Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 635851, 20 pages
doi:10.1155/2011/635851
Research Article
Initial Boundary Value Problem and
Asymptotic Stabilization of the Two-Component
Camassa-Holm Equation
Xiju Zong,1 Xingong Cheng,1 Zhonghua Wang,1 and Zhenlai Han2
1
2
School of Control Science & Engineering, University of Jinan, Jinan 250022, Shandong, China
School of Science, University of Jinan, Jinan 250022, China
Correspondence should be addressed to Xingong Cheng, cse cxg@ujn.edu.cn
Received 9 May 2011; Revised 23 June 2011; Accepted 20 July 2011
Academic Editor: Dirk Aeyels
Copyright q 2011 Xiju Zong et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The nonhomogeneous initial boundary value problem for the two-component Camassa-Holm
equation, which describes a generalized formulation for the shallow water wave equation, on an
interval is investigated. A local in time existence theorem and a uniqueness result are achieved.
Next by using the fixed-point technique, a result on the global asymptotic stabilization problem by
means of a boundary feedback law is considered.
1. Introduction
In this paper, we are concerned with the initial boundary value problem and the asymptotic
stabilization of the two-component Camassa-Holm equation on a compact interval by means
of a stationary feedback law acting on the boundary. The two-component Camassa-Holm
equation reads as follows:
ut − uxxt 3uux − 2ux uxx − uuxxx ρ ρx 0,
ρt ρx u ρux 0,
1.1
which was first derived as a bi-Hamiltonian models by Olver and Rosenau, see 1. The
system 1.1 shares many features with the Korteweg-De Vries equation Camassa-Holm
equation and Degasperis-Procesi Equation; for instance, it has a Lax pair formulation, and
it is integrable. In fact, the system 1.1 is related to the first negative flow of the AKNS
hierarchy via a reciprocal transformation 2, 3. In 4, Constantin and Ivanov deviated 1.1
2
Abstract and Applied Analysis
in the context of shallow water waves theory. As well as they showed that it has global strong
solutions and also finite time blow-up solutions. Well-posedness and blow-up results are
obtained in 5, 6.
For ρ ≡ 0, the equation 1.1 becomes the Camassa-Holm equation, which is modeling
the unidirectional propagation of shallow water waves over a flat bottom. Here ut, x
stands for the fluid velocity at time t in the spatial x direction 7–11. The Camassa-Holm
equation is also a model for the propagation of axially symmetric waves in hyperelastic
rods 12, 13. It has a bi-Hamiltonian structure 3 and is completely integrable 7, 14. Also
there is a geometric interpretation of the equation 1.1 in terms of geodesic flow on the
diffeomorphism group of the circle 15, 16. Its solitary waves are peaked 17. They are
orbitally stable and interact like solitons 18, 19.
The Cauchy problem and initial-boundary value problem for the Camassa-Holm
equation have been studied extensively in 20–26 and references within. It has been shown
that this equation is locally well posed 20–23, 26 for some initial data. The advantage of
the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the
Camassa-Holm equation has peaked solitons and models wave breaking 27, 28 by wave
breaking we understand that the wave remains bounded while its slope becomes unbounded
in finite time 29.
0, the Cauchy problems of 1.1 have been discussed in 5, 30, respectively.
For ρ /
Recently, a new global existence result and several new blow-up results of strong solutions
for the Cauchy problem of 1.1 were obtained in 6. And a new local existence result and
several new blow-up results and blow-up rate of strong solutions for the Cauchy problem of
1.1 defined in a torus were obtained in 31. Guan and Yin proved the existence of global
week solutions to 1.1 provided the initial data satisfying some certain conditions, see 32.
As far as the initial boundary value problem and the asymptotic stabilization of the
two-component Camassa-Holm equation on a compact interval are concerned, there are
seldom results yet, to the authors’ knowledge. Our aim of this paper is to prove the existence
of the initial boundary value problem and the asymptotic stabilization of the two-component
Camassa-Holm equation on a compact interval by acting on the boundary feedback law,
precisely,
1 the exact controllability problem: given two states u0 , ρ0 and u1 , ρ1 and a time
T > 0, can one find a certain function vt such that the solution to 1.1 satisfies
uT u1 , ρT ρ1 ? and
2 the stabilizability problem: can one find a stationary feedback law vx, such that
for any state u0 , ρ0 a solution pair ut, ρt to closed-loop system is global?
To explain our boundary formulation of 1.1, let us first introduce some transformation, precisely, m u − uxx and ρ ρ − 1, which lead the system 1.1 to be equivalent to the
system:
mt umx 2mux ρρx ρx 0,
ρt uρx ρux ux 0.
1.2
Abstract and Applied Analysis
3
Let T be a positive number. In the following we take ΩT 0, T × 0, 1. Let vl and vr be in
C0 0, T , R and m0 ∈ L∞ 0, 1, ρ0 ∈ W 1,∞ 0, 1. We set
Γl {t ∈ 0, T | vl t > 0},
Γr {t ∈ 0, T | vr t < 0}.
1.3
In the following, we will always suppose that the sets
Pl {t ∈ 0, T | vl t 0},
Pr {t ∈ 0, T | vr t 0}
1.4
have a finite number of connected components. Finally, let ml , ρl ∈ L∞ Γl × W 1,∞ Γl and
mr , ρr ∈ L∞ Γr × W 1,∞ Γr . The given functions vl , vr , ml , ρl , and mr , ρr will be the boundary
values for the equation; m0 , ρ0 are the initial data. Let now At, x be the auxiliary function
which lifts the boundary values vl and vr and is defined by
1 − ∂xx At, x 0,
At, 0 vl t,
∀t, x ∈ ΩT ,
At, 1 vr t,
∀t ∈ 0, T .
1.5
Setting u θ A, we can further rewrite the system 1.1 as
mt, x 1 − ∂xx θt, x,
θt, 0 θt, 1 0,
mt θ Amx −2m∂x θ A − ρρx − ρx ,
ρt θ Aρx − ρ 1 ∂x θ A,
m0, · m0 ,
ρ0, · ρ0 ,
m·, 0|Γl ml ,
ρ·, 0|Γl ρl ,
m·, 1|Γr mr ,
1.6
1.7
ρ·, 1|Γr ρr .
−ρρ −ρ 0
x
x
Let y mρ , y0 mρ00 , bt, x −2∂x θA
−∂x θA , and the system
0
−∂x θA , ft, x 1.7 can be written as
∂t y θ A∂x y bt, xy ft, x,
y0, · y0 ,
y·, 0|Γl yl ,
y·, 1|Γr yr .
1.8
We first define what we mean by a weak solution to 1.8. Our test functions will be in the
space:
admΩT ψ ∈ C1 ΩT × C1 ΩT | ∀x ∈ 0, 1, ψt, x 0; ∀t ∈ 0, T /Γl ,
ψt, 0 0; ∀t ∈ 0, T /Γr , ψt, 1 0 .
1.9
4
Abstract and Applied Analysis
m0 ∞
ΩT × W 1,∞ ΩT , when θ ∈ L∞ 0, T ; Lip0, 1, a
Definition 1.1. Given
m y0 ∞ ρ0 ∈ L 1,∞
function pair y ρ ∈ L ΩT × W ΩT is a weak solution to 1.8 if y satisfies
ΩT
y∂t ψ yθ A∂x ψ − ybt, xψ − ft, xψ dt dx
−
1
0
y0 ψ0, xdx T
ψt, 1vr tyr t − ψt, 0vl tyl t dt.
1.10
0
It is obvious that C01 ΩT × C01 ΩT ⊂ admΩT ; therefore, a weak solution to 1.8 is
also a solution to 1.8 in the distribution sense.
Definition 1.2 see 33. For t, x ∈ ΩT , let ϕ·, t, x be the C1 maximal solution to
∂s ϕs, t, x a s, ϕs, t, x ,
ϕt, t, x x,
1.11
which is defined on a certain set et, x, ht, x which is closed because 0, 1 is compact
and with possibly et, x and/or ht, x t.
We take into account the influence of the boundaries by introducing the sets:
P t, x ∈ ΩT | ∃s ∈ et, x, ht, x such that ϕs, t, x ∈ {0, 1}, a s, ϕs, t, x 0 ,
∪ s, ϕs, 0, 0 | ∀s ∈ 0, T ∪ s, ϕs, 0, 1 | ∀s ∈ 0, T ,
I {t, x ∈ ΩT /P | et, x 0},
L t, x ∈ ΩT /P | ϕet, xt, x 0 ,
R t, x ∈ ΩT /P | ϕet, xt, x 1 ,
Γl {t ∈ 0, T | at, 0 > 0},
Γr {t ∈ 0, T | at, 1 > 0}.
1.12
The following lemma, see 33, will play an important role in proving the local time existence
theorem and of a uniqueness result of the initial boundary value problem.
Lemma 1.3. Let a ∈ C0 0, T ; C1 0, 1, b, f ∈ L∞ ΩT , y0 ∈ L∞ 0, 1, yl ∈ L∞ Γl , and
yr ∈ L∞ Γr . We will also suppose that the sets:
Pl {t ∈ 0, T | vl t 0},
Pr {t ∈ 0, T | vr t 0}
1.13
Abstract and Applied Analysis
5
have at most a countable number of connected components. Then the function y, defined by the formula
for t, x ∈ P,
for t, x ∈ I,
yt, x 0,
yt, x exp
t
b r, ϕr, t, x dr y0 ϕ0, t, x
0
t
t
exp
0
for t, x ∈ L,
yt, x exp
t
et,x
for t, x ∈ R,
yt, x exp
b r, ϕr, t, x dr f s, ϕs, t, x ds,
t
et,x
et,x
b r, ϕr, t, x dr yl et, x
t
t
s
exp
s
1.14
b r, ϕr, t, x dr f s, ϕs, t, x ds,
b r, ϕr, t, x dr yr et, x
t
t
et,x
exp
s
b r, ϕr, t, x dr f s, ϕs, t, x ds,
is a weak solution of
∂t y θ A∂x y bt, xy ft, x
1.15
and satisfies
y L∞ ΩT ≤ max y0 L∞ ΩT , yl L∞ Γl , yr L∞ Γr T f L∞ ΩT et
b
L∞ ΩT .
1.16
However, if we let ml , ρl ∈ L∞ Γl × L∞ Γl and mr , ρr ∈ L∞ Γr × L∞ Γr , note that f
depends on the unknown ρ which is not a data; therefore Lemma 1.3 does not hold, or rather
Theorem 6 from the Appendix of 33 can not be applied directly to 1.8 or 1.1. Indeed if
θ and A are given, one can solve the equation on ρ the equation on the second component
in 1.8 or 1.1, but this result only guarantee that ρ is in L∞ . Therefore the source term in
the equation on m is not in L∞ anymore but in L∞ 0, T ; W −1,∞ 0, 1, and then Lemma 1.3
cannot be used to solve the transport equation on m the first component equation on 1.8.
One might try to get more regularity on ρ, but in this case more regularity is also needed on
ρ0 , ρl , ρr0 and even on θ and A to get sufficient geometrical assumptions. Then, one might
manage ρ0 ∈ W 1,∞ 0, 1, ρl ∈ W 1,∞ Γl , ρr ∈ W 1,∞ Γr to obtain at least Lipschitz solution of
the scalar transport equation on ρ and then get a weak solution on m.
The rest of this paper is organized as follows. In Section 2, the main results of the
present paper are stated. Section 3 will be devoted to the proofs of a local time existence
theorem and of a uniqueness result of the initial boundary value problem for the system 1.8
or 1.1. The problem of asymptotic stabilization for the system is analyzed, and a feedback
control law will be investigated in Section 4.
6
Abstract and Applied Analysis
2. Main Results
Theorem 2.1. For T > 0, we consider vl ∈ C0 Γl , vr ∈ C0 Γl such that the sets Pl and Pr have only
a finite number of connected components. Let y0 ∈ L∞ 0, 1 × W 1,∞ 0, 1, yl ∈ L∞ Γl × W 1,∞ Γl ,
and yr ∈ L∞ Γr × W 1,∞ Γr . There exist T > 0 and θ, y a weak solution of the system 1.8 (or
1.1) with θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1. Moreover any such solution θ is in fact
1,p
in C0 0, T ; W 2,p 0, 1 ∩ C1 0, T ; W0 0, 1, for all p < ∞. Furthermore the existence time of
a maximal solution is larger than minT , T ∗ , with
ln 1 β/C
T max
,
β>0
2 C1 2 sinh1 β C
C max y0 L∞ ×W 1,∞ , yl L∞ ×W 1,∞ , yr L∞ ×W 1,∞ ,
∗
C1 2.1
1
vl L∞ 0,T , vr L∞ 0,T .
tanh1
In a second step, we will show a weak-strong uniqueness property.
Theorem 2.2. Let θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1, and let y ∈ Lip0, 1 ×
Lip0, 1 be a weak solution of 1.8 (or 1.1); then it is unique in the function space
L∞ 0, T ; C1,1 0, 1 × L∞ ΩT 2 .
Let Al > 2 sinh1, Ar > Al cosh1 sinh2, T > 0 and M a symmetric matrix, and
assume that ρ0 , ρl , and ρr have compact supports in 0, 1/Γl /Γr , respectively. Our feedback
law for 1.8 or 1.1 reads
⎧ ⎪
vl y Al yC0 0,1×C0 0,1 ,
⎪
⎪
⎨ y ∈ C0 0, 1 × C0 0, 1 −→ vr y Ar yC0 0,1×C0 0,1 ,
⎪
⎪
⎪
⎩
ẏl t Myl t.
2.2
Theorem 2.3. For any y0 ∈ C0 0, 12 there exists y, v ∈ C0 ΩT 2 × C2 0, 1 a weak
solution of 1.1 and 2.2 satisfying
∀x ∈ 0, 1,
y0, x y0 x.
2.3
Furthermore any maximal solution of 1.1, 2.2, and 2.3 is global, and if we let
Ar − Al cosh1 sinh2
,
c min Al − 2 sinh1,
sinh1
1
τ
ln
M
2
cyC0 0,1
M
2
,
2.4
Abstract and Applied Analysis
7
then we have
y 0
≤
C 0,1
M
2
.
c1 M
2 t − τ
2.5
Remark 2.4. For ρ ≡ 0, the system 1.1 becomes the classical Camassa-Holm equation, and
the above theorems degenerate those of 33 with k 0.
3. Proofs of the Main Theorems
3.1. Local Existence Theorem
This strategy is borrowed from 33. We want to solve 1.6 1.8 or 1.1. Equation 1.6
is a linear elliptic equation, and with θ fixed 1.8 is a linear transport equation in y, with
boundary data.
Given θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1, we will define y mρ to be
the solution to 1.8, and once we have m in L∞ ΩT , we introduce θ solution of
1 − ∂xx θ m.
3.1
Then F is defined as the operator θ Fθ
Lemma 3.1. The function A defined by 1.5 satisfies
∀t, x ∈ ΩT ,
At, x ∈ C0 0, T ; C∞ 0, 1,
At, x 1
sinhxvr t sinh1 − xvl t,
sinh1
At, x
L∞ 0,T ;C1,1 0,1 ≤
3.2
cosh1 vl L∞ 0,T vr L∞ 0,T .
sinh1
Then for a function θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1, we consider ϕ the
flow of θ A.
Lemma 3.2. The flow ϕ satisfies the following properties.
1 ϕ is C1 with the following partial derivatives:
∂1 ϕs, t, x θ A s, ϕs, t, x ,
∂2 ϕs, t, x −θ A exp
∂3 ϕs, t, x exp
s
t
s
t
∂x θ A r, ϕr, t, x dr ,
∂x θ A r, ϕr, t, x dr .
3.3
8
Abstract and Applied Analysis
2 For all j 1, 2, 3,
T θA
C0 Ω ∂j ϕ 0 ≤ 1 θ A
0
T .
C ΩT e
C
3.4
For t, x ∈ ΩT , ϕ·, t, x is defined on a set et, x, ht, x, here et, x is basically the entrance time
in ΩT of the characteristic curve going through t, x.
3 If et, x > 0, then ϕet, x, t, x ∈ {0, 1}.
4 If ht, x < T , then ϕht, x, t, x ∈ {0, 1}.
For θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1, we define the solution to 1.8
y ∈ L ΩT × W 1,∞ ΩT by
∞
t
⎛
for t, x ∈ I,
⎜exp
yt, x ⎜
⎝
⎞
E1 dr
0
t
0
0
exp
E2 dr
⎟ ⎟y0 ϕ0, t, x
⎠
0
t
0
⎜exp
⎜
⎝
⎛
for t, x ∈ L,
et,x
0
t
exp
t
⎜exp
⎜
⎝
s
et,x
⎞
E1 dr
0
E2 dr
⎟
⎟
⎟yl et, x
⎠
0
t
exp
s
E2 dr
⎟
⎟
⎠
3.5
−ρρx − ρx
ds,
−∂x θ A
⎞
t
t
s
⎞
⎟ −ρρx − ρx
⎟
ds,
⎠
−∂x θ A
E2 dr
⎞
E1 dr
⎛
et,x
0
t
exp
0
⎜exp
⎜
yt, x ⎜
⎝
E1 dr
0
et,x
t
⎛
for t, x ∈ R,
s
t
⎜exp
⎜
yt, x ⎜
⎝
t
⎛
et,x
E1 dr
0
⎛
⎜exp
⎜
⎝
0
t
exp
t
s
0
et,x
E2 dr
⎟
⎟
⎟yr et, x
⎠
⎞
E1 dr
0
t
exp
s
E2 dr
⎟
⎟
⎠
−ρρx − ρx
−∂x θ A
ds
Abstract and Applied Analysis
9
with
E1 −2∂x θ A r, ϕr, t, x ,
E2 −∂x θ A r, ϕr, t, x .
3.6
And we have
1 the function y is the unique weak solution of 1.8 in the sense of Definition 4.1;
thanks to Lemma 1.3. y is in L∞ ∩ W 1,∞ .
Remark 3.3. y is the only weak solution of 1.8, and also y is in L∞ ∩ W 1,∞ which is crucial
for the stabilization problem because of the coupling between the two components of y.
However, rather thanks to the regularity on the boundary data ρ is indeed Lipschitz inside
the zones L, R, and I; it ensures that the transition between those zones should be continuous
under the kind of compatibility conditions between ρ0 , ρl , and ρr ; for example, all three have
a compact support in 0, 1/Γl /Γr .
2 Since y ∈ L∞ ΩT × W 1,∞ ΩT , we immediately get y ∈ W 1,∞ 0, T ; H −1 0, 1 ×
W 2,∞ 0, T ; H −1 0, 1 and satisfies 1.8. Also we can get the estimates:
yt, x ∞
,
≤
C
exp
2T
θ
A
∂
∂
∞
∞
0
x
x
∞
L ΩT L ΩT L ΩT ×L ΩT ∂t yt, x ∞
≤ 3C0 exp 2T ∂x θ
L∞ ΩT ∂x A
L∞ ΩT ,
L 0,T ;H −1 ×L∞ 0,T ;H −1 3.7
× θ
L∞ 0,T ;Lip0,1 A
L∞ 0,T ;Lip0,1 ,
where
C0 max y0 L∞ ×W 1,∞ , yl L∞ ×W 1,∞ , yr L∞ ×W 1,∞
T ρL∞ 1 ρx L∞ 0,T ;Lip0,1 ∂x θ
L∞ ΩT ∂x A
L∞ ΩT .
3.8
3 If t, x ∈ I ∪ L ∪ R and if s, s ∈ et, x, ht, x2 , one has the following property:
⎛
⎜exp
y s, ϕs, t, x ⎜
⎝
t
⎞
E1 dr
0
t
0
0
exp
E2 dr
⎟ ⎟y s , ϕ s , t, x .
⎠
3.9
0
From the elliptic equation we can get
Lemma 3.4. There exists a unique θ ∈ L∞ 0, T ; H01 0, 1 such that
1 − ∂xx θ m
3.10
10
Abstract and Applied Analysis
holds in D 0, 1 for all t, x ∈ ΩT . Furthermore θ ∈ L∞ 0, T ; C1,1 0, 1∩Lip0, T ; H01 0, 1,
since m ∈ L∞ ΩT ∩ Lip0, T ; H −1 0, 1. Moreover we have the bounds:
θ L∞ 0,T ;C1,1 0,1
∂t θ
≤ 1 sinh1
m
L∞ ΩT ≤ 1 sinh1yL∞ ΩT ×W 1,∞ ΩT ,
≤ ∂t m
L∞ 0,T ;H −1 L∞ 0,T ;H01 0,1
≤ ∂t yL∞ 0,T ;H −1 ×W 1,∞ 0,T ;H −1 .
3.11
The proof can be found in 33 and omitted.
Thus, for θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1, the operator F can be defined
as θ Fθ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip0, T ; H01 0, 1.
Let B0 and B1 be positive numbers, then we set
CB0 ,B1 ,T θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip 0, T ; H01 0, 1 | both θ
L∞ 0,T ;C1,1 0,1 ≤ B0 ,
θ
Lip0,T ;H −1 0,1 ≤ B1 .
3.12
Obviously CB0 ,B1 ,T is convex. We will endow CB0 ,B1 ,T with the norm θ
L∞ 0,T ;Lip0,1 .
Lemma 3.5. There exist positive numbers B0 , B1 and T , such that F maps CB0 ,B1 ,T into itself.
Proof. The proceeding of proof is similar to that of 33, Lemma 3, but the constant C0 differs
slightly from that of 33, Lemma 3. Let us first introduce the two following constants:
C0 max y0 L∞ ×W 1,∞ , yl L∞ ×W 1,∞ , yr L∞ ×W 1,∞
T ρL∞ 1 ρx L∞ 0,T ;Lip0,1 ∂x θ
L∞ ΩT ∂x A
L∞ ΩT ,
C1 3.13
1
vl L∞ 0,T , vr L∞ 0,T .
tanh1
Estimates 3.7, and 3.11 on y and θ now read,
yt, x ∞
≤ C0 exp 2T ∂x θ
L∞ ΩT C1 ,
L ΩT ×W 1,∞ ΩT ∂t yt, x ∞
≤
3C
exp
2T
θ
∂
∞ Ω C1
0
x
−1
∞
−1
L
L 0,T ;H ×L 0,T ;H T
× θ
L∞ 0,T ;Lip0,1 C1 ,
θ
L∞ 0,T ;C1,1 0,1
∂t θ
L∞ 0,T ;H01 0,1
≤ 1 sinh1yt, xL∞ ΩT ×W 1,∞ ΩT ,
≤ ∂t yt, xL∞ 0,T ;H −1 ×L∞ 0,T ;H −1 .
3.14
Abstract and Applied Analysis
11
Combining those estimates we get for all θ ∈ CB0 ,B1 ,T :
θ
≤ C0 1 sinh1 exp 2T ∂x θ
L∞ ΩT C1
∂t θ ≤ C0 1 sinh1 exp2T B0 C1 ,
≤ 3C0 exp 2T ∂x θ
L∞ ΩT C1
θ
L∞ 0,T ;Lip0,1 C1
L∞ 0,T ;C1,1 0,1
L∞ 0,T ;H01 0,1
3.15
≤ 3C0 exp2T B0 C1 B0 C1 .
To obtain θ ∈ CB0 ,B1 ,T , it is sufficient that
C0 1 sinh1 exp2T B0 C1 ≤ B0 ,
B0 3C0 exp2T B0 C1 B0 C1 ≤ B1 .
3.16
Once we have chosen T and B0 , it is easy to choose B1 to satisfy the second inequality.
For the first one we just choose B0 sufficiently large and then T close to 0. More precisely,
C max y0 L∞ ×W 1,∞ , yl L∞ ×W 1,∞ , yr L∞ ×W 1,∞ ,
B0 > C1 sinh1,
T≤
ln B0 /C1 sinh1
,
2B0 C1 3.17
we complete the proof by taking B0 /C1 sinh1 C β, β > 0.
Lemma 3.6. CB0 ,B1 ,T is compact with respect to the norm · L∞ 0,T ;Lip0,1 .
The proof is very similar to that appeared in 33 and omitted.
Lemma 3.7. For y ∈ H s × H s−1 , s > 2, fy is bounded on bounded sets in H s × H s−1 . Therefore,
fy is bounded on bounded sets in L∞ × L∞ by the embedding theorem.
The proof is very similar to that appeared in 33 and omitted.
Lemma 3.8. The operator F : CB0 ,B1 ,T → CB0 ,B1 ,T is continuous with respect to the norm
θ
L∞ 0,T ;Lip0,1 .
Proof. Take a sequence {θn } which tends to θ with respect to · L∞ 0,T ;Lip0,1 , set θn Fθn
and θ Fθ, denote by ϕn the flow of θn A and ϕ the flow of θ A, and we have that ϕn → ϕ
locally in C1 as n → ∞, thanks to Proposition A.4 in 33. What we will need to do is to show
that mn → m in L1 0, 1 as n → ∞ and ρn → ρ in L1 0, 1 as n → ∞.
Let t ∈ 0, T , having supposed that Pl and Pr have only a finite number of connected
components, we can assume, reducing t if necessary that vl and vr do not change sign on 0, t.
Since the characteristics of ϕn and ϕ may or may not cross before time t, we only consider the
case that ϕt, 0, 0 ≤ ϕn t, 0, 0 ≤ ϕt, 0, 1 ≤ ϕn t, 0, 1, without loss of generality. The other
12
Abstract and Applied Analysis
cases are proved in the same way. We first point out that since θn ∈ CB0 ,B1 ,T we have a bound
for {yn } in L∞ ΩT . Now
1
0
!
!
!yt, x − yn t, x!dx
ϕt,0,0 ϕ t,0,0 ϕt,0,1 ϕ t,0,1 1
n
n
ϕt,0,0
0
ϕn t,0,0
ϕt,0,1
ϕn t,0,1
!
!
!yt, x − yn t, x!dx
3.18
I1 I2 I3 I4 I5 .
Since ϕn t, 0, 0 → ϕt, 0, 0 as n → ∞ and ϕn t, 0, 1 → ϕt, 0, 1 as n → ∞ and thanks to
the uniform bound on yn L∞ , we see that both I2 and I4 tend to 0 when n goes to infinity.
For I1 we have
I1 ϕt,0,0
!
!
!yt, x − yn t, x!dx
0
!⎛
⎞
t
!
!
ϕt,0,0 !⎜exp
E1 dr
0
⎟
⎟
!⎜
et,x
t
⎟yl et, x
!⎜
!⎝
⎠
0
!
0
exp
E
dr
2
!
et,x
t
et,x
t
⎛
⎜exp
⎜
⎝
⎛
s
0
t
⎜exp
⎜
−⎜
⎝
E1 dr
en t,x
0
t
exp
E1n dr
0
s
⎞
⎟ −ρρx − ρx
⎟
ds
⎠
−∂
A
θ
x
E2 dr
⎞
0
t
exp
3.19
en t,x
E2n dr
⎟
⎟
⎟yl en t, x
⎠
t
!
⎞
!
!
0
⎜exp E1n dr
⎟ −ρρx − ρx
!
s
⎜
⎟
ds!!dx,
−
t
⎝
⎠
−∂x θn A
!
en t,x
0
exp E2n dr
!
t
⎛
s
where
E1n −2∂x θn A r, ϕn r, t, x ,
E2 −∂x θn A r, ϕn r, t, x .
3.20
Thanks to the boundedness on the f
L∞ ×L∞ Lemma 3.7 and Proposition A.2 of 33, if
t, x ∈ P we have ϕn t, 0, 0 → ϕt, 0, 0 as n → ∞. This implies that if yl was continuous,
since we have a uniform bound on θn L∞ 0,T ;C1,1 0,1 the dominated convergence theorem
would provide: I1 → 0. I3 → 0 and I5 → 0 which can be obtained by using the same
method. Therefore, for y0 , yl and yr continuous we have yt, · − yn t, ·
L1 → 0.
Abstract and Applied Analysis
13
From the inequality 56 in 33, we obtain
yt, · 1
≤ y0 L1 0,1×L1 0,1 yl L1 0,t∩Γl ×L1 0,t∩Γl
L 0,1×L1 0,1
yr L1 0,t∩Γr ×L1 0,t∩Γr ρL1 0,1×L1 0,1 1 ρx L1 0,1×L1 0,1
∂x θ A
L1 0,1 ∂x θ A
L∞ ΩT exp 3t
∂x θ A
L∞ ΩT ,
yn t, · 1
≤ y0 L1 0,1×L1 0,1 yl L1 0,t∩Γl ×L1 0,t∩Γl
L 0,1×L1 0,1
yr L1 0,t∩Γr ×L1 0,t∩Γr ρn L1 0,1×L1 0,1 1 ρn,x L1 0,1×L1 0,1
∂x θn A
L1 0,1 ∂x θn A
L∞ ΩT exp 3t
∂x θn A
L∞ ΩT .
3.21
So the general case of convergence yt, · − yn t, ·
L1 → 0 follows from the density of C0 in
L1 and the uniform bound on θn L∞ 0,T ;Lip0,1 .
Now only the restriction on t remains; we recall that until now we supposed that vl
and vr did not change sign on 0, t. If vl and vr do not change sign on 0, t1 and then on
t1 , t, we have
yt1 , · − yn t1 , · 1 −→ 0.
L
3.22
Let yn the solution of
∂t yn θn A∂x yn bn t, xyn fn t, x,
yn t1 , · yt1 ,
yn ·, 0|Γl yl ,
yn ·, 1|Γr yr .
3.23
We can conclude that as n → ∞,
yt, · − yn t, ·
L1 0,1×L1 0,1
≤ yt, · − yn t, ·L1 0,1×L1 0,1 yn t, · − yn t, ·L1 0,1×L1 0,1
≤ yt1 , · − yn t1 , ·L1 0,1×L1 0,1 exp 3t − t1 ∂x θn A
L∞ ΩT yt, · − yn t, ·L1 0,1×L1 0,1
≤ yt1 , · − yn t1 , ·L1 0,1×L1 0,1 exp 3t − t1 ∂x θn A
L∞ ΩT yt, · − yn t, ·L1 0,1×L1 0,1 −→ 0.
3.24
14
Abstract and Applied Analysis
Thus the convergence in L1 0, 1 propagates on each interval where vl and vr do not
change sign; thanks to the hypothesis on Pr and Pl we have that for all t ∈ 0, T ,
yt, · − yn t, ·
L1 0,1 → 0, as n → ∞. Combining this first convergence result with the
uniform bound and using the dominated convergence theorem in the time variable, we obtain
y − yn L1 ΩT → 0 which implies that
θ − θn L1 0,T ;W 2,1 0,1
−→ 0.
3.25
From the compactness of CB0 ,B1 ,T , we get that θn → θ holds in CB0 ,B1 ,T .
All the above lemmas result in the application of Schauders fixed point theorem to F
and we get a solution
θ ∈ L∞ 0, T ; C1,1 0, 1 ∩ Lip 0, T ; H01 0, 1 .
3.26
From the construction of F and from Proposition A.8 in 33 the additional regularity
properties of any solution θ,
1,p
θ ∈ C0 0, T ; W 2,p 0, 1 ∩ C1 0, T ; W0 0, 1 ,
∀p < ∞.
3.27
3.2. Uniqueness
In this subsection, we will show that the solution to the system 1.6 and 1.8 is unique;
be two solutions of 1.6 and 1.8 for the same initial and
that to say, given y, θ and y,
θ
boundary data, we will get y y and θ θ.
x, Ft, x ft, x − ft,
x, then
Let Y y − y,
Θ θ − θ, Bt, x bt, x − bt,
∈ L∞ ΩT and Y ∈ L∞ ΩT × L∞ ΩT is
Θ ∈ Lip0, T ; H01 0, 1. And we have M m − m
the solution to
∂t Y θ A∂x Y bY Bt, xy − Θ∂x y Ft, x,
Y 0, · 0,
Using the lemma again with bt, x we get
for t, x ∈ P,
for t, x ∈ I,
Y ·, 0|Γl 0,
Y ·, 1|Γr 0.
−2∂x θA
0
0
−∂x θA
and f Bt, xy − U∂x y Ft, x,
Y t, x 0,
Y t, x t
t
exp
0
s
b r, ϕr, t, x dr f s, ϕs, t, x ds,
3.28
Abstract and Applied Analysis
for t, x ∈ L,
for t, x ∈ R,
Y t, x Y t, x 15
t
t
et,x
exp
s
t
t
et,x
exp
s
b r, ϕr, t, x dr f s, ϕs, t, x ds,
b r, ϕr, t, x dr f s, ϕs, t, x ds.
3.29
Since Ut, ·
L∞ 0,1 ≤ C Y t, ·
L∞ ×L∞ for some positive constant C , and y, ∂x y
bounded, we see that for some C > 0,
3.30
ft, · ∞ ∞ ≤ C Y t, ·
L∞ ×L∞ .
L ×L
And since bt, x is bounded, we get that
Y t, ·
L∞ ×L∞ ≤ C
s
0
Y s, ·
L∞ ×L∞ ds.
3.31
Then we complete the proof of the uniqueness by using Gronwall’s lemma.
4. Stabilization
4.1. Preliminary Results
The equilibrium state that we want to stabilize is y 0, θ A 0. A natural idea is
using Lyapunov indirection method to investigate whether the linearized system around the
equilibrium state is stabilizable or not. Its stabilization would provide a local stabilization
result on the nonlinear system. Unfortunately, the linearized system is not stabilizable, for
the state of the linearized system around the equilibrium state is constant. We see that the
sign of θ A controls the geometry of the characteristics, and the sign of ∂x θ A controls
the ingredient information of y along the characteristics. We will use the return method that
Coron introduced in 34. We would like our feedback law, vl y Al y
C0 0,1×C0 0,1 ,
vr y Ar y
C0 0,1×C0 0,1 , to provide θ A ≥ 0 and ∂x θ A ≥ 0. However, there is
a difficulty in the stabilization problem. It needs not to be true that the transition between
those zones is continuous rather thanks to the regularity on the boundary data ρ is indeed
Lipschitz inside the zones L, R, and I. To achieve this target, we have to prescribe yl , and we
just need to make a continuous transition at t, x 0, 0 and let yl asymptotically converge
in time; we assume that compatibility conditions hold; precisely, ρ0 , ρl and ρr have compact
supports in 0, 1/Γl /Γr , respectively. To achieve this target, we have to prescribe yl , and we
just need to make a continuous transition at t, x 0, 0 and let yl asymptotically converge
in time. This is guaranteed by
∂t yl Myl ,
4.1
where M, symmetric matrix, is the unique matrix solution to the matrix function:
P M MT P −Q,
4.2
16
Abstract and Applied Analysis
for some symmetric positive-definite matrices, P and Q. Indeed, let V t, yl ylT P yl be the
Lyapunov candidate, and that yl asymptotically converges in time is equivalent to that the
time derivative of the V , V̇ ylT P M MT P yl is strictly negative. A fixed-point strategy
will be used again to prove the existence of a solution to the closed-loop system. We begin by
defining the domain of the operator.
Definition 4.1. Let X be the space of g, N ∈ C0 0, T ; 0, 12 × C0 0, T satisfying
1 g0, x y0 x, gt, 0 y0 0eMt ,
2 gt, ·
C0 0,1 ≤ Nt,
3 Nt is nonincreasing and N0 ≤ y0 C0 0,1×C0 0,1 .
Lemma 4.2. The domain X is nonempty, convex, bounded, and closed with respect to the uniform
topology.
Mt
, y0 C0 0,12 eMt ∈
Taking M λ,0
0,λ , λ < 0, satisfying 4.1 and 4.2, and y0 xe
X, so X is nonempty.
Now for y, Nt ∈ X we define θ̆ and Ă as the solutions of
mt, x 1 − ∂xx θ̆t, x,
ŭt, 0 θ̆t, 1 0,
∀t, x ∈ ΩT ,
1 − ∂xx Ăt, x 0,
Ăt, 0 Al Nt,
Ăt, 1 Ar Nt,
4.3
∀t, x ∈ ΩT .
That is,
∀t, x ∈ ΩT ,
ŭt, x −
x
sinhx − ξmt, ξdξ,
0
Nt
Ăt, x sinhxAr sinh1 − xAl .
sinh1
4.4
Thus, we have the estimates:
!
!
!
!
!θ̆t, x! ≤ 2 sinh1yC0 0,1×C0 0,1 ,
!
!
!
!
!∂x θ̆t, x! ≤ 2 cosh1yC0 0,1×C0 0,1 ,
!
!
!
!
!∂xx θ̆t, x! ≤ 1 2 sinh1yC0 0,1×C0 0,1 ,
!
!
!∂x Ăt, x! ≥ Ar − 2 cosh1 Nt,
sinh1
!
!
!Ăt, x! ≥ Al Nt.
4.5
Abstract and Applied Analysis
17
And in turn,
θ̆ Ă t, x ≥ Al − 2 sinh1yC0 0,1×C0 0,1 ,
4.6
Ar − 2 cosh1Al − sinh2 y 0
∂x θ̆ Ă t, x ≥
.
C 0,1×C0 0,1
sinh1
4.7
Now, if ϕ is the flow of θ̆ Ă, e is C1 and since ŭ Ă ≥ 0, ϕ·, t, x is nondecreasing. Thus we
can define the entrance time and then the operator S as follows.
"
Let et, x min{s ∈ 0, T | ϕs, t, x 0}, for t, x ∈ 0, T × 0, 1, Sy, N y,
N
with
1 if x ≥ ϕt, 0, 0,
t
⎛
⎞
exp
E
dr
0
1
⎜
⎟ 0
⎟y0 ϕ0, t, x
t
yt,
x ⎜
⎝
⎠
0
exp E2 dr
t
0
t
⎛
⎜exp
⎜
⎝
s
0
⎞
E1 dr
0
0
t
exp
s
E2 dr
⎟
⎟
⎠
4.8
−ρρx − ρx
−∂x θ A
ds,
2 if x ≤ ϕt, 0, 0,
t
⎛
⎜exp
yt,
x ⎜
⎝
⎞
E1 dr
0
t
0
0
⎛
exp
t
E2 dr
⎟ Met,x
⎟e
y0 0
⎠
0
⎞
t
exp
E
dr
0
1
⎜
⎟ −ρρx − ρx
s
⎜
⎟
t
ds,
⎝
⎠
−∂
A
θ
et,x
x
0
exp E2 dr
4.9
s
3 Nt yt, ·
C0 0,1×C0 0,1 .
From Lemma 1.3 we know that y is the weak solution of
∂t y θ A∂x y
bt, xy ft, x,
y0,
· 0,
yt,
0 eMt y0 0.
4.10
18
Abstract and Applied Analysis
Lemma 4.3. (1) The operator S maps X to X.
(2) The family SX is uniformly bounded and equicontinuous.
(3) S is continuous with respect to the uniform topology.
The proof is very similar to 33 except for here the state y is a two-component vector
and the proof is omitted.
We can apply Schauder’s fixed-point theorem to S and get y, N fixed point of S.
4.2. Stabilization and Global Existence
From 4.6 and 4.7, for all t, x ∈ ΩT ,
θ̆ Ă t, x ≥ cyr, ·C0 0,1×C0 0,1 ,
∂x θ̆ Ă t, x ≥ cyr, ·C0 0,1×C0 0,1 ,
4.11
y is the solution of the transport equation 1.8 and it satisfies
t
⎛
⎜exp
yt, x ⎜
⎝
t
s
s
⎞
E1 dr 0
⎛
⎜exp
⎜
⎝
0
t
exp
t
r
0
E1 dr
s
E2 dr
⎟ ⎟y s, ϕs, t, x
⎠
⎞
4.12
0
⎟
⎟fr, xdr.
t
⎠
exp E2 dr r
We get for t ≥ s
t
!
!
!
!
!yt, x! ≤ exp − cyr, · 0
dr y!s, ϕs, t, x!.
C 0,1×C0 0,1
s
This implies that |yt, x| 4.13
#
|mt, x|2 |ρt, x|2 decreases along the characteristics
strictly for the times where yt, · /
0. But we have also imposed yt, 0 ys, 0eMt−s ;
therefore |yt, x| also decreases along x 0. This already shows, thanks to the existence
theorem, that a maximal solution of the closed loop system is global. To get a more precise
statement, we consider all the characteristics between time t and s, and we obtain
$
t
%
M
s−r
yt, · 0
≤ max e 2
exp − cyr, ·C0 0,1×C0 0,1 dr
C 0,1×C0 0,1
r∈s,t
s
× yt, ·C0 0,1×C0 0,1 .
4.14
Abstract and Applied Analysis
19
Now we define
gr e
Then
M
2 s−r
t
exp − cyr, ·C0 0,1×C0 0,1 dr .
s
g r cyt, ·C0 0,1×C0 0,1 − M
2 gr,
4.15
4.16
and we know that as long as the quantity yt, ·
C0 0,1×C0 0,1 is not equal to zero, it strictly
decreases. So if y0 C0 0,1×C0 0,1 > M
2 /c, for t small enough yt, ·
C0 0,1×C0 0,1 ≥
M
2 /c, and we have
yt, · 0
≤ e−
M
2 t y0 C0 0,1×C0 0,1 ,
C 0,1×C0 0,1
4.17
which implies yτ, ·
C0 0,1×C0 0,1 ≤ M
2 /c. This provides for τ ≤ s ≤ t, the inequality
yt, ·
C0 0,1×C0 0,1
≤ e−
&t
s
ys, ·C0 0,1×C0 0,1 .
c
yr,·
C0 0,1×C0 0,1 dr 4.18
And we conclude with a classical comparison principle for ODES.
Acknowledgments
The authors would like to express their gratitude to the reviewer and editor for the valuable
comments and suggestions. The first author would like to express his gratitude to Professor
Adrain Constantin for leading his attention to this new system and Professor Zhaoyang
Yin for suggesting the present problem. This work was supported by the National Natural
Scientific Foundation of China Grants 11026112 and 61074021. This work is partially
supported by the Shandong Province Young and Middle aged Scientists Research Award
Fund Grant no. BS2011SF001.
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International Journal of
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International Journal of
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Optimization
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