© Hindawi Publishing Corp.
OBSERVABILITY AND UNIQUENESS THEOREM
FOR A COUPLED HYPERBOLIC SYSTEM
BORIS V. KAPITONOV and JOEL S. SOUZA
(Received23 December 1998)
Abstract.We deal with the inverse inequality for a coupled hyperbolic system with dis-sipation. The inverse inequality is an indispensable inequality that appears in the Hilbert Uniqueness Method(HUM), to establish equivalence of norms which guarantees uniqueness and boundary exact controllability results. The term observability is due to the mathemati-cian Ho (1986) who usedit in his works relating it to the inverse inequality. We obtain the inverse inequality by the Lagrange multiplier method under certain conditions.
Keywords and phrases. Observability, inverse inequality, uniqueness theorem, unique continuation.
2000 Mathematics Subject Classification. Primary 35L05, 35L65.
1. Introduction. Several approaches are known concerning the principle of unique continuation. One of them consists in the classical principle of identity for analytical function, that is, holomorphic (analytic) functions which are defined in some region can frequently be extended to holomorphic functions in some larger region. These extensions are uniquely determined by the given functions (cf. [6, 7, 10]). Another extension process, introducedby Holgren, is basedon the resolution of the homoge-neous boundary value problem for the wave equation, that is, given{f (x),f1(x)}in
Ᏸ(Ω)×Ᏸ(Ω), let us consider the homogeneous boundary value problem:
utt−∆u=0, inQ=Ω×]0,T [,
u(x,0)=f (x), ut(x,0)=f1(x),inΩ,
u=0, onΣ=Γ×]0,T [.
(1.1)
It is well known that in this case (1.1) has a strong solution and
∂u
∂ν ∈L2(Σ). (1.2)
Then we define inᏰ(Ω)×Ᏸ(Ω)the quadratic form:
f ,f1
Ᏸ(Ω)×Ᏸ(Ω)=
Σ0 ∂u
∂ν
2 dΓdt
1/2
, (1.3)
implies thatuis zero inQ, i.e., the seminorm (1.3) is, indeed, a norm inᏰ(Ω)×Ᏸ(Ω). This is true due to the Holgren’s theorem (cf. [3, 8]).
Our purpose in this paper is to establish a result of unique continuation in the direction of the Holgren’s theorem based on a result of Ruiz [11]. We present this approach as follows.
LetΩbe a bounded domain inRnwith boundaryΓ=∂Ω. In the cylinderQ=Ω×]0,T [
we consider the initial boundary value problem for the wave equation (1.1). We are interestedin the following result: there exists an open subsetᏰ⊂Ωsuch that there exists a timeT >0 anda constantC >0, so that
Ω
ut2+|∇u|2dx≤CT 0
Ᏸ
ut2dx dt+u2
L2(Ω×(0,T ))
,
∀f ,f1∈H1
0(Ω)×L2(Ω).
(1.4)
Lions (cf. [9]) provedthat this estimate holds ifΩis of classC2, andᏰis a
neigh-borhoodof the part of the boundaryΓ0=Γ0(x0)= {x∈Γ;(x−x0,ν(x)) >0}, where x0is some point ofRn, andν(x)is the unit outer normal.
Bardos, Lebeau andRauch [2] showedthat, whenΩis of classC∞, inequality (1.4)
holds ifᏰsatisfies some “geometric control property,” that is: there exists someT >0 such that every ray of geometric optics intersects the setᏰ×(0,T ).
The estimate (1.4) implies the following unique continuation result: ifv∈H1(Ω× ]0,T [)is solution of
vtt−∆v+b(x)v=0, inΩ×]0,T [,
v=0, onΣ=Γ×]0,T [, v=0, inᏰ×]0,T [,
(1.5)
whereb∈L∞(Ᏸ×]0,T [), then
v≡0, inΩ×]0,T [, (1.6)
see Zuazua [12].
2. Problem formulation. In the cylinderΩ×]0,T [we consider the following initial boundary value problem for the coupled hyperbolic system with dissipation
∂2u ∂t2 −
∂ ∂xi
A∂x∂u
i
+B(x)∂v∂t =0, (2.1a)
∂2v ∂t2−
∂ ∂xi
A∂x∂v
i
−B(x)∂u∂t =0, (2.1b)
uu(x,t(x,00))==f (x),f1(x), vv(x,t(x,00))==g(x),g1(x), inΩ, (2.1c)
u=0, v=0, onΣ=Γ×]0,T [, (2.1d)
whereu=(u1,...,um),(v1,...,vm),A=A∗andB(x)=B∗(x)are square matrices of
We assume that
n
i=1
Aζi·ζi≥a0 n
i=1
ζi2, a0>0,whereζ
i=ζi1,...,ζim
∈Rn (2.2)
andB(x)η·η≥0, x∈Ω, η∈Rm. We denote byᏴthe real Hilbert space of quadruples
{u,v,u1,v1}ofm-compoundvector-functions such that
u,v∈H1(Ω)m, u|
Γ=v|Γ=0, u1,v1∈L2(Ω)m. (2.3) The inner product inᏴis defined by the formula
u,v,u1,v1,f ,g,f1,g1Ᏼ=
Ω
A∂u ∂xi·
∂f ∂xi+A
∂v ∂xi·
∂g
∂xi+u1·f1+v1·g1
dx.
(2.4)
We defined an unbounded operatorᏭinᏴgiven by:
D(Ꮽ)=u,v,u1,v1∈Ᏼ; u,v∈H2(Ω)∩H1
0(Ω)m, u1,v1∈H01(Ω)m
,
Ꮽu,v,u1,v1=
u1,v1,∂x∂
i
A∂x∂u
i
−Bv1,∂x∂
i
A∂x∂v
i
+Bu1
foru,v,u1,v1∈Ᏸ(Ꮽ).
(2.5)
In standard way, we can check that the domainᏰ(Ꮽ∗)of the adjoint operator is
given by:
D(Ꮽ∗)=f ,g,f1,g1∈Ᏼ; f ,g∈H2(Ω)∩H1
0(Ω)m, f1,g1∈H01(Ω)m
,
Ꮽ∗f ,g,f1,g1= −f1,g1, ∂ ∂xi
A∂x∂f
i
−Bg1,∂x∂
i
A∂x∂g
i
+Bf1
forf ,g,f1,g1∈D(Ꮽ∗).
(2.6)
Thus, the operatorᏭ is skew selfadjoint and from Stone’s theorem it follows that it generates a 1-parameter group of unitary operatorsᐁ(t)inᏴ. Moreover,ᐁ(t)is strongly continuous intandᐁ(t){f ,g,f1,g1}is strongly differentiable with respect totfor
f ,g,f1,g1∈D(Ꮽ) and dtdᐁ(t)f ,g,f1,g1=Ꮽᐁ(t)f ,g,f1,g1. (2.7)
Furthermore,ᐁ(t)takesD(Ꮽ)ontoD(Ꮽ)are commutes withᏭ.
It follows that if{u,v,u1,v1} =ᐁ(t){f ,g,f1,g1}, thenu,v is strong solution to problem (2.1), withut=u1, vt=v1, andpossesses the following regularity:
u,v,ut,vt∈C0[0,T ];D(Ꮽ)∩C1[0,T ];Ᏼ. (2.8)
Now, we introduce the following notation:
φ(u)=
n
i=1 A∂u
∂xi·
∂u
∂xi, E(t)=
Ω
ut2+vt2+φ(u)+φ(v)
Since the operatorᐁ(t)is unitary, we have
E(t)≡ᐁ(t)f ,g,f1,g12Ᏼ=f ,g,f1,g12Ᏼ≡E(0), ∀t∈R. (2.10) We also observe that, forᏲ= {f ,g,f1,g1} ∈Ᏼ,ᐁ(t)Ᏺis a weak solution inᏴto the abstract Cauchy problem
d dt
u,v,u1,v1=u1,v1,Au,Av (2.11)
in the following sense: T
0
ᐁ(t)Ᏺ,dΦdt
Ᏼ+
ᐁ(t)Ᏺ,A∗Φ Ᏼ
dt= −Ᏺ,Φ(0)Ᏼ, (2.12)
for everyΦ∈L2(0,T;D(A∗)),Φ
t∈L2(0,T;Ᏼ),Φ(T )=0.
3. Observability and uniqueness theorem. In this section, we obtain the inverse inequality by using the Lagrange multiplier methodwith certain conditions over an arbitrary neighborhoodᏰ⊂Ω.
It is easy to verify that the unperturbedwave equation of (2.1a) and(2.1b) is varia-tional andthat the expression
ᏸut,∂x∂u i
=1
2
n
i=1 A∂u
∂xi·
∂u ∂xi−
1 2ut
2 (3.1)
is its Lagrangian.
The invariance of the elementary action ᏸdx1···dxndt under the 1-parameter
group of dilations in all variables with infinitesimal operator
t∂ ∂t+
xi−xi0
∂ ∂xi−
n−1 2 uq
∂
∂uq (3.2)
leads to a conservation law (cf. [1, E. Noether’s theorem, page 88]). Now let{f ,g,f1,g1} ∈D(Ꮽ)andletu(x,t),v(x,y)be a solution of problem (2.1).
After integration by parts, over the cylinderΩ×]0,T [, that conservation law we then obtain the identity:
T E(T )+
Ω
2(∇ϕ,∇)u·ut+2(∇ϕ,∇)v·vt+C(x)u·ut+C(x)v·vtdx t=T
t=0
= T
0
Γ
∂v ∂ν
φ(u)+φ(v)dΓdt
+ T
0
Ω
2(∇ϕ,∇)v·But−2(∇ϕ,∇)u·Bvt+Cv·But
−CBvt·u+12∆C(Au·u+Av·v)+(u)+(v)
dx dt,
(3.3)
where the pair(u,v)is a solution of (2.1);ϕ(x),C(x)are smooth functions inΩ, and
We observe that identities of the kind in (3.3) are common in [4, 5]. We have that 2(∇ϕ,∇)v·But≤εB(∇ϕ,∇)v·(∇ϕ,∇)v+1
εBut·ut,
2(∇ϕ,∇)u·Bvt≤εB(∇ϕ,∇)u·(∇ϕ,∇)u+1
εBvt·vt, Cv·But−CBvt·u=2Cv·But−∂t∂ (Cv·Bu)
≤ −∂t∂(Cv·Bu)+ε|C|Bv·v+1ε|C|But·ut.
(3.5)
From these inequalities we obtain T
0
Ω 2(∇ϕ,∇)v·But−2(∇ϕ,∇)u·Bvt+Cv·But−CBvt·u
dx dt
≤
ΩCv·Budx t=0
t=T+
1
ε
T
0
Ω (1+|C|)But·ut+Bvt·vt
dx dt
+εC0B,C(x),ϕ,a0,ΩT 0
Ω
φ(u)+φ(v)dx dt.
(3.6)
From the identity T
0
ΩBvt·vtdx dt= T
0
ΩBut·utdx dt−
Ω
ut·vt+Auxj·vxj
dxt=T
t=0, (3.7)
it follows that T
0
ΩBvt·vtdx dt≤E(0)+ T
0
ΩBut·utdx dt. (3.8) So, using this inequality, from (3.6) we obtain
T
0
Ω 2(∇ϕ,∇)v·But−2(∇ϕ,∇)u·Bvt+Cv·But−CBvt·u dx dt ≤1 ε T 0
Ω (2+|C|)But·ut
dx dt+C0+1 ε
E(0)+εC0T E(0),
Ω 2(∇ϕ,∇)u·ut+2(∇ϕ,∇)v·vt+C(x)u·ut+C(x)v·vt
dxt=T
t=0
≤C1|∇ϕ|,C(x),a0,ΩE(0).
(3.9)
We then arrive at the following inequality:
T E(0)+εC0T E(0)−C1E(0)−
c0+1ε
E(0)
≤ T 0 Γ ∂ϕ ∂ν
φ(u)+φ(v)dΓ+1 ε T 0 Ω
(2+|C|)But·vtdx dt
+ T 0 Ω 1
2∆C(Au·u+Av·v)+(u)+(v)
dx dt.
(3.10)
Now for an arbitrary pointx0∈Rnwe find:
LetᏰ⊂Ωbe an arbitrary neighborhoodofΓ0. There exists a functionϕ(x)such that (i) (∂ϕ/∂ν)≤0 onΓ,
(ii) ϕ0(x)≥µ >0,inΩ\Ᏸ,whereϕ0(x)=inf∂2ϕ/∂x i∂xj
ηiηj.
SettingC(x)=∆ϕ−2ϕ0(x)+µ, we have
(u)=φ(u)∆φ+1−C(x)+ut21+C(x)−∆ϕ−2ϕxixjAuxiuxi ≤φ(u)∆ϕ+1−C(x)−2ϕ0φ(u)+ut21+C(x)−∆ϕ
=(1−µ)φ(u)+1−2ϕ0(x)+µut2,
T
0
Ω
(u)+(v)dx dt
≤ T
0
Ω
(1−µ)φ(u)+φ(v)+1+µ−2ϕ0(x)ut2+vt2
dx dt
≤(1−µ)
T
0
Ω\Ᏸ
ut2
+vt2+φ(u)+φ(v)dx dt
+ T
0
Ᏸ (1−µ)
φ(u)+φ(v)+1+µ−2ϕ0ut2+vt2
dx dt
=(1−µ)
T
0
ΩE(t)dt+ T
0
Ᏸ2
µ−ϕ0ut2+vt2
dx dt
≤(1−µ)T E(0)+2µ+supϕ0
T
0
Ᏸ
ut2+vt2
dx dt. (3.12)
We assume that
B(x)u·u≥b0|u|2, inᏰ, (3.13)
then
T
0
Ᏸ
vt2dx dt≤ 1 b0
T
0
ᏰB(x)vt·vtdx dt ≤b01
T
0
ΩB(x)vt·vtdx dt
≤b01E0+b01 T
0
ΩB(x)ut·utdx dt,
(3.14)
andsimilarly, T
0
Ᏸ
ut2dx dt≤ 1 bo
T
0
ΩB(x)ut·utdx dt. (3.15) From these inequalities we obtain
T
0
Ω
(u)+(v)dx dt
≤(1−µ)T E(0)+b02µ+supϕ0E(0)
+b04µ+supϕ0
T
0
ΩBut·utdx dt.
Inequalities (3.10) and(3.16) leadto
Tµ−εC0E(0)−
C0+C1+1ε+b02µ+supϕ0
E(0)
≤1
ε(2+sup|C|)+
4
b0
µ+supϕ0T 0
ΩBut·utdx dt
+ T
0
Ω 1
2∆C(Au·u+Av·v)dx dt.
(3.17)
In the general case, we can choose functionϕ(x)in the following way:
ϕ(x)=1
2x−x0
2, x∈Ω\Ᏸ, (3.18)
andextendit inᏰsatisfying the property (i). Then,
∆C=∆∆ϕ−2ϕ0(x)+µ≡0, inΩ\Ᏸ. (3.19)
Thus, from (3.17), we arrive at the inequality:
Tµ−εC0E(0)+
C0+C1+1ε+b02µ+supϕ0
E(0)
≤ 1
ε(2+sup|C|)+
4
b0
µ+supϕ0
T
0
ΩBut·utdx dt
+C2(ϕ,A)
T
0
u2
L2(Ᏸ)+v2L2(Ᏸ)
dt.
(3.20)
Now we can assume that there exists a functionϕ(x)which satisfy (i), (ii), and (iii) ∆2ϕ−2∆ϕ0(x)≤0,inΩ.
(We can construct some examples of these functions under certain assumptions onΩ
andᏰ.) In this case, from (3.17), it follows
Tµ−εC0E(0)−
C0+C1+1ε+b02µ+supϕ0
E(0)
≤ 1
ε
2+sup|C|+b04µ+supϕ0
T
0
ΩBut·utdx dt.
(3.21)
We chooseεsuch thatµ > εC0(we can setε=(µ/2C0)), and
T >2µC0+2C1µ +4µC02 +b04
1+µ1supϕ0
≡T0. (3.22)
Then (3.20) and(3.21) can be rewritten in the following way:
µ
2
T−T0E(0)≤C T
0
ΩBut·utdx dt+C2(ϕ,A) T
0
u2
L2(Ᏸ)v2L2(Ᏸ)
dt, (3.23)
µ
2
T−T0E(0)≤C
T
0
4. Conclusion. We arrive at the following assertion: in the cylinderΩ×]0,T [, we consider the initial boundary value problem for the following coupled hyperbolic sys-tem with dissipation:
∂2u ∂t2 −
∂ ∂xi
A∂x∂u
i
+ᐄᏰ(x)B(x)∂v∂t =0,
∂2v ∂t2−
∂ ∂xi
A∂x∂v
i
−ᐄᏰ(x)B(x)∂u∂t =0,
uu(x,t(x,00))==f (x),f1(x), vv(x,t(x,00))==g(x),g1(x), inΩ,
u=v=0, onΣ=Γ×]0,T [.
(4.1)
HereA=A∗, B(x)=B∗(x), B(x)η·η≥b0|η|2,b0>0; Aζ
i·ζi≥a0|ζi|2, a0>0, B∈
L∞(Ᏸ), and
ᐄᏰ(x)= 1, x∈
Ᏸ,
0, x∈Ᏸ, (4.2)
whereᏰis an arbitrary neighborhood of the boundary (or a part of the boundary). We have that:
(1) IfᏰis an arbitrary neighborhoodofΓ0= {x∈Γ;(x−x0,ν(x))≥0}, then there
existsT0>0 such that, forT > T0, the following inequality holds:
T−T0
Ω
ut2+v
t2+φ(u)+φ(v)dx
≤C1
T
0
Ᏸ
ut2dx dt+C2T 0
u2
L2(Ᏸ)+v2L2(Ᏸ)
dt.
(4.3)
(2) IfᏰandΩare such that there exists a functionϕ(t)with properties (i), (ii), and (iii), then, forT > T0, we have
T−T0
Ω
ut2+v
t2+φ(u)+φ(v)dx≤C
T
0
Ᏸ
ut2dx dt. (4.4)
Remark4.1. Let us consider the coupled system:
∂2u ∂t2−
∂ ∂xi
A∂x∂u
i
+G(x)u+ᐄᏰ(x)B(x)∂v∂t =0,
∂2v ∂t2−
∂ ∂xi
A∂x∂v
i
+G(x)v−ᐄᏰ(x)B(x)∂u∂t =0,
(4.5)
whereA,B, andᏰare defined above,G=G∗, withG(x)η·η≥0, x∈Ω; andsupport (G)⊆Ᏸ¯, G∈W1,∞(Ᏸ).
These new terms considered add to the right-hand side of the main identity (3.3) (andthe rest of the formulas) the following expression:
T
0
Ω ϕxiGxiu·u+(∆ϕ+1−C)Gu·u+ϕxiGxiv·v+(∆ϕ+1−C)Gv·v
dx dt.
(In this case, E(t) =Ω[|u|2+ |v|2+φ(u)+φ(v)+Gu·u+Gv·v]dx.) After the substitutionC=∆ϕ−2ϕ0+µ, this expression takes the form
(1−µ)
T
0
Ω(Gu·u+Gv·v)dx dt
+ T
0
Ω ϕxiGxiu·u+2ϕ0Gu·u+ϕxiGxiv·v+2ϕ0Gv·v
dx dt.
(4.7)
Thus, we obtain under the same assumption: (1) In the first case,
(T−T0)
Ω
ut2+v
t2+φ(u)+φ(v)+Gu·u+Gv·v
dx
≤C1
T
0
Ᏸ
ut2dx dt+C2T 0
u2
L2(Ᏸ)+v2L2(Ᏸ)
dt.
(4.8)
(2) In the secondcase:
T−T0 E(t)≤C1
T
0
Ᏸ
ut2dx dt+C3T 0
u2
L2(Ᏸ)+v2L2(Ᏸ)
dt, (4.9)
i.e., in this situation only the first case makes sense.
Thus, we have the following results of unique continuation, i.e., ifᏰ⊂Ωis an arbi-trary neighborhoodofΓ0satisfying (i) and(ii), with support(G)⊆Ᏸ¯, andthe pair(u,v)
is a strong solution of the initial boundary value problem for the coupled system (4.5) with homogeneous Dirichlet condition and
u≡v≡0, inᏰ×]0,T [, (4.10)
then
u≡v≡0, inΩ×]0,T [. (4.11)
Acknowledgement. Kapitonov was supportedby Russian Fundof
Fundamen-tal Research, project 94-01-0078; andSouza was supportedby CNPq grant Proc.
20.0692/97.6, Brazil.
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Boris V. Kapitonov: Institute of Mathematics, Russian Academy of Sciences and Uni-versidade Federal de Santa Catarina, Russia
Joel S. Souza: Departamento de Matemática da Universidade Federal de Santa Cata-rina C.P.476, CEP88040-900, Florianópolis, SC, Brazil
