# Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

Authors: B. Chentouf; N. Smaoui; A. Alalabi
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.


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## Body

## 1. Introduction

In various problems of fluid dynamics and physics, the evolution of small amplitude long waves is described by the so-called dispersive equations which are governed by nonlinear partial differential equations (PDEs) [1–5]. This topic has attracted the efforts of many researchers from different disciplines to study the dynamics and the control problem of these dispersive equations. The analysis of such equations usually combines rigorous techniques of modern analysis and physics promoting numerous interdisciplinary research articles in the field.In this paper, we will explore the nonlinear adaptive stabilization of the following form of the dispersive equation called the modified generalized Korteweg–de Vries–Burgers (MGKdVB) equation:(1)∂u∂t+γ1uα∂u∂x−ν∂2u∂x2+μ∂3u∂x3+γ2∂4u∂x4=0,x∈0,1,t≥0,subject to the boundary conditions(2)u0,t=0,(3)∂2u∂x20,t=0,(4)∂u∂x1,t=f1t,(5)∂2u∂x21,t=f2t,and the initial condition(6)ux,0=u0x.Here, the variableu=ux,t is a real-valued function that may represent the displacement of the medium or the velocity and depends on two real variables x and t, where x∈0,1 is the distance in the direction of propagation of the medium and t≥0 is the elapsed time. Furthermore, α is assumed throughout this article as a positive integer and the physical parameters γ1, ν, γ2, and µ are positive real numbers which represent nonlinearity, dissipation (diffusion), and dispersion coefficients, respectively. Last, f1t and f2t are nonlinear boundary control functions to be designed so that the solutions of the whole system of equations are stable.Recently, the derivation of the MGKdVB equation (1) when α=3 has been presented by means of the long-wave approximation and perturbation method [6]. Furthermore, the existence and uniqueness of solutions of the MGKdVB equation (1) have been investigated in [7] where linear boundary conditions are considered. Since our main concern in the current work is the adaptive stabilization of the MGKdVB equation (1), we shall assume the existence of a unique regular solution ux,t of the MGKdVB equation (1).As several variants of equation (1) have been actively and continuously used, we are going to highlight some results obtained in the literature. For instance, setting μ=0, α=1, γ1=1, and γ2=0 reduces equation (1) to the well-known Burgers equation. During the last decades, many researchers devoted their time to study this equation including the adaptive and nonadaptive boundary control problem [1, 8–16]. In 1999, the boundary control of the Burgers equation was tackled by designing a nonlinear adaptive control law when the viscosity is unknown [9]. In 2001, an adaptation law for Burgers equation, where the viscosity is unknown was proposed, and the L2- and H1-global stability of the solution were proven [12]. In [14, 15], the generalized Burgers equation (α≥1, μ=0, γ1=1, and γ2=0) is considered under the action of an adaptive control when all the parameters of the equation are unknown. Then, the L2-regulation of the solution to the steady state solution was achieved. Furthermore, the global control of the generalized Burgers equation was also studied both analytically and numerically. The distributed and boundary stabilization problems of the Burgers equation have been considered in [16].The so-called Kuramoto–Sivashinsky (KS) is obtained by settingμ=0, γ1=γ2=1, α=1, and ν<0 in equation (1). Such an equation was first derived in 1978 by Kuramoto [3] to model a chaotic state in a chemical reaction system. It was also derived in 1980 by Sivashinsky [4] to understand the dynamics of flame front propagation. Since then many researchers further explored the KS equation [17–32]. For instance, assuming a space variable x∈ℝ, the Cauchy problem of the KS equation has been proven to admit a unique smooth and exponentially decaying solution that is continuously dependent on its initial data [17]. In turn, a distributed output feedback control has been put forward to obtain the global stabilization of the KS equation [22]. Such a stabilization result has been improved by proposing appropriate boundary controls [23]. In [24], an adaptive control law when the parameters of the KS equation are unknown has been designed and the global asymptotic stability and the convergence of solutions to zero have been established.In addition, settingμ>0, γ1=γ2=1, α=1, and ν<0 in equation (1), the MGKdVB equation reduces to the generalized Kuramoto–Sivashinsky (GKS) equation which was used to model turbulence. This equation has received considerable attention by many researchers [19, 20, 26, 27, 30]. For instance, Kudryashov [19] investigated several classes of analytical solutions of the GKS equation and Guo and Xiang [20] proved the existence and uniqueness of the global and smooth solution for the periodic initial value problem of the GKS equation. In 2002, Iosevich and Miller [26] analyzed the case where the dispersive effects cannot be neglected and obtained solutions to the KdV equation as the limit of the GKS equation when the dispersive and dissipative terms are large. Later on, Larkin [27] considered the GKS equation in a bounded domain as a model of long waves in a viscous fluid flow down on an inclined plane and proved that the solutions to a mixed problem for the KdV equation may be obtained as singular limits of solutions to a corresponding mixed problem for the KS equation.In turn, whenγ1=1 and γ2=0 in equation (1), the MGKdVB equation leads to the generalized Korteweg–de Vries–Burgers (GKdVB) equation. This equation is useful in modeling many physical phenomena such as the unidirectional propagation of planar waves [33], strain waves and longitudinal deformation in a nonlinear elastic rod [34], and pressure waves in a liquid-gas bubble mixture [5]. Note that there has been a growing interest in the analysis and control of the GKdVB equation [2, 33, 35–41]. Specifically, long-time behavior of periodic solutions of the GKdVB equation has been investigated [2]. Then, the global solutions of the periodic GKdVB equation when α<4 are obtained and the periodic initial value problem is shown to be well-posed for sufficiently large initial data in the case when α≥4 [33]. In the context of control systems, nonadaptive boundary control laws have been proposed in [38, 40, 41, 42]. Smaoui and Al-Jamal [38] designed three nonadaptive boundary control laws to show that the dynamics of the GKdVB equation is globally exponentially stable in L20,1 and globally asymptotically stable and semiglobally exponentially stable in H10,1 when α is a positive integer. Then, the controllers developed in [38] have been improved when α is even and when α is odd [41]. Later on, Smaoui et al. [40] designed three different nonadaptive boundary control laws, when α is a positive real number, and showed numerically for certain values of α that the proposed controllers outperform those designed in [38]. On the other hand, the adaptive boundary control of the GKdVB equation was discussed in [39, 42] when either the kinematic viscosity ν or the dynamic viscosity µ is unknown or when both viscosities ν and µ are unknown.Up to our knowledge, the adaptive boundary stabilization problem of the MGKdVB equation has not been examined in the literature. In fact, as previously discussed, the adaptive control stabilization of the MGKdVB equation (1) has been dealt with in the following specific cases: μ=γ2=0 in [9, 12, 14, 15], γ2=0 in [39, 42], and μ=0 in [24].In contrast to [9, 12, 14, 15, 24, 39, 42], the novelty of this paper is to consider the adaptive boundary control of the MGKdVB equation (1) by designing four different nonlinear boundary control laws without assuming the nullity of the physical parameters and depending whether these parameters are known or unknown (i.e., when ν is unknown or γ1 is unknown or ν and γ1 are unknown or when ν, γ1, µ, and γ2 are unknown). Furthermore, an exhaustive list of numerical simulations will be presented for each proposed control to support and validate the theoretical outcome.This paper is arranged as follows. In Section2, the first adaptive boundary nonlinear control law is proposed for the MGKdVB equation when the parameter ν is unknown. In Section 3, the second adaptive nonlinear boundary control law is designed for the MGKdVB equation when the parameter γ1 is unknown. In Section 4, the third adaptive nonlinear boundary control law is designed for the MGKdVB equation when the parameters ν and γ1 are unknown. In Section 5, the fourth adaptive nonlinear boundary control law is designed for the MGKdVB equation when the parameters ν, μ, γ1, and γ2 are unknown. Also, the global exponential stability of the solutions in L20,1 is presented analytically and numerically for each of the designed controller proposed in Sections 2–5. Section 6 compares the rates of convergence of the solutions of the four adaptive presented controllers provided in Sections 2–5. Finally, some concluding remarks are presented in Section 7.


## 2. Design of the First Nonlinear Adaptive Controller

In this section, we present the first nonlinear adaptive boundary control law for the MGKdVB equation. A control law for the MGKdVB equation is proposed when the parameterν is unknown. The next theorem illustrates the first result of our nonlinear adaptive boundary control law.Theorem 1.
The MGKdVB equation given by equation (1), subject to the boundary conditions stated in equations (2)–(5) and the initial condition u0∈L20,1, is globally exponentially stable in L20,1, provided that the following nonlinear control law is proposed:(7)f1t=−η1u1,t,(8)f2t=−γ1α+2μ+γ2η1uα+11,t−γ2μ+γ2η1∂3u∂x31,t+μη12/2+μη2μ+γ2η1u1,t,where(9)dη1dt=r1u21,t,r1>0,(10)dη2dt=r2u21,t,r2>0.Proof 1.
Let(11)Vt=12∫01u2x,tdx,be a Lyapunov function candidate.
DifferentiatingVt with respect to time, we obtain(12)dVdt=∫01∂u∂tx,tux,tdx.
Referring to equation (1), one can write equation (12) as follows:(13)dVdt=∫01uν∂2u∂x2−μ∂3u∂x3−γ1uα∂u∂x−γ2∂4u∂x4dx=ν∫01u∂2u∂x2dx−μ∫01u∂3u∂x3dx−γ1∫01uα+1∂u∂xdx−γ2∫01u∂4u∂x4dx.
Integrating by parts and using the boundary conditions (2)–(5), we obtain(14)dVdt≤−ν∫01∂u∂x2dx+νu1,tf1t−μu1,tf2t+μ2f12t−γ1α+2uα+21,t−γ2u1,t∂3u∂x31,t+γ2f1tf2t.
Using control laws (7) and (8), we obtain(15)dVdt≤−ν∫01∂u∂x2dx−νη1u21,t−μη2u21,t.
Applying Poincaré's inequality gives(16)dVdt≤−ν∫01u2dx−νη1u21,t−μη2u21,t.
Now letEt be the following energy function:(17)Et=Vt+12vr1vη1−a2+12μr2μη2−b2,a≥0,b≥0.
It is clear from this equation thatEt≥0,∀t≥0. Differentiating Et with respect to time yields:(18)dEdt=dVdt+1r1dη1dtνη1−a+1r2dη2dtμη2−b.
By virtue of inequality (16) and equations (9) and (10), we deduce from equation (18) that(19)dEdt≤−ν∫01u2dx−au21,t−bu21,t.
Since−au21,t−bu21,t≤0, for all t≥0 then equation (19) reduces to(20)dEdt≤−ν∫01u2dx≤0,∀t≥0.
Thereby,Et≤E0,∀t≥0 and hence both of η1 and η2 are bounded for all t>0. This implies that u1,t∈L20,∞. Invoking inequality (16), we obtain(21)dVdt≤−2νVt−νη1+μη2u21,t.
Applying Gronwall–Bellman’s inequality, we obtain(22)Vt≤V0e−2νt+C∫0te−2νt−τu21,τdτ,where C=supt>0νη1t+μη2t. Since u1,t∈L20,∞, we conclude that Vt approaches zero as t⟶∞. Therefore, ux,t exponentially tends to zero as t⟶∞.In the next subsection, we give a numerical presentation of the dynamical behavior of the MGKdVB equation by applying the first nonlinear adaptive boundary control law presented in equations (7) and (8).


### 2.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) under the presence of the controls f1t and f2t as presented by equations (7) and (8) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.From equations (7) and (8), one can observe that the first adaptive control law proposed in Theorem 1 does not require the preknowledge of the kinematic viscosity ν which is assumed to be unknown. Nevertheless, for simulation purposes, the value of v is set to be 0.01 in the MGKdVB equation (i.e., equation (1)). The dynamic viscosity µ is chosen to be 0.001. The parameters γ1 and γ2 are set to be 1 and 0.0005, respectively. Moreover, r1 and r2 are chosen to equal 0.2. The initial values of η1 and η2 are set to be such as η10=0.2 and η20=0.1. Moreover, let the initial datum ux,0=sinπx. Figures 1(a)–1(d) show a 3D view of the solution of the MGKdVB equation utilizing the control law presented in Theorem 1 for several values of α. Figure 2 shows the L2-norm of the solutions. It can be noticed from these figures that as α increases, the decay rate of the solution to the steady state solution decreases. This is due to the effect of the nonlinear term which causes the instability in the behavior of the MGKdVB equation.Figure 1
A 3D view of the dynamics of the MGKdVB equation utilizing the first nonlinear adaptive controllerμ=0.001, γ1=1, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 2
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the first nonlinear adaptive controller μ=0.001, γ1=1, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures3 and 4 depict the behavior of the functions η1 and η2 that are utilized in the first control law. Figure 3 shows that, for 0<t≤5, η˙1 decreases as the value of α increases from 2 to 4 and increases after t≃5. Figure 4 indicates that η˙2 behaves similar to η˙1 as α increases. The choice of the feedback gains r1 and r2 will definitely affect the values of η1 and η2 and therefore the stability of the solution. Fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero as can be deduced from inequality (21).Figure 3
η
1 over time for various values of α when applying the first nonlinear adaptive controller.Figure 4
η
2 over time for various values of α when applying the first nonlinear adaptive controller.It should be noted that the parameters of the controllersf1t and f2t are η1, η2, γ1, γ2, µ, and α. For given values of γ1, γ2, µ, and α, the parameters η1 and η2 can be computed from the uncoupled system of ODE equations (9) and (10) once the value at the right boundary u1,t is known, the control gains r1 and r2 are chosen, and the initial conditions η10 and η20 are fixed. When these parameters are evaluated, the controllers f1t and f2t can be easily computed using equations (7) and (8).


## 2.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) under the presence of the controls f1t and f2t as presented by equations (7) and (8) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.From equations (7) and (8), one can observe that the first adaptive control law proposed in Theorem 1 does not require the preknowledge of the kinematic viscosity ν which is assumed to be unknown. Nevertheless, for simulation purposes, the value of v is set to be 0.01 in the MGKdVB equation (i.e., equation (1)). The dynamic viscosity µ is chosen to be 0.001. The parameters γ1 and γ2 are set to be 1 and 0.0005, respectively. Moreover, r1 and r2 are chosen to equal 0.2. The initial values of η1 and η2 are set to be such as η10=0.2 and η20=0.1. Moreover, let the initial datum ux,0=sinπx. Figures 1(a)–1(d) show a 3D view of the solution of the MGKdVB equation utilizing the control law presented in Theorem 1 for several values of α. Figure 2 shows the L2-norm of the solutions. It can be noticed from these figures that as α increases, the decay rate of the solution to the steady state solution decreases. This is due to the effect of the nonlinear term which causes the instability in the behavior of the MGKdVB equation.Figure 1
A 3D view of the dynamics of the MGKdVB equation utilizing the first nonlinear adaptive controllerμ=0.001, γ1=1, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 2
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the first nonlinear adaptive controller μ=0.001, γ1=1, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures3 and 4 depict the behavior of the functions η1 and η2 that are utilized in the first control law. Figure 3 shows that, for 0<t≤5, η˙1 decreases as the value of α increases from 2 to 4 and increases after t≃5. Figure 4 indicates that η˙2 behaves similar to η˙1 as α increases. The choice of the feedback gains r1 and r2 will definitely affect the values of η1 and η2 and therefore the stability of the solution. Fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero as can be deduced from inequality (21).Figure 3
η
1 over time for various values of α when applying the first nonlinear adaptive controller.Figure 4
η
2 over time for various values of α when applying the first nonlinear adaptive controller.It should be noted that the parameters of the controllersf1t and f2t are η1, η2, γ1, γ2, µ, and α. For given values of γ1, γ2, µ, and α, the parameters η1 and η2 can be computed from the uncoupled system of ODE equations (9) and (10) once the value at the right boundary u1,t is known, the control gains r1 and r2 are chosen, and the initial conditions η10 and η20 are fixed. When these parameters are evaluated, the controllers f1t and f2t can be easily computed using equations (7) and (8).


## 3. Design of the Second Nonlinear Adaptive Controller

In this section, we present the second nonlinear adaptive boundary control law for the MGKdVB equation. A control scheme for the MGKdVB equation is proposed when the parameterγ1 is unknown. Furthermore, numerical results will be also discussed to reveal the effectiveness of this control law.The following theorem illustrates the result of our second nonlinear adaptive boundary control.Theorem 2.
The MGKdVB equation given by equation (1) with the boundary conditions given by equations (2)–(5) and the initial condition u0∈L20,1 is globally exponentially stable in the L20,1, by considering the following nonlinear control law:(23)f1t=−η1νu1,t,(24)f2t=μη122vγ2η1+νμu1,t−νγ2γ2η1+νμ∂3u∂x31,t+νη2γ2η1+νμu2α+11,t,where(25)dη1dt=r1u21,t,r1>0,(26)dη2dt=r2u2α+21,t,r2>0.Proof 2.
Consider(27)Vt=12∫01u2x,tdx,and differentiate Vt with respect to time and arguing as for estimate (14), we obtain(28)dVdt≤−ν∫01∂u∂x2dx+νu1,tf1t−μu1,tf2t+μ2f12t−γ1α+2uα+21,t−γ2u1,t∂3u∂x31,t+γ2f1tf2t.
Using the control law given by equations (23) and (24), inequality (28) becomes(29)dVdt≤−ν∫01∂u∂x2dx−η1u21,t−η2u2α+21,t−γ1α+2uα+21,t.
By using Poincaré’s inequality, we obtain(30)dVdt≤−ν∫01u2dx−η1u21,t−η2u2α+21,t−γ1α+2uα+21,t.
The last term of inequality (30) can be bounded as follows:(1)
Ifuα+21,t≤0, then u1,t≤0, and in this case α is odd. Therefore,(31)−γ1α+2uα+21,t=γ1α+2uα+11,t−γ1α+2u1,t≥0.By using Young’s inequality, we obtain(32)−γ1α+2uα+21,t=γ1α+2uα+11,t−γ1α+2u1,t≤γ12α+2u2α+21,t+γ12α+2u21,t.(2)
Ifuα+21,t≥0, then(33)−γ1α+2uα+21,t≤γ1α+2uα+11,tγ1α+2u1,t≤γ12α+2u2α+21,t+γ12α+2u21,t.
Thus, in both cases, we have(34)−γ1α+2uα+21,t≤γ12α+2u2α+21,t+γ12α+2u21,t.
Combining (30) and (34) gives(35)dVdt≤−ν∫01u2dx−η1u21,t−η2u2α+21,t+γ12α+2u2α+21,t+γ12α+2u21,t.
Now, let us define a nonnegative energy functionEt as follows:(36)Et=Vt+12r1η1−γ12α+22+12r2η2−a−γ12α+22,where a>0.
DifferentiatingEt, we obtain(37)dEdt=dVdt+1r1dη1dtη1−γ12α+2+1r2dη2dtη2−a−γ12α+2.
Using inequality (35) and equations (25) and (26), we have(38)dEdt≤−ν∫01u2dx−η1u21,t−η2u2α+21,t+γ12α+2u2α+21,t+γ12α+2u21,t+u21,tη1−γ12α+2+u2α+21,tη2−a−γ12α+2.
The latter leads to(39)dEdt≤−ν∫01u2dx−au2α+21,t.
Noting that−au2α+21,t≤0 for all t≥0, the estimate (39) gives(40)dEdt≤−ν∫01u2dx,that is,(41)dEdt≤−2νVt.
Hence,Et≤E0,∀t≥0. One can conclude that η1 and η2 are bounded functions for all t>0 and thus u1,t∈L20,∞∩L2α+20,∞ when α is a positive integer.
Using inequality (35), we have(42)dVdt≤−2νVt−η1−γ12α+2u21,t−η2−γ12α+2u2α+21,t.
Exploiting the Gronwall–Bellman’s inequality, we obtain(43)Vt≤V0e−2νt+C∫0te−2νt−τu21,τ+u2α+21,τdτ,where C=maxsupt>0η1t−γ1/2α+2,supt>0η2t−γ1/2α+2.
Using the fact thatu1,t∈L20,∞∩L2α+20,∞, we conclude that Vt⟶0 exponentially as t⟶∞. Hence, ux,t exponentially tends to zero as t⟶∞.In the next subsection, we give a numerical presentation of the dynamical behavior of the MGKdVB equation when using the second nonlinear adaptive boundary control law presented in equations (23) and (24).


### 3.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(8)) with the controls f1t and f2t as presented by equations (23) and (24) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.It follows from equations (23) and (24) that the preknowledge of the parameter γ1 is not required in the second adaptive control law proposed in Theorem 2. However, the value of γ1 is set to be 1 in the MGKdVB equation (i.e., equation (1)) for simulation purposes. The kinematic viscosity ν is chosen to be 0.01 and the dynamic viscosity µ is set to be 0.001. The parameter γ2 is chosen to be 0.0005. Moreover, r1 and r2 are chosen to equal 0.2. The initial values of η1 and η2 are set to be such as η10=η20=1 and ux,0=sinπx. Figures 5(a)–5(d) show a 3D view of the solution of the MGKdVB equation under the action of the control law presented in Theorem 2 for different values of α. Moreover, the L2-norm of these solutions is presented in Figure 6. It can be concluded from the figures that as α increases, the convergence rate of the solution to the steady state solution becomes smaller. This is due to the presence of the nonlinear term uα∂u/∂x which dominates over the other terms in the MGKdVB equation.Figure 5
A 3D view of the dynamics of the MGKdVB equation utilizing the second nonlinear adaptive controllerν=0.01, μ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 6
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the second nonlinear adaptive controller μ=0.001, ν=0.01, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures7 and 8 present the behavior of the functions η1 and η2 that act in the second control law. Indeed, Figure 7 shows that the function η1 grows as the value of α increases from 1 to 4. Unlike Figure 7, Figure 8 shows that the function η2 decreases as the value of α increases from 1 to 4. Also, it should be noted that the choice of the feedback gains r1 and r2 has an impact on the values of η1 and η2 and consequently on the stability of the solution. One can observe that fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero, as shown in estimate (42).Figure 7
η
1 over time for various values of α when applying the second nonlinear adaptive controller.Figure 8
η
2 over time for various values of α when applying the second nonlinear adaptive controller.


## 3.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(8)) with the controls f1t and f2t as presented by equations (23) and (24) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.It follows from equations (23) and (24) that the preknowledge of the parameter γ1 is not required in the second adaptive control law proposed in Theorem 2. However, the value of γ1 is set to be 1 in the MGKdVB equation (i.e., equation (1)) for simulation purposes. The kinematic viscosity ν is chosen to be 0.01 and the dynamic viscosity µ is set to be 0.001. The parameter γ2 is chosen to be 0.0005. Moreover, r1 and r2 are chosen to equal 0.2. The initial values of η1 and η2 are set to be such as η10=η20=1 and ux,0=sinπx. Figures 5(a)–5(d) show a 3D view of the solution of the MGKdVB equation under the action of the control law presented in Theorem 2 for different values of α. Moreover, the L2-norm of these solutions is presented in Figure 6. It can be concluded from the figures that as α increases, the convergence rate of the solution to the steady state solution becomes smaller. This is due to the presence of the nonlinear term uα∂u/∂x which dominates over the other terms in the MGKdVB equation.Figure 5
A 3D view of the dynamics of the MGKdVB equation utilizing the second nonlinear adaptive controllerν=0.01, μ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 6
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the second nonlinear adaptive controller μ=0.001, ν=0.01, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures7 and 8 present the behavior of the functions η1 and η2 that act in the second control law. Indeed, Figure 7 shows that the function η1 grows as the value of α increases from 1 to 4. Unlike Figure 7, Figure 8 shows that the function η2 decreases as the value of α increases from 1 to 4. Also, it should be noted that the choice of the feedback gains r1 and r2 has an impact on the values of η1 and η2 and consequently on the stability of the solution. One can observe that fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero, as shown in estimate (42).Figure 7
η
1 over time for various values of α when applying the second nonlinear adaptive controller.Figure 8
η
2 over time for various values of α when applying the second nonlinear adaptive controller.


## 4. Design of the Third Nonlinear Adaptive Controller

In this section, we present the third nonlinear adaptive boundary control law for the MGKdVB equation. A control scheme for the MGKdVB equation is proposed when the parametersν and γ1 are unknown. The following theorem illustrates the result of our third nonlinear adaptive boundary control law.Theorem 3.
The MGKdVB equation given by equation (1) with the boundary conditions given by equations (2)–(5) and the initial condition u0∈L20,1 is globally exponentially stable in the L20,1, by considering the following nonlinear control law:(44)f1t=0,(45)f2t=−γ2μ∂3u∂x31,t+η1u1,t+η2u2α+11,t,where(46)dη1dt=r1u21,t,r1>0,(47)dη2dt=r2u2α+21,t,r2>0.Proof 3.
Let(48)Vt=12∫01u2x,tdx,be a Lyapunov function candidate. Differentiating Vt and utilizing a similar reasoning as in Section 2, we have the following inequality:(49)dVdt≤−ν∫01∂u∂x2dx+vu1,tf1t−μu1,tf2t+μ2f12t−γ1α+2uα+21,t−γ2u1,t∂3u∂x31,t+γ2f1tf2t.
Using the control law given by equations (44) and (45), the above inequality yields:(50)dVdt≤−ν∫01∂u∂x2dx−μη1u21,t−μη2u2α+21,t−γ1α+2uα+21,t.
Thanks to Poincaré’s inequality, we obtain(51)dVdt≤−ν∫01u2dx−μη1u21,t−μη2u2α+21,t−γ1α+2uα+21,t.
Arguing as for inequality (34), the last term in (51) can be bounded as follows:(52)−γ1α+2uα+21,t≤γ12α+2u2α+21,t+γ12α+2u21,t.
Thus, inequality (51) gives(53)dVdt≤−ν∫01u2dx−μη1u21,t−μη2u2α+21,t+γ12α+2u2α+21,t+γ12α+2u21,t,that is,(54)dVdt≤−ν∫01u2dx−μη1−γ12α+2u21,t−μη2+γ12α+2u2α+21,t.
Now, let us define a nonnegative energy functionEt as follows:(55)Et=Vt+12μr1μη1−a−γ12α+22+12μr2μη2−b−γ12α+22,where a,b>0.
DifferentiatingEt with respect to time, we obtain(56)dEdt=dVdt+1r1dη1dtμη1−a−γ12α+2+1r2dη2dtμη2−b−γ12α+2.
Using inequality (54) and equations (46) and (47), we have(57)dEdt≤−ν∫01u2dx+μη1−a−γ12α+2u21,t−μη1−γ12α+2u21,t+μη2−b−γ12α+2u2α+21,t−μη2−γ12α+2u2α+21,t.
Inequality (57) leads to(58)dEdt≤−ν∫01u2dx−au21,t−bu2α+21,t.
Noting that−au21,t−bu2α+21,t≤0, inequality (58) reduces to(59)dEdt≤−2νVt.
Hence,Et≤E0,∀t≥0. One can conclude that η1 and η2 are bounded functions for all t>0 and thus u1,t∈L20,∞∩L2α+20,∞ when α is a positive integer.
From inequality (54), we have(60)dVdt≤−2νVt−μη1−γ12α+2u21,t−μη2−γ12α+2u2α+21,t.
Exploiting the Gronwall–Bellman’s inequality, we have(61)Vt≤V0e−2νt+C∫0te−2νt−τu21,τ+u2α+21,τdτ,where C=maxsupt>0μη1t−γ1/2α+2,supt>0μη2t−γ1/2α+2.
Using the fact thatu1,t∈L20,∞∩L2α+20,∞, we can show that Vt tends to zero as t⟶∞. Hence, ux,t exponentially tends to zero as t⟶∞.


### 4.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) in the presence of the control f1t and f2t as presented by equations (59) and (60) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.We deduce from equations (44) and (45) that the third adaptive control law presented in Theorem 3 is independent of the values of the kinematic viscosity ν and the parameter γ1 which are assumed to be unknown. For simulation purposes, the values of ν and γ1 are chosen to be 0.01 and 1, respectively. The dynamic viscosity µ is set to be 0.001 and the parameter γ2 is chosen to equal 0.0005. Moreover, r1 and r2 are chosen to be 0.2. The initial values of η1 and η2 are set to be such as η10=η20=1, and ux,0=sinπx. Figures 9(a)–9(d) show a 3D view of the solution of the MGKdVB equation utilizing the control law presented in Theorem 3 for different values of α. Figure 10 shows the L2-norm of these solutions. It demonstrates how the nonlinear term uα∂u/∂x affects the dynamics of the MGKdVB equation. When the value of α increases, the solution takes a longer time to approach the steady state solution.Figure 9
A 3D view of the dynamics of the MGKdVB equation utilizing the third nonlinear adaptive controllerμ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 10
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the third nonlinear adaptive controller μ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures11 and 12 depict the behavior of the functions η1 and η2 that are utilized in the third control law. Figure 11 indicates that the function η1 gets bigger as α increases from 1 to 4. Figure 12 indicates that η2 decreases as α increases unlike the behavior of η1. Again similar to the previous two controllers, the choice of the feedback gains r1 and r2 will definitely affect the values of η1 and η2 and therefore the stability of the solution. Fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero as can be deduced from inequality (54).Figure 11
η
1 over time for various values of α when applying the third nonlinear adaptive controller.Figure 12
η
2 over time for various values of α when applying the third nonlinear adaptive controller.


## 4.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) in the presence of the control f1t and f2t as presented by equations (59) and (60) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.We deduce from equations (44) and (45) that the third adaptive control law presented in Theorem 3 is independent of the values of the kinematic viscosity ν and the parameter γ1 which are assumed to be unknown. For simulation purposes, the values of ν and γ1 are chosen to be 0.01 and 1, respectively. The dynamic viscosity µ is set to be 0.001 and the parameter γ2 is chosen to equal 0.0005. Moreover, r1 and r2 are chosen to be 0.2. The initial values of η1 and η2 are set to be such as η10=η20=1, and ux,0=sinπx. Figures 9(a)–9(d) show a 3D view of the solution of the MGKdVB equation utilizing the control law presented in Theorem 3 for different values of α. Figure 10 shows the L2-norm of these solutions. It demonstrates how the nonlinear term uα∂u/∂x affects the dynamics of the MGKdVB equation. When the value of α increases, the solution takes a longer time to approach the steady state solution.Figure 9
A 3D view of the dynamics of the MGKdVB equation utilizing the third nonlinear adaptive controllerμ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 10
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the third nonlinear adaptive controller μ=0.001, γ2=0.0005, r1=0.2, r2=0.2, and u0x=sinπx.Figures11 and 12 depict the behavior of the functions η1 and η2 that are utilized in the third control law. Figure 11 indicates that the function η1 gets bigger as α increases from 1 to 4. Figure 12 indicates that η2 decreases as α increases unlike the behavior of η1. Again similar to the previous two controllers, the choice of the feedback gains r1 and r2 will definitely affect the values of η1 and η2 and therefore the stability of the solution. Fixing the initial conditions, η10 and η20, and increasing the values of r1 and r2 will increase the values of η1 and η2 which in turn speeds up the convergence of the solution to zero as can be deduced from inequality (54).Figure 11
η
1 over time for various values of α when applying the third nonlinear adaptive controller.Figure 12
η
2 over time for various values of α when applying the third nonlinear adaptive controller.


## 5. Design of the Fourth Nonlinear Adaptive Controller

In this section, we present the fourth nonlinear adaptive boundary control law for the MGKdVB equation. A control scheme for the MGKdVB equation is proposed when all the parameters are unknown. The following theorem presents the result of our fourth nonlinear adaptive boundary control law.Theorem 4.
The MGKdVB equation given by equation (1) with the boundary conditions given by equations (2)–(5) and the initial condition u0∈L20,1 is globally exponentially stable in the L20,1-sense, under the presence of the following nonlinear control law:(62)f1t=0,(63)f2t=1+η1∂3u∂x31,t+η2u1,t+η3u2α+11,t,where(64)dη1dt=r1u1,t∂3u∂x31,t,r1>0,(65)dη2dt=r2u21,t,r2>0,(66)dη3dt=r3u2α+21,t,r3>0.Proof 4.
Letting(67)Vt=12∫01u2x,tdx,be a Lyapunov function candidate and differentiating Vt, we obtain as in the previous sections the following inequality:(68)dVdt≤−ν∫01∂u∂x2dx+νu1,tf1t−μu1,tf2t+μ2f12t−γ1α+2uα+21,t−γ2u1,t∂3u∂x31,t+γ2f1tf2t.
Using the control law given by equations (44) and (45), the above inequality yields:(69)dVdt≤−ν∫01∂u∂x2dx−μu1,t∂3u∂x31,t−μη1u1,t∂3u∂x31,t−μη2u21,t−μη3u2α+21,t−γ1α+2uα+21,t−γ2u1,t∂3u∂x31,t,ordVdt≤−ν∫01∂u∂x2dx+u1,t∂3u∂x31,t−μ−μη1−γ2−μη2u21,t−μη3u2α+21,t−γ1α+2uα+21,t.
In light of (34), the last term of inequality (69) can be expanded as follows:(70)−γ1α+2uα+21,t≤γ12α+2u2α+21,t+γ12α+2u21,t.
Thus, inequality (69) can be written as follows:(71)dVdt≤−v∫01∂u∂x2dx−u1,t∂3u∂x31,tμ+μη1+γ2−μη2−γ12α+2u21,t−μη3−γ12α+2u2α+21,t.
Applying Poincaré’s inequality in (71), we obtain(72)dVdt≤−v∫01u2dx−u1,t∂3u∂x31,tμ+μη1+γ2−μη2−γ12α+2u21,t−μη3−γ12α+2u2α+21,t.
Now, we define an energy functionEt as follows:(73)Et=Vt+12μr1μ+μη1+γ22+12μr2μη2−a−γ12α+22+12μr3μη3−b−γ12α+22,where a,b>0.
Taking the derivative ofEt, we obtain(74)dEdt=dVdtt+1r1dη1dtμ+μη1+γ2+1r2dη2dtμη2−a−γ12α+2+1r3dη3dtμη3−b−γ12α+2.
Thanks to inequality (72) and equations (64)–(66), we have(75)dEdt≤−v∫01u2dx−u1,t∂3u∂x31,tμ+μη1+γ2−μη3−γ12α+2u2α+21,t−μη2−γ12α+2u21,t+u1,t∂3u∂x31,tμ+μη1+γ2+u21,tμη2−a−γ12α+2+u2α+21,tμη3−b−γ12α+2.
The above inequality gives(76)dEdt≤−v∫01u2dx−au21,t−bu2α+21,t.
Noting that−au21,t−bu2α+21,t≤0, inequality (76) yields:(77)dEdt≤−v∫01u2dx.
One can conclude thatEt≤E0,∀t≥0. Hence, η1, η2, and η3 are bounded functions for all t>0. Thus, from (64)–(66), we obtain(78)u1,t∈L20,∞∩L2α+10,∞.
From inequality (72), we have(79)dVdt≤−2vVt−u1,t∂3u∂x31,tμ+μη1+γ2−μη2−γ122α+2u21,t−μη3−γ12α+2u2α+21,t.
Exploiting Gronwall–Bellman’s inequality, we obtain(80)dVdt≤V0e−2vt+∫0te−2vt−τu1,τ∂3u∂x31,τμ+μη1+γ2−μη2−γ122α+2u21,τ−μη3−γ12α+2u2α+21,τdτ.
Thus,(81)dVdt≤V0e−2vt+C∫0te−2vt−τu1,τ∂3u∂x31,τ+u21,τ+u2α+21,τdτ,where(82)C=maxsupt>0μη2t−γ12α+2,supt>0μη3t−γ12α+2,supt>0μ+μη1t+γ2.
Hence, using the fact thatu1,t∈L20,∞∩L2α+10,∞, one can show that Vt tends to zero as t⟶∞. Therefore, ux,t exponentially tends to zero as t⟶∞.


### 5.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) in the presence of the control f1t and f2t as presented by equations (62) and (63) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.The fourth adaptive control law designed in equations (62) and (63) is independent of the values of the kinematic viscosity v, the dynamic viscosity μ, γ1, and γ2. Although these parameters are assumed to be unknown, we set in our numerical simulations the values of v, μ, γ1, and γ2 as 0.01, 0.001, 1, and 0.0005, respectively. Moreover, r1, r2, and r3 are chosen to equal 0.2, 0.2, and 0.5, respectively, while the initial values of η1, η2, and η3 are set to be such as η10=η20=η30=1 and ux,0=sinπx. A 3D view of the solution of the MGKdVB equation coupled with the control law presented in Theorem 4 is shown in Figures 13(a)–13(d) for several values of α. Figure 14 shows the L2-norm of these solutions.Figure 13
A 3D view of the dynamics of the MGKdVB equation utilizing the fourth nonlinear adaptive controllerr1=0.2, r2=0.2, r3=0.4, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 14
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the fourth nonlinear adaptive controller r1=0.2, r2=0.2, r3=0.4, and u0x=sinπx.Figures15–17 show the behavior of the functions η1, η2, and η3 that are given in the fourth control law. For instance, it can be concluded from Figure 15 that the function η1 increases as the value of α increases from 1 to 4. Furthermore, Figure 16 indicates that η2 also increases as α increases from 2 to 4. Also, one can notice from Figure 17 that η3 decreases as α increases. Again similar remarks to the previous controllers can be made here; the choice of the feedback gains r1, r2, and r3 will definitely affect the values of η1, η2, and η3 and therefore the stability of the solution. Fixing the initial conditions, η10, η20, and η30, and increasing the values of r1, r2, and r3, will definitely increase the values of η1, η2, and η3 which in turn increase the convergence rate of the solution to zero, as can be shown from inequality (69).Figure 15
η
1 over time for various values of α when applying the fourth nonlinear adaptive controller.Figure 16
η
2 over time for various values of α when applying the fourth nonlinear adaptive controller.Figure 17
η
3 over time for various values of α when applying the fourth nonlinear adaptive controller.Obviously, the parameters of the controllerf2t are η1, η2, η3, and α. Fixing α, the parameters η1, η2, and η3 can be computed from the uncoupled system of ODE equations (64)–(66) once the value at the right boundary u1,t and ∂3u/∂x31,t are known, the control gains r1, r2, and r3 are chosen, and the initial conditions η10, η20, and η30 are fixed. Then, the controller f2t can be simply computed using equation (63). This clearly shows that the computations in the fourth adaptive boundary controller are also reasonable.Remark 1.
The design of the four adaptive boundary controllers proposed in Sections2–5 involves low-computational complexity since it is computationally bounded by the time it takes to solve the uncoupled system of first order differential equations.


## 5.1. Numerical Solutions

Numerical solutions for the MGKdVB equation (i.e., equations (1)–(6)) in the presence of the control f1t and f2t as presented by equations (62) and (63) were simulated using COMSOL Multiphysics software. The solutions are computed for α=1,2,3, and 4.The fourth adaptive control law designed in equations (62) and (63) is independent of the values of the kinematic viscosity v, the dynamic viscosity μ, γ1, and γ2. Although these parameters are assumed to be unknown, we set in our numerical simulations the values of v, μ, γ1, and γ2 as 0.01, 0.001, 1, and 0.0005, respectively. Moreover, r1, r2, and r3 are chosen to equal 0.2, 0.2, and 0.5, respectively, while the initial values of η1, η2, and η3 are set to be such as η10=η20=η30=1 and ux,0=sinπx. A 3D view of the solution of the MGKdVB equation coupled with the control law presented in Theorem 4 is shown in Figures 13(a)–13(d) for several values of α. Figure 14 shows the L2-norm of these solutions.Figure 13
A 3D view of the dynamics of the MGKdVB equation utilizing the fourth nonlinear adaptive controllerr1=0.2, r2=0.2, r3=0.4, and u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)Figure 14
TheL2-norm of ux,t, ux,t, over time for various values of α when applying the fourth nonlinear adaptive controller r1=0.2, r2=0.2, r3=0.4, and u0x=sinπx.Figures15–17 show the behavior of the functions η1, η2, and η3 that are given in the fourth control law. For instance, it can be concluded from Figure 15 that the function η1 increases as the value of α increases from 1 to 4. Furthermore, Figure 16 indicates that η2 also increases as α increases from 2 to 4. Also, one can notice from Figure 17 that η3 decreases as α increases. Again similar remarks to the previous controllers can be made here; the choice of the feedback gains r1, r2, and r3 will definitely affect the values of η1, η2, and η3 and therefore the stability of the solution. Fixing the initial conditions, η10, η20, and η30, and increasing the values of r1, r2, and r3, will definitely increase the values of η1, η2, and η3 which in turn increase the convergence rate of the solution to zero, as can be shown from inequality (69).Figure 15
η
1 over time for various values of α when applying the fourth nonlinear adaptive controller.Figure 16
η
2 over time for various values of α when applying the fourth nonlinear adaptive controller.Figure 17
η
3 over time for various values of α when applying the fourth nonlinear adaptive controller.Obviously, the parameters of the controllerf2t are η1, η2, η3, and α. Fixing α, the parameters η1, η2, and η3 can be computed from the uncoupled system of ODE equations (64)–(66) once the value at the right boundary u1,t and ∂3u/∂x31,t are known, the control gains r1, r2, and r3 are chosen, and the initial conditions η10, η20, and η30 are fixed. Then, the controller f2t can be simply computed using equation (63). This clearly shows that the computations in the fourth adaptive boundary controller are also reasonable.Remark 1.
The design of the four adaptive boundary controllers proposed in Sections2–5 involves low-computational complexity since it is computationally bounded by the time it takes to solve the uncoupled system of first order differential equations.


## 6. Comparison of the Nonlinear Adaptive Controllers Proposed in Sections2–5

In this section, we compare the performances of the nonlinear controllers presented in Sections2–5 for several values of α. This comparison illustrates the efficiency of the adaptive control laws when either ν or γ1 is unknown, when both ν and γ1 are unknown, and when all the parameters are unknown.TheL2-norm of the solutions ux,t of the MGKdVB equation will be used for comparison. Figures 18(a)–18(d) show the L2-norm of ux,t over time for several values of α when u0x=sinπx. These figures show that, for all values of α, the solutions of the MGKdVB equation obtained utilizing the first control law given when v is unknown takes a longer time to reach the steady state solution than the other control laws. A careful look at the figures also demonstrates that, for α=1,2, and 3, solutions of the MGKdVB equation obtained utilizing the second, third, and fourth control laws seem to have a similar decay rate. Figure 18(d) indicates that, when α=4, the second control law which is proposed when γ1 is unknown outperforms the other control laws.Figure 18
TheL2-norm of ux,t, ux,t, over time for various values of α; comparison between the behavior of the equation for the fourth nonlinear adaptive controller, u0x=sinπx. (a) α=1; (b) α=2; (c) α=3; (d) α=4.
(a)
(b)
(c)
(d)


## 7. Concluding Remarks

The adaptive boundary stabilization problem of the MGKdVB equation (1) was tackled in this paper. Four different nonlinear adaptive control schemes were introduced for this equation when the physical parameters ν, μ, γ1, and γ2 are unknown and positive and when α is a positive integer. The global exponential stability of the solution in L20,1 has been established analytically and numerically. In addition, the rates of convergence of the four presented controllers were compared.For future work, we should investigate the adaptive and nonadaptive stabilization of the MGKdVB equation (1) under the presence of a time-delay in the boundary control. It would be also interesting to investigate the same problem when equation (1) itself is subject to the effect of the time delay in one of the terms, especially the nonlinear one uα∂u/∂x.



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