Regionally efficient and strategic actuators

On: 29 May 2013, At: 16:45 Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Systems Science

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/tsys20

Regionally efficient and strategic actuators

L. Afifi , A. Chafia & A. El Jai Published online: 26 Nov 2010.

To cite this article: L. Afifi , A. Chafia & A. El Jai (2002): Regionally efficient and strategic actuators, International Journal of Systems Science, 33:1, 1-12

To link to this article: http://dx.doi.org/10.1080/002077202317216884

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or

systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the

contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and

drug doses should be independently verified with primary sources. The publisher shall not be liable for

any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused

arising directly or indirectly in connection with or arising out of the use of this material.

Regionally e cient and strategic actuators

L. Afifi { , A. Chafiai { and A. El Jai {

In this paper, we introduce and characterize the notions of regional remediability and regional e cient actuators. We study their relationship with regional controllability and regional strategic actuators. We also determine the set of regionally remediable disturbances and, for each disturbance , we give the optimal control which ensures its regional compensation .

1. Introduction

This work, which is motivated by environment and par- ticularly pollution problems, is an extension to the regional case, of the remediability and e cient actuator notions introduced by A®® et al. (1998) for a class of linear distributed and disturbed systems. The well- known disturbance problems, the so-called disturbance rejection or decoupling problems, have been studied, particularly for ®nite-dimensional systems (Otsuka 1991, Pandol® 1986, Malabre and Rabah 1993, Senamel et al. 1995, Rabah and Malabre 1997).

However, in this work, where spatial variables, actua- tors and sensors play an important role, the considered problem is di€ erent. The aim is, ®rst, to show how to determine, with respect to the sensors, regionally e - cient actuators ensuring the regional compensation of any known or unknown disturbance. This is the basic notion of regional remediability which we shall study in local, global, exact and weak aspects and which is, as will be shown, a weaker notion than regional controll- ability. We also determine the set of regionally remedi- able disturbances and we give, using an extension of the Hilbert uniqueness method (HUM) (Lions 1988), the optimal control which ensures regionally the exact com- pensation of a disturbance on the system.

The regional aspect of these notions is motivated by the fact that we may have remediability on a region

! » O but not on the whole geometrical support O of the considered system and, even if we have remediability

on O, the cost is reduced if we are interested only in a subregion ! of O.

More precisely, we suppose that O is an open and bounded subset of

ⁿ

with a su ciently regular boundary G ˆ @O , and we consider, without loss of gen- erality, a class of disturbed linear systems described by the following state equation:

… S

P

† zz _ … t † z … 0 †

ˆ ˆ

Az … t † ‡ f … t † ‡ Bu … t †; 0 < t < T z

₀

;

»

… 1 † where A generates a strongly continuous semigroup

… S … t ††

t50

on the state space X ˆ L

²

…O† , B 2 L…U ; L

²

…O†† , u 2 L

²

… 0; T ; U† ; U is a Hilbert space (control space), f is a disturbance term generally unknown (A®® and El Jai 1994, 1995), and the system

… S

_P

† is augmented by the regional output equation

… E † y

^!

… t † ˆ Ci

_!

p

_!

z … t †; … 2 † where C 2 L… L

²

…O† ; Y † , Y is a Hilbert space (observa- tion space), p

_!

is the restriction operator de®ned by

p

_!

: L

²

…O† ¡! L

²

…!†;

z 7¡! p

_!

z ˆ z

_j!

; … 3 † i

_!

is the adjoint operator of p

_!

, which is de®ned by

i

_!

: L

²

…!† ¡! L

²

…O†;

z 7¡! i

_!

z ˆ z in ! 0 elsewhere:

( … 4 †

The observation y

^!

only depends on the state p

_!

z of the system in the region !.

In the case where f ˆ 0 and u ˆ 0 (i.e. the normal case), we have

y

^!

… t † ˆ Ci

_!

p

_!

S … t † z

0

but, if f 6ˆ 0 and u ˆ 0, then

International Journal of Systems ScienceISSN 0020±7721 print/ISSN 1464±5319 online#2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/00207721.html

Received 27 April 1999. Revised 8 November 1999. Accepted 8 November 1999.

{Faculty of Sciences. University Hassan II Ain Chock, BP 5366- MaaÃrif Casablanca, Morocco.

{Systems Theory Laboratory, University of Perpignan, 52 avenue de villeneuve, 66860 Perpignan Cedex, France.

Downloaded by [University of Sydney] at 16:45 29 May 2013

y

^!

… t † ˆ Ci

_!

p

_!

S … t † z

₀

‡

…

t 0

Ci

_!

p

_!

S … t ¡ s † f … s † ds 6ˆ Ci

_!

p

_!

S … t † z

₀

:

We study the existence of an input operator B (actua- tors) whose choice depends on the output operator C (sensors), which ensures ®nite time regional compensa- tion of any disturbance, that is for every f 2 L

²

… 0; T; L

²

…O†† , there exists u 2 L

²

… 0; T ; U† such that

…

T 0

Ci

_!

p

_!

S … T ¡ s † f … s † ds

‡

…

T 0

Ci

_!

p

_!

S … T ¡ s † Bu … s † ds ˆ 0 or

®® ®

®

…

T 0

Ci

_!

p

_!

S … T ¡ s † f … s † ds

‡

…

T 0

Ci

_!

p

_!

S … T ¡ s † Bu … s † ds ®® ® ® < ° for a given ° > 0:

This work is organized as follows. In } 2, we recall the notions of regional controllability, observability, strategic actuators and sensors. In } 3, we de®ne and characterize the notion of exact and weak regional remediability, as well as regional e cient actuators. In } 4, we study the relationship between regional controll- ability and regional remediability, and hence between regional strategic actuators and regional e cient actua- tors, with application to a di€ usion process. We study, in } 5, the problem of regional exact remediability with minimum energy using an extension of the HUM. Then, we characterize the set of disturbances which are exactly remediable on !, and we give the optimal control which exactly compensates on !, an arbitrary disturbance of the considered system.

It should be noted that these approaches may be extended to other systems.

2. Preliminaries

2.1. Regional controllability

We consider the system described by the following equation:

… S † zz _ … t † z … 0 †

ˆ ˆ

Az … t † ‡ Bu … t †; 0 < t < T ; z

₀

:

»

… S † has a unique weak solution on ‰ 0; T Š , denoted z

_u

…:†

and given by

z

_u

… t † ˆ S … t † z

₀

‡

…

t 0

S … t ¡ s † Bu … s † ds:

Let H be the operator de®ned by H: L

²

… 0; T; U† ¡! L

²

…O†

u 7¡! Hu ˆ

…

T 0

S … T ¡ s † Bu … s † ds: … 5 † We have

z

_u

… T † ˆ S … T † z

₀

‡ Hu; … 6 † then

p

_!

z

u

… T † ˆ p

_!

S … T † z

0

‡ p

_!

Hu: … 7 †

2.1.1. De®nition and characterization . In this section, we recall some de®nitions and characterization s related to regional controllability (El Jai et al. 1995). The pre-

®x !- denotes the fact that it is regional on !.

De®nition 2.1: The system … S † is said to be

(i) !-exactly controllable (or exactly controllable in !) on ‰ 0; T Š if, for all z

₀

2 L

²

…O† , z

_d

2 L

²

… ! † , there exists u 2 L

²

… 0; T; U† such that p

_!

z

_u

… T † ˆ z

_d

and (ii) !-weakly controllable (or weakly controllable in !)

on ‰ 0; T Š if, for all z

0

2 L

²

…O† , z

d

2 L

²

…!† and ° > 0, there exists u 2 L

²

… 0; T ; U† such that

k p

_!

z

_u

… T † ¡ z

_d

k

L²…!†

< °: &

Let B

^?

, S

^?

…:† , H

^?

be the adjoint operators of B, S …:†

and H respectively and U

^?

be the dual space of U . We have the following result (El Jai et al. 1995).

Proposition 2.1: The system … S † is (i) !-exactly controllable on ‰ 0; T Š

() Im … p

_!

H † ˆ L

²

…!† … 8 † () 9®

!

> 0 such that k z

^?

k

L²…!†

4 ®

!

k B

^?

S

^?

… T ¡ † i

_!

z

^?

k

L²…0;T;U^?†

;

8 z

^?

2 L

²

…!† … 9 † () the operator

M

_!

ˆ p

_!

HH

^?

i

_!

: L

²

…!† ¡! L

²

…!†

is coercive … 10 †

and

(ii) !-weakly controllable on ‰ 0; T Š

Downloaded by [University of Sydney] at 16:45 29 May 2013

() Im … p

_!

H † ˆ L

²

… ! † … 11 † () ker … H

^?

i

_!

† ˆ f 0 g … 12 † () the operator

M

_!

ˆ p

_!

HH

^?

i

_!

is positive definite: … 13 † Regional exact controllability implies regional weak controllability, but the converse is not true (El Jai et al.

1995).

2.1.2. Regional controllability and actuators. We suppose that L

²

…O† has an orthonormal basis of eigen- functions …’

nj

†

ⁿ⁵¹

jˆ1;rn

of A such that

A’

_nj

ˆ ¶

n

’

nj

for j ˆ 1; . . . ; r

_n

; n 5 1 with ¶

n

& ¡1:

In the case of p actuators …O

i

; g

_i

†

iˆ1;p

(El Jai and Pritchard 1988), we have U ˆ

^p

and

B :

^p

¡! L

²

…O†

u … t † 7¡! Bu … t † ˆ X

^p

iˆ1

g

_i

u

_i

… t †;

where u ˆ … u

₁

; . . . ; u

_p

†

^T

2 L

²

… 0; T;

^p

† , g

_i

2 L

²

…O

i

† with O

i

ˆ supp … g

_i

† » O for i ˆ 1; p and O

i

\ O

j

ˆ 1 for i 6ˆ j. The adjoint operator B

^?

of B is de®ned by

B

^?

z ˆ …h g

₁

; z i; . . . ; h g

_p

; z i†

^T

for z 2 L

²

…O†;

where in the general case N

^T

is the transposed matrix of N, h:; :i ˆ h:; :i

L²…O†

is the inner product in L

²

…O† and h:; :i

_!

is the inner product in L

²

…!† . For k 2 L

²

…O† , if D ˆ supp … k † , we have h k; :i = h k; :i

L²…O†

= h k; :i

L²…D†

. The system … S † becomes

zz _ … t † ˆ Az … t † ‡ X

^p

iˆ1

g

_i

u

_i

… t †; 0 < t < T ; z … 0 † ˆ z

₀

:

… 14 †

De®nition 2.2: The actuators …O

i

; g

_i

†

iˆ1;p

are said to be

!-strategic (or strategic in !), if the corresponding system … S † is !-weakly controllable. &

We have the following proposition (El Jai et al. 1995).

Proposition 2.2: The actuators …O

i

; g

_i

†

iˆ1;p

are !- strategic, if and only if

\

n51

ker ‰ M

_n

®

n

… ! †Š ˆ f 0 g ; … 15 † where

M

n

ˆ …h g

i

; ’

nj

i†

^iˆ1;p

jˆ1;rn

… 16 †

and

®

n

…!† ˆ

®

n;1

…!†

.. .

®

n;r_n

…!†

0 B @

1

C A ; … 17 †

with

®

n;j

…!† ˆ …®

nj;km

…!††

_fk51;mˆ1;r_ng

and

®

nj;km

… ! † ˆ h ’

nj

; ’

km

i

!

:

In the case of pointwise actuators … b

i

; ¯

b_i

†

iˆ1;p

, we have z

_u

…:† 2 L

²

… 0; T; V † , where V is a Hilbert space such that V

⁰

» L

²

…O† » V with continuous injections. The char- acterization of !-strategic pointwise actuators is similar to (15) with M

_n

ˆ …’

nj

… b

_i

††

^iˆ1;p

jˆ1;rn

. 2.2. Regional observability

The observability is a dual notion of controllability.

The results given in this part may be deduced from } 2.1 by duality. Let us consider the system

… S

0

† zz _ … t † z … 0 †

ˆ ˆ

Az … t † ; 0 < t < T;

z

₀

2 X:

»

The initial state z

₀

is supposed to be unknown in !, that is p

_!

z

₀

is unknown. The regional observability is the possibility to reconstruct p

_!

z

₀

from the output equation

… E

0

† y

^!0

… t † ˆ K … t † i

_!

p

_!

z

0

; … 18 † with K … t † ˆ CS … t † , we have

K : L

²

…O† ¡! L

²

… 0; T; Y † and the adjoint operator K

^?

of K is de®ned by

K

^?

: L

²

… 0; T; Y

^?

† ¡! L

²

…O†

¼ 7¡! K

^?

¼ ˆ

…

T 0

S

^?

… t † C

^?

¼… t † dt: … 19 † De®nition 2.3: The system … S

₀

† together with the output … E

₀

† , (or … S

₀

† ‡ … E

₀

† ) is said to be

(i) !-exactly observable (or exactly observable in !) on

‰ 0; T Š , if Im … p

_!

K

^?

† ˆ L

²

…!† and

(ii) !-weakly observable (or weakly observable in !) on

‰ 0; T Š , if Im … p

_!

K

^?

† ˆ L

²

…!† , or equivalently ker … Ki

_!

† ˆ f 0 g: … 20 †

&

We have the following proposition.

Proposition 2.3: The system … S

₀

† ‡ … E

₀

† is (i) !-exactly observable on ‰ 0; T Š

Downloaded by [University of Sydney] at 16:45 29 May 2013

() 9 ®

_!

> 0 such that k z k

L²…!†

4 ®

!

k Ki

_!

z k

L²…0;T;Y†

; 8 z 2 L

²

…!† … 21 † () operator

C

!

ˆ p

_!

K

^?

Ki

_!

: L

²

…!† ¡! L

²

…!†

is coercive … 22 †

and

(ii) !-weakly observable on ‰ 0; T Š () the operator

C

!

ˆ p

_!

K

^?

Ki

_!

is positive definite: … 23 † Regional exact observability implies regional weak observability, but the converse is not true.

If … S

0

† ‡ … E

0

† is !-weakly observable on ‰ 0; T Š , then z

^!0

may be given by

z

^!₀

ˆ … p

_!

K

^?

Ki

_!

†

^¡1

p

_!

K

^?

y

^!₀

… 24 † with y

^!₀

de®ned by (18). The operator … p

_!

K

^?

Ki

_!

†

^¡1

p

_!

K

^?

is the generalized inverse of Ki

_!

.

If the output of the system is given by q zone sensors

… D

_io

; h

_i

†

iˆ1;q

with h

_i

2 L

²

…O† , D

_i

ˆ supp … h

_i

† » O for

i ˆ 1; . . . ; q and D

_i

\ D

_j

ˆ 1 if i 6ˆ j (El Jai and

Pritchard 1988), then the operator C is de®ned by C : L

²

…O† ¡!

^q

;

z 7¡! Cz ˆ …h h

₁

; z i; . . . ; h h

_q

; z i†

^T

; and

C

^?

³ ˆ X

^q

iˆ1

³

i

h

_i

for ³ ˆ …³

1

; . . . ; ³

q

†

^T

2

^q

we have

Ci

_!

p

_!

z ˆ …h h

₁

; z i

!

; . . . ; h h

_q

; z i

!

†

^T

with measure … D

i

\ !† > 0 for i ˆ 1; . . . ; q (if measure

… D

i₀

\ !† ˆ 0, then the observation h h

i₀

; z i

!

corre- sponding to the sensor … D

_i0

; h

_i0

† is null).

De®nition 2.4: The sensors are said to be !-strategic if the corresponding system … S

₀

† ‡ … E

₀

† is !-weakly obser-

vable. &

We have the following proposition.

Proposition 2.4: The sensors … D

_i

; h

_i

†

iˆ1;q

are !-strategic if and only if

\

n51

ker ‰ G

n

®

n

… ! †Š ˆ f 0 g … 25 † where ®

n

…!† is the matrix de®ned in (17) and

G

_n

ˆ …h h

_i

; ’

nj

i†

^iˆ1;q

jˆ1;rn

:

If the state of system is su ciently regular (continue), then in the case of q pointwise sensors, we have a similar characterization to (25) with G

_n

ˆ …’

nj

… c

_i

††

_j^iˆ1;q

ˆ1;rn

, where c

_i

are the sensors locations.

3. Regional remediability

3.1. The considered system and notations

We consider the disturbed system … S

_P

† described by the state equation (1) and augmented by the output equation (2). The state of system … S

P

† , denoted z

u;f

, is given by

z

u;f

… t † ˆ S … t † z

0

‡

…

t 0

S … t ¡ s † Bu … s † ds ‡

…

t 0

S … t ¡ s † f … s † ds:

Then

z

_u;f

… T † ˆ S … T † z

₀

‡ Hu ‡ z

_f

ˆ S … T † z

₀

‡ Hu ‡ H Hf ~ ; … 26 † where H H ~ is the linear operator de®ned by

H ~

H : L

²

… 0; T ; L

²

…O†† ¡! L

²

…O†

f 7¡! z

f

ˆ H Hf ~ ˆ

…

T 0

S … T ¡ s † f … s † ds:

… 27 †

Then

y

^!_u;f

… T † ˆ Ci

_!

p

_!

z

_u;f

… T †

ˆ Ci

_!

p

_!

S … T † z

₀

‡ Ci

_!

p

_!

Hu ‡ y

^!_f

; … 28 † with

y

^!_f

ˆ Ci

_!

p

_!

H Hf ~ : … 29 † Let R

_!

be the linear operator de®ned by

R

_!

: L

²

… 0; T; L

²

…O†† ¡! Y

f 7¡! R

_!

f ˆ y

^!f

ˆ Ci

_!

p

_!

H Hf ~ : … 30 † The adjoint operator R

^?_!

of R

_!

is given by

R

^?_!

: Y

^?

¡! L

²

… 0; T ; L

²

…O††;

³ 7¡! R

^?_!

³ ˆ H H ~

^?

i

_!

p

_!

C

^?

³ ˆ S

^?

… T ¡ † i

_!

p

_!

C

^?

³:

… 31 †

We have

y

^!_u;f

… T † ˆ Ci

_!

p

_!

S … T † z

0

‡ Ci

_!

p

_!

Hu ‡ R

_!

f : … 32 †

Downloaded by [University of Sydney] at 16:45 29 May 2013

3.2. Remediability

In the case without disturbance and no control, the system … S

_P

† is given by

zz _ … t † ˆ Az … t †; 0 < t < T;

z … 0 † ˆ z

₀

; and the regional observation by

y

^!

… t † ˆ Ci

_!

p

_!

S … t † z

₀

:

However, if the system is disturbed by a term f 2 L

²

… 0; T; L

²

…O†† , the observation becomes

y

^!

… t † ˆ Ci

_!

p

_!

S … t † z

0

‡

…

t 0

Ci

_!

p

_!

S … t ¡ s † f … s † ds 6ˆ Ci

_!

p

_!

S … t † z

₀

:

We introduce a control term Bu in order to compensate regionally and in ®nite time this disturbance that is such that

y

^!_u;f

… T † ˆ Ci

_!

p

_!

S … T † z

₀

‡ Ci

_!

p

_!

Hu ‡ R

_!

f ˆ Ci

_!

p

_!

S … T † z

₀

or equivalently

Ci

_!

p

_!

Hu ‡ R

_!

f ˆ 0:

We have the following de®nitions.

De®nition 3.1:

(i) We say that … S

_P

† ‡ … E † is f ¡ !-remediable exactly on ‰ 0; T Š if there exists u 2 L

²

… 0; T; U† such that

Ci

_!

p

_!

Hu ‡ R

_!

f ˆ 0: … 33 † (ii) We say that … S

_P

† ‡ … E † is f ¡ !-remediable weakly on ‰ 0; T Š if, for every ° > 0, there exists u 2 L

²

… 0; T; U† such that

k Ci

_!

p

_!

Hu ‡ R

_!

f k < °: … 34 †

&

De®nition 3.2: We say that … S

P

† ‡ … E † is !-exactly (or

!-weakly) remediable on ‰ 0; T Š if, for every f 2 L

²

… 0; T; L

²

…O†† , … S

P

† ‡ … E † is f ¡ !-remediable exactly (respectively weakly) on ‰ 0; T Š . &

The disturbance f is considered as an element of L

²

… 0; T ; X † , but it can be in a greater or a smaller space F . As the regularity of z

_u;f

… : † depends on f and on the control term Bu, in the general case, we have z

_u;f

…:† 2 L

²

… 0; T ; V † where V is a Hilbert space such that V

⁰

» X » V, with continuous injections (X is iden- ti®ed with its dual). The de®nitions remain the same by replacing L

²

… 0; T; X † by F or X by V.

3.3. Characterization results

3.3.1. Regional remediability. From de®nition 3.1, we deduce the following proposition.

Proposition 3.1:

(i) The system … S

P

† ‡ … E † is f ¡ !-remediable exactly on ‰ 0; T Š if and only if

R

_!

f 2 Im … Ci

_!

p

_!

H †: … 35 † (ii) The system … S

P

† ‡ … E † is f ¡ !-remediable weakly

on ‰ 0; T Š if and only if

R

_!

f 2 Im … Ci

_!

p

_!

H †: … 36 † Let us remark that, for f ˆ ¡ Bu, we have R

_!

f ˆ y

^!_f

ˆ ¡ Ci

_!

p

_!

Hu; then

Im … Ci

_!

p

_!

H † » Im … R

_!

† : … 37 † The characterization of the regional exact remediability is given in the following proposition.

Proposition 3.2: There is equivalence between (i) … S

P

† ‡ … E † is !-exactly remediable on ‰ 0; T Š , (ii)

Im … R

_!

† » Im … Ci

_!

p

_!

H † … 38 † and

(iii) 9®

!

> 0 such that 8 ³ 2 Y

^?

k S

^?

… T ¡ † i

_!

p

_!

C

^?

³k

L²…0;T;L²…O††

4 ®

!

k B

^?

S

^?

… T ¡ † i

_!

p

_!

C

^?

³k

L²…0;T;U^?†

: … 39 † Proof:

(i) () (ii) This is shown by de®nition.

(ii) () (iii) This derives from (31), and

… Ci

_!

p

_!

H †

^?

ˆ H

^?

i

_!

p

_!

C

^?

ˆ B

^?

S

^?

… T ¡ † i

_!

p

_!

C

^?

… 40 † and the equivalence between

Im … F † » Im … G † and

9® > 0 such that k F

^?

z

^?

k

X^?

4 ®k G

^?

z

^?

k

Y^?

; 8 z

^?

2 Z

^?

for any re¯exive Banach spaces X, Y, Z, and F 2 L… X; Z † , G 2 L… Y; Z † . &

For B ˆ I , that is in the case of an action on all the domain O , the system … S

_P

† ‡ … E † is !-exactly remedi- able, for every ! » O and every output operator C (this result derives from (39) in proposition 3.2).

Downloaded by [University of Sydney] at 16:45 29 May 2013

For C ˆ I, we have y

^!_u;f

… T † ˆ z

^!_u;f

… T †

ˆ i

_!

p

_!

S … T † z

₀

‡ i

_!

p

_!

Hu ‡ i

_!

p

_!

H Hf ~ ; then

z

^!_u;f

… T † ˆ i

_!

p

_!

S … T † z

₀

() i

_!

p

_!

H Hf ~ ˆ ¡ i

_!

p

_!

Hu () p

_!

H Hf ~ ˆ ¡ p

_!

Hu:

If there exists u 2 L

²

… 0; T; U† such that p

_!

Hu ‡ p

_!

H Hf ~ ˆ 0, then … S

_P

† ‡ … E † is f ¡ !-remediable exactly on ‰ 0; T Š . The converse is true if C is injective.

We have H

^?

i

_!

p

_!

C

^?

ˆ B

^?

S

^?

… T ¡ † i

_!

p

_!

C

^?

; then using (31) we have

H

^?

i

_!

p

_!

C

^?

ˆ B

^?

R

^?_!

… 41 † For the characterization of regional weak remediability, we have the following proposition.

Proposition 3.3: There is equivalence between (i) … S

_P

† ‡ … E † is !-weakly remediable on ‰ 0; T Š , (ii)

Im … R

_!

† » Im … Ci

_!

p

_!

H † … 42 † and

(iii)

ker … B

^?

R

^?_!

† ˆ ker … R

^?_!

† … 43 † Proof:

(i) () (ii) This arises by de®nition.

(ii) () (iii) This is proved by considering orthogonal spaces and using (37) and (41). &

If, for ! » O , there is an operator P

_!

such that y

^!_:

ˆ P

_!

y

^O_:

, that is the regional observation y

^!_:

may be obtained from the global observation y

^O_:

, then the exact remediability on O , implies the exact remediability on !.

This is possible in particular case where C is injective or if

Im … C † ˆ Im … Ci

_!

p

_!

† Im … Ci

_!^c

p

_!^c

†;

where !

^c

ˆ On!.

In the case of p actuators …O

i

; g

_i

†

iˆ1;p

, the character- ization of the regional exact remediability is given in the following proposition deriving from (39) and proposi- tion 3.2.

Proposition 3.4: … S

_P

† ‡ … E † is !-exactly remediable on

‰ 0; T Š if and only if there exists ®

!

> 0 such that

…

T

0

k S

^?

… T ¡ s † i

_!

p

_!

C

^?

³ k

²L²…O†

ds 4 ®

_!

X

^p

iˆ1

…

T

0

h g

_i

; S

^?

… T ¡ s † i

_!

p

_!

C

^?

³i

²

ds;

8³ 2 Y

^?

: … 44 † Let …’

nj

†

ⁿ⁵¹

jˆ1;rn

be an orthonormal basis of eigenfunctions of A in L

²

…O† , and …¶

n

†

n51

…¶

n

with multiplicity r

_n

) the associated eigenvalues. The semi- group … S … t ††

t50

generated by A is given by

S … t † z ˆ X

n51

e

^¶nt

X

^rn

jˆ1

h z; ’

_nj

i’

nj

: … 45 †

It is easy to show the following results.

Corollary 3.1: … S

_P

† ‡ … E † is !-exactly remediable on

‰ 0; T Š , if and only if there exists ®

!

> 0 such that 8 ³ 2 Y

^?

we have

X

n51

1

2¶

n

… e

^2¶nT

¡ 1 † X

^rn

jˆ1

h’

nj

; C

^?

³i

²!

4 ®

_!

X

^p

iˆ1

…

T 0

X

n51

e

^{¶n…T¡s†}

X

^rn

jˆ1

h’

nj

; C

^?

³i

_!

h g

_i

; ’

nj

i

Á !

2

ds:

… 46 † If the output is given by means of q sensors

… D

_i

; h

_i

†

iˆ1;q

, we have the following corollary.

Corollary 3.2: … S

P

† ‡ … E † is !-exactly remediable on

‰ 0; T Š if and only if there exists ®

!

> 0 such that 8 ³ ˆ …³

1

; . . . ; ³

q

†

^T

2

^q

X

n51

1

2¶

n

… e

^2¶nT

¡ 1 † X

^rn

jˆ1

X

^q

lˆ1

³

l

h h

_l

; ’

nj

i

_!

Á !

2

4 ®

!

X

^p

iˆ1

…

T 0

X

n51

e

^{¶n…T¡s†}

X

^rn

jˆ1

h g

_i

; ’

nj

i X

^q

lˆ1

³

l

h h

_l

; ’

nj

i

!

Á !

2

ds:

… 47 † This corollary, may be used as a technical tool for characterizing the exact remediability.

3.3.2. Regional e cient actuators . One of the major interests of this work is to characterize the structures of actuators, which may lead to regionally remediable systems. Such actuators will be said to be regional e - cient. We introduce hereafter the notion of !-e cient actuators, by analogy with !-strategic actuators.

Downloaded by [University of Sydney] at 16:45 29 May 2013

De®nition 3.3: The actuators …O

i

; g

_i

†

iˆ1;p

are said to be

!-e cient if the excited system … S

_P

† together with the output … E † is !-weakly remediable. &

In the case of p actuators and a semigroup given by (45), we have the following characterization result.

Proposition 3.5: The actuators …O

i

; g

_i

†

iˆ1;p

are !-e - cient if and only if

ker … p

_!

C

^?

† ˆ \

n51

ker … M

n

f

^!n

†; … 48 † where M

_n

is given by (16) and f

_n^!

is de®ned by

f

_n^!

: Y

^?

¡!

^rn

;

³ 7¡! f

_n^!

…³† ˆ …h C

^?

³; ’

n1

i

!

; . . . ; h C

^?

³; ’

nrn

i

!

†

^T

:

… 49 † Proof: Let ³ 2 Y

^?

; we have

R

^?_!

³ ˆ X

n51

e

^{¶n…T¡ †}

X

^rn

jˆ1

h i

_!

p

_!

C

^?

³; ’

nj

i’

nj

and, by analyticity, we obtain R

^?_!

³ ˆ 0 () X

^rn

jˆ1

h i

_!

p

_!

C

^?

³; ’

nj

i ’

nj

ˆ 0; 8 n 5 1 () h C

^?

³; ’

nj

i

!

ˆ 0; 8 n 5 1; 8 j ˆ 1; r

n

() p

_!

C

^?

³ ˆ 0:

Then

ker … R

^?_!

† ˆ ker … p

_!

C

^?

† : … 50 † On the other hand

B

^?

R

^?_!

³ ˆ X

n51

e

^{¶n…T¡ †}

X

^rn

jˆ1

h g

_l

; ’

nj

ih i

_!

p

_!

C

^?

³; ’

nj

i

Á !

T

lˆ1;p

and also, by analyticity, we have B

^?

R

^?_!

³ ˆ 0

() X

^rn

jˆ1

h g

_l

; ’

nj

ih C

^?

³; ’

nj

i

!

ˆ 0; 8 n 5 1; 8 l ˆ 1; p:

According to (49), we have B

^?

R

^?_!

³ ˆ 0

() M

n

f

n^!

… ³ † ˆ 0; 8 n 5 1;

then

ker … B

^?

R

^?_!

† ˆ \

n51

ker … M

_n

f

_n^!

† … 51 † The result derives from the equality (43). &

Now, if the output is given by q zone sensors

… D

_i

; h

_i

†

14i4q

with h

_i

2 L

²

… D

_i

†; D

_i

ˆ supp … h

_i

† » O;

measure … D

_i

\ !) > 0, the characterization of !-e cient actuators is given by the following.

Proposition 3.6: The actuators …O

i

; g

i

†

iˆ1;p

are !-e - cient if and only if

\

n51

ker … M

_n

G

^T_n;!

† ˆ f 0 g; … 52 † where

G

_n;!

ˆ …h h

_i

; ’

nj

i

!

†

^iˆ1;q

jˆ1;rn

: Proof: For ³ ˆ …³

1

; . . . ; ³

q

†

^T

2

^q

, we have

p

_!

C

^?

³ ˆ X

^q

iˆ1

³

i

p

_!

h

i

:

Since the functions … h

_i

†

iˆ1;q

are linearly independent, because D

_i

\ D

_j

ˆ 1 for i 6ˆ j, and measure

… D

_i

\ ! † > 0, then … p

_!

h

_i

†

iˆ1;q

are linearly independent and consequently

ker … p

_!

C

^?

† ˆ f 0 g;

using (50), we have

ker … R

^?_!

† ˆ f 0 g:

On the other hand, B

^?

R

^?_!

³ ˆ X

n51

e

^{¶n…T¡ †}

X

^rn

jˆ1

h g

_l

; ’

nj

i X

^q

iˆ1

³

i

h h

_i

; ’

nj

i

!

Á !

T

lˆ1;p

;

then

B

^?

R

^?_!

³ ˆ 0 () X

^rn

jˆ1

h g

_l

; ’

nj

i X

^q

iˆ1

³

i

h h

_i

; ’

nj

i

!

ˆ 0;

8 l ˆ 1; p; 8 n 5 1 () M

_n

G

^T_n;!

³ ˆ 0; 8 n 5 1 and hence

ker … B

^?

R

^?_!

† ˆ \

n51

ker … M

n

G

^Tn;!

† : … 53 † The result derives from equality (43). &

Then we deduce the following corollaries.

Corollary 3.3: If there exists n

₀

5 1 such that

rank … M

_n0

G

^T_n0;!

† ˆ q; … 54 † then the actuators …O

i

; g

_i

†

iˆ1;p

are !-e cient.

Proof: This proof derives from (52) and ker … M

_n0

G

^T_n0;!

† ˆ f 0 g using (54). &

We also have

Corollary 3.4: If there exists n

₀

5 1 such that

Downloaded by [University of Sydney] at 16:45 29 May 2013

rank … G

^T_n0;!

† ˆ q … 55 † and

rank … M

_n0

† ˆ r

_n0

; … 56 † then the actuators …O

i

; g

_i

†

iˆ1;p

are !-e cient.

Proof: This proof is immediate with the conditions (55) and (56), which imply that ker … M

_n0

G

^T_n0;!

† ˆ f 0 g , and the result derives from (52). &

The following are clear.

(a) The conditions (55) and (56) ˆ) (54).

(b) The conditions (55) and (56) ˆ) q 4 r

n₀

4 p.

(c) The condition (54) ˆ) q 4 p.

(d) The hypothesis p 5 sup

_n51

… r

_n

† is necessary for actuators … D

_i

; g

_i

†

iˆ1;p

to be !-strategic (El Jai et al.

1995), but it is not necessary for them to be !-e - cient.

(e) The condition q 4 p is not necessary for actuators to be !-e cient.

Indeed, in the case of one actuator …O

1

; g

1

† and q sensors … D

_i

; h

_i

†

iˆ1;q

, with q > 1, we have

M

n

ˆ …h g

1

; ’

nj

i†

jˆ1;rn

of dimension … 1; r

n

†;

G

^T_n;!

ˆ …h h

_i

; ’

nj

i

_!

†

^jˆ1;rn

iˆ1;q

of dimension … r

_n

; q †:

Then

M

_n

G

^T_n;!

ˆ X

^rn

jˆ1

h g

₁

; ’

nj

ih h

_l

; ’

nj

i

!

Á !

lˆ1;q

of dimension … 1; q †

and, using proposition 3.6, …O

1

; g

₁

† is !-e cient if and only if T

n51

ker … M

_n

G

^T_n;!

† ˆ f 0 g: Then, if there exists n

₁

; n

₂

; . . . ; n

_m

such that n

_i

6ˆ n

_j

for i 6ˆ j and

\

iˆ1;m

ker … M

_ni

G

^T_ni;!

† ˆ f 0 g; … 57 † then …O

1

; g

₁

† is !-e cient. In the particular case when m ˆ q, the condition (57) is equivalent to

X

^rn1

jˆ1

h g

1

; ’

n₁j

ih h

1

; ’

n₁j

i

!

X

^rn1

jˆ1

h g

1

; ’

n₁j

ih h

q

; ’

n₁j

i

!

.. .

. . .

.. . X

^rnq

jˆ1

h g

₁

; ’

nqj

ih h

₁

; ’

nqj

i

!

X

^rnq

jˆ1

h g

₁

; ’

nqj

ih h

_q

; ’

nqj

i

!

6ˆ 0: … 58 †

Remark 3.1:

(i) One can consider the problem of regional remedia- bility in ! from T

₀

. This consists in studying the existence of a control term Bu reducing the regional observation to the normal case, from T

₀

, that is for all T 5 T

₀

. In the case of a semi-group given by (45), we show by analyticity that the previous result (on regional remediability, regional e cient actua- tors, their characterizations , etc.) remains true from T

₀

, but that this is not generally true.

(ii) Let us also note that in the pointwise actuators case, the state p

_!

z of the system in ! is in a space V such that V

⁰

» L

²

…!† » V . The characterizations are similar to those obtained for zone actua-

tors. &

4. Regional remediability and regional controllability In this section, we study the relationship between regional controllability and regional remediability, and therefore between !-strategic actuators and !-e cient actuators.

Proposition 4.1: If … S † is !-exactly (or weakly) control- lable on ‰ 0; T Š , then … S

_P

† ‡ … E † is !-exactly (or weakly) remediable on ‰ 0; T Š .

Proof: For the exact remediability, we consider

³ 2 Y

^?

; then we have

k S

^?

… T ¡ † i

_!

p

_!

C

^?

³k

²L²…0;T;L²…O††

ˆ

…

T

0

k S

^?

… T ¡ s † i

_!

p

_!

C

^?

³k

²L²…O†

ds 4

…

T

0

k S

^?

… T ¡ s † i

_!

k

²

ds k p

_!

C

^?

³ k

²L²…!†

4 M

_!

k p

_!

C

^?

³k

²L²…!†

;

with M

_!

> 0, because the semigroup is bounded on

‰ 0; T Š . Since … S † is !-exactly controllable, there exists

®

⁰_!

> 0 such that

k p

_!

C

^?

³k

L²…!†

4 ®

⁰!

k B

^?

S

^?

… T ¡ † i

_!

p

_!

C

^?

³k

L²…0;T;U^?†

and, consequently, there exists ®

!

ˆ M

_!

…®

⁰!

†

²

> 0 such that

k S

^?

… T ¡ † i

_!

p

_!

C

^?

³ k

²L²…0;T;L²…O††

4 ®

!

k B

^?

S

^?

… T ¡ † i

_!

p

_!

C

^?

³k

²L²…0;T;U^?†

: The result derives from proposition 3.2.

For the weak remediability, let ³ 2 ker … H

^?

i

_!

p

_!

C

^?

† ; we have H

^?

i

_!

p

_!

C

^?

³ ˆ 0; then p

_!

C

^?

³ ˆ 0, because ker … H

^?

i

_!

† ˆ f 0 g . Since R

^?_!

ˆ S

^?

… T ¡ † i

_!

p

_!

C

^?

, then

³ 2 ker … R

^?_!

† and consequently ker … H

^?

i

_!

p

_!

C

^?

† ˆ

Downloaded by [University of Sydney] at 16:45 29 May 2013

ker … B

^?

R

^?_!

† » ker … R

^?_!

† ; then ker … B

^?

R

^?_!

† ˆ ker … R

^?_!

† ; we have therefore the regional weak remediability. &

From the previous results, we deduce that !-strategic actuators are !-e cient. The converse of proposition 4.1 is not true. This is illustrated in the following example.

Example 4.1: We consider a di€ usion process described by the following system:

… S

P

†

@z

@t … x; t † ˆ Az … x; t † ‡ f … x; t † ‡ Bu … t † in O Š 0; T ‰;

z … x; 0 † ˆ 0 in O;

z … x; t † ˆ 0 on @O Š 0; T ‰;

8 >

> >

> >

<

> >

> >

> :

… 59 †

with Az ˆ ¢z for z 2 D … A † ˆ H

²

…O† \ H

₀¹

…O† , f 2 L

²

… 0; T; L

²

…O†† and u 2 L

²

… 0; T; U† . (S) is aug- mented by the output equation

… E † y … t † ˆ Ci

_!

p

_!

z …:; t †: … 60 † (i) Firstly for B ˆ I; f ˆ 0 and ! ˆ O , the system (59) is !-exactly controllable on H

₀¹

… ! † , but not on L

²

… ! † . However, inequality (39) is veri®ed on L

²

… ! † for B ˆ I, then … 59 † ‡ … 60 † is !-exactly reme- diable for every output operator C.

(ii) Consider now the particular case where O ˆŠ 0; 1 ‰ and Bu … t † ˆ g …:† u … t † , the system (59) can be written in the form

@z

@t … x; t † ˆ @

²

z

@¹

²

…¹; t † ‡ g …¹† u … t † ‡ f …¹; t † in Š 0; 1 ‰ Š 0; T ‰ ; z … ¹; 0 † ˆ z

₀

… ¹ † in Š 0; 1 ‰ ;

z …¹; t † ˆ 0 on f 0; 1 g Š 0; T ‰;

… 61 †

together with the output equation (60). Consider an actuator … I; g † , with I »Š 0; 1 ‰ and g 2 L

²

… I † , a sensor

… J ; h † , with J »Š 0; 1 ‰ and h 2 L

²

… J † and an unknown disturbance f 2 L

²

… 0; T; L

²

…Š 0; 1 ‰†† .

For n 5 1, we have ’

n

… ¹ † ˆ 2

¹⁼²

sin … np¹ † and

¶

n

ˆ ¡ n

²

p

²

. As we assume that h 6ˆ 0, there exists n

0

such that h h; ’

n₀

i 6ˆ 0. Thus we have the following results.

The actuator … I; g † is O -e cient if h g; ’

_n0

i 6ˆ 0, or equivalently

…

I

g … ¹ † sin … n

0

p¹ † d¹ 6ˆ 0 … 62 † by corollary 3.3.

For ! ˆ O and g ˆ ’

n0

, the actuator …Š 0; 1 ‰; g † is O -e cient but not O -strategic, because the condition

…

I

g … ¹ † sin … np¹ † d¹ 6ˆ 0; 8 n 5 1; … 63 † is not satis®ed.

For ! ˆŠ 0;

¹₂

‰ , h ˆ ’

1

and g ˆ ’

2

, the actuator

…Š 0; 1 ‰ ; g † is not O -e cient because h g; ’

n

ih h; ’

n

i ˆ 0 8 n 5 1;

but it is !-e cient since, for n

₀

ˆ 2, we have h g; ’

₂

ih h; ’

₂

i

!

ˆ h’

1

; ’

2

i

!

6ˆ 0;

then corollary 3.3 gives the result. &

Let us note that the results are analogous in the two- dimensional case.

5. Regional exact remediability with minimum energy In this section, we consider the problem which consists to compensate a disturbance by considering a minimum energy control.

5.1. Problem statement

Under the same hypothesis as in } 1, consider the system … S

P

† together with the output … E † :

… S

P

† zz _ … t † z … 0 †

ˆ ˆ

Az … t † ‡ f … t † ‡ Bu … t †; 0 < t < T ; z

₀

;

»

… 64 †

… E † y

^!

… t † ˆ Ci

_!

p

_!

z … t †: … 65 † For z

₀

2 L

²

…O† and f 2 L

²

… 0; T ; L

²

…O†† , the regional exact remediability consists in ®nding a control u 2 L

²

… 0; T ; U† such that

y

^!_u;f

… T † ˆ Ci

_!

p

_!

S … T † z

₀

or equivalently

Ci

_!

p

_!

Hu ‡ y

^!_f

ˆ 0:

Now the problem that we consider consists in exploring among such controls, the control that is of minimum energy. That is to say, given a region ! and a disturb- ance f , does a control which realizes the f ¡ !-exact remediability and minimizing the cost function

J … u † ˆ k u k

²L²…0;T;U†

exist? This will be given in theorem 5.1.

Firstly let us show results which will be used to proof theorem 5.1.

5.2. Preliminaries results

For ³ 2 Y

^?

² Y, the mapping k³k

_F!

ˆ

…

T

0

k B

^?

S

^?

… T ¡ s † i

_!

p

_!

C

^?

³k

²_U^?

ds

1=2

… 66 †

Downloaded by [University of Sydney] at 16:45 29 May 2013