Abstract
Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries
equation posed on a finite interval
1 Introduction
The Korteweg–de Vries (KdV) equation
was first derived by Boussinesq in [2, (283 bis)] and by Korteweg and de Vries in [14], 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 [22].
Rosier studied in [20] the following nonlinear Neumann boundary control problem for the KdV equation with homogeneous Dirichlet boundary conditions, posed on a finite spatial interval:
where
By means of multiplier technique and the Hilbert uniqueness method
(HUM) [15], Rosier proved that (1.2)
is exactly controllable if and only if the length
of the spatial domain is not critical,
i.e.,
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
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
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 [5] 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
Consider the case when there is no control, i.e.,
where the boundary conditions are homogeneous. For the Lyapunov function
we have
Thus,
which gives the local asymptotic stability around the origin for the nonlinear equation (1.6). However, when
This paper is devoted to investigating the local asymptotic stability of
corresponding to
Stability: for every
(Local) attractivity: there exists
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
Theorem 1.1.
Consider the KdV equation
(1.6) with
such that, with
the following three properties hold for every solution y to (1.6)
with
Local exponential attractivity of G:
(1.13)where
Local invariance of G: If
If
(1.14)
In particular,
Remark 1.2.
It can be derived from [9, Theorem 1 and comments] that, for
every
That is, besides the origin, there also exist other steady states of the
nonlinear KdV equation (1.6). Therefore,
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 [19] and Kelley [12],
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 [10].
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 [21], Chu, Coron and
Shang studied in [6]
the local asymptotic stability property of (1.6) with the critical
length
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 Preliminaries
2.1 Some properties for the linearized equation of (1.6) around the origin
The origin
Let
with
Then the linearized equation (1.9) can be written as an evolution
equation in
The following lemma can be immediately obtained.
Lemma 2.1.
Proof.
By calculation, we get
with
Hence we get the existence of
The following proposition is proved.
Proposition 2.2 ([20, Proposition 3.1]).
Moreover,
If
Lemma 2.3.
There exists a unique pair of conjugate eigenvalues of
Moreover, the corresponding eigenfunctions of
respectively, where C is an arbitrary constant, and
with
Remark 2.4.
The equations satisfied by
and
Remark 2.5.
We have
and, with the definition of Θ given in (2.3),
From the results in Lemma 2.1, Proposition 2.2 and Lemma 2.3, we obtain the following corollary.
Corollary 2.6.
Let us define
where
with
2.2 Some properties of the KdV equation (1.6)
By considering equation (1.6) as a special
case (with
Definition 2.7.
Let
such that, for every
one has
A solution to the Cauchy problem
(1.6) on
such that, for every
Then by considering equation (1.6) as a special case of [8, (A.1)]
(with
Proposition 2.8.
Let
Proposition 2.9.
Let
for all
Let us also mention that for every solution y to (1.6) on
This can be easily seen by multiplying the first equation of (1.6) with y, integrating on
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
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 “
By (1.13), (2.12) and the fact that M is of finite dimension, there exists an increasing sequence of positive real numbers
Let
Let
By (3.5) and (3.6), there exists
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 [21], the existence of a center manifold for (1.6) was proved for the first critical length, i.e.,
Theorem 3.1.
There exist
Local exponential attractivity of G:
where
Local invariance of G: If
3.2 Dynamics on the local center manifold
In this section we study the dynamics of (1.6) on
Let
then Ω is a bounded open subset of
Since
From (2.4), (2.9), (3.7) and (3.8), we have
Hence, in the sense of distributions on
Similarly, in the sense of distributions on
Hence, if we define
then
Note that, by (1.10) and (3.9),
We now estimate g close to
Using Definition 2.7 with
Letting
We expand g in a neighborhood of
such that
As usual, by (3.16), we mean that, for every
Similar definitions are used in (3.17), (3.18) and later on.
We now expand the left-hand side of (3.14) in terms of
For the functions
where the constant
Looking successively at the terms in
Since (3.23)–(3.25) must hold
for every
In Appendix A, we derive the unique functions
with
Let us now study the local asymptotic stability property of
Method 1: Normal form.
Let
and it follows from (3.10) and (3.29) that, as
where
and
We can rewrite (3.34) as
and it is known from [11, pp. 45, 47] that (3.37) has the following Poincaré normal form:
where
According to (3.35) and (3.36), through a simple computation, we have
Using (3.40) and (3.41), the formula of ρ provided by (3.39) gives
It follows that we can derive the Poincaré normal form of the reduced equation on the local center manifold (3.38). Moreover, in Cartesian coordinates, (3.38) is
where
In polar coordinates, set
We have, as
Now it is clear to see from (3.42) that the origin
A straightforward computation leads to the existence of
which concludes the proof of Theorem 1.1.
Method 2: Lyapunov function.
Let us start with a formal motivation. Recall that, by (1.8) and with E defined in (1.7), we have, along the trajectories of (1.6),
with
where
and one may hope to absorb
We follow this strategy together with the approximation of g previously found. For
and
Then, using (2.4), (2.5) and (3.26)–(3.28) (compare with (3.14)), one gets that, along the trajectories of (3.10), for
Then, using (3.45) with
with
Let us emphasize that, even if “along the trajectories of (3.10)” might be misleading,
Using (3.9), we get the existence of
We can now define our Lyapunov function
From (3.48), we have the existence of
Let us assume for the moment that, for every
Then, by homogeneity, there exists
for all
From (3.49) and (3.52), we get the existence of
Moreover, straightforward estimates show that there exists
which, together with (3.53), proves the existence of
It only remains to prove (3.50). From Appendix A,
one gets that
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
From (3.54) and Appendix A (see in particular (A.7)–(A.9)), we have
Hence, the two resulting polynomials do not have a common root. Thus, (3.50) is proved.
Remark 3.2.
It follows from our proof of Theorem 1.1 that the decay rate stated in (1.14) is optimal in the following sense: there exists
For the Lyapunov approach, let us point out that, decreasing if necessary
4 Conclusion and future works
In this article, we have proved that for the
critical case of
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.
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)
Funding statement: 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).
A On the solution a, b and c to equations (3.26), (3.27) and (3.28)
Set
and
First, adding each equation of (3.28) to the corresponding
equation of (3.26), we have
the following ODE for
Second, subtracting each equation of (3.28) from the corresponding equation of (3.26), we obtain
which gives
Substitute (A.5) into (3.27), then
the following ODE for
where the boundary conditions follow from (2.4), (2.5), (3.27), (A.2), (A.4) and (A.5).
Employing the method of undetermined coefficients, we get the following (unique) solution to the nonhomogeneous ODE (A.3):
where the fundamental solutions
and the constants are
and
Here,
and each
and
Similarly, the method of undetermined coefficients gives the following (unique) solution to the nonhomogeneous ODE system (A.6):
where the fundamental solutions
with
and the constants are
and
Here, the matrix
with
Each
and
Therefore, we derive from (A.1) that
and
From (A.5), we obtain
Acknowledgements
We would like to thank Shengquan Xiang for useful comments on a preliminary version of this article.
References
[1] P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported. Volume 2, John Wiley & Sons, Chichester (1989), 1–38. 10.1007/978-3-322-96657-5_1Search in Google Scholar
[2] J. Boussinesq, Essai sur la théorie des eaux courantes, Mém. Prés. P. Div. Sav. de Paris 13 (1877), 666–680. Search in Google Scholar
[3] J. Carr, Applications of Centre Manifold Theory, Appl. Math. Sci. 35, Springer, New York, 1981. 10.1007/978-1-4612-5929-9Search in Google Scholar
[4] E. Cerpa, Exact controllability of a nonlinear Korteweg–de Vries equation on a critical spatial domain, SIAM J. Control Optim. 46 (2007), no. 3, 877–899. 10.1137/06065369XSearch in Google Scholar
[5] E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, 457–475. 10.1016/j.anihpc.2007.11.003Search in Google Scholar
[6] J. Chu, J.-M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg–de Vries equation with critical lengths, J. Differential Equations 259 (2015), no. 8, 4045–4085. 10.1016/j.jde.2015.05.010Search in Google Scholar
[7] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. 136, American Mathematical Society, Providence, 2007. Search in Google Scholar
[8] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 367–398. 10.4171/JEMS/13Search in Google Scholar
[9] G. G. Doronin and F. M. Natali, An example of non-decreasing solution for the KdV equation posed on a bounded interval, C. R. Math. Acad. Sci. Paris 352 (2014), no. 5, 421–424. 10.1016/j.crma.2014.02.001Search in Google Scholar
[10] M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer, London, 2011. 10.1007/978-0-85729-112-7Search in Google Scholar
[11] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Math. Soc. Lecture Note Ser. 41, Cambridge University Press, Cambridge, 1981. Search in Google Scholar
[12] A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations 3 (1967), 546–570. 10.1016/0022-0396(67)90016-2Search in Google Scholar
[13] H. K. Khalil, Nonlinear Systems, Macmillan, New York, 1992. Search in Google Scholar
[14] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), no. 5, 422–443. 10.1080/14786449508620739Search in Google Scholar
[15] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. Search in Google Scholar
[16] R. Nagel, Spectral and asymptotic properties of strongly continuous semigroups, Semigroups of Linear and Nonlinear Operations and Applications (Curaçao 1992), Kluwer, Dordrecht (1993), 225–240. 10.1007/978-94-011-1888-0_12Search in Google Scholar
[17] L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Texts Appl. Math. 7, Springer, New York, 2001. 10.1007/978-1-4613-0003-8Search in Google Scholar
[18] G. A. Perla Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg–de Vries Equation with localized damping, Quart. Appl. Math. 60 (2002), no. 1, 111–129. 10.1090/qam/1878262Search in Google Scholar
[19] V. A. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1297–1324. Search in Google Scholar
[20] L. Rosier, Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33–55. 10.1051/cocv:1997102Search in Google Scholar
[21] N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations 198 (2004), no. 2, 381–421. 10.1016/j.jde.2003.10.006Search in Google Scholar
[22] G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., John Wiley & Sons, New York, 1999. 10.1002/9781118032954Search in Google Scholar
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