Abstract
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: 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 , the equation (1.1) becomes the Camassa-Holm equation, which is modeling the unidirectional propagation of shallow water waves over a flat bottom. Here stands for the fluid velocity at time in the spatial 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]).
For , 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 and and a time , can one find a certain function such that the solution to (1.1) satisfies , ? and(2)the stabilizability problem: can one find a stationary feedback law , such that for any state a solution pair to closed-loop system is global?
To explain our boundary formulation of (1.1), let us first introduce some transformation, precisely, and , which lead the system (1.1) to be equivalent to the system: Let be a positive number. In the following we take . Let and be in and , . We set In the following, we will always suppose that the sets have a finite number of connected components. Finally, let and . The given functions , , , , and , will be the boundary values for the equation; , are the initial data. Let now be the auxiliary function which lifts the boundary values and and is defined by Setting , we can further rewrite the system (1.1) as Let , , , , and the system (1.7) can be written as We first define what we mean by a weak solution to (1.8). Our test functions will be in the space:
Definition 1.1. Given , when , a function pair is a weak solution to (1.8) if satisfies
It is obvious that ; therefore, a weak solution to (1.8) is also a solution to (1.8) in the distribution sense.
Definition 1.2 (see [33]). For , let be the maximal solution to which is defined on a certain set (which is closed because is compact) and with possibly and/or .
We take into account the influence of the boundaries by introducing the sets: 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 , , , , and . We will also suppose that the sets: have at most a countable number of connected components. Then the function , defined by the formula is a weak solution of and satisfies
However, if we let and , note that depends on the unknown 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 and are given, one can solve the equation on (the equation on the second component in (1.8) (or (1.1))), but this result only guarantee that is in . Therefore the source term in the equation on is not in anymore but in , and then Lemma 1.3 cannot be used to solve the transport equation on (the first component equation on (1.8)). One might try to get more regularity on , but in this case more regularity is also needed on ,, and even on and to get sufficient geometrical assumptions. Then, one might manage , , to obtain at least Lipschitz solution of the scalar transport equation on and then get a weak solution on .
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.
2. Main Results
Theorem 2.1. For , we consider , such that the sets and have only a finite number of connected components. Let , , and . There exist and a weak solution of the system (1.8) (or (1.1)) with . Moreover any such solution is in fact in , for all . Furthermore the existence time of a maximal solution is larger than , with
In a second step, we will show a weak-strong uniqueness property.
Theorem 2.2. Let , and let be a weak solution of (1.8) (or (1.1)); then it is unique in the function space .
Let , , and a symmetric matrix, and assume that , and have compact supports in , respectively. Our feedback law for (1.8) (or (1.1)) reads
Theorem 2.3. For any there exists a weak solution of (1.1) and (2.2) satisfying Furthermore any maximal solution of (1.1), (2.2), and (2.3) is global, and if we let then we have
Remark 2.4. For , the system (1.1) becomes the classical Camassa-Holm equation, and the above theorems degenerate those of [33] with .
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 fixed (1.8) is a linear transport equation in , with boundary data.
Given , we will define to be the solution to (1.8), and once we have in , we introduce solution of Then is defined as the operator
Lemma 3.1. The function defined by (1.5) satisfies
Then for a function , we consider the flow of .
Lemma 3.2. The flow satisfies the following properties. (1) is with the following partial derivatives:(2)For all ,,, For , is defined on a set , here is basically the entrance time in of the characteristic curve going through . (3) If , then .(4) If , then .
For , we define the solution to (1.8) by with And we have(1) the function is the unique weak solution of (1.8) in the sense of Definition 4.1; thanks to Lemma 1.3. is in .
Remark 3.3. is the only weak solution of (1.8), and also is in which is crucial for the stabilization problem because of the coupling between the two components of . However, rather thanks to the regularity on the boundary data is indeed Lipschitz inside the zones , , and ; it ensures that the transition between those zones should be continuous under the kind of compatibility conditions between , , and ; for example, all three have a compact support in .
(2)Since , we immediately get and satisfies (1.8). Also we can get the estimates:where(3) If and if , one has the following property:
From the elliptic equation we can get
Lemma 3.4. There exists a unique such that holds in for all . Furthermore , since . Moreover we have the bounds:
The proof can be found in [33] and omitted.
Thus, for , the operator can be defined as .
Let and be positive numbers, then we set Obviously is convex. We will endow with the norm .
Lemma 3.5. There exist positive numbers , and , such that maps into itself.
Proof. The proceeding of proof is similar to that of [33, Lemma 3], but the constant differs slightly from that of [33, Lemma 3]. Let us first introduce the two following constants:
Estimates (3.7), and (3.11) on and now read,
Combining those estimates we get for all :
To obtain , it is sufficient that
Once we have chosen and , it is easy to choose to satisfy the second inequality. For the first one we just choose sufficiently large and then close to 0. More precisely,
we complete the proof by taking .
Lemma 3.6. is compact with respect to the norm .
The proof is very similar to that appeared in [33] and omitted.
Lemma 3.7. For , , is bounded on bounded sets in . Therefore, is bounded on bounded sets in by the embedding theorem.
The proof is very similar to that appeared in [33] and omitted.
Lemma 3.8. The operator is continuous with respect to the norm .
Proof. Take a sequence which tends to with respect to , set and , denote by the flow of and the flow of , and we have that locally in as , thanks to Proposition in [33]. What we will need to do is to show that in as and in as .
Let , having supposed that and have only a finite number of connected components, we can assume, reducing if necessary that and do not change sign on . Since the characteristics of and may or may not cross before time , we only consider the case that , without loss of generality. The other cases are proved in the same way. We first point out that since we have a bound for in . Now
Since as and as and thanks to the uniform bound on , we see that both and tend to 0 when goes to infinity.
For we have
where
Thanks to the boundedness on the (Lemma 3.7) and Proposition of [33], if we have as . This implies that if was continuous, since we have a uniform bound on the dominated convergence theorem would provide: . and which can be obtained by using the same method. Therefore, for , and continuous we have .
From the inequality (56) in [33], we obtain
So the general case of convergence follows from the density of in and the uniform bound on .
Now only the restriction on remains; we recall that until now we supposed that and did not change sign on . If and do not change sign on and then on , we have
Let the solution of
We can conclude that as ,
Thus the convergence in propagates on each interval where and do not change sign; thanks to the hypothesis on and we have that for all , , as . Combining this first convergence result with the uniform bound and using the dominated convergence theorem in the time variable, we obtain which implies that
From the compactness of , we get that holds in .
All the above lemmas result in the application of Schauders fixed point theorem to and we get a solution
From the construction of F and from Proposition A.8 in [33] the additional regularity properties of any solution ,
3.2. Uniqueness
In this subsection, we will show that the solution to the system (1.6) and (1.8) is unique; that to say, given and be two solutions of (1.6) and (1.8) for the same initial and boundary data, we will get and .
Let , , , , then . And we have and is the solution to Using the lemma again with and , we get
Since for some positive constant , and , bounded, we see that for some , And since is bounded, we get that
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 , . 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 controls the geometry of the characteristics, and the sign of controls the ingredient information of along the characteristics. We will use the return method that Coron introduced in [34]. We would like our feedback law, , , to provide and . 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 is indeed Lipschitz inside the zones , , and . To achieve this target, we have to prescribe , and we just need to make a continuous transition at and let asymptotically converge in time; we assume that compatibility conditions hold; precisely, , and have compact supports in , respectively. To achieve this target, we have to prescribe , and we just need to make a continuous transition at and let asymptotically converge in time. This is guaranteed by where , symmetric matrix, is the unique matrix solution to the matrix function: for some symmetric positive-definite matrices, and . Indeed, let be the Lyapunov candidate, and that asymptotically converges in time is equivalent to that the time derivative of the , 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 be the space of satisfying(1), ,(2),(3) is nonincreasing and .
Lemma 4.2. The domain is nonempty, convex, bounded, and closed with respect to the uniform topology.
Taking , , satisfying (4.1) and (4.2), and , so is nonempty.
Now for we define and as the solutions of That is,
Thus, we have the estimates: And in turn, Now, if is the flow of , is and since , is nondecreasing. Thus we can define the entrance time and then the operator as follows.
Let , for , with(1) if ,(2)if ,(3).
From Lemma 1.3 we know that is the weak solution of
Lemma 4.3. (1) The operator maps to .
(2) The family is uniformly bounded and equicontinuous.
(3) is continuous with respect to the uniform topology.
The proof is very similar to [33] except for here the state is a two-component vector and the proof is omitted.
We can apply Schauder’s fixed-point theorem to and get fixed point of .
4.2. Stabilization and Global Existence
From (4.6) and (4.7), for all , is the solution of the transport equation (1.8) and it satisfies We get for
This implies that decreases along the characteristics (strictly for the times where ). But we have also imposed ; therefore also decreases along . 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 and , and we obtain
Now we define
Then and we know that as long as the quantity is not equal to zero, it strictly decreases. So if , for small enough , and we have which implies . This provides for , the inequality 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).