The Korteweg–de Vries (KdV) equation
was first derived by Boussinesq in [2, (283 bis)] and by Korteweg and de Vries in , for describing the propagation of small amplitude long water waves in a uniform channel. This equation is now commonly used to model unidirectional propagation of small amplitude long waves in nonlinear dispersive systems. An excellent reference to help understand both physical motivation and deduction of the KdV equation is the book by Whitham .
Rosier studied in  the following nonlinear Neumann boundary control problem for the KdV equation with homogeneous Dirichlet boundary conditions, posed on a finite spatial interval:
where , the state is , and denotes the controller. The equation comes with one boundary condition at the left end-point and two boundary conditions at the right end-point. Rosier first considered the first-order power series expansion of around the origin, which gives the following corresponding linearized control system:
By means of multiplier technique and the Hilbert uniqueness method (HUM) , Rosier proved that (1.2) is exactly controllable if and only if the length of the spatial domain is not critical, i.e., , where denotes the following set of critical lengths:
Then, by employing the Banach fixed point theorem, he derived that the nonlinear KdV control system (1.1) is locally exactly controllable around 0 provided that . In the cases with critical lengths , Rosier demonstrated in  that there exists a finite dimensional subspace M of which is unreachable for the linear system (1.2) when starting from the origin. In , Coron and Crépeau treated a critical case of (i.e., taking in ), where k is a positive integer such that (see [7, Theorem 8.1 and Remark 8.2])
Here, the uncontrollable subspace M for the linear system (1.2) is one-dimensional. However, through a third-order power series expansion of the solution, they showed that the nonlinear term always allows to “go” in small time into the two directions missed by the linearized control system (1.2), and then, using a fixed point theorem, they deduced the small-time local exact controllability around the origin of the nonlinear control system (1.1). In , Cerpa studied the critical case of , where
In this case, the uncontrollable subspace M for the linear system (1.2) is of dimension 2, and Cerpa used a second-order expansion of the solution to the nonlinear control system (1.1) to prove the local exact controllability in large time around the origin of the nonlinear control system (1.1) (the local controllability in small time for this length L is still an open problem). Furthermore, Cerpa and Crépeau considered in  the cases when the dimension of M for the linear system (1.2) is higher than 2. They implemented a second-order expansion of the solution to (1.1) for the critical lengths for any , and implemented an expansion to the third order if for some . They showed that the nonlinear term always allows to “go” into all the directions missed by the linearized control system (1.2) and then proved the local exact controllability in large time around the origin of the nonlinear control system (1.1).
Consider the case when there is no control, i.e., in (1.1), which gives the following initial-boundary-value KdV problem posed on a finite interval :
where the boundary conditions are homogeneous. For the Lyapunov function
Thus, is stable (see (P1) below for the definition of stable) for the KdV equation (1.6). Moreover, it has been proved in  that, if , then 0 is exponentially stable for the corresponding linearized equation around the origin:
which gives the local asymptotic stability around the origin for the nonlinear equation (1.6). However, when , Rosier pointed out in  that equation (1.9) is not asymptotically stable. Inspired by the fact that the nonlinear term introduces the local exact controllability around the origin into the KdV control system (1.1) with , we would like to discuss whether the nonlinear term could introduce local asymptotic stability around the origin for (1.6).
This paper is devoted to investigating the local asymptotic stability of for (1.6) with the critical length
corresponding to and in (1.3). Let us recall that this local asymptotic stability means that the following two properties are satisfied.
Stability: for every , there exists such that implies
(Local) attractivity: there exists such that implies
As mentioned above, the stability property (P1) is implied by (1.8). Our main concern is thus the local attractivity property (P2). We prove the following theorem, where the precise definition of a solution to (1.6) is given in Definition 2.7, and the precise definition of the finite dimensional vector space when is given in (2.8).
Consider the KdV equation (1.6) with . There exist , , and a map , where is the orthogonal of M for the -scalar product, satisfying
such that, with
the following three properties hold for every solution y to (1.6) with :
Local exponential attractivity of G :
where denotes the distance between and G :
Local invariance of G : If , then for all .
If , then there exists such that
In particular, is locally asymptotically stable in the sense of the -norm for (1.6).
It can be derived from [9, Theorem 1 and comments] that, for every , there are nonzero stationary solutions with the period of L to the following ordinary differential equation (ODE):
That is, besides the origin, there also exist other steady states of the nonlinear KdV equation (1.6). Therefore, is not globally asymptotically stable for (1.6): Property (P2) does not hold for arbitrary .
Our proof of Theorem 1.1 relies on the center manifold approach. This center manifold is G in Theorem 1.1. Center manifold theory plays an important role in studying dynamic properties of nonlinear systems near “critical situations”. The center manifold theorem was first proved for finite dimensional systems by Pliss  and Kelley , and the readers could refer to [13, 17] for more details of this theory. Analogous results are also established for infinite dimensional systems, such as partial differential equations (PDEs) [3, 1] and functional differential equations . The center manifold method usually leads to a dimension reduction of the original problems. Then, in order to derive stability properties (asymptotic stability or unstability) of the full nonlinear equations, one only needs to analyze the reduced equation (restricted on the center manifold). When dealing with the infinite dimensional problems, this method can be extremely efficient if the center manifold is finite dimensional. Following the results on existence, smoothness and attractivity of a center manifold for evolution equations in , Chu, Coron and Shang studied in  the local asymptotic stability property of (1.6) with the critical length for any positive integer k such that (1.4) holds. They proved the existence of a one-dimensional local center manifold. By analyzing the resulting one-dimensional reduced equation, they obtained the local asymptotic stability of 0 for (1.6). For , we get, following , the existence of a two-dimensional local center manifold. It is predictable that the two-dimensional local center manifold introduces more complexity than the one-dimensional local center manifold case.
The organization of this paper is as follows. In Section 2, some basic properties of the linearized KdV equation (1.9) and the KdV equation (1.6) are given. Then, in Section 3, we recall a theorem on the existence of a local center manifold for the KdV equation (1.6) and analyze the dynamics on the local center manifold. Theorem 1.1 follows from this analysis. In Section 4, we present the conclusion and some possible future works. We end this article with Appendix A containing computations which are important for the study of the dynamics on the center manifold.
2.1 Some properties for the linearized equation of (1.6) around the origin
The origin is an equilibrium of the initial-boundary-value nonlinear KdV problem (1.6). In this subsection, we derive some properties for the linearized KdV equation (1.9) around the origin of (1.6) posed on the finite interval , where for which there exists a unique pair satisfying (1.5).
Let be the linear operator defined by
Then the linearized equation (1.9) can be written as an evolution equation in :
The following lemma can be immediately obtained.
exists and is compact on . Hence, , the spectrum of , consists of isolated eigenvalues only: , where denotes the set of eigenvalues of .
By calculation, we get
Hence we get the existence of and that, by the Sobolev embedding theorem, this operator is compact on Therefore, , the spectrum of , consists of isolated eigenvalues only. ∎
The following proposition is proved.
Proposition 2.2 ([20, Proposition 3.1]).
generates a -semigroup of contractions on , that is, for any given initial data , is the mild solution of the linearized equation (1.9), and
Moreover, for every .
If for all , then it follows directly from the ABLP theorem (Arendt–Batty–Lyubich–Phong)  that the semigroup is asymptotically stable on . Since we only have for all , the main concern needs to be put on the eigenvalues on the imaginary axis and their corresponding eigenfunctions. Following the proofs of [6, Lemma 2.6] and [20, Lemma 3.5] yields the next lemma.
There exists a unique pair of conjugate eigenvalues of on the imaginary axis, that is,
Moreover, the corresponding eigenfunctions of with respect to are
respectively, where C is an arbitrary constant, and are two nonzero real-valued functions:
The equations satisfied by and are
and, with the definition of Θ given in (2.3),
is the unique eigenvalue pair of on the imaginary axis, and all the other eigenvalues of have negative real parts which are uniformly bounded away from the imaginary axis, i.e., there exists such that any of the nonzero eigenvalues of has a real part which is less than .
Let us define
2.2 Some properties of the KdV equation (1.6)
Let and . A solution to the Cauchy problem (1.6) on is a function
such that, for every and for every satisfying
A solution to the Cauchy problem (1.6) on is a function
such that, for every , y restricted to is a solution to (1.6) on .
Then by considering equation (1.6) as a special case of [8, (A.1)] (with and ), the following two propositions about the existence and uniqueness of the solutions to (1.6) follow directly from [8, Propositions 14 and 15].
Let . There exist and such that, for every with , there exists at least one solution y to equation (1.6) on which satisfies
Let . There exists such that, for each pair of solutions , corresponding to each pair of initial conditions , to equation (1.6) on , the following inequalities hold:
for all .
Let us also mention that for every solution y to (1.6) on or on
This can be easily seen by multiplying the first equation of (1.6) with y, integrating on and performing integration by parts. One then gets, if y is smooth enough,
which gives (2.12). The general case follows from a smoothing argument. As a consequence of Proposition 2.8, Proposition 2.9 and (2.12), one sees that (1.6) has one and only one solution defined on if .
3 Existence of a center manifold and dynamics on this manifold
Let us start this section by recalling why, as it is classical, the property “ is locally asymptotically stable in the sense of the -norm for (1.6)” stated at the end of Theorem 1.1 is a consequence of the other statements in this theorem. For convenience, let us recall the argument. Let be such that and let y be the solution to (1.6). It suffices to check that
Let . By (3.4), there exists such that
which, together with (2.12), implies that
which concludes the proof of (3.1).
The remaining parts of this section are organized as follows. We first recall in Section 3.1 a theorem (Theorem 3.1) on the existence of a local center manifold for (1.6). Then in Section 3.2 we analyze the dynamics of (1.6) on this center manifold and deduce Theorem 1.1 from this analysis.
3.1 Existence of a local center manifold
In [6, Theorem 3.1], following , the existence of a center manifold for (1.6) was proved for the first critical length, i.e., . The same proof applies for our L (i.e., ) and allows us to get the following theorem.
Local exponential attractivity of G :
where denotes the distance between and G :
Local invariance of G : If , then for all .
3.2 Dynamics on the local center manifold
In this section we study the dynamics of (1.6) on with
then Ω is a bounded open subset of which contains . Let , and let y be the solution of (1.6) on for the initial data . It follows from (2.12) and Theorem 3.1 that for every . Hence we can define, for , by requiring that
Hence, in the sense of distributions on ,
Similarly, in the sense of distributions on ,
Hence, if we define , by
We now estimate g close to . Let be such that
As usual, by (3.16), we mean that, for every , there exists such that
where the constant is defined by
Let us now study the local asymptotic stability property of for (3.10). We propose two methods for that. The first one is a more direct one, which relies on normal forms for dynamical systems on . The second one, which relies on a Lyapunov approach related to the physics of (1.6), is less direct. However, there is a reasonable hope that this second method can be applied to other critical lengths for which the dimension of M is larger than 2.
Method 1: Normal form.
Let . Then
where are polynomials in of degree To be more precise, we have
We can rewrite (3.34) as
In polar coordinates, set
We have, as ,
Now it is clear to see from (3.42) that the origin is asymptotically stable for (3.10) if and is not stable if . From (2.1)–(2.3), (3.30)–(3.33) and Appendix A, we can obtain all the coefficients (). Then, using Matlab, it follows that
A straightforward computation leads to the existence of such that, at least if is small enough, one has for the solution to (3.42),
which concludes the proof of Theorem 1.1.
Method 2: Lyapunov function.
with . It is therefore natural to consider the following candidate for a Lyapunov function:
where is small enough. Indeed, one then gets
and one may hope to absorb with and get on , at least in a neighborhood of 0.
We follow this strategy together with the approximation of g previously found. For , let (see (3.16))
Then, using (3.45) with (which, by (2.4), (2.5), (3.26)–(3.28), (3.43) and (3.44), satisfies ), along the trajectories of (3.10), we have from (2.6), (2.7), (3.15) and (3.29)–(3.33) that the right-hand side of (3.45) is , and
with defined by
Let us emphasize that, even if “along the trajectories of (3.10)” might be misleading, is just a function of . It is the same for , , which appear below. Using (1.11) and (3.9), we have, along the trajectories of (3.10),
We can now define our Lyapunov function . Let . Let be defined by
Let us assume for the moment that, for every ,
Then, by homogeneity, there exists such that
Moreover, straightforward estimates show that there exists such that, for every ,
It only remains to prove (3.50). From Appendix A, one gets that , then (3.50) holds if . Let us now deal with the case . If we divide both polynomials in the two equations on the left-hand side of (3.50) by , then the two resulting polynomials have a common root if and only if their resultant is zero. This resultant is the determinant of the Sylvester matrix S:
Straightforward computations show that
Correction added on 10 September 2018 after online publication: Mistakes within (3.54) have been corrected. This would not influence the later calculations since we use a software to calculate directly the value of the determinant rather than using the general formula.
Hence, the two resulting polynomials do not have a common root. Thus, (3.50) is proved.
4 Conclusion and future works
In this article, we have proved that for the critical case of , is locally asymptotically stable for the KdV equation (1.6). First, we recalled that the equation has a two-dimensional local center manifold. Next, through a second-order power series approximation at of the function g defining the local center manifold, we derived the local asymptotic stability of on the local center manifold and obtained a polynomial decay rate for the solution to the KdV equation (1.6) on the center manifold.
Since the KdV equation (1.6) also has other (periodic) steady states than the origin (see Remark 1.2), it remains an open and interesting problem to consider the (local) stability property of these steady states for the KdV equation (1.6). Furthermore, it remains to consider all the other critical cases with a two-dimensional (local) center manifold as well as all the last remaining critical cases, i.e., when the equation has a (local) center manifold with a dimension larger than 2.
Employing the method of undetermined coefficients, we get the following (unique) solution to the nonhomogeneous ODE (A.3):
where the fundamental solutions , , are
and the constants are
and each is the matrix formed by replacing the l-th column of with a column vector , where
Similarly, the method of undetermined coefficients gives the following (unique) solution to the nonhomogeneous ODE system (A.6):
where the fundamental solutions , , are
and the constants are
Here, the matrix is defined by
Each is the matrix formed by replacing the l-th column of with a column vector , where
Therefore, we derive from (A.1) that
From (A.5), we obtain
We would like to thank Shengquan Xiang for useful comments on a preliminary version of this article.
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About the article
Published Online: 2016-10-14
Published in Print: 2018-11-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11401021
Award identifier / Grant number: 11301387
Funding Source: European Research Council
Award identifier / Grant number: 266907 (CPDENL)
The authors were supported by European Research Council advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7). In addition, J. Chu was supported by the National Natural Science Foundation of China (no. 11401021) and the Doctoral Program of Higher Education (no. 20130006120011). P. Shang was supported by the National Natural Science Foundation of China (no. 11301387) and the Doctoral Program of Higher Education (no. 20130072120008).