GRADIENT REMEDIABILITY IN LINEAR DISTRIBUTED PARABOLIC SYSTEMS ANALYSIS, APPROXIMATIONS AND SIMULATIONS

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GRADIENT REMEDIABILITY IN LINEAR DISTRIBUTED PARABOLIC SYSTEMS

ANALYSIS, APPROXIMATIONS AND SIMULATIONS

S. Rekkab* and S. Benhadid

Mathematics Department, Faculty of Exact Sciences, University Mentouri, Constantine, Algeria

Received: 28 April 2017 / Accepted: 22 August 2017 / Published online: 01 September 2017

ABSTRACT

The aim of this paper is the introduction of a new concept that concerned the analysis of a

large class of distributed parabolic systems. It is the general concept of gradient remediability.

More precisely, we study with respect to the gradient observation, the existence of an input

operator (gradient efficient actuators) ensuring the compensation of known or unknown

disturbances acting on the considered system. Then, we introduce and we characterize the

notions of exact and weak gradient remediability and their relationship with the notions of

exact and weak gradient controllability. Main properties concerning the notion of gradient

efficient actuators are considered. The minimum energy problem is studies, and we show how

to find the optimal control, which compensates the disturbance of the system. Approximations

and numerical simulations are also presented.

Keywords: actuators efficient; disturbance; gradient; parabolic systems; remediability;

sensors.

Author Correspondence, e-mail: rekkabsoraya@yahoo.fr doi: http://dx.doi.org/10.4314/jfas.v9i3.18

ISSN 1112-9867

1. INTRODUCTION

Various real problems can be formulated within certain concepts of distributed systems

analysis. These concepts consist of a set of notions as controllability, detectability,

observability, remediability…, which enable a better knowledge and understanding of the

system to be obtained. These concepts have been studies at different degrees (exact, weak,

extended). Systems analysis can be done from a purely theoretical viewpoint [1,2]. However,

the study may be also become concrete, in some sense, by using the structures of actuators

and sensors. Thus, one can study the different concepts of controllability via actuators

structures [3-5] or the different concepts of observability via sensors structures [6-8].

An extension of these concepts that is very important in practical applications is that of

gradient controllability [9], gradient detectability [10] and gradient observability [11-13].

These concepts are of great interest from a more practical and control point of view since

there exist systems that cannot be controllable but gradient controllable or that cannot be

observable but gradient observable or that cannot be detectable but gradient detectable and

they provide a means to deal with some problems from the real world, for example in the

thermic isolation problem it may be that the control is only required to vanish the

temperature-gradient before crossing the wall.

Hence, with the same preoccupation, we introduce in this paper, a new concept that is gradient

remediability of distributed parabolic systems. We recall that the notion of remediability

consists in studying the existence of a convenient input operator (efficient actuators), ensuring

the remediability of any disturbance acting in the considered systems. This problem is

particularly motivated by pollution problems and so called space compensation or

remediability problem. The notions of remediability and efficient actuators are introduced and

studied first for discrete systems and there for continuous systems of a finite time horizon and

for other situations (regional and asymptotic cases, internal or boundary actions of

disturbances) [14-16].

This paper is organized as follows: In the second paragraph, we start by presenting the

notations and some preliminary material. After, we recall the notions of exact and weak

of exact and weak gradient remadiability. We show how to find an input operator (actuators)

with respect to the output (gradient observation) that ensures the gradient remediability of a

disturbed parabolic system. By analogy with the relation between the remediability and the

controllability examined in the finite time case, it is then natural to study, in the paragraph 4,

the relationship between the gradient remediability and the gradient controllability. We show

that the gradient remediability is weaker and more supple than the gradient controllability of

the parabolic systems, that is to say, if any parabolic systems are gradient controllable, then it

is gradient remediable. The fifth paragraph recall the notion of gradient strategic actuators and

we give a characterization of gradient remediability which shows that the gradient

remediability of any system may depend on the structure of the actuators and sensors. Then

we introduce and we characterize the notion of gradient efficient actuators. In paragraph 6,

using an extension of Hilbert Uniqueness Method (H.U.M), we examine the problem of

gradient remediability with minimum energy, and we give the optimal control that

compensate an arbitrary disturbance. In the last paragraph, approximations and numerical

results are presented.

2. CONSIDERED SYSTEM

Let  be an open and bounded regular subset of IRn



n1,2,3



with a smooth boundary  . ForT 0 , we denoted by Q



0,T



, 



0,T



. Consider a parabolic system defined by









 









 





                  0 , 0 , , , , 0 t y x y x y Q t x f t Bu t x y A t x t y  (1)

Where A is a second order linear differential operator with compact resolvent and which generates a strongly continuous semi-group



S

 

t



_t0on the Hilbert space L2

 

 .



S*

 

t



_t0 is considered for the adjoint semi-groupe of



S

 

t



_t0 .



U X



u L



TU



B

L

, ,  2 0, ; where U is a Hilbert space representing the control space and

 

 H₀1

The system (1) admits a unique solution yC



0,T;H1₀

 





C1



0,T;L2

 





[17] given by

 



 



_







 



_





  

t t ds s f s t S ds s Bu s t S y t S t y 0 0 0

and it is augmented by the output equation

 

t C y

 

t

z   (2) where CL





L2

 





n,O



, O is a Hilbert space (observation space). In the case of an observation on



0,T



with

q

sensors, we take generally q

IR O  .The operator



is defined by

 



 



                     n n x y x y x y y y L H , , , : 2 1 2 1 0 

while



* its adjoint operator.

3. GRADIENT REMEDIABILITY

The system (1) is disturbed by the force f assumed unknown and excited by a control u that will be chosen so as to compensate for the disturbance f . In the autonomous case,

without disturbance



f 0



and without control



u0



, this same system is written













 





                0 , 0 , , , 0 t y x y x y Q t x y A t x t y  (3)

The system (3) admits a unique solution yC



0,T;H1₀

 





C1



0,T;L2

 





[17] given by

 

 

0

y t S t

y  then the observation on



0,T



is normal and it is given by z

 

t CS

 

t y0. But if the control u0and the disturbance f 0, the observation noted z_u,f is disturbed such that

 

t  z_u,f 

 

 

_





  

 

_





  

t t ds s f s t S C ds s Bu s t S C y t S C 0 0 0

The problem consists to study the existence of an input operator B (actuators), with respect to the output operator C (sensors), ensuring at the timeT , the gradient remediability of any

disturbance on the system, that is:

For any f L2



0,T;X



, there exists uL2



0,T;U



such that

 

 

0 , T C ST y

z_{u f}   this is

equivalent to CHuCFf 0where H and F are two operators defined by







  



    T ds s Bu s T S Hu u X U T L H 0 2 ; , 0 :







  



    T ds s f s T S Ff f X X T L F 0 2 ; , 0 :

This leads to the following definitions

Definition 1:

1) We say that the system (1) augmented by the output equation (2) is exactly gradient f -

remediable on



0,T



, if there exists a control uL2



0,T;U



such that 0    Hu C Ff C

2) We say that the system (1) augmented by the output equation (2) is weakly gradient f -

remediable on



0,T



, if for every



0,there exists a control uL2



0,T;U



such that

     O Ff C Hu C

3) We say that the system (1) augmented by the output equation (2) is exactly (resp. weakly)

gradient remediable on



0,T



, if for every f L2



0,T;X



the system (1)–(2) is exactly (resp. weakly) gradient f - remediable.

Proposition 1:

Let us consider f L2



0,T;X



.

1) The system (1)–(2) is exactly gradient f - remediable on



0,T



if and only if



C H



Ff

C Im  Im

2) The system (1)–(2) is weakly gradient f - remediable on



0,T



if and only if



C H



Ff

C Im  Im

Proof:

exists uL2



0,T;U



such that CHuCFf 0 , that is



u



C Hu1 H C Hu C Ff C        with



u₁ u



u₁L2



0,T;U



then H C Ff C Im  .

Conversely, we assume that CFf ImCH, then there exists uL2



0,T;U



such that

Hu C Ff

C   that is CFf CHu0 this gives CFf CH



u



0. We put



T U



L u

u₁   2 0, ; where the system (1)–(2) is exactly gradient f - remediable.

2) We assume that the system (1)–(2) is weakly gradient f - remediable on



0,T



, then



T U



L u 0, ; , 0   2   such that     O Hu C Ff C that is 0,uL2



0,T;U



such that CFf CH



u



_O  . We put u u L2



0,T;U



1   ,

then  0,u₁L2



0,T;U



such that    

O Hu C Ff C ₁ , this gives



C H



Ff C Im  .

Conversely, we assume that CFf Im



CH



, then  0,u₁L2



0,T;U



such that



 



Ff C Hu _O

C ₁ . We put u₁  u L2



0,T;U



, then 0,uL2



0,T;U



such that CFf CHu_O  where the system (1)–(2) is weakly gradient f - remediable. Proposition 2:

1) The system (1)–(2) is exactly gradient remediable on



0,T



if and only if

H C F

C Im  Im

2) The system (1)–(2) is weakly gradient remediable on



0,T



if and only if

H C F C Im  Im Proof:

1) We assume that the system (1)–(2) is exactly gradient remediable on



0,T



then



T X



L f  2 0, ;

 the system (1)–(2) is exactly gradient f - remediable on



0,T



and from the Proposition 1 we have, f L2



0,T;X



: CFf ImCH , this gives

H C F

C  Im 

Im .

Conversely, we assume that ImCF  ImCH and we show that the system (1)–(2) is exactly gradient remediable on



0,T



. Let f _L2



0,T;X



H C F

C  Im 

Im we have CFfImCH then there exists uL2



0,T;U



such that CFf CHu that is to say CFf CHu0 this gives CFf CH



u



0. We put u₁  u L2



0,T;U



where the system (1)–(2) is exactly gradient remediable. 2) We assume that the system (1)–(2) is weakly gradient remediable on



0,T



thenf L2



0,T;X



the system (1)–(2) is weakly gradient f - remediable on



0,T



and from the Proposition.1 we have, f L2



0,T;X



: CFf ImCH , this gives

H C F

C  Im 

Im .

Conversely, we assume that ImCF  ImCHand we show that the system (1)–(2) is weakly gradient remediable on



0,T



. Let f L2



0,T;X



then CFfImCF. Since

 F

C

Im ImCH then CFf Im



CH



this leads to 0, u L2



0,T;U





 

 such

that CFf CHu  by putting u₁  u  L2



0,T;U



this gives  0,u₁L2



0,T;U



such that    

O

Hu C Ff

C ₁ where the system

(1)–(2) is weakly gradient remediable on



0,T



.

4. GRADIENT REMEDIABILITY AND GRADIENT CONTROLLABILITY

By analogy with the relation between the remediability and the controllability examined in the

finite time case, it is then natural to study, in this paragraph, the relationship between the

gradient remediability and the gradient controllability. Firstly, we recall the definitions of

exact and weak controllability.

Definition 2:

1) The system (1) is said to be exactly gradient controllable on



0,T



if yd 



L2

 





n there exist a control uL2



0,T;U



such thaty

 

T  yd .

2) The system (1) is said weakly gradient controllable on



0,T



if yd 



L2

 





n , 



0 there exist a control uL2



0,T;U



such that

 

          _{L } n d y T y ₂ .

Proposition 3:

1) If the system (1)–(2) is exactly gradient controllable on



0,T



, then it is exactly gradient remediable on



0,T



.

2) If the system (1)–(2) is weakly gradient controllable on



0,T



, then it is weakly gradient remediable on



0,T



.

Proof:

1) We assume that the system (1)–(2) is exactly gradient controllable and let yd S

 

T y0

then there exists uL2



0,T;U



such that y

 

T S

 

T y0 that is to say

 

0

 

0 y T S Ff Hu y T S   

 this leads to HuFf 0 then let

0   

Hu C Ff

C and then the system (1) – (2) is exactly gradient remediable.

2) We assume that the system (1)–(2) is weakly gradient controllable and let yd S

 

T y0

then  0, u L2



0,T;U



  such that

 

 

           _{L } n y T S T y 0 ₂ that is   _{ }       _{L } n Ff

Hu 2 . Since the operator C is continue, consequently we have

  n L O k Hu Ff Ff C Hu C       _      

 2 with k 0 where the system (1)–(2) is

weakly gradient remediable.

5. GRADIENT REMEDIABILITY, SENSORS AND ACTUATORS

We suppose that the system (1) is excited by p zone actuators



_i,g_i



_1ip,g_iL2

 

_i ,

 





_i supp g_i in this case the control space is U IRp and the operator:

 



   

 



_

     

     p i i i i p p t u x g x Bu t u t u t u t u X IR B 1 2 1 , , , :   

Its adjoint is given by

p T p p z IR g z g z g z B _           , , , , , , 2 2 1 1 * _ (4)

Also suppose that the output of the system (2) is given by qsensors



D_i,h_i



_1iq,h_i L2

 

D_i , being the spatial distribution,D_i supph_i , for i1 , ,qand D i Dj  fori  , then j the operator C is defined by:

 





 

 

 

 

T n i Dq i q n i D i n i D i q n t y h t y h t y h t Cy IR L C         







   1 2 1 2 1 1 1 2 , , , , , , :  its adjoint is given by C with for*  



₁,₂,,_q



T IRq

 

 

 

 

 

 

_       







   q i i i i D q i i i i D q i i i i D x h x x h x x h x C 1 1 1 * , , ,         (5) Lemma 1 [18]:

Let V,W and Z be reflexive Banach spaces, PL



V,Z



andQL



W,Z



. Then the following properties are equivalent:

i. Im P ImQ

ii.  0 such that , * '

' * * ' * Z z z Q z P W V *   

We have the following characterizations: Proposition 4:

1) The system (1)–(2) is exactly gradient remediable on



0,T



if and only if there exists

0 



such that for everyIRq, we have





 





           _p IR T L X T L B S T C C T S ; , 0 2 * * * * ' ; , 0 2 * * * . .







2) The system (1) – (2) is weakly gradient remediable on



0,T



if and only if



* * * *





* * *



ker

ker B F  C  F  C

Proof:

1) It follows from the fact that F**C* S*



T .



*C* and that H**C*B*S*



T.



*C*

and since the Proposition 2, we put PCF and QCH and using the Lemma 1. 2) We assume that the system (1)–(2) is weakly gradient remediable on



0,T



and we show

that ker



B*F**C*



ker



F**C*



. Let IRq such that B*F**C* 0, and we have









       . . * * * * * T S B H T S F

then, B*F**C* 0H**C* 0 this gives ker



H * *C*



and we have















 

 ker * * *

ImC H H C . Since the hypothesis and the Proposition 2, we have

H C F C Im  Im then ImCF 



ker



H**C*





 f L2



0,T;X



:CFf









 * * *

kerH C  CFf, 0 because ker



H * *C*



, this gives







ImCF



 ker



F**C*



, where the result.

Conversely, assume that ker



B*F**C*



ker



F**C*



and we show that

H C F

C Im 

Im . Let f L2



0,T;X



such that f  ImCF , we have











 

 ker * * *

ImC H H C . For every IRq such that H**C*0 , that is 0 * * * *   C F

B we have F**C* 0 because ker



B*F**C*



ker



F**C*



then CFf, 0, where the result. Corollary 1:

The system (1)–(2) is exactly gradient remediable on



0,T



if and only if  0 such that

q IR   , we have









 





   _    p i T L i T X ds g S T s C ds C s T S i 1 ₀ 2 * * * 0 2 ' * * * 2 ,    Proof:

Since the Proposition 4, the system (1)–(2) is exactly gradient remediable on



0,T



if and only if there exists 0 such that for every q

IR   , we have





 





  2 ; , 0 * * * * 2 ' ; , 0 * * * 2 2 . . C _{L T X} B S T C _{L T IR}p T S       

by using (4) the formula of the operator B*, we have





_







      p i T i T X ds g S T s C ds C s T S 1 0 2 * * * 0 2 ' * * * ,   

where the result.

In the following, without loss of generality we consider, the system (1) with a dynamics Aof the form

   

 



 



 

 





   1 0 2 1 2 1 , ,w w y D A H H y Ay m r j j m L j m m m   where

 

1 1    m m r j j m

w is an orthogonal basis in H1₀

 

 of eigenvectors of A orthonormal in

 



2

L , associated to eigenvalues _m0 with a multiplicity r . Then, the operator _m A generates on the Hilbert space L2

 

 a strongly continuous semi-group



S

 

t



_t0 given by [1,19]:

 

 





    m r j j m L j m m t m _{y w w} e y t S 1 2 1 ,  (6) Corollary 2:

The system (1)–(2) is exactly gradient remediable on



0,T



if and only if  0 such that

q IR   , we have





_{ }  











           _{ }           p i T j m i m r j j m m s T m m m r j n L mj T m m ds w g w C e w C e 1 0 2 1 * 1 1 1 2 2 * 2 , , , 1 2 1       Proof:

Since the Corollary 1, the system (1)-(2) is exactly gradient remediable on



0,T



if and only if there exists



0 such that for every IRq, we have





_







       p i T i i T X ds g S T s C ds C s T S 1 0 2 * * * 0 2 ' * * * ,   

By using (6) the formula of the operator S and since it is auto-adjoint, we obtain





   









_  _      T _r j j m m s T T L ds e C w ds C s T S m m 0 1 2 * * 1 2 0 2 * * * , 2   





_



     m r j j m T m m m w C e 1 2 * 2 1 , 1 2 1    and





   









                  p i T j m i m r j j m m s T m p i T i L i S T s C ds e C w g w ds g 1 0 2 1 * * 1 1 0 2 2 * * * , , ,   

where the result.

Corollary 3:

The system (1)–(2) is exactly gradient remediable on



0,T



if and only if  0 such that

q IR   , we have





      _x ds w h w g e x w h e p i m r j n k q l k mj l l i L j m i m s T m T m m r j n k q l k L D_l mj l l T m m 2 1 1 1 1 2 1 0 1 1 1 1 2 2 2 , , , 1 2 1















                               Proof:

Since the Corollary 2, the system (1)-(2) is exactly gradient remediable on



0,T



if and only if there exists



0 such that for every IRq, we have





_{ }  











           _{ }           p i T j m i m r j j m m s T m m m r j n L mj T m m ds w g w C e w C e 1 0 2 1 * 1 1 1 2 2 * 2 , , , 1 2 1      

and by using (5) the formula of the operator C , we have *

   n L n mj mj mj q l l l l D q l l l l D q l l l l D n L mj x w x w x w h h h w C       _          _                                                







2 2 1 1 1 1 2 * , ,          Dl L n k q l k mj l l x w h 2 1 1 ,

 

     

where the result.

This characterization shows that the remediability of a system may depend on the structure of

the actuators and sensors.

By analogy with the concept of gradient strategic actuator, we introduce the notion of gradient

efficient actuator, as follows:

Definition 3:

The actuators



_i,g_i



_1ip,g_i L2

 

_i are said to be gradient efficient if the system (1)–(2) so excited is weakly gradient remediable.

The actuators gradient efficient define actions with the structure (spatial distribution, location

the following characterization of the gradient efficient actuators.

Form1, let M_m be the matrix of order



p r_m



defined by M



g w



i p

j i j m i m  , ,1 

and 1 j r_m and let G_m be the matrix of order



q r_m



defined by

q i x w h G j i n k k j m i m _           



 1 , , 1 and 1 j r_m. Proposition 5:

The actuators



_i,g_i



_1ip,g_iL2

 

_i are gradient efficient if and only if







m m



m f M C ker ker 1 * *  

  . Where, for IRqand m1,

 

  m f





T rm m r m m m C w C w IR w C    * *, ₁ , * *, ₂ ,, * *, Proof:

We assume that the actuators



_i,g_i



_1ip,g_iL2

 

_i are gradient efficient and we show

that







m m



m f M C ker ker 1 * *  

  . Since the Proposition 4, the system (1)-(2) is weakly gradient remediable on



0,T



if and only if ker



B*F**C*



ker



F**C*



. Let q

IR   , we have









* * * * * * * * . C T S B C F B                                               













               m r j L p mj P L mj m T m m r j L mj L mj m T m m r j L mj L mj m T m w g w C e w g w C e w g w C e 1 2 2 * * 1 . 1 2 2 2 2 * * 1 . 1 1 2 1 2 * * 1 . , , , , , ,        and we have m1,

 

                                   







         m r j L p mj P L mj m r j L mj L mj m r j L mj L mj m m w g w C w g w C w g w C f M 1 2 2 * * 1 2 2 2 2 * * 1 1 2 1 2 * * , ; , ; , ;      If we assume that



_{m m}



m f M ker 1     , then





  , 0,



1,2, ,



, 1 ; 1 , ker 1 * * 2          



  g w i p m w C m f M m r j mj i L mj m m     

_

_

1 , , , 2 , 1 , 0 , , 1 1 * * .       





   m p i w g w C e m m r j mj i j m T m _  



* * * *



* * * * ker 0 B F C C F B         where







* * * *



1 ker ker M_mf_m B F C m     that is







* * * *



1 ker ker M_mf_m B F C m     .

On the other hand, we have for every q

IR   ,





  j m m m r j j m T m _{C w w} e C T S C F





         1 1 * * . * * * * * * , .    We assume that



* * *



ker F  C   , then *_* * _0  C F





  0 , . 1 1 * * . * * * * * *       





   j m m m r j j m T m _{C w w} e C T S C F    0 , 1 1 * * _  



  j m m m r j j m w w C   *C* 0ker



*C*



. Then,



* * *





* *



ker

ker F  C   C . If we assume thatker



*C*



, then *C* 0 that is   0 , 1 1 * * . * * *    





   j m m m r j j m T m _{C w w} e C F   



* * *



ker F  C   , then



* *





* * *



ker

ker C  F  C that is ker



*C*



ker



F**C*



.

Where the result.

Corollary 4:

If there exists m₀ 1 such that

q G

rank mT₀  (7)

then the actuators



_i,g_i



_1ip,g_iL2

 

_i are efficient if and only if ker





 

0

1   T m m m G M  . Proof: Let q IR   , then



_{m m}T



m G M ker 1   







MmGmT



 0, m1 p i m x w h w g q l l m r j n k k j m l j m i, , 0, 1, 1, , 1 1 1         





   

  m i p w C w g m r j L mj j m i, , 0, 1, 1, , 1 2 * * _       



 





0, 1  M_mf_m  m



m m



m f M ker 1     this gives





  T m m m G M ker 1 



m m



m f M ker 1   .

On the other hand,ker



*C*



 *C*0, then for m that appears in the hypothesis ₀

and by using (5) the formula of the operator C , we obtain *

  0 1 1 2 0 0 * * , , 1 , 0 , , _m q l l D L n k k j m l l j m j r x w h w C        





    T m T m G G 0 0 0 ker    and

since rankG_mT₀ q , then kerG_mT₀ 

 

0 this gives  0. That is ker



 C* *





 

0

Finally, the proof follows directly from the Proposition 5. Corollary 5:

If there exists m₀ 1 such that rank



G_mT



q

0 and if



M G



q rank _{m m}T  0 0 (8) Or



Mm₀



rm₀ rank  (9)

then the actuators



_i,g_i



_1ip,g_iL2

 

_i are gradient efficient. Proof:

Assume that there exists m₀ 1 such that rank



M_n0G_nT₀



q. The matrix



M_m0G_mT₀



is of order p q . From the theorem of rank to matrices [20], we haverank



M_m0G_mT₀



dim



ker



M_m0G_mT₀





q, and then dim



ker



M_m0G_mT₀





0 witch is equivalent toker





 

0 0 0  T m m G M ker





 

0 1    T m m m G M

 . Since the Corollary 4, that is equivalent to the gradient efficient of the actuators



_i,g_i



_1ip,g_iL2

 

_i .

Now, we suppose that rank



G_mT₀



q and





0

0 m

m r

M

rank  . The matrix



G_mT₀



is of order rm₀ q . By using the theorem of rank to matrices [20], we have



G







G





q

rank _mT  _mT 

0

0 dim ker then, dim



ker



0





0

T m

G . That is equivalent to





 

0

ker G_mT₀  (10) The same, the matrix



Mm₀



is of order p rm₀. By using the theorem of rank for matrices

[19], we have













0 0 0 dim ker m m m M r M

rang   . And from (9), we obtain







ker



0

dim Mm₀  which is equivalent to





 

0

ker Mm₀  (11)

On the other hand, let



M_m G_mT



0 0 ker   , then





0 0 0  T m m G M which gives





0 0 0   T m m G M . From (11), we obtain 0 0 T m

G and from (10), we obtain  0 then ker





 

0

0 0  T m m G M

and then, ker





 

0

1   T m m m G M

 which is equivalent, from the Corollary 4, to the gradient efficient of the actuators



_i,g_i



_1ip,g_iL2

 

_i . Remark 1:

1) The condition (8) q  p.

2) The condition q  p is not necessary for actuators to be gradient efficient. Indeed, in the case of a single actuator



₁,g



and of q sensors



D_i,h_i



_1iq , with q1 ,





m r j j m m g w M    1 , is of order



1r_m



and q i m r j n k k j m i T m x w h G                



1 1 1 , is of order



r_m q



consequently q i m r j n k k j m i j m T m m x w h w g G M               



1 1 1 , , is of order



1q



.

From the Corollary 4, if there exists m₀ 1 such that rankGT_m q

0 , then



1,g



is

gradient efficient if and only if ker





 

0

1   T m m m G M

 . Then there exists n₁,n₂,,n_m such that n i nj for i  and j





 

0 ker , 1   T i n i n m i G M  (12) then



₁,g



is gradient efficient. In particular ifm  , the condition (12) is equivalent to q

0 , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1         









        q n r j n k k j q n q j q n q n r j n k k j q n j q n n r j n k k j n q j n n r j n k k j n j n x w h w g x w h w g x w h w g x w h w g    

6. GRADIENT REMEDIABILITY WITH MINIMUM ENERGY

For f L2



0,T;X



, we study the existence and the unicity of an optimal control



P



IR T L

u 2 0, ; ensuring, at the timeT, the gradient remediability of the disturbance f

such that CHuCFf  0. That is the set defined by







 2 0, ; /    0



 u L T IR C Hu C Ff

D P (13)

is non empty.

We consider the function

 

2 _{ }

; , 0 2 2 P q _{L T IR} IR u Ff C Hu C u J     

The considered problem becomes J

 

u

D u

min . For its resolution, we will use an extension of the Hilbert Uniqueness Methods (H.U.M). ForIRq, let us note





2 1 0 2 * * * * *           



T IR ds C s T S B  _P  *is a semi-norm on q

IR . If the condition (7) is verified then it is a norm if and only if the system (1)-(2) is weakly gradient remediable on



0,T



. The corresponding inner product is

given by B S



T s



C B S



T s



C ds T



     0 * * * * * * * * * , ,   

and the operator  :IR q IRq defined by











         T ds C s T S BB s T S C C HH C 0 * * * * * * *   

Then, we have the following proposition:

Proposition 6:

If the condition (7) is verified, then _*is a norm on IRqif and only if the system (1)-(2) is weakly gradient remediable on



0,T



and the operator is invertible.

Proof :





0 2 1 0 2 * * * * *            



T IR ds C s T S B  _P 





_{ } 0 ; , 0 * * * * 2     P IR T L C s T S B 





* *



* *





* *





* * * *



* * ker . ker 0 . C B S T C B F C T S B             But ker



B*F**C*







m m



m f M ker 1 

 (see the Proof of Proposition 5) and we have also





  T m m m G M ker 1 



m m



m f M ker 1 

 (see the Proof of Corollary 4) then,ker



B*F**C*







m mT



m G M ker 1   this gives 



_{m m}T



m G M ker 1 

 and since the

Corollary 4 we obtain the result.

On the other hand the operator  is symmetric, indeed

q IR q IR C HH C    ,   ** * ,  _{q IR}q IR C HH C          , * * * ,

and positive definite, indeed

 









0 , 0 , , , , 2 * 0 * * * * * * * * ; , 0 * * * * * * * * * 2               



          for ds C s T S B C s T S B C H C H C HH C T IR IR T L IR IR P P q q

and then is invertible.

We give hereafter the expression of the optimal control ensuring the gradient remediability of

a disturbance f at the timeT. Proposition 7:

For f L2



0,T;X



, there exists a unique q f IR

 such that _f CFf and the

control

 

 

_f f B S C u  * * * * .

.   verifiesCHu_f CFf 0. Moreover, it is optimal and

0, ;  * 2 _{T IR} f L P f u_   . Proof :

From the Proposition 6, the operator  is invertible then, for f L2



0,T;X



, there exists a

unique q

f IR

Ff C f    and if we put

 

 

_f f B S C u  * * * * . .   , we obtain









_f T f f C S T s BB S T s C  ds C Hu         

_

0 * * * *       C Ff C Hu_f CHu_f CFf 0.

The set D defined by (13) is closed, convex and not empty. For u D , we

have

 

2 _{ } ; , 0 2 _{T IR}P L u u

J  . J is strictly convex on D, and then has a unique minimum at

D u * , characterized by   0; , ; , 0 * * 2   P IR T L u v u v D. For v D, we have  P

 

 

 P f f v u _{L T IR} B S C f v B S C f _{L T IR} u ; , 0 * * * * * * * * ; , 0 2 2 . , . , _         0 ,     _q IR f f C Hv   Since u*is unique, then u* u_f and

f u is optimal with  

 

  2 * 2 ; , 0 * * * * 2 ; , 0 2 2 _{T IR} . f _{L T IR} f L P P f B S C u_      .

6. APPROXIMATIONS AND NUMERICAL SIMULATIONS

This section concerns approximations and numerical simulations of the problem of gradient

remediability. First we give an approximation of _f as a solution of a finite dimension linear system Ax b and then the optimal control

f

u , with a comparison between the

corresponding observation noted z_u,f and the normal case. 7.1 Approximations

 Coefficients of the system: For i, j1,let

q IR j i j i e e

a   , where

 

e_{i 1iq}is the canonical basis of IRq, we have











      T i i C S T s BB S T s C e ds e 0 * * * *

       



 





                         n k _{L Dj} j k l m n k _{L Di} i k h m L h m L l m M m m r l N m m r h p m m T m m j i h x w h x w w g w g e a 1 2 1 ' ' 2 ' 2 ' 2 1 1 '1 ' 1 1 ' ' , , , , 1          and f CFf q IR j j C Ff e b   ,

For N sufficiently large, we have

   

_{ }

 





         T L h m s T m N m m r h n k _{L Dj} j k h m j h e f s w ds x w b 0 2 ' ' 1 ' ' 1 1 2 ' , , 

 The optimal control: In this part, we give an

approximation of the optimal control

f u_ which is defined by

 





_f f B S T C u_ * * * * . .    .

Its function coordinates u_{j, f}

 

. are given by

 

j





f f j g S T C u _ * * * , .  , .     i k h m N m m r h L j h m j T m n k q i f i h x w w g e , ' , 1 ' ' 1 2 ' . ' 1 1 ,   

   

       

for a large integer N .

 Cost: The minimum energy (cost) is defined by





2 1 0 2 * * * * ; , 0 2 _         

_

      T p IR f p IR T L f B S T s C ds u    2 1 1 0 2 1 ' ' 1 ' ' 1 , 1 , ,                   

   

      p j T N m m r h h m j k h m i n k f i q i s T m _{g w ds} x w h e 

for N sufficiently large.

 The corresponding observation: The observation corresponding to the control is given by

 

 

 

 

_







 

 

_





  

t t f f f u t C S t y C St s Bu s ds C St s f sds z 0 0 0 ,   Its coordinates

 

q j f f u j z         1 ,

, _ . are obtained for a large integer N , as follows:

 

_{ }    

_{ }

 

_{ }



_



_



                      N m r h n k t h m s t j k h m N m r h n k p i t i s t j k h m h m i N m r h k L D h m j L h m n k t f u j m m m f m m j m f ds w s f e h x w ds s u e h x w w g x w h w y e t z 1 ' 1 1 ₀ ' ' 1 ' 1 1 1 ₀ , ' ' 1 ' 1 ' ' 0 1 , , ' ' ' ' ' 2 2 ' , , , , , ,     

7.2 Numerical simulations

We consider without loss of generally the following diffusion system









   

 









 









                      



  T t y x y x y T t x f t u x g t x y t x t y p i i i i , 0 0 , 0 , , 0 , , , 0 1  

with 

 

0,1 and a Dirichlet boundary condition. In this case, the functions w_m

 

. are defined byw_m

 

x  2sin



mx



;m1. The associated eigenvalues are simple and given by_mm22; m1. Then in the case of:

 an initial state: y0

 

. 0,

 a sensor:



D,h



with D

 

0,1 and h

 

x  2x2



q1



 an efficient actuator:



,g



with 

 

0,1 and g

 

x 2x3



p1



 a disturbance function: defined by

 

, 240 10 ; 0

        t e t x f x t

For

M

 N



1

and T 70, we obtain numerical results illustrating the theoretical results established in previous sections. Hence, in figure1, we give the representations of the discrete

observation z_u,f corresponding to the control uu_f and the disturbance f and z_0,0

which represent the normal observation, that is u0 and f 0.

This figure show that for t sufficiently large



t50



, the disturbance f is compensate by the control optimal

f

u_ at the time T



T 70



that is, we have z _f

 

t z

 

t

f

u_ ,  0,0 .

The optimal control u_f ensuring the gradient remediability of the disturbance f , is

represented in figure 2.

Fig.2.Representation of the optimal control

f

u_ (blue line)

8. CONCLUSION

In this paper, which is an extension of previous works to the analysis of the gradient of a

large class of parabolic systems, new notions of weak and exact gradient remediability are

introduced and characterized. The relation between the notion of gradient remediability and

the notion of gradient controllability is also studied. We have shown that a parabolic system

is gradient remediable if it is gradient controllable. Furthermore, we have shown that the

exact and weak gradient remediability of a system may depend on the structure and the

number of the actuators and sensors. Using an extension of the Hilbert Uniqueness Method,

we have shown how to find the optimal control ensuring the gradient remediability of the

known or unknown disturbance. The results of illustrative examples and numerical

approximations are acceptable.

These results are developed for a class of discrete linear distributed parabolic systems, but

the considered approach can be extended to other class of systems with a convenient choice