Regional gradient compensation with minimum energy Текст научной статьи по специальности «Математика»

2019 Математика и механика № 61

UDC 517.9 MSC 35B20; 35R06; 35R15; 49J20

DOI 10.17223/19988621/61/3

S. Rekkab, H. Aichaoui, S. Benhadid

REGIONAL GRADIENT COMPENSATION WITH MINIMUM ENERGY

In this paper we interest to the regional gradient remediability or compensation problem with minimum energy. That is, when a system is subjected to disturbances, then one of the objectives becomes to find the optimal control which compensates regionally the effect of the disturbances of the system, with respect to the regional gradient observation. Therefore, we show how to find the optimal control ensuring the effect compensation of any known or unknown disturbance distributed only on a subregion of the geometrical evolution domain, with respect to the observation of the gradient on any given subregion of the evolution domain and this in finite time. Under convenient hypothesis, the minimum energy problem is studied using an extension of the Hilbert Uniqueness Method (HUM). Approximations, numerical simulations, appropriate algorithm, and illustrative examples are also presented.

Keywords: gradient, optimal control; regional remediability; disturbance; efficient actuators.

1. Introduction

The control problem of distributed parameter systems arises in engineering applications and many different contexts, which are characterized by a spatiotemporal evolution. Systems analysis consists of a set of concepts as controllability, observability, remediability,...that allows a better understanding of those systems and consequently enables to conduct them in a better way. Moreover, the analysis itself has to deal not with the whole domain, but with its specific subdomain of interest. Thus, motivated by practical applications, El Jai and Zerrik have introduced and studied the so-called regional analysis [1-5]. Such analysis aims to analyze or control a system in which an objective function is defined only on a prescribed subregion. Therefore, the system dynamics is defined in the whole the domain Q , whilst the objective is focused on a given subregion ю, where roc Q . An extension of this study that is very important in practical applications is that of regional analysis of the gradient developed in [6-10]. This study is 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 problem from the real world. With the same preoccupation, the regional gradient remediability and regionally efficient gradient actuators are introduced and characterized recently for linear distributed systems in [11].

In this work, we show how to find practically the optimal control (convenient regionally gradient efficient actuators) ensuring the gradient compensation regionally, based on a result of characterization obtained in our previous work [11]. In addition, it constitutes also an extension of our previous work [12] to the regional case.

This paper is organized as follows. In the second section, we start by presenting the considered problem. After, we recall the definitions of exact and weak regional gradient

remediability, the notion of regional gradient efficient actuators, and a characterization which shows that the regional gradient remediability of any system may depend on the structure of the actuators and sensors.

In section 3, under a condition on the structure of the actuators and the weak regional gradient remediability hypothesis, using an extension of Hilbert Uniqueness Method (HUM), we examine the problem of gradient remediability with minimum energy regionally. Then, we give the optimal control which compensates an arbitrary disturbance.

In the last section, approximations, simulations, and numerical results are presented.

2. Considered problem, definitions, and characterization

Let Q be an open and bounded subset of IRn ((n = 1,2,3) with a regular boundary dQ. Fix T > 0 and let denoted by Q = Qx]0,T[, Z = dQx ]0,T[ . Consider the system described by the parabolic equation

(x,t) = Ay(x,t) + Bu(t) + f (x,t) - Q,

y(x,0) = y0 (x) - Q, (2.1)

.y(I, t) = 0 - Z,

where A is a second order linear differential operator which generates a strongly continuous semi-group (S (t))t>0 on the Hilbert space L2 (Q). (S* (t))t is considered

for the adjoint semi-group of (S(t))t>0. B e L(U,X),u e L2 (0,T;U) where U is a Hilbert space representing the control space and X = Hi (Q) the state space. The disturbance term f e L2 (0, T;X) is generally unknown.

In system (2.1), the disturbance function f has a space support which can be, in practical applications, a part ra of the domain Q (roc Q). The system (2.1) admits a unique solution y e C(0, T;Hl0 (Q)) fl C1 (0, T;L2 (Q)) [13] given by

t t yu,f (t) = S(t)y0 +|S(t-5)Bu(s)ds + |S(t-s) f (s)ds .

0 0

For roc Q an open subregion of Q with a positive Lebesgue measure, we consider the operators

Xro : (L2 (Q))n ^(L2 (ro))n , and Xro : L2 (Q) ^ L2 (ro),

y ^ y|ro, y ^ y|ro ,

while their adjoints denoted by x^, and XX1, respectively, are defined by

X*ro : (L2 (ro))n ^ (L2 (Q))n , and Xro : L2 (ro) ^ L2 (Q),

* iy on ro, _* iy on ro,

y ^Xroy = ¡0 on Q\ro, y roy = ¡0 on Q\ro.

Consider also the operator V defined by

V: H 0 (O)^(l2 (O))",

y , ^ ,..., ^ ],

^ d x1 d x2 d xn j

while V* its adjoint operator.

The system (2.1) is augmented by the regional output equation

< f (t) = ClwVyu J (t), (2.2)

where C e L ((l2 (ra)) , O), O is a Hilbert space (observation space). In the case of

an gradient observation on [0, T] with q sensors, we take generally O = IRq . In the autonomous case, without disturbance (f = 0) and without control (u = 0), the gradient observation in ra is given by

<0 (t) = CXraVS(t)y0, it is then normal. However, if f ^ 0 and u ^ 0, the regional gradient observation is disturbed.

The problem consists to study the existence of an input operator B (actuators), with respect to the output operator C (sensors), ensuring the gradient compensation at finite time T , of any disturbance acting on the system, that is to say: For any f e L2 (0, T; X), there exists u e L2 (0, T;U), such that

<f (t) = CXraVS(T)y0,

this is equivalent to

C XraVHu + C XraVFf = 0, where H and F are two operators defined by

H : L2 (0, T;U) ^ X, F: L2 (0, T;X) ^ X,

t and t

u ^ Hu = |S(T -s)Bu (s)ds, f ^ Ff = JS(T - s)f (s)ds.

00

This leads to the following definitions. Definition 1.

1) We say that the system (2.1) augmented by the output equation (2.2) is exactly regionally f-remediable in ra, if there exists a control u e L2 (0, T;U), such that

CXraVHu + C XraVFf = 0.

2) We say that the system (2.1) augmented by the output equation (2.2) is weakly regionally f-remediable in ra on [0, T], if for every e> 0, there exists a control

u e L2 (0, T;U) such that

\\CXraVHu + CxwVFf\\mq < 0.

3) We say that the system (2.1) augmented by the output equation (2.2) is regionally exactly (resp. weakly) remediable in ra, if for every f e L2 (0,T;X) the system (2.1) -(2.2) is exactly (resp. weakly)f-remediable in ra .

We suppose that the system (2.1) is excited by p zone actuators

(Q,,g, )<,.< ,g, e L2 (Q,), Q, era , Vi = 1,...,p , in this case the control space is

U = IRp and the operator B is given by

B: IRp ^ X,

p

U (t) = (U1 (t), U2 (t^^ , Up (t)) ^ Bu = XXO, (x) g/ (x )ui (t).

i =1

Its adjoint is given by

B* Z = (< gl, 4 qM g 2 , 4 Q2,.",( gp , Z) Q p )T £ IRP .

Also suppose that the output of the system (2.1) is given by q sensors (D,, h, )1</.<q, hi e L2 (D,), being the spatial distribution, Di = supph, era , for i = 1,..., q and Di H Dj = for i ^ j, then the operator C is defined by

C: (L2 (ra))" ^ IRq, y (t) M. Cy (t ) = fj^<h1, yi (t)) d1 , ]C<h2, y, (t)> Di,..., ±(hq , y, (t)) L

i=1 i=1 i=1

q

, i.

its adjoint is given by C* with for0 = (01,02,. • •, 6q) e IR

C*0 = Xd, (x)0ihi (x),£xd, (x)QA (x),...,£xD, toeA (x)1 e (l2 (ra))" .

V i=1 i=1 i =1 /

We recall the following notion of the regionally gradient efficient actuator [11].

Definition 2. The actuators (Q, , gi )1<i< , gi e L2 (Q, ) are said to be regionally

gradient efficient, if the system (2.1) - (2.2) so excited is weakly regional gradient remediable.

For m > 1, let Mm be the matrix of order (p x rm) defined by

Mm =((g/, Wmj)L2(Q/ ))/j ,1 < - < p and 1 < j < rm

and let Gm be the matrix of order (q x rm) defined by

G„ =

( "I dw x ^

m, -x1

k=A dxk Il2(D„ A]

, 1 < i < q and 1 < j < rm

Corollary 1 [11]. If there exists m0 > 1, such that

rank Gi = q, (2.3)

then the actuators {fli, gi )<.< , gi e L2 {fli) are regionally gradient efficient if and only if

n ker{MmGTm ) = {0}.

m>1 v '

3. Regional gradient remediability with minimum energy

Under a condition (2.3) and the weak regional gradient remediability hypothesis, we study in this section the problem of the exact regional gradient remediability with minimal energy.

For f e L2 {0, T; X), we study the existence and the unicity of an optimal control u e L2 {0,T; U) ensuring, at the time T , the exact regional gradient remediability of the disturbance f , such that

C X<BVHu + C laVFf = 0.

That is the set defined by

D = {u e L2 {0,T;IRp)/ClaVHu + CX<BVFf = 0} (3.1)

is non empty.

We consider the function

J {u ) = llCXfflVHu + CUVFfflRq +1MlL2 {0,T ;IRP ) .

The considered problem becomes

min J {u).

ueD

For its resolution, we will use an extension of the Hilbert Uniqueness Methods (HUM).

For QeIRq , let us note

1

INI * =f f| |b*s * {t - * )Vx:Ce| ]Rpds 12.

V 0 y

The corresponding inner product is given by

<e, a) * = J( B*S * {T - s )V*x*C*e, B* S * {T - s )vYfflC*a )ds

0

and the operator A : IRq ^ IRq defined by

Ae = c1wVhh V^c *e = c1wV] s {t - s )bb* s* {t - s) v*£c *e ds.

0

Then, we have the following proposition:

Proposition 1. If the condition (2.3) is satisfied, then | ||* is a norm on IRq if and only if system (2.1) - (2.2) is weakly regional gradient remediable on [0, T] and the operator A is invertible.

Proof. We have

' t

1 ^ 2

\BS- (T -, )v*x;C*e| 2Rpds j = 0 ^¡BS- (T -, )Vx;C-e|L,(0TJR,) = 0

■ B*S* (T -.) V*X;C*e = 0 ^ e e ker(B*S* (T - .)V*X;C*) = ker^V^C* ). But ker (bWxic* )= H ker (Mf),

v ' m>1 v '

where, for m > 1,

f: :eeIRq ^fm(e)=(v*x*C*e,Wm1),{V*Cc%Wm2),..,(V*Cc%Wmm)) eIRrm . Indeed, let0e IRq , we have

b* F VYraC *e=b* s* (T-.)v*x;c *e =

m>1 j=1

m

£ -)^(V*x;C *e, Wm^Jg2, Wm/)

m>1 j=1

/L2(Q2)

rm

jV*X*.C wj („„j p)

Vm>1 j=1 • ' y

and we have Vm > 1

Mmfm (e) =

rm

y.yx*rac *e; w:j!l2 (q)\ g1, j (Q1)

rm

Wmj^ L2 (Q)(g 2, j (Q2)

rm

£(v*x;c*e; WmjljL2(Q^gP , Wmj/jL(Qp)

If we assume thate e H ker(Mmf , then

m>1

eeker(Mmfmffl), Vm>1^£(V*x;C*e;Wm^^g,w,)=0,Vie{1,2,...,p},Vm>1

rm

£ ^ (T -)^V*x;C*e, Wmj^i^g/, Wmj^ = 0, Vi e{1,2,..., p}, Vm > 1 j=1

■ B*F*V*x*raC*e = 0 ^ e e ker(B*FV^C*), H ker(Mmf:)e ker(B*FV^C*),

m>1 j=1

where

that is

n ker(MmC) = ker{bWYc*)

m>1 '

and we have also n ker{MmGTm) = n ker[Mmf®). Indeed, let e eIRq , then

m>1 ' m>1 V '

ee n ker {MmGTm) MmGTm )e = 0, Vm > 1,

11 \ m m t \mm/ ' '

m>1 ' v '

q rm I n dw ■ \

,WmMh,0/ = 0,Vm > 1,V/ = 1,...,p ,

i=i j=i \ k=i dxk ii2(d,)

« , Wmj^ (vYC Wmjj jl (Q) = 0, Vm > 1, V/ = 1,..., p ,

j=1 1 J

« (Mm/:)e = 0, Vm > 1 обе П ker ( M/ ).

\ m*** m f ' \ m*** m f

y ' m>1

Where ker(5*FV£C*)= П ker(MmGTm ), this gives 0е П ker(MmGTm ) and

m>1 m>1

since the Corollary 1, we obtain the result.

On the other hand, the operator A is symmetric, indeed,

<Ae, c)mq = (cXfflvHH Vx;c *e, a)R =(e, cx(dvhh V^c *a)R = {e,Ao)mq

and positive definite, indeed,

(Ae, e)mq = (cXfflvHH V^c *e, e)R = (hvyc*e, hvvc*e\ 2,

\ /L2{0,T;IRP)

j(B*S* (T -s) vY„Ce,b*S* (t-s) v^Ce)^ ds

> o, /or e * о

and then Л is invertible. □

We give hereafter the expression of the optimal control ensuring the regional gradient remediability of a disturbance / at the time T .

Proposition 2. For / е L2 (0, T ; X ), there exists a unique 0 / е IRq such that

ле / = -c xfflw

and the control

/ ()=

verifies

Ue/ (.) = B*S* (,)VYC e/

C XfflVHMe / + C XfflW = 0.

Moreover, it is optimal and

h /111 (0,T ; IRP )=lle /l I*.

Proof. From Proposition 1, the operator A is invertible; then, for f e L2 (0, T;X), there exists a unique e f e IRq, such that Ae f = -CxraVFf and, if we put

0/ ,.,= ()V*

T

ue (.) = B*S* (.) vyfflC 0/, we obtain

A0/ = C%fflVjS(T - ,)BB*S* (T - ,) vYfflC*0/ ds = CX„VH^

0

^ -C%aVF/ = ClwVHu0f ^ ClwVHu0f + CXfflVF/ = 0.

The set D defined by (4.1) is closed, convex, and not empty. For u e D, we have J(u) = ^ufLi. T.irp). J is strictly convex on D , and then has a unique minimum

at u e D , characterized by

/ * * \

(u , v - u ) 2, „> > 0; Vv e D .

\ IL2 (0,T ; IRP )

For v e D, we have

/u0 , v - u0 \„ ^ = / B*S* (.) V*x„C*0 ,, v - B*S* (.) V*x„C *0 f\ 2,

\ 0/' 0//l2 (0,t;irp ) \ W W / ' L2 (0,T ; IRP )

= (0 /, C loyHv - A0 /)ir, =

Since u* is unique, then u* = u0/ and u0/ is optimal with

l . 11' „, = i |b*s * (.) V*Yic *0 /1„,=110 J I2. □

X IL2 (0,T; JRP) =1B S (.)V x&)C e flL2 (c,T; IR?) = P f 11* .

4. Approximations and numerical simulations

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

First, we give an approximation of e f as a solution of a finite dimension linear

system Ax = b and then the optimal control ue , with a comparison between the

corresponding observation noted z^ f and z^ the observation corresponding to the

ef

autonomous case.

4.1. Approximations Coefficients of the system: For i, j > 1, let aij = (^Ae,, e^RRq, where (e, )K is the canonical basis of IRq , we have

Ae, = CxfflV{S (T - s) BB\S* (T - s) V^C*e,. ds .

0

And for M, N sufficiently large, we have

m m n rm, p f j\m +v)T

-1

m=1 l=1 m '=1 h=1 T=1

' dw

X

V m'

s /dw

(gx , Wm^L2 gx , ^^ 'h )_

L2 (QT)

=A dxk

L2 (D, )k=

ml

"A dxk

(4.1)

2 (j )

and A6f = -CXfflVFf, then b} = -(CXfflVFf, . For N sufficiently large, we have

N m n Idw

bj «-ZZZ , hj/ i ^{T-S)(f (s), Wm^>L2 („) . (4.2)

m'=1 h=1 k=1 \ ^k Il2 (Dj )o

The optimal control: In this part, we give an approximation of the optimal control

ue which is defined by ue (.) = B*S* (T - .)V*%*C*9 f. Its function coordinates

f f J

e (.) are given by

J,e f

N rm ' n q

uj ,e f (.) = ( gj, S * (T - .)v*£c *e f)

ldwm'h

X£X£e,f^(T-)( gj, Wmh)L

=1 h=1 k=1 i=1

:(qj a dxk

(4.3)

L2 (D )

for a large integer N.

Cost: The minimum energy (cost) is defined by

( T

iil2 (o,t ; 1RP )

|b* S * (T - s )vyœc *e flRPds

V o

T f

e^m (T-s )e. Ih , —Wm!h.

i,f \ i ' PK,-

Si ssss

; =1 o v m '=1 h=1 k=1 i=1

k iL2 (D, )

{gj , Wm'^L

( J )

ds

for N sufficiently large.

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

t t zl f {t) = CxfflVS {t)/ + CXfflV{ S{t - s)Buef {s)ds + CXfflV{S {t - s) f {s) ds . (4.4)

0 0

Its coordinates I z®u f {.)) are obtained for a large integer N as follows:

V ef'J !\< j<q

H< j<q

n m n

z f , f (' exm"{y0, wmh)L, J h

m '=1 h=1 k=1

dw

m'h

dx.

k >L2 (Dj )

N rm ' n P

dw

/ _I

SSSSgi,*mh)L2(0t) ^,hj) ie^t-s\ef (s)ds (4.5)

m'=1 h=1 k=1 i=1 \ dxk Il2(dj )0 f

N rm n

S, ' J i (s) wm.^L2(.) ds.

m'=1 h=1 k=1 \ k iL2 (dj )o

+

4.2. Numerical simulations We recall the problem considered above:

(P) jFind u e L2 (0,T;U), such that

( )|cXfflVHu + CXfflVFf = 0.

Based on Proposition 2 and using the above results, we can develop an algorithm which allows us to determine a sequence of controls which converges to the solution u* of (P). The output is given by (4.4) - (4.5). Algorithm

Step 1: Data: the domain Q , the subregion ra , the final time T, the initial state y0, the disturbance function f , the sensors (D, h), the efficient gradient actuators (a, g), and the accuracy threshold e.

Step 2: Choose a low truncation order M = N .

Step 3: Compute z0°o: the output, when f = 0 and u = 0 (an autonomous case). Step 4: Compute z^f : the output, when f ^ 0 and u = 0 (a disturbed case).

Step 5: Solve a finite dimension linear system A9 = b, where these coefficients are given by (4.1) - (4.2).

Step 6: Calculate u given by (4.3).

Step 7: Compute z^f: the output when f ^ 0 and u ^ 0 (a disturbed and controlled case, that is to say a compensate case).

Step 8: If |zra, f - z^ 01|l2 < e, then stop. Otherwise,

Step 9: M ^ M +1 and N ^ N +1 and return to step 3. Step 10: the optimal control u corresponds to the solution u* of (P). Now, we give a numerical example, which illustrates the efficiency of the approach given in the above section.

Illustrative example. We consider without loss of generally the following diffusion system

dy (x, t) = Ay (x,t) + £XQg. (x)u,. () + f (x,t) Qx]0,T[

y (x,0) = y0 (x) Q

y (I, t ) = 0 5Qx]0, T [

with Q = ]0,1[ and a Dirichlet boundary condition. In this case, the functions wm (.) are defined by

wm (x) = V2sin(mnx);m > 1.

The associated eigenvalues are simple and given by

Xm = -m2n2 ; m > 1. Let ra = ]0.15,0.25[ (ra c Q) be the geometrical support or the disturbance. Then in the case of: - an initial state: y0 (.) = 0,

- a sensor: (D,h) with D = ra and h(x) = V2x2 (q = 1),

- an efficient actuator: (ct,g) with ct = ra and g (x) = 2x3 (p = 1),

- a disturbance function: defined by f (x) = 240 cos x .

For M = N = 10 and T = 0.5 , we obtain numerical results illustrating the theoretical results established in previous sections. Hence, in Fig. 1, we give the representations of the

discrete observation zu f corresponding to the control u = uQ and the disturbance f

f

and z0 0, which represent the normal observation, that is u = 0 and f = 0.

Time

Fig. 1. Representation of zuf (line 1), z0f (line 2) and z0,0 (line 3)

This figure shows that the disturbance f is compensated by the optimal control u = Me at the time T that is, we have z® f (T) = z® 0 (T).

The optimal control ue ensuring the regional gradient remediability of the disturbance f is represented in Fig. 2.

Time

0 0.1 0.2 0.3 0.4 0.5 0.6

Table 1 shows that if we want to eliminate the effect of the disturbance in a short time T , the cost increases.

Table 1

Evolution cost with respect to the finite time T

T Cost

0.3 1.06-105

0.4 1.05-105

0.5 1.03-105

1 9.46-104

2 8.99-104

3 8.89-104

5 8.85-104

10 8.84-104

100 8.84-104

Conclusions

Under a condition on the sensors and the weak regional gradient remediability hypothesis, we have studied the problem of exact regional gradient remediability with minimal energy. That is to say, when a system is subjected to disturbances, we have shown how to find the optimal control, which compensate the effect of the disturbance that can be located in a given subregion of the space domain, with respect to the regional gradient observation and this using an extension of the Hilbert Uniqueness Method. Illustrative examples, numerical approximations, and results are also presented. These results are developed for a class of discrete linear distributed parabolic systems, but the considered approach can be extended to regional or bounded gradient remediability of other class of systems with a convenient choice of space.

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Received: March 25, 2019

Soraya REKKAB (Doctor, Faculty of Exact Sciences, Mentouri University, Constantine, Algeria) E-mail : rekkabsoraya@gmail.com

Houda AICHAOUI (Researcher, Faculty of Exact Sciences, Mentouri University, Constantine, Algeria.) E-mail : aichaoui_houda@hotmail.fr

Samir BENHADID (Doctor, Faculty of Exact Sciences, Mentouri University, Constantine, Algeria.) E-mail: ihebmaths@yahoo.fr

Реккаб С., Айчаой Х., Бенхадид С. ЛОКАЛЬНАЯ ГРАДИЕНТНАЯ КОМПЕНСАЦИЯ ПРИ МИНИМУМЕ ЭНЕРГИИ. Вестник Томского государственного университета. Математика и механика. 2019. № 61. С. 19-31

DOI 10.17223/19988621/61/3

Ключевые слова: градиент, оптимальное управление, локальная восстановимость, возмущение, эффективные актюаторы.

Рассматривается проблема локальной градиентной восстановимости или компенсации при минимальных затратах энергии. Иными словами, при возмущении системы одной из задач становится отыскание оптимального управления, которое локально компенсирует результат возмущения системы по отношению к локальному градиентному измерению. Таким образом, показано, как найти оптимальное управление, обеспечивающее компенсацию любого известного или неизвестного возмущения, распределённого лишь на части области геометрического роста, по отношению к измерению градиента на любой заданной подобласти области роста за конечное время. Проблема минимума энергии исследуется при удобных предположениях с помощью обобщенного метода единственности Гильберта. Представлены также приближения, численное моделирование, соответствующий алгоритм и иллюстративный примеры.

Rekkab S., Aichaoui H., Benhadid S. (2019) REGIONAL GRADIENT COMPENSATION WITH MINIMUM ENERGY. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics]. 61. pp. 19-31

DOI 10.17223/19988621/61/3