Initial Boundary Value Problem and Asymptotic Stabilization of the Two-Component Camassa-Holm Equation

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 andm0 ∈ L∞ 0, 1 , ρ0 ∈W1,∞ 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 × W1,∞ Γl and mr, ρr ∈ L∞ Γr ×W1,∞ Γr . The given functions vl, vr ,ml, ρl, andmr , ρr will be the boundary values for the equation; m0, ρ0 are the initial data. Let now A t, x be the auxiliary function which lifts the boundary values vl and vr and is defined by 1 − ∂xx A t, x 0, ∀ t, x ∈ ΩT , A t, 0 vl t , A t, 1 vr t , ∀t ∈ 0, T . 1.5 Setting u θ A, we can further rewrite the system 1.1 as m t, x 1 − ∂xx θ t, x , θ t, 0 θ t, 1 0, 1.6 mt θ A mx −2m∂x θ A − ρρx − ρx, ρt θ A ρx − ( ρ 1 ) ∂x θ A , m 0, · m0, m ·, 0 |Γl ml, m ·, 1 |Γr mr, ρ 0, · ρ0, ρ ·, 0 |Γl ρl, ρ ·, 1 |Γr ρr. 1.7 Let y (m ρ ) , y0 (m0 ρ0 ) , b t, x ( −2∂x θ A 0 0 −∂x θ A ) , f t, x ( −ρρx−ρx −∂x θ A ) , and the system 1.7 can be written as ∂ty θ A ∂xy b t, x y f t, x , y 0, · 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: adm ΩT { ψ ∈ C1 ΩT × C1 ΩT | ∀x ∈ 0, 1 , ψ t, x 0; ∀t ∈ 0, T /Γl, ψ t, 0 0; ∀t ∈ 0, T /Γr , ψ t, 1 0 } . 1.9 4 Abstract and Applied Analysis Definition 1.1. Given y0 (m0 ρ0 ) ∈ L∞ ΩT × W1,∞ ΩT , when θ ∈ L∞ 0, T ; Lip 0, 1 , a function pair y (m ρ ) ∈ L∞ ΩT ×W1,∞ ΩT is a weak solution to 1.8 if y satisfies ∫∫ ΩT [ y∂tψ y θ A ∂xψ − yb t, x ψ − f t, x ψ ] dtdx


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: ρ t ρ x u ρu x 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 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 u t, 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][21][22][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 .For ρ / 0, the Cauchy problems of 1.1 have been discussed in 5, 30 , respectively.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 u 0 , ρ 0 and u 1 , ρ 1 and a time T > 0, can one find a certain function v t such that the solution to 1.1 satisfies u T u 1 , ρ T ρ 1 ?and 2 the stabilizability problem: can one find a stationary feedback law v x , such that for any state u 0 , ρ 0 a solution pair u t , ρ t to closed-loop system is global?
To explain our boundary formulation of 1.1 , let us first introduce some transformation, precisely, m u − u xx and ρ ρ − 1, which lead the system 1.1 to be equivalent to the system: m t um x 2mu x ρρ x ρ x 0, ρ t uρ x ρu x u x 0.

Abstract and Applied Analysis 3
Let T be a positive number.In the following we take Ω T 0, T × 0, 1 .Let v l and v r be in In the following, we will always suppose that the sets The given functions v l , v r , m l , ρ l , and m r , ρ r will be the boundary values for the equation; m 0 , ρ 0 are the initial data.Let now A t, x be the auxiliary function which lifts the boundary values v l and v r and is defined by

1.5
Setting u θ A, we can further rewrite the system 1.1 as A , and the system 1.7 can be written as 1.8 We first define what we mean by a weak solution to 1.8 .Our test functions will be in the space: 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.
and y r ∈ L ∞ Γ r .We will also suppose that the sets: , 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 ρ 0 , ρ l , ρ r0 and even on θ and A to get sufficient geometrical assumptions.Then, one might manage 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.

Main Results
Theorem 2.1.For T > 0, we consider v l ∈ C 0 Γ l , v r ∈ C 0 Γ l such that the sets P l and P r have only a finite number of connected components.Let There exist T > 0 and θ, y a weak solution of the system 1.8 (or for all p < ∞.Furthermore the existence time of a maximal solution is larger than min T, T * , with

2.1
In a second step, we will show a weak-strong uniqueness property.
Let A l > 2 sinh 1 , A r > A l cosh 1 sinh 2 , T > 0 and M a symmetric matrix, and assume that ρ 0 , ρ l , and ρ r have compact supports in 0, 1 /Γ l /Γ r , respectively.Our feedback law for 1.8 or 1.1 reads

2.4
Abstract and Applied Analysis 7 then we have 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.

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 ; C 1,1 0, 1 ∩ Lip 0, T ; H 1 0 0, 1 , we will define y m ρ to be the solution to 1.8 , and once we have m in L ∞ Ω T , we introduce θ solution of Then F is defined as the operator θ F θ Lemma 3.1.The function A defined by 1.5 satisfies

Lemma 3.2. The flow ϕ satisfies the following properties.
1 ϕ is C 1 with the following partial derivatives:

3.3
2 For all j 1, 2, 3, For t, x ∈ Ω T , ϕ •, t, x is defined on a set e t, x , h t, x , here e t, x is basically the entrance time in Ω T of the characteristic curve going through t, x .
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 ρ 0 , ρ l , and ρ r ; for example, all three have a compact support in 0, 1 /Γ l /Γ r .

Lemma 3.5.
There exist positive numbers B 0 , B 1 and T , such that F maps C B 0 ,B 1 ,T into itself.
Proof.The proceeding of proof is similar to that of 33, Lemma 3 , but the constant C 0 differs slightly from that of 33, Lemma 3 .Let us first introduce the two following constants:

3.13
Estimates 3.7 , and 3.11 on y and θ now read,

3.14
Combining those estimates we get for all θ ∈ C B 0 ,B 1 ,T :

3.15
To obtain θ ∈ C B 0 ,B 1 ,T , it is sufficient that

3.16
Once we have chosen T and B 0 , it is easy to choose B 1 to satisfy the second inequality.For the first one we just choose B 0 sufficiently large and then T close to 0. More precisely, 3.17 The proof is very similar to that appeared in 33 and omitted.
Lemma 3.7.For y ∈ H s × H s−1 , s > 2, f y is bounded on bounded sets in H s × H s−1 .Therefore, f y 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 : Proof.Take a sequence {θ n } which tends to θ with respect to • L ∞ 0,T ;Lip 0,1 , set θ n Fθ n and θ Fθ, denote by ϕ n the flow of θ n A and ϕ the flow of θ A, and we have that ϕ n → ϕ locally in C 1 as n → ∞, thanks to Proposition A.4 in 33 .What we will need to do is to show that m n → m in L 1 0, 1 as n → ∞ and ρ n → ρ in L 1 0, 1 as n → ∞.Let t ∈ 0, T , having supposed that P l and P r have only a finite number of connected components, we can assume, reducing t if necessary that v l and v r do not change sign on 0, t .Since the characteristics of ϕ n and ϕ may or may not cross before time t, we only consider the case that ϕ t, 0, 0 ≤ ϕ n t, 0, 0 ≤ ϕ t, 0, 1 ≤ ϕ n t, 0, 1 , without loss of generality.The other cases are proved in the same way.We first point out that since θ n ∈ C B 0 ,B 1 ,T we have a bound for {y n } in L ∞ Ω T .Now 1 0 y t, x − y n t, x dx ϕ t,0,0 0 ϕ n t,0,0 ϕ t,0,0 ϕ t,0,1 ϕ n t,0,0 ϕ n t,0,1 ϕ t,0,1 1 ϕ n t,0,1 y t, x − y n t, x dx

3.18
Since ϕ n t, 0, 0 → ϕ t, 0, 0 as n → ∞ and ϕ n t, 0, 1 → ϕ t, 0, 1 as n → ∞ and thanks to the uniform bound on y n L ∞ , we see that both I 2 and I 4 tend to 0 when n goes to infinity.
For I 1 we have where

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 ϕ n t, 0, 0 → ϕ t, 0, 0 as n → ∞.This implies that if y l was continuous, since we have a uniform bound on θ n L ∞ 0,T ;C 1,1 0,1 the dominated convergence theorem would provide: I 1 → 0. I 3 → 0 and I 5 → 0 which can be obtained by using the same method.Therefore, for y 0 , y l and y r continuous we have y t, • − y n t, From the inequality 56 in 33 , we obtain

3.21
So the general case of convergence y t, • − y n t, • L 1 → 0 follows from the density of C 0 in L 1 and the uniform bound on θ n L ∞ 0,T ;Lip 0,1 .Now only the restriction on t remains; we recall that until now we supposed that v l and v r did not change sign on 0, t .If v l and v r do not change sign on 0, t 1 and then on t 1 , t , we have Let y n the solution of

3.23
We can conclude that as n → ∞,

14
Abstract and Applied Analysis Thus the convergence in L 1 0, 1 propagates on each interval where v l and v r do not change sign; thanks to the hypothesis on P r and P l we have that for all t ∈ 0, T , y t, • − y n t, • L 1 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 − y n L 1 Ω T → 0 which implies that From the compactness of C B 0 ,B 1 ,T , we get that All the above lemmas result in the application of Schauders fixed point theorem to F and we get a solution From the construction of F and from Proposition A.8 in 33 the additional regularity properties of any solution θ, 3.27

Uniqueness
In this subsection, we will show that the solution to the system 1.6 and 1.8 is unique; that to say, given y, θ and y, θ be two solutions of 1.6 and 1.8 for the same initial and boundary data, we will get y y and θ θ.

3.28
Using the lemma again with b t, x

3.29
Since U t, for some positive constant C , and y, ∂ x y bounded, we see that for some C > 0, And since b t, x is bounded, we get that Then we complete the proof of the uniqueness by using Gronwall's lemma.

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, v l y A l y C 0 0,1 ×C 0 0,1 , v r y A r y C 0 0,1 ×C 0 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 y l , and we just need to make a continuous transition at t, x 0, 0 and let y l asymptotically converge in time; we assume that compatibility conditions hold; precisely, ρ 0 , ρ l and ρ r have compact supports in 0, 1 /Γ l /Γ r , respectively.To achieve this target, we have to prescribe y l , and we just need to make a continuous transition at t, x 0, 0 and let y l asymptotically converge in time.This is guaranteed by where M, symmetric matrix, is the unique matrix solution to the matrix function: for some symmetric positive-definite matrices, P and Q.Indeed, let V t, y l y T l Py l be the Lyapunov candidate, and that y l asymptotically converges in time is equivalent to that the time derivative of the V , V y T l PM M T P y l 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 ∈ C 0 0, T ; 0, 1 2 × C 0 0, T satisfying 1 g 0, x y 0 x , g t, 0 y 0 0 e Mt , 2 g t, Lemma 4.2.The domain X is nonempty, convex, bounded, and closed with respect to the uniform topology.
Taking M λ,0 0,λ , λ < 0, satisfying 4.1 and 4.2 , and y 0 x e Mt , y 0 C 0 0,1 2 e Mt ∈ X, so X is nonempty.Now for y, N t ∈ X we define θ and Ȃ as the solutions of

4.3
That is,

4.4
Thus, we have the estimates:

4.5
And in turn, θ Ȃ t, x ≥ A l − 2 sinh 1 y C 0 0,1 ×C 0 0,1 , 4.6 Now, if ϕ is the flow of θ Ȃ, e is C 1 and since ȗ Ȃ ≥ 0, ϕ •, t, x is nondecreasing.Thus we can define the entrance time and then the operator S as follows.Let e t, x min{s ∈ 0, T | ϕ s, t, x 0}, for t, x ∈ 0, T × 0, 1 , S y, N y, N with (2) The family S X 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.

4.18
And we conclude with a classical comparison principle for ODES.
obtain at least Lipschitz solution of the scalar transport equation on ρ and then get a weak solution on m.