# Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation

Authors: Nejib Smaoui; Rasha H. Al-Jamal
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system
of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory.


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## Body

## 1. Introduction

The generalized Korteweg-de Vries-Burgers
(GKdVB) equation(1.1)ut−νuxx+μuxxx+uαux=0,x∈[0,2π],t≥0,u(x,0)=u0(x),where ν,μ≥0,   and α is a positive integer, is one of the simplest
partial differential equations that displays nonlinearity, with fixed level of
dissipation and dispersion. It has depicted many phenomena, for example strain
wave and longitudinal deformation in a nonlinear elastic rod [1]. If α=1, and ν=0 in (1.1), the GKdVB equation becomes the
classical KdV equation which was derived in 1872 by Boussinesq and Korteweg and
de Vries to model the unidirectional propagation of waves in many physical
systems [2, 3]. If α=1,
the GKdVB equation becomes the KdVB equation which was used as a model for long
waves in shallow water [4] and as model of unidirectional propagation of planar
waves [5]. If α=1 and μ=0,
the GKdVB equation becomes the well-known Burgers equation [6]. If α=2,
the GKdVB equation becomes a model of some physical phenomena [7]. Moreover, the importance of
the GKdVB equation for larger values of α was discussed by Benjamin et al. [7] and Bona et al. [8].Recently, the control problem of the KdVB equation,
KdV equation, and Burgers equation has been treated by many investigators; see
[9–19] to name a few. Since the infinite-dimensional nature
of the PDE models for fluid flow processes can be considered as an obstacle for
the synthesis of practically implementable output feedback controllers,
researchers have been motivated to develop model reduction techniques for the
derivation of low-dimensional ordinary differential equation (ODE) models that
mimic the dynamics of the PDE models [20–23].In this paper, we present a distributed
control scheme for GKdVB equation with periodic boundary conditions and the
following initial condition:(1.2)u0(x)=f(x)=e−10(0.4x−1)2,using a reduction technique
known by the Karhunen-Loéve (K-L) Gelerkin procedure. Our approach which is
based on the K-L procedure is different from the one carried by Rosier and
Zhang [15]. We derive
a system of ODEs that mimics the dynamics of the GKdVB equation, and show that
the system of ODEs has the same qualitative structure to that of the GKdVB
equation. Then, we apply state feedback controllers on the ODE system to force
the dynamics of the GKdVB equation to follow a certain behavior.The paper is organized as follows. In Section2, numerical simulations of the GKdVB equation are obtained using
pseudospectral Fourier Galerkin method. Then, Karhunen-Loève decomposition is used on the numerical
simulation data to extract the coherent structures for α=1 and α=2.
Section 3 presents the K-L Galerkin projection method used on the GKdVB
equation to extract a system of ODEs that mimics the dynamics of the GKdVB
equation. Section 4 introduces three feedback linearization control schemes
used on the obtained system of ODEs to enhance the convergence rate to the
steady-states. Numerical results are shown in each section to illustrate the
presented theory, and some concluding remarks are given in Section 5.


## 2. The Karhunen-Loève Decomposition

The Karhunen-Loève (K-L) decomposition is a very useful and
powerful statistical technique that is used in many applications. In the
literature, the K-L decomposition is known by different names such as the
principal component analysis (PCA) [24] , the empirical orthogonal functions [25], the quasiharmonic modes
[26], the singular
value decomposition (SVD) or the proper orthogonal decomposition (POD)
[27], and the Hoteling
transform [28]. The
method was mainly used for data compression and feature identification. Among
the many applications that utilized the Karhunen-Loève (K-L) decomposition, the K-L decomposition
was used in fluid dynamics [27, 29], in the analysis of two-dimensional Navier-Stokes
(N-S) equation [22, 30–32], and in the study of flames [33].In this section, we use the K-L decomposition
on the numerical simulation data of the GKdVB equation in order to extract the
most energetic eigenfunctions or coherent structures that span the data set in
an optimal way. Since K-L decomposition is heavily used in this section, we
briefly describe the steps involved in this decomposition.First, we collect the data which represents a
numerical solution of the GKdVB equation at different time steps. That is, we
have the following data{Xi}i=1M,
with Xi=[xi1xi2⋯xiN]T at the ith time step, where M is the number of vectors and N is the number of entries in a vector.Next, the meanX¯ of the data is computed such
that(2.1)X¯=1M∑i=1MXi.The mean is then subtracted from each Xi, i=1,…,M.
The resulting vectors(2.2)X^i=Xi−X¯,i=1,…,M,are called the caricature vectors which have zero mean.Based on the snapshot method [29], the covariance matrix,
which is a way to measure how the data is spread out, is computed. The (i,j) element of the covariance matrix C is given by(2.3)cij=1M〈X^i,X^j〉,i,j=1,…,M,where 〈⋅,⋅〉 denotes the usual Euclidean inner product.The eigenfunctions are computed as
follows:(2.4)ψk=∑i=1Mϕi[k]X^i,k=1,…,M,where ϕi[k] is the ith component of the kth eigenvector. These eigenfunctions form an
optimal basis for the decomposition of the data set(2.5)X^(x,t)=∑i=1Mai(t)ψi(x),where ai(t) are the coefficients calculated from the
projections of the sample vector onto an eigenfunction and are calculated
as(2.6)ai=〈X^⋅ψi〉〈ψi⋅ψi〉,i=1,…,M.The energy of the data is
defined as follows:(2.7)E=∑i=1Mλi, where {λi,i=1,…,M} is the set of eigenvalues that corresponds to
the set of eigenfunctions {ψi,i=1,…,M}.
To each eigenfunction, an energy percentage Ek is assigned based on the eigenfunction's associated
eigenvalue:(2.8)Ek=λkE. Finally, the data can be regenerated from the optimal
basis by the following representation:(2.9)X(x,t)≈X¯+∑i=1Mai(t)ψi(x).In this section, the K-L decomposition is used
to analyze the solution of the GKdVB equation (1.1) subject to periodic boundary
conditions and the initial condition given by (1.2). The numerical solution u(x,t) of the GKdVB equation is obtained by a
pseudospectral Fourier Galerkin procedure. Then, u(x,t) is expanded in terms of the K-L eigenfunctions ψn as follows:(2.10)u(x,t)=∑n=1Man(t)ψn(x),where(2.11)ψn(x)=∑k=−HHck,neikxare the K-L eigenfunctions and
where H depends on the spatial discretization of ψ.


## 2.1. The K-L Decomposition for the Caseα=1

Figure1(a) shows the solution u(x,t) of the GKdVB equation (1.1) with α=1 as it evolves to the steady-state solution for
the initial condition given by (1.2); the time t was chosen to be 35 seconds; dt=0.001 second; ν=0.5; and μ=0.01(a) A 3D
landscape plot of the simulated solution of the GKdVB equation whenα=1, ν=0.5, μ=0.01,
and f(x)=e−10(0.4x−1)2.
(b) The most energetic eigenfunctions of the solution of the GKdVB equation in
(a). (c) The data coefficients associated with the eigenfunctions in (b). (d) A
3D landscape plot of the approximated solution using (3.2).
(a)(b)(c)(d)The K-L decomposition was applied on the
numerical solution mentioned above. Two eigenfunctions capturing 99.6% of the
energy were obtained (see Figure1(b)). The first eigenfunction captures 95.4%
of the energy and the second one captures 4.2% of the energy. Figure 1(c)
presents the data coefficients that are obtained using (2.6). Figure 1(d) depicts
the approximated solution of the GKdVB equation using two eigenfunctions.Comparing Figures1(a) and 1(d), one can
conclude that two eigenfunctions are sufficient to capture the dynamics of the
GKdVB equation when α=1.


## 2.2. The K-L Decomposition for the Caseα=2

Numerical experiments show that when the
degree of nonlinearity of the GKdVB equation increases by increasing the
parameterα,
the solution of the GKdVB equation evolves faster to the steady-state solution,
and this is due to the small size of the initial condition used.Figure2(a) shows the solution u(x,t) of the GKdVB equation (1.1) with α=2 as it evolves to the steady-state solution for
the initial condition given by (1.2), and when ν=0.5,
and μ=0.01.(a) A 3D landscape plot of the simulated solution of
the GKdVB equation whenα=2, ν=0.5, μ=0.01,
and f(x)=e−10(0.4x−1)2.
(b) The most energetic eigenfunctions of the solution of the GKdVB equation in
(a). (c) The data coefficients associated with the eigenfunctions in (b). (d) A
3D landscape plot of the approximated solution using (3.2).
(a)(b)(c)(d)Applying the K-L decomposition on the
numerical solution mentioned above, two eigenfunctions capturing 99.7% of the
energy were obtained (see Figure2(b)). The first eigenfunction captures 95.4%
of the energy and the second one captures 4.3% of the energy. Figure 2(c) presents
the corresponding data coefficients obtained, and Figure 2(d) depicts the
approximated solution of the GKdVB equation using the above eigenfunctions.When comparing Figures2(a) and 2(d), one can
conclude that the K-L decomposition was able to capture the large-scale
dynamics of the GKdVB equation with only two eigenfunctions.


## 3. The K-L Galerkin Projection

In order to extract a system of ODEs that
mimics the dynamics of the original GKdVB equation (1.1), we first write the
original PDE as follows:(3.1)∂u∂t=D(u), with given
initial and boundary conditions, where “D” is a differential operator,
and u(x,t) is an approximation solution that can be
written in the following form:(3.2)u(x,t)=∑i=1Kai(t)ψi(x). In (3.2), ai(t) is the ith solution of the system of ODEs and can be
computed in a way that minimize the residual error produced by the approximate
solution above, ψi(x) is the ith eigenfunction from the K-L decomposition,
and K is the number of the most energetic
eigenfunctions.The system of ODEs can be derived by projecting
the normalized eigenfunctions onto the PDE as follows:(3.3)a˙i(t)=〈D(∑i=1Kai(t)ψi(x)),ψi(x)〉,i=1,…,K with initial
condition(3.4)ai(0)=〈u(x,0),ψi(x)〉,i=1,…,K, where u(x,0) is known from the original PDE.


## 3.1. The K-L Galerkin Projection for the Caseα=1

Using (3.2) on the GKdVB equation (1.1) with α=1,(3.5)ut=νuxx−μuxxx−uux, and choosing
the numbers of eigenfunctions K=2,
we obtain the following:(3.6)∑i=12a˙i(t)ψi(x)=ν∑i=12ai(t)ψi′′(x)−μ∑i=12ai(t)ψi′′′(x)−(∑i=12ai(t)ψi(x))(∑j=12aj(t)ψj′(x)),where a˙i(t) is the derivative with respect to time and ψi'(x) is the first derivative with respect to x.
Now, taking the Euclidean inner product of the above equation with ψk, k=1,2 and using the orthogonality property of ψi's:(3.7)〈ψi,ψj〉=∫02πψi(x)ψj(x)dx={0,ifi≠j,1,ifi=j,we obtain the following system of ODEs:(3.8)a˙k(t)=ν∑i=12ai(t)〈ψk,ψi′′〉−μ∑i=12ai(t)〈ψk,ψi′′′〉−∑i=12∑j=12ai(t)aj(t)〈ψk,ψiψj′〉,k=1,2.Substituting the eigenfunctions obtained for the case α=1 in (3.8), we get(3.9)a˙1=−0.0054188768νa1−0.0175592926a1a2+0.0806774017νa2+0.0208940165μa2+0.0047853181a22,a˙2=0.0806774017νa1−0.0208940165μa1+0.0175592854a12−0.0047853181a1a2−1.3396124415νa2.The solution of the above system
can be obtained numerically using any ODE solver. Using MATLAB ODE solver, a
solution of the system was found for the initial conditions a1(0)=0.006 and a2(0)=−0.1 (see  Figure 3).Generated solutions ofa1 and a2 from the K-L Galerkin ODE system when   α=1 with the initial conditions  a1(0)=0.006 and a2(0)=−0.1.
(a)(b)(c)Using (3.2) with the data coefficients computed
above and the normalized eigenfunctions obtained from the K-L decomposition, we
obtain an approximate solution of the GKdVB equation. Figure 4 shows the
approximated solution of the GKdVB equation when α=1.Figure 4
Approximated solution of the GKDVB equation generated by the K-L Galerkin ODE system whenα=1 with the initial conditions  a1(0)=0.006 and a2(0)=−0.1.Comparing Figure1(d) with Figure 4, one can
conclude that the Galerkin projection method gives a reasonable approximation
of the solution of the GKdVB equation when α=1.


## 3.2. The K-L Galerkin Projection for the Caseα=2

Using the procedure illustrated in Section3.1
above, a system of ODEs is obtained for the case α=2:(3.10)a˙k(t)=ν∑i=12ai(t)〈ψk,ψi′′〉−μ∑i=12ai(t)〈ψk,ψi′′′〉−∑i=12∑j=12ai2(t)aj(t)〈ψk,ψi2ψj′〉−2∑i≠j=12ai(t)aj2(t)〈ψk,ψiψjψj′〉. Substituting
the eigenfunctions obtained for the case α=2 in (3.10), we get(3.11)a˙1=−0.00613709a12a2+0.00428373a1a22−0.00374423a23+0.00830501μa2−0.0066366916νa1+0.0918063477νa2,a˙2=0.00613709a13−0.00428373a12a2+0.00374423a1a22−0.00830501μa1+0.0918063477νa1−1.3533719227νa2.Figure 5 shows the general
behavior of the solution of system (3.11) computed by the MATLAB ODE solver with
the initial conditions a1(0)=35 and a2(0)=−30,
and Figure 6 presents an approximation of the solution of the GKdVB equation
when α=2.
Comparing Figure 2(d) with Figure 6, one can deduce that the system of ODEs
computed by the Galerkin projection method mimics the dynamics of the GKdVB
equation when α=2.Generated solutions ofa1 and a2 from the K-L Galerkin ODE system when  α=2 with the initial conditions  a1(0)=35 and a2(0)=−30.
(a)(b)(c)Figure 6
Approximated solution of the GKDVB equation generated by the K-L Galerkin ODE system whenα=2  with the initial conditions   a1(0)=35 and a2(0)=−30.


## 4. Feedback Linearization Control Scheme for the GKdVB Equation

In this section, we analyze the GKdVB equation
using distributed control. The idea of using distributed control on PDEs was
investigated in [20, 34–38]. Three schemes are introduced in this section.


## 4.1. A Feedback Linearization Control Scheme for the GKdVB Equation

The GKdVB equation with distributed control
can be written as(4.1)∂u∂t=ν∂2u∂x2−μ∂3u∂x3−uα∂u∂x+∑i=1nabivi(t), where bi(x) is the actuator distribution function, vi(t) is the ith input, and na is the number of actuators which will be
chosen to be 2.As  mentioned before in (3.2), the solution of
the GKdVB equation can be expressed in terms of the K-L eigenfunctions ψi, i=1,2.
Hence, (4.1) becomes(4.2)∑i=12a˙i(t)ψi(x)=ν∑i=12ai(t)ψi′′(x)−μ∑i=12ai(t)ψi′′′(x)−(∑i=12ai(t)ψi(x))α(∑i=12ai(t)ψi′(x))+∑i=1nabivi(t).Using the Galerkin projection method, the
GKdVB equation is transformed into the following system of
ODEs:(4.3)a˙k(t)=ν∑i=12ai(t)〈ψk,ψi′′〉−μ∑i=12ai(t)〈ψk,ψi′′′〉−g(t)+∑i=12βikvi(t), where(4.4)g(t)=〈(∑i=12ai(t)ψi(x))α(∑i=12ai(t)ψi′(x)),ψk〉,k=1,2,βik=∫02πψkbidx. Define(4.5)ηk(t)=ν∑i=12ai(t)〈ψk,ψi′′〉−μ∑i=12ai(t)〈ψk,ψi′′′〉−g(t),(4.6)wk(t)=∑i=12βikvi(t),k=1,2, then the system
of ODEs (4.3) becomes(4.7)a˙k=ηk(t)+wk(t),k=1,2.Proposition 4.1.
Letξ1 and ξ2 be two positive real numbers, then the
following feedback linearization control scheme: (4.8)wk(t)=−ηk(t)−ξkak(t),k=1,2,renders the system of ODEs in (4.7)
exponentially stable.Proof.
Substituting (4.8) in (4.7), we
get(4.9)a˙k(t)=−ξkak(t),k=1,2,or(4.10)ak(t)=ak(0)e−ξkt,k=1,2, since ξ1,ξ2>0,
then ak(t) will converge exponentially to zero as t→∞.
Therefore, the system of ODEs (4.7) with the controller given in (4.8) is
exponentially stable.Remark 4.2.
We
can force the solution of the system to converge to any desired fixed point.
This can be achieved by making the following change of variables:(4.11)a˜k(t)=ak(t)+ck,k=1,2,where (c1,c2) correspond to the coordinates of the desired
fixed point.Now, from (4.6), we have the
following:(4.12)(w1(t)w2(t))=(β11β21β12β22)⋅(v1(t)v2(t)).Using the
result of the above proposition, system (4.12) can be written by(4.13)(−η1(t)−ξ1a1(t)−η2(t)−ξ2a2(t))=(β11β21β12β22)⋅(v1(t)v2(t)). Provided that
the coefficients βij,
for (i,j=1,2),
are well chosen (i.e., β11β22−β12β21≠0), then the controllers v1(t) and v2(t) in (4.1) are determined by(4.14)(v1(t)v2(t))=(β11β21β12β22)−1⋅(−η1(t)−ξ1a1(t)−η2(t)−ξ2a2(t)).The solution of the controlled system of ODEs
given by (4.7) can be calculated easily by any ODE solver. The above controller
was tested numerically using the MATLAB ODE solver on the GKdVB equation for α=1 and α=2 with the initial condition given by (1.2).
Figure 7 shows the plots of the solutions' (a1(t) and a2(t)) profiles of the above controller for
different values of ξ's; these profiles are also compared to the
solution of the ODE system produced by the Galerkin projection when α=1.
It should be noted that, when the value of ξ increases, the solution evolves faster to the
fixed point, and a better control is obtained. We present in Figure 8 a 3D
landscape of the approximated solution of the GKdVB equation when α=1 with ξ1 and ξ2 being chosen to be 5.The solutions
of the controlled system of ODEs compared to the solutions of the uncontrolled
ODE system whenα=1, a1(0)=0.006, a2(0)=−0.1 with the initial condition given by (1.2).
(a)(b)(c)Figure 8
A 3D landscape of the approximated controlled solution
of the GKdVB equation whenα=1, ξ1=ξ2=5, with the initial condition given by (1.2).The solutions of the controller given by (4.7)
for different values of ξ's were compared numerically to the solution
of the ODE system produced by the Galerkin projection when α=2 (see Figure 9). Figure 10 depicts a 3D
landscape approximation solution of the GKdVB equation when α=2 with ξ1 and ξ2 being chosen to be 0.2.The solutions
of the controlled system of ODEs compared to the solutions of the uncontrolled
ODE system whenα=2, a1(0)=35, a2(0)=−30, with the initial condition given by (1.2).
(a)(b)(c)Figure 10
A 3D landscape of the approximated controlled solution
of the GKdVB equation whenα=2, ξ1=ξ2=0.2, with the initial condition given by (1.2).


## 4.2. Another Control Scheme for the GKdVB Equation

In the control scheme given by (4.8), all the
terms of ηk, k=1,2,
were canceled, whereas some of the elements of ηk,   k=1,2,
have a stabilizing effect on the dynamics and hence there is no need to cancel
them. In this section, we design another version of the feedback controller
given by (4.8).As mentioned before, the solution of the GKdVB
equation can be expressed in terms of the K-L eigenfunctionsψi, i=1,2, as follows:(4.15)u(x,t)=∑n=12∑l=−hhan(t)cl,neilx, and h is an integer which depends on the spatial
discretization of ψn's. Using the above representation of the K-L
eigenfunctions in (4.3), we get the following system of ODEs:(4.16)a˙k(t)=ν∑i=12∑l=−hh−l2cl,icl,kai(t)+μ∑i=12∑l=−hhil3cl,icl,kai(t)−g(t)+∑i=12βikvi(t). Let(4.17)A0=(∑l=−hhl2cl,12∑l=−hhl2cl,1cl,2∑l=−hhl2cl,2cl,1∑l=−hhl2cl,22),a(t)=(a1(t)a2(t)),f˜(t)=(f˜1(t)f˜2(t)),w(t)=(w1(t)w2(t)), where f˜k(t) is(4.18)f˜k(t)=μ∑i=12∑l=−hhil3cl,icl,kai(t)−g(t), where k=1,2.
Then, the system of ODEs can be written as(4.19)a˙(t)=−νA0a(t)+f˜(t)+w(t). The matrix A0 can be easily computed, and it is easy to
check that the matrix A0 is positive definite.Proposition 4.3.
The controller(4.20)w(t)=−f˜(t)renders the system of ODEs exponentially stable.Proof.
Substituting (4.20) in (4.19), we obtain
the following:(4.21)a˙(t)=−νA0ak(t) or(4.22)a(t)=a(0)e−νA0t, since the
matrix “−A0” is negative definite, then a(t) converges exponentially to zero as t→∞.
Therefore, the system of ODEs (4.19) with the controller given by (4.20) is exponentially
stable.Using the result of the above proposition, then the
system(4.23)(w1(t)w2(t))=(β11β21β12β22)⋅(v1(t)v2(t)) can be
presented as(4.24)(−f˜1(t)−f˜2(t))=(β11β21β12β22)⋅(v1(t)v2(t)). Provided that
the coefficients βij,
for (i,j=1,2),
are well chosen (i.e., β11β22−β12β21≠0), then the controllers v1(t) and v2(t) in (4.1) are determined as
follows:(4.25)(v1(t)v2(t))=(β11β21β12β22)−1⋅(−f˜1(t)−f˜2(t)).


## 4.3. A Control Scheme for the GKdVB Equation Using a Single Actuator

Sections4.2 and 4.3 addressed the control
problem of the GKdVD equation when the system uses two actuators (i.e., two
control inputs). This section discusses the control of the GKdVB equation when
only one actuator is used.The GKdVB equation with one actuator can be
written as(4.26)∂u∂t=ν∂2u∂x2−μ∂3u∂x3−uα∂u∂x+b(x)v(t),0≤x≤2π, where v(t) is the control input and b(x) acts to distribute the control throughout the
spatial domain [0,2π].The derivation of the system of ordinary
differential equations (ODEs) based on K-L Galerkin projection results
in(4.27)a˙k(t)=ν∑i=12ai(t)〈ψk,ψi′′〉−μ∑i=12ai(t)〈ψk,ψi′′′〉−〈(∑i=12ai(t)ψi(x))α(∑i=12ai(t)ψi′(x)),ψk〉+βkv(t),where(4.28)βk=∫02πψk(x)b(x)dx,k=1,2. Hence, the behavior of the GKdVB equation for the case α=2 in (4.26) can be approximated by the following
system of ODEs:(4.29)a˙1(t)=−0.00613709a12a2+0.00428373a1a22−0.00374423a23+0.00830501μa2−0.0066366916νa1+0.0918063477νa2+β1v(t),a˙2(t)=0.00613709a13−0.00428373a12a2+0.00374423a1a22−0.00830501μa1+0.0918063477νa1−1.3533719227νa2+β2v(t).Proposition 4.4.
The state feedback controller(4.30)v(t)=−k1a1(t)−k2a2(t)with the design parameters k1&k2 such that k1β1>0,  k2β2>0, and k1β2=k2β1 renders the ODE system in (4.29) asymptotically
stable.Proof.
Consider the following Lyapunov function candidate:(4.31)V(t)=12(a12(t)+a22(t)).Note that V(t)>0 if (a1(t),a2(t))≠(0,0) and V(t)=0 if (a1(t),a2(t))=(0,0).
Taking the derivative of V(t) with respect to time and using (4.29) and (4.30),
it follows that(4.32)V˙=a˙1a1+a˙2a2=(−0.00613709a12a2+0.00428373a1a22−0.00374423a23+0.00830501μa2−0.0066366916νa1+0.0918063477νa2+β1v(t))a1+(0.00613709a13−0.00428373a12a2+0.00374423a1a22−0.00830501μa1+0.0918063477νa1−1.3533719227νa2+β2v(t))a2≤−(β1a1+β2a2)(k1a1+k2a2)≤−β2k2(a2+β2k1+β1k22β2k2a1)2≤0.Note that V˙ is negative definite. Hence, by Lyapunov
theory, the controller scheme given by (4.30) guarantees the asymptotic stability
of the GKdVB equation.


## 5. Concluding Remarks

In this paper, we have analyzed the control
problem of the GKdVB equation subject to periodic boundary conditions by applying
a distributed control strategy. The Karhunen-Loève Galerkin method was used to produce systems
of ODEs which mimic the dynamics of the GKdVB equation forα=1 and α=2.
Then, we used three state feedback linearization control schemes on the system
of the ODEs that render it exponentially stable. Simulation results are
presented to show the effectiveness of the developed control schemes.For future work, we will look into the
development of adaptive and optimal control schemes for the GKdVB equation, and
the design of boundary controllers for different values ofα.



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*Source: 621672-2008-06-23.xml*