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

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.

1. Introduction

The generalized Korteweg-de Vries-Burgers (GKdVB) equation𝑢_𝑡−𝜈𝑢_𝑥𝑥+𝜇𝑢_𝑥𝑥𝑥+𝑢^𝛼𝑢_𝑥=0,𝑥∈[0,2𝜋],𝑡≥0,𝑢(𝑥,0)=𝑢₀(𝑥), (1.1)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:𝑢₀(𝑥)=𝑓(𝑥)=𝑒^{−10(0.4𝑥−1)2}, (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 Section 2, 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 {𝑋_𝑖}^𝑀_𝑖=1, with 𝑋_𝑖=[𝑥¹_𝑖𝑥²_𝑖⋯𝑥^𝑁_𝑖]^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 mean 𝑋 of the data is computed such that1𝑋=𝑀^𝑀_𝑖=1𝑋_𝑖. (2.1)The mean is then subtracted from each 𝑋_𝑖, 𝑖=1,…,𝑀. The resulting vectors𝑋_𝑖=𝑋_𝑖−𝑋,𝑖=1,…,𝑀, (2.2)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 by𝑐_𝑖𝑗=1𝑀⟨𝑋_𝑖,𝑋_𝑗⟩,𝑖,𝑗=1,…,𝑀, (2.3)where ⟨⋅,⋅⟩ denotes the usual Euclidean inner product.

The eigenfunctions are computed as follows:𝜓_𝑘=^𝑀_𝑖=1𝜙_𝑖^[𝑘]𝑋_𝑖,𝑘=1,…,𝑀, (2.4)where 𝜙_𝑖^[𝑘] is the 𝑖th component of the 𝑘th eigenvector. These eigenfunctions form an optimal basis for the decomposition of the data set𝑋(𝑥,𝑡)=^𝑀_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥), (2.5)where 𝑎_𝑖(𝑡) are the coefficients calculated from the projections of the sample vector onto an eigenfunction and are calculated as𝑎_𝑖=⟨𝑋⋅𝜓_𝑖⟩⟨𝜓_𝑖⋅𝜓_𝑖⟩,𝑖=1,…,𝑀. (2.6)The energy of the data is defined as follows:𝐸=^𝑀_𝑖=1𝜆_𝑖, (2.7) where {𝜆_𝑖,𝑖=1,…,𝑀} is the set of eigenvalues that corresponds to the set of eigenfunctions {𝜓_𝑖,𝑖=1,…,𝑀}. To each eigenfunction, an energy percentage 𝐸_𝑘 is assigned based on the eigenfunction's associated eigenvalue:𝐸_𝑘=𝜆_𝑘𝐸. (2.8) Finally, the data can be regenerated from the optimal basis by the following representation:𝑋(𝑥,𝑡)≈𝑋+^𝑀_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥). (2.9)

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 𝜓_𝑛 as follows:𝑢(𝑥,𝑡)=^𝑀_𝑛=1𝑎_𝑛(𝑡)𝜓_𝑛(𝑥), (2.10)where𝜓_𝑛(𝑥)=^𝐻_𝑘=−𝐻𝑐_𝑘,𝑛𝑒^𝑖𝑘𝑥 (2.11)are the K-L eigenfunctions and where 𝐻 depends on the spatial discretization of 𝜓.

2.1. The K-L Decomposition for the Case 𝛼=1

Figure 1(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)

(b)

(c)

(d)

Figure 1 

(a) A 3D landscape plot of the simulated solution of the GKdVB equation when 𝛼=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).

The K-L decomposition was applied on the numerical solution mentioned above. Two eigenfunctions capturing 99.6% of the energy were obtained (see Figure 1(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 Figures 1(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 Case 𝛼=2

Numerical experiments show that when the degree of nonlinearity of the GKdVB equation increases by increasing the parameter 𝛼, 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.

Figure 2(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)

(b)

(c)

(d)

Figure 2 

(a) A 3D landscape plot of the simulated solution of the GKdVB equation when 𝛼=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).

Applying the K-L decomposition on the numerical solution mentioned above, two eigenfunctions capturing 99.7% of the energy were obtained (see Figure 2(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 Figures 2(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:𝜕𝑢𝜕𝑡=𝐷(𝑢), (3.1) 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:𝑢(𝑥,𝑡)=^𝐾_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥). (3.2) In (3.2), 𝑎_𝑖(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, 𝜓_𝑖(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:𝑎̇_𝑖(𝑡)=⟨𝐷(^𝐾_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥)),𝜓_𝑖(𝑥)⟩,𝑖=1,…,𝐾 (3.3) with initial condition𝑎_𝑖(0)=⟨𝑢(𝑥,0),𝜓_𝑖(𝑥)⟩,𝑖=1,…,𝐾, (3.4) where 𝑢(𝑥,0) is known from the original PDE.

3.1. The K-L Galerkin Projection for the Case 𝛼=1

Using (3.2) on the GKdVB equation (1.1) with 𝛼=1,𝑢_𝑡=𝜈𝑢_𝑥𝑥−𝜇𝑢_𝑥𝑥𝑥−𝑢𝑢_𝑥, (3.5) and choosing the numbers of eigenfunctions 𝐾=2, we obtain the following:𝑎̇²_𝑖=1𝑖(𝑡)𝜓_𝑖(𝑥)=𝜈²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖^′′(𝑥)−𝜇²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖^′′′(𝑥)−(²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥))(²_𝑗=1𝑎_𝑗(𝑡)𝜓^′_𝑗(𝑥)), (3.6)where 𝑎̇_𝑖(𝑡) is the derivative with respect to time and 𝜓^′_𝑖(𝑥) is the first derivative with respect to 𝑥. Now, taking the Euclidean inner product of the above equation with 𝜓_𝑘, 𝑘=1,2 and using the orthogonality property of 𝜓_𝑖's:⟨𝜓_𝑖,𝜓_𝑗⟩=₀^2𝜋𝜓_𝑖(𝑥)𝜓_𝑗(𝑥)𝑑𝑥=0,if𝑖≠𝑗,1,if𝑖=𝑗, (3.7)we obtain the following system of ODEs:𝑎̇_𝑘(𝑡)=𝜈²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′⟩−𝜇²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′′⟩−²²_𝑖=1𝑗=1𝑎_𝑖(𝑡)𝑎_𝑗(𝑡)⟨𝜓_𝑘,𝜓_𝑖𝜓^′_𝑗⟩,𝑘=1,2. (3.8)Substituting the eigenfunctions obtained for the case 𝛼=1 in (3.8), we get𝑎̇₁=−0.0054188768𝜈𝑎₁−0.0175592926𝑎₁𝑎₂+0.0806774017𝜈𝑎₂+0.0208940165𝜇𝑎₂+0.0047853181𝑎²₂,𝑎̇₂=0.0806774017𝜈𝑎₁−0.0208940165𝜇𝑎₁+0.0175592854𝑎²₁−0.0047853181𝑎₁𝑎₂−1.3396124415𝜈𝑎₂. (3.9)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).

(a)

(b)

(c)

Figure 3 

Generated solutions of 𝑎1 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.

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 when 𝛼=1 with the initial conditions 𝑎1(0)=0.006 and 𝑎2(0)=−0.1.

Comparing Figure 1(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 Case 𝛼=2

Using the procedure illustrated in Section 3.1 above, a system of ODEs is obtained for the case 𝛼=2:𝑎̇_𝑘(𝑡)=𝜈²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′⟩−𝜇²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′′⟩−²²_𝑖=1𝑗=1𝑎²_𝑖(𝑡)𝑎_𝑗(𝑡)⟨𝜓_𝑘,𝜓²_𝑖𝜓^′_𝑗⟩−2²_{𝑖≠𝑗=1}𝑎_𝑖(𝑡)𝑎²_𝑗(𝑡)⟨𝜓_𝑘,𝜓_𝑖𝜓_𝑗𝜓^′_𝑗⟩. (3.10) Substituting the eigenfunctions obtained for the case 𝛼=2 in (3.10), we get𝑎̇₁=−0.00613709𝑎²₁𝑎₂+0.00428373𝑎₁𝑎²₂−0.00374423𝑎³₂+0.00830501𝜇𝑎₂−0.0066366916𝜈𝑎₁+0.0918063477𝜈𝑎₂,𝑎̇₂=0.00613709𝑎³₁−0.00428373𝑎²₁𝑎₂+0.00374423𝑎₁𝑎²₂−0.00830501𝜇𝑎₁+0.0918063477𝜈𝑎₁−1.3533719227𝜈𝑎₂. (3.11)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.

(a)

(b)

(c)

Figure 5 

Generated solutions of 𝑎1 and 𝑎2 from the K-L Galerkin ODE system when 𝛼=2 with the initial conditions 𝑎1(0)=35 and 𝑎2(0)=−30.

Figure 6 

Approximated solution of the GKDVB equation generated by the K-L Galerkin ODE system when 𝛼=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𝜕𝑢𝜕𝜕𝑡=𝜈²𝑢𝜕𝑥²𝜕−𝜇³𝑢𝜕𝑥³−𝑢^𝛼𝜕𝑢+𝜕𝑥^𝑛𝑎_𝑖=1𝑏_𝑖𝑣_𝑖(𝑡), (4.1) where 𝑏_𝑖(𝑥) is the actuator distribution function, 𝑣_𝑖(𝑡) is the 𝑖th input, and 𝑛_𝑎 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 𝜓_𝑖, 𝑖=1,2. Hence, (4.1) becomes𝑎̇²_𝑖=1𝑖(𝑡)𝜓_𝑖(𝑥)=𝜈²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖^′′(𝑥)−𝜇²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖^′′′(𝑥)−(²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥))^𝛼(²_𝑖=1𝑎_𝑖(𝑡)𝜓^′_𝑖(𝑥))+^𝑛𝑎_𝑖=1𝑏_𝑖𝑣_𝑖(𝑡). (4.2)

Using the Galerkin projection method, the GKdVB equation is transformed into the following system of ODEs:𝑎̇_𝑘(𝑡)=𝜈²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′⟩−𝜇²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′′⟩−𝑔(𝑡)+²_𝑖=1𝛽^𝑘_𝑖𝑣_𝑖(𝑡), (4.3) where𝑔(𝑡)=⟨(∑²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥))^𝛼(∑²_𝑖=1𝑎_𝑖(𝑡)𝜓^′_𝑖(𝑥)),𝜓_𝑘𝛽⟩,𝑘=1,2,_𝑘𝑖=∫₀^2𝜋𝜓_𝑘𝑏_𝑖𝑑𝑥. (4.4) Define𝜂_𝑘(𝑡)=𝜈²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′⟩−𝜇²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′′𝑤⟩−𝑔(𝑡),_𝑘(𝑡)=²_𝑖=1𝛽^𝑘_𝑖𝑣_𝑖(𝑡),𝑘=1,2, (4.5)(4.6) then the system of ODEs (4.3) becomes𝑎̇_𝑘=𝜂_𝑘(𝑡)+𝑤_𝑘(𝑡),𝑘=1,2. (4.7)

Proposition 4.1. Let  𝜉₁ and 𝜉₂ be two positive real numbers, then the following feedback linearization control scheme: 𝑤_𝑘(𝑡)=−𝜂_𝑘(𝑡)−𝜉_𝑘𝑎_𝑘(𝑡),𝑘=1,2, (4.8)renders the system of ODEs in (4.7) exponentially stable.

Proof. Substituting (4.8) in (4.7), we get𝑎̇_𝑘(𝑡)=−𝜉_𝑘𝑎_𝑘(𝑡),𝑘=1,2, (4.9)or𝑎_𝑘(𝑡)=𝑎_𝑘(0)𝑒^{−𝜉𝑘𝑡},𝑘=1,2, (4.10) since 𝜉₁,𝜉₂>0, then 𝑎_𝑘(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: ^0𝑥0𝑎^02𝑑𝑐_𝑘(𝑡)=𝑎_𝑘(𝑡)+𝑐_𝑘,𝑘=1,2, (4.11)where (𝑐₁,𝑐₂) correspond to the coordinates of the desired fixed point.

Now, from (4.6), we have the following:𝑤₁𝑤(𝑡)₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⋅𝑣₁𝑣(𝑡)₂(𝑡). (4.12)Using the result of the above proposition, system (4.12) can be written by−𝜂₁(𝑡)−𝜉₁𝑎₁(𝑡)−𝜂₂(𝑡)−𝜉₂𝑎₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⋅𝑣₁𝑣(𝑡)₂(𝑡). (4.13) Provided that the coefficients 𝛽^𝑗_𝑖, for (𝑖,𝑗=1,2), are well chosen (i.e., 𝛽¹₁𝛽²₂−𝛽²₁𝛽¹₂≠0), then the controllers 𝑣₁(𝑡) and 𝑣₂(𝑡) in (4.1) are determined by𝑣₁𝑣(𝑡)₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⁻¹⋅−𝜂₁(𝑡)−𝜉₁𝑎₁(𝑡)−𝜂₂(𝑡)−𝜉₂𝑎₂(𝑡). (4.14)

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.

(a)

(b)

(c)

Figure 7 

The solutions of the controlled system of ODEs compared to the solutions of the uncontrolled ODE system when 𝛼=1, 𝑎1(0)=0.006, 𝑎2(0)=−0.1 with the initial condition given by (1.2).

Figure 8 

A 3D landscape of the approximated controlled solution of the GKdVB equation when 𝛼=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.

(a)

(b)

(c)

Figure 9 

The solutions of the controlled system of ODEs compared to the solutions of the uncontrolled ODE system when 𝛼=2, 𝑎1(0)=35, 𝑎2(0)=−30, with the initial condition given by (1.2).

Figure 10 

A 3D landscape of the approximated controlled solution of the GKdVB equation when 𝛼=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 𝜂_𝑘, 𝑘=1,2, were canceled, whereas some of the elements of 𝜂_𝑘, 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 eigenfunctions 𝜓_𝑖, 𝑖=1,2, as follows:𝑢(𝑥,𝑡)=^2ℎ_{𝑛=1𝑙=−ℎ}𝑎_𝑛(𝑡)𝑐_𝑙,𝑛𝑒^𝑖𝑙𝑥, (4.15) and ℎ is an integer which depends on the spatial discretization of 𝜓_𝑛's. Using the above representation of the K-L eigenfunctions in (4.3), we get the following system of ODEs:𝑎̇_𝑘(𝑡)=𝜈^2ℎ_{𝑖=1𝑙=−ℎ}−𝑙²𝑐_𝑙,𝑖𝑐_𝑙,𝑘𝑎_𝑖(𝑡)+𝜇^2ℎ_{𝑖=1𝑙=−ℎ}𝑖𝑙³𝑐_𝑙,𝑖𝑐_𝑙,𝑘𝑎_𝑖(𝑡)−𝑔(𝑡)+²_𝑖=1𝛽^𝑘_𝑖𝑣_𝑖(𝑡). (4.16) Let𝐴₀=^{ℎ𝑙=−ℎ}𝑙²𝑐^2ℎ_{𝑙,1𝑙=−ℎ}𝑙²𝑐_𝑙,1𝑐_{ℎ𝑙,2𝑙=−ℎ}𝑙²𝑐_𝑙,2𝑐^ℎ_{𝑙,1𝑙=−ℎ}𝑙²𝑐²_𝑙,2,𝑎𝑎(𝑡)=¹𝑎(𝑡)₂,(𝑡)^{0𝑥002𝑑𝑐}𝑓(𝑡)=^{0𝑥002𝑑𝑐}𝑓₁(𝑡)_{0𝑥002𝑑𝑐}𝑓₂𝑤(𝑡),𝑤(𝑡)=¹𝑤(𝑡)₂,(𝑡) (4.17) where ^0𝑥0𝑓^02𝑑𝑐_𝑘(𝑡) is^0𝑥0𝑓^02𝑑𝑐_𝑘(𝑡)=𝜇^2ℎ_{𝑖=1𝑙=−ℎ}𝑖𝑙³𝑐_𝑙,𝑖𝑐_𝑙,𝑘𝑎_𝑖(𝑡)−𝑔(𝑡), (4.18) where 𝑘=1,2. Then, the system of ODEs can be written as𝑎(̇𝑡)=−𝜈𝐴₀𝑎(𝑡)+𝑓^{0𝑥002𝑑𝑐}(𝑡)+𝑤(𝑡). (4.19) 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 𝑤(𝑡)=−𝑓^{0𝑥002𝑑𝑐}(𝑡) (4.20)renders the system of ODEs exponentially stable.

Proof. Substituting (4.20) in (4.19), we obtain the following:𝑎(̇𝑡)=−𝜈𝐴₀𝑎_𝑘(𝑡) (4.21) or𝑎(𝑡)=𝑎(0)𝑒^{−𝜈𝐴0𝑡}, (4.22) 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 system𝑤₁𝑤(𝑡)₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⋅𝑣₁𝑣(𝑡)₂(𝑡) (4.23) can be presented as−𝑓^{0𝑥002𝑑𝑐}₁−(𝑡)_{0𝑥002𝑑𝑐}𝑓₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⋅𝑣₁𝑣(𝑡)₂(𝑡). (4.24) Provided that the coefficients 𝛽^𝑗_𝑖, for (𝑖,𝑗=1,2), are well chosen (i.e., 𝛽¹₁𝛽²₂−𝛽²₁𝛽¹₂≠0), then the controllers 𝑣₁(𝑡) and 𝑣₂(𝑡) in (4.1) are determined as follows:𝑣₁𝑣(𝑡)₂=𝛽(𝑡)¹¹𝛽¹₂𝛽²₁𝛽²₂⁻¹⋅−𝑓^{0𝑥002𝑑𝑐}₁−(𝑡)_{0𝑥002𝑑𝑐}𝑓₂(𝑡). (4.25)

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

Sections 4.2 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𝜕𝑢𝜕𝜕𝑡=𝜈²𝑢𝜕𝑥²𝜕−𝜇³𝑢𝜕𝑥³−𝑢^𝛼𝜕𝑢𝜕𝑥+𝑏(𝑥)𝑣(𝑡),0≤𝑥≤2𝜋, (4.26) 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 in𝑎̇_𝑘(𝑡)=𝜈²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′⟩−𝜇²_𝑖=1𝑎_𝑖(𝑡)⟨𝜓_𝑘,𝜓_𝑖^′′′⟩−⟨(²_𝑖=1𝑎_𝑖(𝑡)𝜓_𝑖(𝑥))^𝛼(²_𝑖=1𝑎_𝑖(𝑡)𝜓^′_𝑖(𝑥)),𝜓_𝑘⟩+𝛽_𝑘𝑣(𝑡), (4.27)where𝛽_𝑘=₀^2𝜋𝜓_𝑘(𝑥)𝑏(𝑥)𝑑𝑥,𝑘=1,2. (4.28) Hence, the behavior of the GKdVB equation for the case 𝛼=2 in (4.26) can be approximated by the following system of ODEs:𝑎̇₁(𝑡)=−0.00613709𝑎²₁𝑎₂+0.00428373𝑎₁𝑎²₂−0.00374423𝑎³₂+0.00830501𝜇𝑎₂−0.0066366916𝜈𝑎₁+0.0918063477𝜈𝑎₂+𝛽₁𝑎̇𝑣(𝑡),₂(𝑡)=0.00613709𝑎³₁−0.00428373𝑎²₁𝑎₂+0.00374423𝑎₁𝑎²₂−0.00830501𝜇𝑎₁+0.0918063477𝜈𝑎₁−1.3533719227𝜈𝑎₂+𝛽₂𝑣(𝑡). (4.29)

Proposition 4.4. The state feedback controller 𝑣(𝑡)=−𝑘₁𝑎₁(𝑡)−𝑘₂𝑎₂(𝑡) (4.30)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:1𝑉(𝑡)=2(𝑎²₁(𝑡)+𝑎²₂(𝑡)). (4.31)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 that𝑉̇=𝑎̇₁𝑎₁+𝑎̇₂𝑎₂=(−0.00613709𝑎²₁𝑎₂+0.00428373𝑎₁𝑎²₂−0.00374423𝑎³₂+0.00830501𝜇𝑎₂−0.0066366916𝜈𝑎₁+0.0918063477𝜈𝑎₂+𝛽₁𝑣(𝑡))𝑎₁+(0.00613709𝑎³₁−0.00428373𝑎²₁𝑎₂+0.00374423𝑎₁𝑎²₂−0.00830501𝜇𝑎₁+0.0918063477𝜈𝑎₁−1.3533719227𝜈𝑎₂+𝛽₂𝑣(𝑡))𝑎₂≤−(𝛽₁𝑎₁+𝛽₂𝑎₂)(𝑘₁𝑎₁+𝑘₂𝑎₂)≤−𝛽₂𝑘₂(𝑎₂+𝛽₂𝑘₁+𝛽₁𝑘₂2𝛽₂𝑘₂𝑎₁)²≤0. (4.32)Note that 𝑉̇ 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 for 𝛼=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 of 𝛼.