• No results found

The controllability for the internally controlled 1 D wave equation via a finite difference method

N/A
N/A
Protected

Academic year: 2020

Share "The controllability for the internally controlled 1 D wave equation via a finite difference method"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

The controllability for the internally

controlled 1-D wave equation via a finite

difference method

Juntao Li

1,2

, Yuhan Zheng

1

and Guojie Zheng

1*

*Correspondence:

guojiezheng@yeah.net 1College of Mathematics and

Information Science, Henan Normal University, Xinxiang, 453007, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, we study the controllability of the semi-discrete internally controlled 1-D wave equation by using the finite difference method. In the discrete setting of the finite difference method, we derive the observability inequality and get the exact controllability for the semi-discrete internally controlled wave equation in the one-dimensional case. Then we also analyze whether the uniform observability inequality holds for the adjoint system ash→0.

MSC: 49K20; 35J65

Keywords: wave equation; finite difference; observability inequality

1 Introduction

In this paper, we discuss the topic related to the controllability for the space semi-discretizations of the internally controlled one-dimensional wave equation. Let us first introduce certain notations and state the controlled system as studied in this paper. Let ω= (a,b) be an open and nonempty subset of (, ),χωrepresent the characteristic func-tion ofω. The controlled wave equation is represented as follows:

⎧ ⎪ ⎨ ⎪ ⎩

∂tty(x,t) –∂xxy(x,t) =χωu(x,t), (x,t)∈(, )×(,T),

y(,t) =y(,t) = , t∈(,T),

y(x, ) =y(x), ∂ty(x, ) =y(x), x∈(, ),

(.)

whereT > , the initial value (y,y) belongs toH

(, )×L(, ) andu(·) is a control function taken from the spaceL(,T;L(, )).

Problem (.) is said to be exactly controllable from the initial value (y,y)∈H (, )×

L(, ) in timeT if there exists a control functionu(·)L(,T;L(, )), such that the solution of (.) satisfies (y(T),∂ty(T)) = (, ). The problem of the controllability of wave equations has also been the object of numerous studies. Extensive related references can be found in [–] and the rich work cited therein.

In this work, we shall mainly focus on the issue of how the controllability property can be achieved under the numerical approximation schemes. Now, we will introduce the nu-merical project by using the finite difference method. GivenN∈N, we defineh=N+. We

(2)

consider the nodal points

x= ; xj=jh, j= , . . . ,N; xN+= ,

which divides [, ] intoN+  subintervalsIj= [xj,xj+],j= , , . . . ,N. We suppose that the nodal pointsxk+, . . . ,xk+pω, andx, . . . ,xk,xk+p+, . . . ,xN ∈(, )\ω, for somek,p∈N withk+pN.

Now, we consider the following finite difference semi-discretization of (.):

⎧ semi-discrete case. Next, we shall rewrite equation (.) by vectors. Let

(3)

whereIp×pis ap×pidentity matrix, andp,k∈N, which are mentioned above (.). The system (.) can be rewritten as follows:

⎧ ⎪ ⎨ ⎪ ⎩

yh(t) +Ahyh(t) =Bhuh(t), t∈(,T),

y(t) =yN+(t) = , t∈(,T),

yh() =yh, yh() =yh,

(.)

wheredenotes derivation with respect to time. In fact, system (.) is in the form of linear ordinary differential equations for an unknown vector functiony(t)h= (y(t), . . . ,yN(t))T, with the boundary conditionsy(t) =yN+(t) = , anduh(t) plays the role of control func-tion. The adjoint system for system (.) can be represented as

⎧ ⎪ ⎨ ⎪ ⎩

φh(t) +Ahφh(t) = , t∈(,T), φ(t) =φN+(t) = , t∈(,T),

φh() =φh, φh() =φh,

(.)

where the initial dateφh= (φ

, . . . ,φN)T, andφh= (φ, . . . ,φN)T. It is easy to check that φ=φN+=  andφ=φN+=  are the compatibility conditions. Throughout the paper, we suppose that the compatibility conditions hold for any initial value.

Taking the boundary conditionsφ(t) =φN+(t) = ,∀t∈[,T], for the solution of (.) we define the energy of the semi-discrete system (.) as

Eh(t) =

h

N

i=

φi(t)+φi+(t) –φi(t)

h

, ∀t∈[,T].

SinceBhis a symmetric matrix, the observability inequality of (.) can be formulated as: To find a constantC(T,h) such that (see [–])

Eh()≤C(T,h)

T

Bhφh(t) 

RNdt, (.)

where

Eh() =

h

N

i=

φi+φi+–φi

h

.

Remark . The energyEh(t) is conserved for the solution of (.). Namely, for anyh>  and solutionφh(t) of (.), we have (see [])

Eh(t) =Eh(), for anyt∈[,T].

In this paper, we will study whether the inequality (.) for adjoint system (.) holds. We are also interested in whether the constantC(T,h) is bounded ash→. The main results of the paper are presented as follows.

Theorem . For any T> ,the observability estimate(.)for the adjoint system(.)

(4)

Remark . Theorem . shows that the semi-discrete system (.) or (.) is controllable for any timeT> .

Theorem . For any T> ,we have

sup

solution of(.)

Eh()

T

Bhφh(t) 

RNdt

→+∞, as h→. (.)

To the best of our knowledge, Infante and Zuazua made the first study of this topic in []. They studied a controllability result for the semi-discrete -D wave equation with boundary control. However, the uniform controllability for the semi-discrete systems in [] cannot be derived as the discretization parameterh→. The main differences be-tween [] and our paper are as follows. In [], the authors focused on a one-dimensional boundary controlled wave equation, and we mainly study the internally controlled -D wave equation. In this case, the controller is more complicated than the case with con-troller on boundary. Regarding other works on this subject, we mention [, , ] and [].

The paper is organized as follows: Section  briefly describes some preliminary results on the finite difference scheme. The proofs of Theorem . and Theorem . are provided in Section .

2 The finite difference scheme

In this section, we will discuss the numerical project by the finite difference method. We consider the numerical problem for (.) in the state spaceRN with the usual Euclidean norm and inner product denoted by · RN and·,·RN, respectively. To this end, we first

introduce some properties for the eigenvalues and eigenvectors of the matrix Ah. The spectrum forAh can be explicitly computed in this case (see []). The eigenvaluesλi(h) (i= , . . . ,N) satisfy

λi(h) = 

hsin 

iπh

, (.)

and the corresponding unit eigenvectors inRN are

whi =whi,, . . . ,wih,NT, and whi,j=

N+ sin(iπjh), j= , . . . ,N. (.) Clearly, the family of eigenvectors

wh,wh, . . . ,whNforms an orthonormal basis ofRN. (.) Indeed, we can easily get the following properties for these eigenvectors.

Lemma .

(i) For any eigenvectorw= (w,w, . . . ,wN)Twith eigenvalueλ(h)of matrixAh,the

following identity holds:

N

j=

wjwj+

h

=λ(h)

N

j=

|wj|, (.)

(5)

(ii) Ifwk= (wk,,wk,, . . . ,wk,N)Tandwl= (wl,,wl,, . . . ,wl,N)Tare eigenvectors associated

to eigenvalueλk,λl,andλk=λl,then we have

N

j=

(wk,jwk,j+)(wl,jwl,j+) = , (.)

wherewk,=wk,N+= ,andwl,=wl,N+= . This lemma is quoted from [].

Lemma . Assume N is large enough so thatωcontains at least two consecutive nodal points.Then

Bhwhi = , for all i= , , . . . ,N.

Proof Since there are more than two consecutive nodal points inω= (a,b), we letl(h) denote the first natural number such thatl(h)h∈(a,b), andm(h) denote the last natural number such thatm(h)h∈(a,b). Then we have

Bhwhi =

, . . . , ,whi,l(h), . . . ,whi,m(h), , . . . , T. (.)

Here,

whi,l(h)=

N+ sin

iπl(h)h=

N+ sin

il(h

N+ 

,

and

whi,l(h)+=

N+ sin

iπl(h) + h=

N+ sin

i(l(h) + )π

N+ 

could not be zero at the same time. Thus, we complete the proof of the lemma.

Especially, we can get the following property for the eigenvectors for the matrixAh, which will play a key role in the proof of the main results.

Proposition . Assume N is large enough so thatωcontains at least two consecutive nodal points.Then there exists a positive constant L which is independent on h such that

Bhwh iRN >L

holds for the unit eigenvectorwhi (i= , . . . ,N)of the matrix Ah.

Proof First of all, we claim that there exists a positive numberMwhich is independent on

rsuch that

b

a

(6)

To this end, we calculate the following integrations:

On the one hand, there exists a positive numberNdepending only onba, such that

b

a

sin(rπt)dt>ba

 , asr>N.

On the other hand, there exists a positive numberMdepending only onN, such that

b

a

sin(rπt)dt>M, as  <rN.

TakingM=min{ba,M}, we see that the claim is correct.

After directly calculating, we obtain

Bhwhi From the theory of classic analysis, we have

(7)

i.e.,

Combining the above inequality with (.), we see that

independent onh, such that

BhwhiRN >L,

for any unit eigenvectorwh

i (i= , . . . ,N).

(8)

Remark . This theorem gives a fundamental property for the unit eigenvectors whi

(i= , . . . ,N) of the discrete Laplacian operator. It shows that the energy for these unit eigenvectors have an uniform lower boundary, which is positive and not dependent onh, in a nonempty and open subsetω⊂(, ).

Now, we need to introduce certain notations. LetXh

denote the spaceRNequipped with the norm · 

u=h

N

j=

|uj|, for anyu= (u,u, . . . ,uN)T∈RN.

LetXhdenote the spaceRN equipped with the norm · ,

u=h

N

j=

ujuj+

h

, for anyu= (u,u, . . . ,uN)T∈RN,

whereu=uN+= . According to the definitions of the discrete norms, the energy can be represented asEh(t) =( φh(t)+ φh(t)), whereφh(t) is the solution of equation (.). Lemma . For any vector u= (u,u, . . . ,uN)T∈RN,we have the following inequality:

λ(h)u≤ u.

This lemma can easily be deduced from Lemma ..

Remark .

(i) According to Lemma ., it is easy to find that the spacesXhandXhare both Banach spaces. In fact,Xh

andXhcan be regarded as the discrete version of the

spaceL(, )andH

(, ), respectively. Thus, Lemma . can be regarded as the

discrete version of Poincaré’s inequality.

(ii) SinceRN×RN is a finite dimensional space, thus all norms of this space are equivalent. In particular, there exist positive numbersC,C, such that

C(z,z)Xh

×Xh

(z,z)RN×RNC(z,z)Xh

×Xh (.)

hold for any(z,z)∈RN×RN.

3 The proof of Theorem 1.1 and Theorem 1.2

3.1 The proof of Theorem 1.1

Proof First of all, we will prove that there exists a positive constantC(T,h) such that the inequality

φh,φhRN×RNC(T,h)

T

Bhφh(t) 

RNdt (.)

(9)

LetT> . We first define a functionF:RN×RNRas

h,φh=

T

Bhφh(t)RNdt,

whereφh(t) is the solution of (.) with initial data (φh,φh). Obviously,F is continuous. Now, we will prove that

minh,φh;φh,φhRN×RN= 

L(h,T) (.)

holds for certain positive constantL(h,T) only depending onhandT.

Suppose that there exists an unit vector (ϕh,ϕh) inRN ×RN such thatF(ϕh

,ϕh) = . Since(ϕh,ϕh)RN×RN= ,ϕhandϕhcould not be zero at the same time. Without loss of

generality, we assume thatϕh= . According to (.), we have

ϕh=

N

j= ϕjwhj

and

ϕh=

N

j= ϕjwhj,

whereNj=|ϕ|j = ϕhRN = . Solving (.), we can deduce that

φh(t) =

N

j=

βj(t)whj, (.)

whereβj(t) =ϕjcos(

λj(h)t) + ϕj

λj(h)

sin(λj(h)t).

From the definition of the functionFand the assumption thatF(ϕh,ϕh) = , we have

 =h,ϕh=

T

Bhφh(t) 

RNdt=

T

N

j=

βj(t)Bhwhj

dt.

Thus,

N

j=

βj(t)Bhwhj = , for anyt∈[,T]. (.)

It follows from Lemma . or Proposition . that Bhwhj =  for anyj= , , . . . ,N. It is obvious that the rank of subspace spanned by{Bhwh, . . . ,BhwhN}is less thanN. Therefore, we can assume thatBhwh, . . . ,Bhwhα, with ≤α<N, are linear independent inRN, and

spanBhwh, . . . ,Bhwhα

=spanBhwh, . . . ,BhwhN

(10)

Hence,

Bhwhq= α

j=

bqjBhwhj, for anyq=α+ , . . . ,N.

For any q=α+ , . . . ,N, there exists at least one scalar bqj(q) (≤j(q)≤α) such that

bqj(q)= . This, together with (.), indicates that

 = α

j=

βj(t)Bhwhj + N

q=α+ βq(t)

α

j=

bqjBhwhj

= α

j=

βj(t) + N

q=α+

βq(t)bqj Bhwhj, for anyt∈[,T]. (.)

According to the linear independence of{Bhwhj}αj=, we can deduce that

βj(t) + N

q=α+

βq(t)bqj= , for anyj= , , . . . ,α, and for anyt∈[,T]. (.)

Takingt= , we get

ϕj + N

q=α+

ϕqbqj= , for anyj= , , . . . ,α.

Differentiating (.) twice and takingt= , we have

λj(h)ϕj + N

q=α+

λq(hqbqj= , for anyj= , , . . . ,α.

By induction, we obtain

λmj (hj+ N

q=α+

λmq(hqbqj= , for anyj= , , . . . ,α, andm∈N+. (.)

It follows from (.) that{λj(h)}Nj=are different from each other. Thus, we can deduce that

ϕj = , for anyj= , , . . . ,α, (.)

and

ϕqbqj= , for anyq=α+ , . . . ,N, andj= , , . . . ,α. Takingj=j(q), we have

(11)

This, together with (.), leads to a contradiction to the assumption thatϕh= . Thus, (.) holds. Note that (.) impliesF(μυ,μυ) =μF(μυ,μυ) for every (υ,υ)RN×RN andμ∈R. Thus, it is obvious that inequality (.) leads to (.).

From (.) with (.), it is easy to obtain the observability inequality (.) of the semi-discrete system (.). This completes the proof of this theorem.

3.2 The proof of Theorem 1.2

Proof Given the initial dataφh=wh By Lemma ., one shows that

Eh() =

This completes the proof of this theorem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JL provided the questions. YZ and GZ gave the proof for the main result together. All authors read and approved the final manuscript.

Author details

1College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China.2College of

Computer and Information Engineering, Henan University of Economics and Law, Zhengzhou, 450002, P.R. China.

Acknowledgements

(12)

(in Science and Technology) in University of Henan Province (14IRTSTHN023), Henan Higher School Funding Scheme for Young Teachers (2012GGJS-063).

Received: 22 March 2016 Accepted: 15 November 2016 References

1. Bardos, C, Lebeau, G, Rauch, J: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim.30, 1024-1065 (1992)

2. Coron, J-M: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007)

3. Zuazua, E: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp. 527-621. Elsevier Science, Amsterdam (2006) 4. Castro, C, Micu, S: Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite

element method. Numer. Math.102, 413-462 (2006)

5. Infante, JA, Zuazua, E: Boundary observability for the space semi-discretization of the 1-D wave equation. ESAIM: Math. Model. Numer. Anal.33, 407-438 (1999)

6. Zuazua, E: Propagation, observation, control and numerical approximation of waves by finite difference method. SIAM Rev.47, 197-243 (2005)

7. Castro, C, Micu, S: Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal.28, 186-214 (2008)

8. Micu, S, Zuazua, E: An introduction to the controllability of partial differential equations. In: Sari, T (ed.) Collection Travaux en Cours Hermannin Quelques Questions de Théorie du Contrôle, pp. 67-150 (2005)

References

Related documents

Specifically, we study two well knwon FLOSS repositories, Debian and SourceForge: it will be studied whether those repositories that experience higher levels of evolutionary

Just as he had insisted in his Preface to the English translation of Homo Academicus in 1988 (Bourdieu, 1984, 1988) that it was a book which was not about ‘homo academicus

He argued that most psychologists, academics and undergraduates alike, see evolutionary psychology as equivalent to social psychology, cognitive psychology, personality

Chapter 6 to Chapter 8 characterise welfare changes for the elderly and children, separately for men and women and boys and girls, in terms of poverty, income, nutrition and

(2007) ‘Storytelling, time and evolution: the role of strategic leadership in complex adaptive systems’, Leadership Quarterly , Vol. (1994) ‘Density-independent selection

court - Bosola in The Duchess of Malfi, Flamineo in The White Devil, De.. Flores in

On the one hand, the allocation regime has continuously allowed open politicisation for state secretaries since 1990 and for all other ranks of civil servants

In this way, both categories of historical gaming outlined here directly reflect the ways history is presented in the world outside games, but this factor needs to be considered far