R E S E A R C H
Open Access
The wave equation with internal control in
non-cylindrical domains
Lizhi Cui
**Correspondence: [email protected]
College of Applied Mathematics, Jilin University of Finance and Economics, Changchun, 130117, China
Abstract
In this paper, we shall be concerned with interior controllability for a one-dimensional wave equation in a domain with moving boundary. When the speed of the moving endpoint is less than a certain constant which is less than the characteristic speed, we obtain exact controllability for this equation.
Keywords: interior controllability; wave equation; non-cylindrical domains
1 Introduction and main results GivenT> . For any <k< , set
αk(t) = +kt, fort∈[,T]. (.)
Also, define the following non-cylindrical domains:
QkT=(y,t)∈R; <y<αk(t),t∈[,T]
,
and for any <m<m<n<n< ,
Q=
(y,t)∈R;mαk(t) <y<nαk(t),t∈[,T]
,
Q=
(y,t)∈R;mαk(t) <y<nαk(t),t∈[,T]
.
Consider the following controlled wave equation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
utt–uyy=Bv, (y,t)∈QkT,
u(,t) = , u(αk(t),t) = , t∈(,T),
u(y, ) =u(y), u
t(y, ) =u(y), y∈(, ),
(.)
wherev∈[H(Q)]is control variable,uis state variable, (u,u)∈L(, )×H–(, ) is
any given initial value andB∈C∞(QkT),
B(y,t) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
= , (y,t)∈QkT\Q,
= , (y,t)∈Q,
∈(, ), (y,t)∈Q\Q.
By [], it is easy to check that (.) has a unique weak solutionu:
u∈C[,T];L,αk(t)
∩C[,T];H–,αk(t)
.
The main purpose of this paper is to study exact controllability of (.) in the following sense.
Definition . Equation (.) is called exactly controllable at the timeT, if for any initial value (u,u)∈L(, )×H–(, ) and any target (u
d,ud)∈L(,αk(T))×H–(,αk(T)),
one can always find a controlv∈[H(Q)]such that the corresponding weak solutionu
of (.) satisfies
u(T) =ud, ut(T) =ud.
The main result of this paper is stated as follows.
Theorem . Suppose that <k˜< , <k<k.˜ For any given T>Tk∗, (.)is exactly con-trollable at time T in the sense of Definition..
Remark . k˜andTk∗will be defined during proof of this theorem.
There is a variety of literature on interior and boundary controllability problems of wave equations in a cylindrical domain (seee.g.[–]). However, there is only little work of wave equations defined in non-cylindrical domains. We refer to [–] for some known results in this respect. In practical situations, many processes evolve in domains whose boundary has moving parts. A simple model,e.g., is the interface of ice water mixture when temperature increases. To study the controllability problem of wave equations with moving boundary or free boundary is very meaningful. In [–], boundary controllability for wave equations with a moving boundary has been obtained. In [], some controllability results for wave equations with Dirichlet boundary conditions in suitable non-cylindrical domains were investigated. In [], in the one-dimensional case, the following condition seems necessary:
∞
αk(t)dt<∞.
the time variablet. Moreover, in the published work, variable coefficients are only depen-dent of space variablexin most cases. Meanwhile, in our paper, the variable coefficients are dependent of space variablexand time variablet. To solve this, motivated by [], the key point is to construct a suitable multiplier different from that in [].
The rest of this paper is organized as follows. In Section , we reduce the controllability problem of (.) to that of a wave equation with variable coefficients in a cylindrical do-main. Section is devoted to proving an observability inequality of a wave equation with variable coefficients in a cylindrical domain.
2 Reduction to controllability problems in a cylindrical domain
When <k< , in order to prove Theorem ., we first transform (.) into a wave equation with variable coefficients in a cylindrical domain in this section. Set
Q= (, )×(,T), Q= (m,n)×(,T), Q=
m,n×(,T).
To this aim, set
x= y αk(t)
and w(x,t) =u(y,t) =uαk(t)x,t
, for (y,t)∈QkT.
Then it is easy to check that (x,t) varies inQ. Also, (.) is transformed into the following equivalent wave equation inQ:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
wtt– [βαkk(x(t,t))wx]x+
γk(x)
αk(t)wtx=Bv(x,t), (x,t)∈Q, w(,t) = , w(,t) = , t∈(,T), w(x, ) =w, w
t(x, ) =w, x∈(, ),
(.)
where
v(x,t) =vαk(t)x,t
=v(y,t), βk(x,t) =
–kx
αk(t)
,
γk(x) = –kx, w=u, w=u+kxux,
B(x,t) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
= , (x,t)∈Q\Q,
= , (x,t)∈Q,
∈(, ), (x,t)∈Q\Q.
(.)
For any given initial value (w,w)∈L(, )×H–(, ) and any controlv∈[H(Q)], (.)
admits a unique weak solution
w∈C[,T];L(, )∩C[,T];H–(, ).
To prove this, we first solve the following system.
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ξtt– [βαk(x,t)
k(t) ξx]x+
γk(x)
αk(t)ξtx= , (x,t)∈Q, ξ(,t) = , ξ(,t) = , t∈(,T), ξ(x, ) =w, ξ
t(x, ) =w, x∈(, ).
(.)
For any given (w,w)∈L(, )×H–(, ), this system has a unique weak solution
ξ∈C[,T];L(, )∩C[,T];H–(, ).
Then in order to obtain interior controllability of (.), we only prove interior controlla-bility result for the following wave equation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ηtt– [βαk(x,t)
k(t) ηx]x+
γk(x)
αk(t)ηtx=Bv(x,t), (x,t)∈Q, η(,t) =η(,t) = , t∈(,T), η(x, ) =ηt(x, ) = , x∈(, ).
(.)
Theorem . Let T>Tk∗.Then,for any target(wd,wd)∈L(, )×H–(, ),there exists
a control v∈[H(Q)]such that corresponding weak solutionηof (.)satisfies
η(T) =wd, ηt(T) =wd.
WriteH= [H(Q
)],F=L(, )×H–(, ) andF=H(, )×L(, ). Define a linear
operatorA:
A:H→F,
Ag=αk(T)ηt(x,T) –kη(x,T) +γk(x)ηx(x,T),αk(T)η(x,T)
, ∀g∈H.
ThenAis surjective is equivalent to interior controllability of the wave equation (.). And Ais surjective is derived from an observability inequality of the form
Az,zH≥Cz,zF, ∀
z,z∈F(C> ), (.) for the dual operatorA:F→HforT>Tk∗.
3 Observability inequality of wave equations with variable coefficients First, we defineA.Ais described by the following homogeneous wave equation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
αk(t)ztt– [βk(x,t)zx]x+γk(x)ztx= , inQ,
z(,t) = , z(,t) = , on (,T),
z() =z, z
t() =z, in (, ),
(.)
wherek∈(, ), (z,z)∈H(, )×L(, ) is any given initial value, andαk,βkandγkare
given in (.). Equation (.) has a unique weak solution,
Set B the adjoint of the extension operator B in (.), and if v∈[H(Q
)], thenB :
H(Q)→H(Q
). HenceAis defined as follows:
Az,z=Bαk(t)z
=αk(t)z(x,t), (x,t)∈Q,∀
z,z∈F,
wherezis the solution of (.). Therefore, (.) is equivalent to the following inequality:
αk(t)zH(Q
)≥Cz
,z
F, ∀
z,z∈F. (.)
In the following, we shall give a proof of (.) by the multiplier method. Define the following weighted energy for (.):
E(t) =
αk(t)zt(x,t)
+βk(x,t)zx(x,t)
dx fort≥,
It follows that
EE() =
z(x)+βk(x, )zx(x)
dx.
We obtain the following lemma (see the detailed proof in []).
Lemma . For any(z,z)∈H(, )×L(, )and t∈[,T],any solution z of (.) sat-isfies the following estimate:
E(t) = αk(t)
E. (.)
Equation (.) is derived with a special multiplier. Set
F(x,k) =βk,x(x,t) βk(x,t)
= –k
x
–kx ≤, (x,k)∈[, ]×(, ).
It is easy to checkFx(x,k) < , (x,k)∈[, ]×(, ). We have
F(,k) = –k
–k≤F(x,k), (x,k)∈[, ]×(, ).
We seeFk(,k) =(––kk) < ,k∈(, ). Therefore, we obtain, for anyη> ,
FF(, –η) =
–( –η)
– ( –η) ≤F(,k), k∈(, –η].
Hence, we derive
F≤F(,k)≤F(x,k)≤, (x,k)∈[, ]×(, –η]. (.)
Assume thatλ∈(, ) and a pointx∈(m,n) to be unknown for now. We require the
Lemma . Assume that p(x)be a solution of first-order linear differential equation
p(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
λ– , x∈(,m), λ, x∈(m,x),
λ+Fp(x), x∈(x,n),
λ– +Fp(x), x∈(n, ),
(.)
then there exist a uniqueλ∈(, )and a unique x∈(m,n)such that p(x)belongs to C[, ]
and satisfies
p() =p() =p(x) = . (.)
Proof By (.), (.) and the constant variation method, it follows that
p(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(λ– )x≤, x∈(,m), (x–x)λ≤, x∈(m,x),
λ F[e
F(x–x)– ]≥, x∈(x
,n), (λ–)
F [e
F(x–)– ]≥, x∈(n, ).
Note thatp(x)∈C[, ], we have
p(m– ) =p(m+ ), p(n– ) =p(n+ ).
From this, we obtain
λλ–=
m x
, x∈(m,n), (.)
λλ+=
eF(n–)–
eF(n–)–eF(n–x), x∈(m,n). (.)
By (.) and (.), we haveλ–andλ+are monotone decreasing and increasing with respect
toxand satisfy
λ–,λ+∈(, ); λ–(m) >λ+(m); λ–(n) <λ+(n).
Hence there exists a uniquex∈(m,n) of the equationλ–=λ+, the corresponding value
λ=λ–=λ+∈(, ).
Remark . It is easy to verify that
M max
≤x≤
In the following, we prove (.) by the above multiplierp(x). Multiplying the first
equa-In the following, we calculate the above three integralsDi(i= , , ), respectively. It is easy
to check that
Write
i(t)
αk(t)p(x)zt(x,t)zx(x,t) +
γk(x)p(x)zx(x,t)
dx. (.)
By (.)-(.), we obtain
T
αk(t)px(x)zt(x,t)
+
p(x) βk(x,t)
x
βk(x,t)
zx(x,t)
dx dt
= T
αk,t(t)p(x)zt(x,t)zx(x,t)dx dt+i() –i(T). (.)
By Lemma ., it follows that
px(x) =λ– , x∈[,m]
and
p(x)≤, x∈[,m]. Then we have
F(x,k)p(x) = βk,x(x,t) βk(x,t)
p(x)≥, (x,t)∈[,m]×[,T]
and
px(x)≤
βk,x(x,t)
βk(x,t)
p(x) +λ– , (x,t)∈[,m]×[,T].
From this, we have
px(x)βk(x,t) –βk,x(x,t)p(x)≤(λ– )βk(x,t), (x,t)∈[,m]×[,T]. (.)
Similarly, we obtain
px(x)βk(x,t) –βk,x(x,t)p(x)≤λβk(x,t), (x,t)∈[m,x]×[,T]; (.)
for anyη> ,k∈(, –η],
px(x)βk(x,t) –βk,x(x,t)p(x)≤λβk(x,t), (x,t)∈[x,n]×[,T], (.)
and
px(x)βk(x,t) –βk,x(x,t)p(x)≤(λ– )βk(x,t), (x,t)∈[n, ]×[,T]. (.)
By (.)-(.), it follows that for anyη> ,k∈(, –η], λ
T
n
m
αk(t)zt(x,t)+βk(x,t)zx(x,t)
+λ–
It follows that
For eachε∈(, –λ), we have
From the above inequality, it follows that
Write
˜
k=min
–λ
p(m) + ( –λ)m,
–λ
p(n) + ( –λ)n, –η
.
By (.), (.), (.) and (.), we derive, for eachk∈(,k),˜
T
n
m
αk(t)zt(x,t)+βk(x,t)zx(x,t)
≥( –λ–M)
T
E(t)dt– M –kE
=
–λ–√M
k ln( +kT) – M –k
E. (.)
Set
Tk∗=e
kM
(–λ–√M)(–k)–
k .
IfT>Tk∗, we have –λ–
√ M
k ln( +kT) –
M
–k> . Also,
T
n
m
αk(t)zt(x,t)+βk(x,t)zx(x,t)
≥
–λ–√M
k ln( +kT) – M –k
E
≥C
–λ–√M
k ln( +kT) – M –k
zH (,)+
zL(,)
. (.)
Equation (.) is deduced by (.).
Remark . We can finally check that
Tlim
k→T
∗
k = max{m, –n}.
It is well known that (.) in the cylindrical domain is interiorly controllable at any time T>T. However, we do not know whetherT∗
k is sharp.
4 Conclusions
In this paper, we consider interior controllability for a one-dimensional wave equation in a domain with moving boundary. When the speed of the moving endpoint is less than a certain constant which is less than the characteristic speed, we obtain exact controllability for this equation. In the future, we hope that we may consider controllability problem of wave equations with free boundary.
Acknowledgements
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
All authors read and approved the final manuscript.
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Received: 23 April 2017 Accepted: 18 July 2017 References
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