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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.

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:

For

The Cauchy problem and initial-boundary value problem for the Camassa-Holm equation have been studied extensively in [

For

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,

the exact controllability problem: given two states

the stabilizability problem: can one find a stationary feedback law

To explain our boundary formulation of (

Given

It is obvious that

For

We take into account the influence of the boundaries by introducing the sets:

Let

However, if we let

The rest of this paper is organized as follows. In Section

For

In a second step, we will show a weak-strong uniqueness property.

Let

Let

For any

For

This strategy is borrowed from [

Given

The function

Then for a function

The flow

For all

If

If

For

the function

Since

where

If

From the elliptic equation we can get

There exists a unique

The proof can be found in [

Thus, for

Let

There exist positive numbers

The proceeding of proof is similar to that of [

Estimates (

Once we have chosen

The proof is very similar to that appeared in [

For

The proof is very similar to that appeared in [

The operator

Take a sequence

Let

For

From the inequality (56) in [

Now only the restriction on

From the compactness of

All the above lemmas result in the application of Schauders fixed point theorem to

From the construction of F and from Proposition A.8 in [

In this subsection, we will show that the solution to the system (

Let

Since

Then we complete the proof of the uniqueness by using Gronwall’s lemma.

The equilibrium state that we want to stabilize is

Let

The domain

Taking

Now for

Thus, we have the estimates:

Let

if

if

From Lemma

(1) The operator

(2) The family

(3)

The proof is very similar to [

We can apply Schauder’s fixed-point theorem to

From (

This implies that

Now we define

Then

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).