Nonlinear Dyn (2008) 51:439–446 DOI 10.1007/s11071-007-9222-5

ORIGINAL ARTICLE

Boundary control of the generalized Korteweg–de Vries–Burgers equation Nejib Smaoui · Rasha H. Al-Jamal

Received: 4 October 2006 / Accepted: 22 January 2007 / Published online: 16 February 2007 C Springer Science + Business Media B.V. 2007 

Abstract This paper considers the boundary control problem of the generalized Korteweg–de Vries– Burgers (GKdVB) equation on the interval [0, 1]. We derive a control law of the form u(0, t) = 1 u x (1, t) = u x x (1, t) + μ(α+2) u α+1 (1, t) = 0, where μ > 0 and α is a positive integer, and prove that it guarantees L 2 -global exponential stability, H 1 global asymptotic stability, and H 1 -semiglobal exponential stability. Numerical results supporting the analytical ones for both the controlled and uncontrolled equations are presented using a finite element method. Keywords Generalized Korteweg–de Vries–Burgers equation . Nonlinear boundary control . Stability

1 Introduction In this paper, the boundary control problem of the generalized Korteweg–de Vries–Burgers (GKdVB) equation ∂u ∂u ∂ 3u ∂ 2u − ν 2 + μ 3 + uα = 0, ∂t ∂x ∂x ∂x t ≥ 0,

x ∈ [0, 1],

N. Smaoui ( ) · R. H. Al-Jamal Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait e-mail: [email protected]

(1)

u(0, t) = 0,

(2)

∂u (1, t) = f 1 (u(1, t)), ∂x

(3)

∂ 2u (1, t) = f 2 (u(1, t)), ∂x2 u(x, 0) = u 0 (x),

(4) (5)

where μ, ν > 0 and α is a positive integer is considered. In Equation (1), the independent variable x represents the medium of propagation, t is proportional to elapsed time, and u(x, t) is a velocity at the point x at time t. In Equations (3)–(5), the boundary functions f 1 and f 2 are considered as control inputs, and u 0 is the initial condition. It should be noted that the GKdVB equation reduces to the Korteweg–de Vries–Burgers (KdVB) equation when α = 1. The KdVB equation is considered to be one of the simplest nonlinear mathematical models displaying the features of both dispersion and dissipation [1, 5, 14, 16]. When ν = 0 and α = 1, the GKdVB equation reduces to the Korteweg–de Vries (KdV) equation which is a nonlinear dispersive partial differential equation that presents a model of propagation of small amplitude along water waves in a uniform channel [9, 21, 28]; and when μ = 0 and α = 1, the KdVB equation reduces to the Burgers equation which models turbulent liquid flow through a channel [2–4, 10, 27]. Recently, the control problem of the KdVB equation [5], KdV equation [20, 21], and Burgers Springer

440

equation [25, 26] has received a lot of attention. Balogh and Krsti´c [5] investigated the KdVB equation using boundary control. In their work, global stability of the solution in the L 2 -sense and global stability in the H 1 sense were proved. Rosier [20, 21] worked on the KdV equation where an exact boundary control of the linear and nonlinear KdV equations was established in [20]; and the control was illustrated numerically in [21]. Smaoui [25, 26] considered boundary and distributed control of the Burgers equation. A boundary control is used in [25] to show the exponential stability of the Burgers equation analytically as well as numerically. In [26], a system of ODEs was constructed to mimic the dynamics of Burgers equation, then a state feedback control scheme was implemented on the system to show that the Burgers solution can be controlled to any desired state. In 1993, Bona and Luo [6] studied the asymptotic behavior of solutions to the GKdVB equation for α ≥ 2. They showed that the GKdVB equation is well posed in certain function spaces. Xia and Huang [29] worked on the spectral methods for a class of GKdVB equations and proved the existence and uniqueness of the global solution of a semi-discrete scheme under some conditions. L¨u and Zhang [17] studied the discrete Fourier spectral approximation of the global attractor of the GKdVB equation. They proved the existence of an approximate attractor to the global attractor using the Banach fixed-point theorem. Numerical techniques were also developed to approximate solutions of the GKdVB equation (see [8, 11, 13]). The paper is organized as follows. In Section 2, a boundary control law to the GKdVB equation using Lyapunov theory is proposed to show the global exponential stability of the solution in L 2 (0, 1) and the semi-global exponential and global asymptotic stability of the solution in H 1 (0, 1). Section 3 presents some numerical results using finite element techniques to show the effectiveness of the developed control schemes. Finally, some concluding remarks are given in Section 4.

Nonlinear Dyn (2008) 51:439–446

control problem of the GKdVB equation has not been treated elsewhere. Since Bona and Luo [6] showed that the GKdVB equation is well posed in certain function spaces, we will assume that the GKdVB equation has a solution u(x, t) ∈ Cb1 ([0, ∞); L 2 (0, 1)) ∩ C([0, ∞); H 3 (0, 1)), where Cb1 ([0, ∞); L 2 (0, 1)) is the set of all continuously differentiable bounded functions defined on [0, ∞) with values in L 2 (0, 1) and C([0, ∞); H 3 (0, 1)) is the set of all continuous functions defined on [0, ∞) with values in H 3 (0, 1). In this section, we propose a boundary control law for the GKdVB equation and prove its global exponential stability in L 2 (0, 1) and its global asymptotic and semi-global exponential stability in H 1 (0, 1) for all values of α. 2.1 Exponential stability in L 2 (0, 1) Theorem 2.1. Let α be a positive integer. The GKdVB Equation (1) with boundary conditions given by Equations (2)–(4) is globally exponentially stable in L 2 (0, 1) under the following control law: f 1 (u(1, t)) = 0, f 2 (u(1, t)) = −

(6) 1 u α+1 (1, t). μ(α + 2)

Proof: If we take the L 2 inner product of Equation (1) with 2u(x, t), we obtain  1  1 2u(x, t)u t (x, t) d x − 2νu(x, t)u x x (x, t) d x 0

 +

0 1

2μu(x, t)u x x x (x, t) d x 0

 +

1

2u α+1 u x (x, t) d x = 0.

0

Since  1  2u(x, t)u t (x, t) d x = 0

Springer

1 0

d 2 u (x, t) d x dt

d = ||u(x, t)||2 , dt

2 The boundary control problem of the GKdVB equation The boundary control problem of the KdVB equation (α = 1 in Equation (1)) was treated by many investigators [5, 7, 12, 15, 16, 18, 19, 22–24, 30, 31]. However, up to the knowledge of the authors, the boundary

(7)

and  1 u(x, t)u x x (x, t) d x 0

1 = u(x, t)u x (x, t)0 − ||u x (x, t)||2 ,

(8)

Nonlinear Dyn (2008) 51:439–446

441

and 

1

0

Equation (11) becomes 1 u(x, t)u x x x (x, t) d x = u(x, t)u x x (x, t)0 

−

d u(x, t)2 ≤ −2νu(x, t)2 . dt

 1 d  2 u x (x, t) d x 0 2 dx 1 1   1 = u(x, t)u x x (x, t) − u 2x (x, t) , 2 0 0 1

Integrating both sides of (13) with respect to time, we get u(x, t)2 ≤ u 0 (x)2 e−2νt .

1

(14)

Hence, one can write,

and 

(13)

u α+1 (x, t)u x (x, t) d x =

0

 × 0

1

u(x, t) ≤ u 0 (x) e−νt .

1 α+2

1 d α+2 1 (u (x, t)) d x = u α+2 (x, t)0 , dx α+2

then Equation (8) becomes

(15)

Therefore, u(x, t) converges to zero exponentially as t → ∞, since ν > 0. Thus, the solution of GKdVB Equation (1) is globally exponentially stable in L 2 (0, 1). 

1  2   d u(x, t)2 − 2ν u(x, t)u x (x, t)0 − u x (x, t) dt 1 1

  1 2  + 2μ u(x, t)u x x (x, t) − u x (x, t) 2 0 0

1  1 +2 (9) u α+2 (x, t) = 0. α+2 0

Remark 1. It can be easily shown that if α is an even positive integer, then the GKdVB equation is globally exponentially stable in L 2 (0, 1) under the following control law:

Substituting the boundary conditions given by Equations (2)–(4) and the control law given by Equations (6) and (7) into Equation (9), one obtains

where a ≥ 0.

d 2 u(x, t)2 + 2νu x (x, t)2 − u α+2 (1, t) dt (α + 2) + μu 2x (0, t) +

2 u α+2 (1, t) = 0. α+2

(10)

2.2 Asymptotic stability and semi-global exponential stability in H 1 (0, 1) The following lemmas are needed to prove the next theorem:

u(x, t)∞ ≤ cu x (x, t). for some positive constant c.

Since μ > 0, Equation (10) leads to d u(x, t)2 ≤ −2νu x (x, t)2 . dt

f 2 (u(1, t)) = au α−1 (1, t),

Lemma 2.1. Let u(x, t) ∈ H 1 (0, 1) and u(0, t) = 0, then

Thus, d u(x, t)2 = −2νu x (x, t)2 − μu 2x (0, t). dt

f 1 (u(1, t)) = 0,

Proof: By Agmon’s inequality, we have (11) u(x, t)∞ ≤ cu(x, t)1/2 u x (x, t)1/2 ,

Using the Poincar´e inequality u(x, t) ≤ u x (x, t),

(12)

then using the Poincare inequality, we obtain the desired result.  Springer

442

Nonlinear Dyn (2008) 51:439–446

Lemma 2.2. Let u(x, t) ∈ H 2 (0, 1), then the following inequality holds:  1 c 2u α (x, t)u x (x, t)u x x (x, t) d x ≤ u x (x, t)2α+2 ν 0 + νu x x (x, t)2 .

d νt (e u(x, t)2 )+2νeνtu x (x, t)2≤ νeνtu(x, t)2 . dt (19) Integrating (19) with respect to t, we get  t t eνt u(x, τ )2 0 + 2ν eντ u x (x, τ )2 dτ ≤ ν 

for some positive constant c. ×

Proof: 

1



α

2u (x, t)u x (x, t)u x x (x, t) d x ≤ 2 max |u(x, t)| 

1

×

Using Cauchy–Schwartz inequality, the inequality αβ ≤ 2ν1 α 2 + ν2 β 2 , and Lemma (2.1), we obtain the desired inequality.  Theorem 2.2. Let α be a positive integer. The GKdVB Equation (1) with boundary conditions given by Equations (2)–(4) is globally asymptotically stable and semi-globally exponentially stable in H 1 (0, 1) under the control law given by Equations (6) and (7). Proof: Integrating (11) with respect to t, we obtain  t u(x, t)2 − u 0 (x)2 + 2ν u x (x, τ )2 dτ ≤ 0, 0

0

1 u x (x, τ )2 dτ ≤ u 0 (x)2 . 2ν

Using Poincar´e inequality, (16) becomes  t 1 u 0x (x)2 . u x (x, τ )2 dτ ≤ 2ν 0



t

u x (x, τ )2 dτ ≤

0

M u 0x (x)2 . 2ν

Now multiplying (11) by eνt , we get Springer

(20) Using (13), we have following inequality: eνt

(16)

(17)

(18)

d u(x, t)2 ≤ −2νeντ u(x, t)2 . dt

(21)

Integrating (21) with respect to t, we obtain  t  t νt d 2 e eντ u(x, τ )2 dτ u(x, τ ) dτ ≤ −2ν dτ 0 0  t t d = −2eντ u(x, τ )2 0 + 2 eντ u(x, τ )2 dτ , dτ 0

−

Since u(x, t) ∈ Cb1 ([0, ∞); L 2 (0, 1)) ∩C([0, ∞); H (0, 1)), then there exists M > 0, such that u x (x, t)2α ≤ Mu x (x, t)2 . Thus  t u x (x, τ )2α dτ ≤ M ×

d u(x, τ )2 dτ , dτ

 t  t d 2ν eντ u x (x, τ )2 dτ ≤ − eντ u(x, t)2 dτ . dτ 0 0

or

3

0

eντ

or

|u x (x, t)u x x (x, t)| d x.

0

or  t

t

eντ u(x, τ )2 dτ = eντ u(x, τ )2 |t0

0

0≤x≤1

0

0

−

α

0

t



t

d u(x, τ )2 dτ ≤ dτ 0 t − 2eντ u(x, τ )2 0 ≤ 2u 0 (x)2 . eνt

Using (22) in (20), we get  t 1 eνt u x (x, τ )2 dτ ≤ u 0 (x)2 . ν 0

(22)

(23)

Now, if we take the L 2 inner product of Equation (1) with 2u x x (x, t), we obtain  1  1 2u x x (x, t)u t (x, t) d x − 2νu 2x x (x, t) d x 0

 +

0

1

2μu x x (x, t)u x x x (x, t) d x 0

 +

0

1

2u α u x (x, t)u x x (x, t) d x = 0.

(24)

Nonlinear Dyn (2008) 51:439–446



443

The following equalities are true 1

0

or

1 2u x x (x, t)u t (x, t) d x = 2u x (x, t)u t (x, t)0 

−

1 d u 2α+2 (1, t) u x (x, t)2 ≤ dt μ(α + 2)2 c + u x (x, t)2α+2 . ν

1

2u x (x, t)u xt (x, t) d x, 0



1 = 2u x (x, t)u t (x, t)0  1  d 2 − u x (x, t) d x, 0 dt 1 d = 2u x (x, t)u t (x, t)0 − u x (x, t)2 , dt 1

0

Multiplying (27) by eνt , we get d νt (e u x (x, t)2 ) ≤ ν eνt u x (x, t)2 dt 1 + u 2α+2 (1, t)eνt μ(α + 2)2 c + u x (x, t)2α+2 eνt . ν

2νu 2x x (x, t) d x = 2νu x x (x, t)2 ,

and 

(27)

(28)

Since 

1

0

 = 0

1

≤ 0

Using the boundary conditions (2)–(4) with the control given by Equations (6) and (7), Equation (24) becomes d − u x (x, t)2 − 2νu x x (x, t)2 dt 1 + u 2α+2 (1, t) − μu 2x x (0, t) μ(α + 2)2  1 + 2u α u x (x, t)u x x (x, t) d x = 0, (25) 0

or d u x (x, t)2 = −2νu x x (x, t)2 dt 1 + u 2α+2 (1, t) − μu 2x x (0, t) μ(α + 2)2  1 + 2u α u x (x, t)u x x (x, t) d x. 0

u s (s, t) ds 

1  d  2 μ u x x (x, t) d x = μu 2x x (x, t)0 . dx

2

x

u 2 (x, t) =

2μu x x (x, t)u x x x (x, t) d x

0 1

u 2x (x, t) d x = u x (x, t)2 ,

then (28) becomes d νt (e u x (x, t)2 ) ≤ ν eνt u x (x, t)2 dt  1 c + + u x (x, t)2α+2 eνt . μ(α + 2)2 ν

(30)

Integrating (30) with respect to t, we obtain  t t d ντ (e u x (x, τ )2 ) dτ ≤ ν eντ u x (x, τ )2 dτ dτ 0 0   t 1 c + + u x (x, τ )2α+2 eντ dτ . μ(α + 2)2 ν 0



Using (23), we obtain

(26)

eνt u x (x, t)2 ≤ u 0x (x)2 + u 0 (x)2   t 1 c + + u x (x, τ )2α+2 eντ dτ . μ(α + 2)2 ν 0

Using Lemma (2.2), we obtain d u x (x, t)2 ≤ −2νu x x (x, t)2 dt 1 + u 2α+2 (1, t) − μu 2x x (0, t) μ(α + 2)2 c + u x (x, t)2α+2 + νu x x (x, t)2 , ν

(29)

(31) Now, using Gronwall–Bellman’s inequality, we obtain eνt u x (x, t)2 ≤ (u 0x (x)2 + u 0 (x)2 ) e

(

1 μ(α+2)2

+ νc )

t 0

u x (x,τ )2α dτ

.

(32) Springer

444

Nonlinear Dyn (2008) 51:439–446

Using Poincar´e inequality and (18), Equation (32) becomes M

eνt u x (x, t)2 ≤ 2u 0x (x)2 e 2ν

(

1 μ(α+2)2

+ νc )u 0x (x)2

Hence, the solution of the GKdVB equation is globally asymptotically and semi-globally exponentially  stable in H 1 (0, 1).

, (33)

3 Numerical results

.

The COMSOL Multiphisics software, which is based on the finite element method, is used on all numerical simulations presented herein. Equation (1) with homogeneous boundary condition (i.e., the uncontrolled system or the open loop system) was simulated with the initial condition u(x, 0) = sin(π x). Figure 1 depicts

or M

u x (x, t)2 ≤ 2u 0x (x)2 e 2ν

(

1 μ(α+2)2

+ νc )u 0x (x)2 −νt

e

(34)

Fig. 1 Time evolution of the uncontrolled GKdVB equation when ν = 0.1, μ = 0.1, α = 1, and u(x, 0) = sin(π x)

1

0.8

u(x,t)

0.6

0.4

0.2

0 1 0.8

0.7 0.6

0.6

0.5 0.4

0.4

0.3 0.2

0.2 x

Fig. 2 Time evolution of the controlled GKdVB equation when ν = 0.1, μ = 0.1, α = 1, and u(x, 0) = sin(π x)

0

0.1

t

0

1

0.8

u(x,t)

0.6

0.4

0.2

0 1 0.8

0.7 0.6

0.6

0.5 0.4

0.4 x

0.3 0.2

0.2 0

Springer

0.1 0

t

Nonlinear Dyn (2008) 51:439–446

445

Fig. 3 Time evolution of the uncontrolled GKdVB equation when ν = 0.1, μ = 0.01, α = 2, and u(x, 0) = sin(π x)

1

0.8

u(x,t)

0.6

0.4

0.2

0 1 0.8

2.5 0.6

2 1.5

0.4 x

Fig. 4 Time evolution of the controlled GKdVB equation when ν = 0.1, μ = 0.01, α = 2, and u(x, 0) = sin(π x)

1

0.2

0.5 0

0

t

1

0.8

u(x,t)

0.6

0.4

0.2

0 1 0.8

2.5 0.6

2 1.5

0.4 x

1

0.2

0.5 0

the time evolution of the solution u(x, t) with μ = 0.1, ν = 0.1 and α = 1. As we can see from Fig. 1, the uncontrolled solution seems to converge to a nontrivial steady-state solution. However, when the boundary control is applied on the second derivative as in Equation (4), while keeping the first derivative boundary condition in Equation (3) equal to 0 at x = 1, the solution converges to zero (see Fig. 2). Equation (1) was also simulated for the case α = 2 with ν = 0.1, μ = 0.01 and u(x, 0) = sin(π x). Again, the numerical solution seems to converge to a nontrivial solution before applying the boundary control inputs

0

t

given by Equations (6) and (7) (see Fig. 3). On the other hand, applying the control inputs at the boundary, as shown in Equations (3) and (4), forces the solution of the GKdVB equation to converge to the zero solution as presented by Fig. 4. 4 Concluding remarks We have used a boundary control to analyze the exponential stability of the generalized KdVB in L 2 sense, the asymptotic stability in H 1 , and the semi-global exponential stability in H 1 . Also, we have presented some Springer

446

numerical results based on finite element technique to support and reinforce the analytical ones. Up to the knowledge of the authors, the boundary control problem of the GKdVB equation was treated here for the first time and the design of other controllers to speed up the convergence rate to the zero dynamics will be the subject of future research work. References 1. Armaou, A., Christofides, P.D.: Wave suppression by nonlinear finite-dimensional control. Chem. Eng. Sci. 55, 2627– 2640 (2000) 2. Atwell, J.A., King, B.B.: Stabilized finite element methods and feedback control for Burgers’ equation. In: Proceedings of the American Control Conference, Chicago, IL (2000) 3. Baker, J.A., Armaou, A., Christofides, P.D.: Nonlinear control of incompressible fluid flows: application to Burgers equation and 2D chaneel flow. J. Math. Anal. Appl. 252, 230–255 (2000) 4. Balogh, A., Krsti´c, M.: Global boundary stabilization and regularization of Burgers’ equation. In: Proceedings of the American Control Conference, San Diego, CA, pp. 1712– 1716 (1999) 5. Balogh, A., Krsti´c, M.: Boundary control of the Korteweg–de Vries–Burgers equation: further results on stabilization and well-posedness, with numerical demonstration. IEEE Trans. Autom. Control 45(9), 1739–1745 (2000) 6. Bona, J.L., Luo, L.: Decay of solutions to nonlinear dispersive wave equations. Differ. Int. Equations 6, 961–980 (1993) 7. Bona, J.L., Schonbek, M.E.: Traveling-wave solutions to the Korteweg–de Vries–Burgers equation. Proc. R. Soc. Edinburgh 101A, 207–226 (1985) 8. Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Computations of blow-up and decay for periodic solutions of the generalized Korteweg–de Vries–Burgers equation. Appl. Numer. Math. 10, 335–355 (1992) 9. Bracken, P.: Some methods for generating solutions to the Korteweg–de Vries equation. Physica A 335, 70–78 (2004) 10. Christofides, P.D., Armaou, A.: Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. Syst. Control Lett. 39, 283–294 (2000) ` 11. Edel’man, I.Ya.: Propagation of nonlinear waves on a porous medium with two-phase saturation by a liquid and a gas. Fluid Dyn. 31(4), 552–559 (1996) 12. Gao, P., Zhao, Y.: Boundary stabilization for the general Korteweg–de Vries–Burgers equation. Acta Anal. Funct. Appl. 5(2), 110–118 (2003) 13. Karakashian, O., McKinney, W.: On the approximation of solutions of the generalized Korteweg–de Vries–Burgers equation. Math. Comput. Simul. 37(4/5), 405–416 (1994) 14. Karch, G.: Self-similar lage time behavior of solutions to Korteweg–de Vries–Burgers equation. Nonlinear Anal. 35, 199–219 (1999)

Springer

Nonlinear Dyn (2008) 51:439–446 15. Komornik, V., Russell, D.L., Zhang, B.Y.: Stabilization de l’´equation de Korteweg–de Vries. C. R. Acad. Sci. Paris Ser. I Math. 312(11), 841–843 (1991) 16. Liu, W.-J., Krstic, M.: Controlling nonlinear water waves: boundary stabilization of the Korteweg–de Vries–Burgers equation. In: Proceedings of the American Control Conference, San Diego, CA, pp. 1637–1641 (1999) 17. L¨u, S., Zhang, F.: The spectral method for long time behavior of generalized KdV–Burgers equations. Math. Numer. Sin. 21(2), 129–138 (1999) 18. Naumkin, P.I., Shishmarev, I.A.: A problem on the decay of step-like data for the Korteweg–de Vries–Burgers equation (Russian). Funct. Anal. Appl. 25(1), 16–25 (1991) 19. Nishihara, K., Rajopadhye, S.V.: Asymptotic behavior of solutions to the Korteweg–de Vries–Burgers equation. Differ. Integral Equations 11(1), 85–93 (1998) 20. Rosier, L.: Exact boundary controllability for the Korteweg– de Vries equation on a bounded domain. ESAIM Conrol Optim. Cal. Var. 2, 33–55 (electronic) (1997) 21. Rosier, L.: Exact boundary controllability for the linear Korteweg–de Vries equation – a numerical study. ESAIM Proc. 4, 255–267 (1998) 22. Russell, D.L., Zhang, B.Y.: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31(3), 659–676 (1993) 23. Russell, D.L., Zhang, B.Y.: Smoothing and decay properties of solutions of the Korteweg–de Vries equation on a periodic domain with point dissipation. J. Math. Anal. Appl. 190(2), 449–488 (1995) 24. Russell, D.L., Zhang, B.Y.: Exact controllability and stabilizability of the Korteweg–de Vries equation. Trans. Am. Math. Soc. 348(9), 3643–3672 (1996) 25. Smaoui, N.: Nonlinear boundary control of the generalized Burgers equation. Nonlinear Dyn. J. 37(1), 75–86 (2004) 26. Smaoui, N.: Boundary and distributed control of the viscous Burgers equation. J. Comput. Appl. Math. 182, 91–104 (2005) 27. Smaoui, N.: Analyzing the dynamics of the forced Burgers equation. J. Appl. Math. Stochastic Anal. 13(3), 269–285 (2000) 28. Taha, T.R.: A parallel algorithm for solving higher KdV equation on hypercube. In: Proceedings of the Fifth IEEE Distributed Memory Computing Conference, Charleston, SC, pp. 564–567 (1990) 29. Xia, Z., Huang, F.: Fourier spectral methods for a class of generalized KdV–Burgers equations with variable coefficients. In: Proceedings of the First World Congress, 4 volumes, Tampa, FL, August 19–26 (1992) 30. Zhang, B.Y.: Boundary stabilization of the Korteweg–de Vries equation. In: Control Estimation of Distributed Parameter Systems: Nonlinear Phenomena (Vorau, Austria, 1993). Int. Ser. Numer. Math. 118, 371–389 (1994) 31. Zhang, L.H.: Decay of solutions of a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. Proc. R. Soc. Edinburgh Sect. A 124(2), 263–271 (1994)