)

:

(3

:

19) On the other hand, since

²

L

²(0

T H

⁰¹()) and

Z

r

dx

=

Z

;

d

; = 0

then (0

^;

^r

0) ²

L

¹(0

T

^H). Thus, by the classical theory of semi- groups, the nonhomogeneous problem (3.18) has a unique solution with

(

ww

⁰

)²

C

(0

T

]^H)

:

Moreover, the solution can be expressed as

(

ww

⁰

) =

S

(

t

)(

w

(0)

w

⁰(0)

(0))+

Z

t

0

S

(

t

^;

)(0

^;

^r

0)

d

where

S

(

t

) denotes the strongly continuous semigroup of contractions gen- erated by the thermoelastic system. By Theorem 2.1, we have

k

S

(

t

)^k

e

^{(1;! t)=2}

⁸

t

0

then we deduce from (3.19)

E

(

wt

)2^k

S

(

t

)^k2

E

(

w

0)+ 2

Z

t

0

k

S

(

t

^;

)^kk

^r

(

)^k0

d

²

2

e

^{1;! t}

E

(

w

0)+ 2

^{2Z t}

0

e

^{1;! (t;)}

d

^{Z t}

0

kr

(

)^k20

d

2

e

^{1;! t}

E

(

w

0)+ 4

²

²

²

e

²

!

²

E

(

vT

)

:

(3

:

20)

Set

u

=

w

^;

v

=

+

and =^;

m

(

w

⁰+

v

⁰)

(3

:

21)

then

u

satises

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

u

^{00 ;}

u

^;(

+

)^rdiv

u

+

^r

= 0 in

Q

^0;

+

div

u

⁰= 0 in

Q

= 0 on

u

= 0 on ¹

@u @

^{+ (}

+

)div(

u

)

+

am

u

= on ²

u

(0) =

u

⁰(0) = 0 (0) = 0 in

u

(

T

) =

w

(

T

)^;

v

⁰

u

⁰(

T

) =

w

⁰(

T

)^;

v

¹ in

:

We dene an operator by

(

v

⁰

v

¹) = (

w

(

T

)

w

⁰(

T

))

:

Then it is clear that is a linear operator from

W

into

W

. Moreover, by (3.16) and (3.20), we have

k(

v

⁰

v

¹)^k2W

(note denition (2

:

1) of the norm of

W

)

E

(

wT

)

2

e

^{1;! T}

E

(

w

0)+ 4

²

²

²

e

²

!

²

E

(

vT

) (since

w

(0) =

v

(0)

w

⁰(0) =

v

⁰(0) (0) = 0)

= 2

e

^{1;! T}

E

(

v

0)+ 4

²

²

²

e

²

!

²

E

(

vT

)

h2

e

^{2(1;! T)}+ 4

²

²

²

e

²

!

²

i

E

(

vT

) (use (3

:

16))

=^h2

e

^{2(1;! T)}+ 4

²

²

²

e

²

!

²

i

k(

v

⁰

v

¹)^k2W

:

Therefore,

k^k22

e

^{2(1;! T)}+ 4

²

²

²

e

²

!

²

:

Let

T

⁰ be large enough so that (2.33) holds if

T

T

⁰. Then ^;

I

is an isomorphism from

W

onto

W

. Thus, for any (

u

⁰

u

¹) ²

W

, there exists a unique (

v

⁰

v

¹)²

W

such that

(

u

⁰

u

¹) = (

v

⁰

v

¹)^;(

v

⁰

v

¹)

= (

w

(

T

)

w

⁰(

T

))^;(

v

⁰

v

¹)

= (

u

(

T

)

u

⁰(

T

))

:

⁽³

:

22)

Consequently, we have constructed a control function =^;

m

(

w

⁰+

v

⁰) solving the partial controllability problem (2.30).

On the other hand, multiplying the rst equation of (3.15) by

v

ⁱ⁰ and integrating over

Q

, we obtain

Z

2

m

^j

v

^0j2

d

=

E

(

vT

)^;

E

(

v

0)

:

(3

:

23) Multiplying the rst equation and second equation of (3.18) by

w

ⁱ⁰ and

^, respectively, and integrating over

Q

, we deduce from (3.19) and (3.20) that (the following

c

's denoting various constants that may depend on

T

)

E

(

wT

)+

Z

Q

jr

^j2

dxdt

+

Z

2

m

^j

w

^0j2

d

=

E

(

w

0)^;

^Z

Q

w

^0r

dxdt

E

(

w

0)+

cE

(

vT

)+

Z

T

0

E

(

w

)

d

cE

(

w

0)+

cE

(

vT

)

:

(3

:

24)

Noting

E

(

v

0) =

E

(

w

0), we deduce from (3.16), (3.23) and (3.24) that

Z

2 j j

2

m

d

⁼

Z

2

m

^j

v

⁰+

w

^0j2

d

cE

(

vT

)

which, combined with (3.22), implies (2.34) since ^;

I

is an isomorphism from

W

onto itself.

References

1] W.J. Liu: Partial exact controllability and exponential stability in higher dimensional linear thermoelasticity, ESAIM: Control Optim. Calc. Var.,³, 1998, 23{48.