Adaptive Stabilization of the Korteweg-de Vries-Burgers Equation with Unknown Dispersion

Abstract

This paper studies the adaptive control problem of the Korteweg-de Vries-Burgers equation. Using the Lyapunov function method, we prove that the closed-loop system including the parameter estimator as a dynamic component is globally 𝐿² stable. Furthermore, we show that the state of the system is regulated to zero by developing an alternative to Barbalat's lemma which cannot be used in the present situation. The closed-loop system is shown to be well posed.

1. Introduction

In this paper, we are concerned with the problem of boundary adaptive control of the KdVB equation: 𝑢_𝑡−𝜀𝑢_𝑥𝑥+𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥𝑢=0,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=𝑢_𝑥()𝑢1,𝑡=0,𝑡>0,_𝑥𝑥(0,𝑡)=𝜑₀𝑢,𝑡>0,_𝑥𝑥(1,𝑡)=𝜑₁𝑢(),𝑡>0,𝑥,0=𝑢⁰(𝑥),0<𝑥<1, (1.1) where the viscosity parameter 𝜀>0. The dispersion parameter 𝛿>0 is unknown, 𝜑₀ and 𝜑₁ are control inputs, and 𝑢⁰(𝑥) is an initial state in an appropriate function space. When 𝜀=0, the KdVB equation becomes the KdV equation; when 𝛿=0, it becomes the Burgers equation.

The problem of control of the Burgers, KdV, and KdVB equations has received extensive attention for several decades [1–7]. In [2], Liu and Krstic obtained the adaptive control of the Burgers equation. Up to now, it seems not to have many discussions on the adaptive control of the KdV and KdVB equation. In this paper we establish a Barbalat-like lemma [8] and use the Lyapunov function method to prove that the system of the KdVB equation is globally 𝐿² stable under the boundary conditions. Using Banach fixed point theorem, we proved the well-posedness of the KdVB equation under the given boundary condition.

The rest of the paper is organized as follows. We present our main results in Section 2. In Section 3, we establish the alternative to Barbalat’s lemma. In Section 4, we prove that the KdVB equation with the previous adaptive boundary feedbacks is globally 𝐿² stable. By the alternative to Barbalat’s lemma, we show the regulation of the solution. In Section 5, we establish the global existence and uniqueness of the solution with help of the Banach fixed point theorem.

We now introduce some notations used throughout the paper. 𝐻^𝑠(0,1) denotes the usual Sobolev space [9] for any 𝑠∈𝑅. For 𝑠≥0, 𝐻^𝑠₀(0,1) denotes the completion of 𝐶^∞₀(0,1) in 𝐻^𝑠(0,1), where 𝐶^∞₀(0,1) denotes the space of all infinitely differentiable functions on (0,1) with compact support in (0,1). The 𝐻^𝑚 norm is defined in the usual way, 𝑚=0,1,2,…. The norm on 𝐿²(0,1) is denoted by ‖⋅‖. It is easy to see that ‖𝑢‖²≤2‖𝑢‖²_𝐻2. (1.2) Let 𝑋 be the Banach space and 𝑇>0. We denote by 𝐶^𝑛([0,𝑇];𝑋) the space of 𝑛 times continuously differentiable functions defined on [0,𝑇] with values in 𝑋. We denote by (⋅,⋅) the scalar product of 𝐿²(0,1).

2. Main Result

For notational convenience, in what follows, we denote 𝜂₀=𝜂₀−1,𝜂6𝛿¹=𝜂₁−1,6𝛿 (2.1) where 𝜂₀and𝜂₁ will be used as estimates of 1/6𝛿.

Consider the system 𝑢_𝑡−𝜀𝑢_𝑥𝑥+𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥𝑢=0,_𝑥(0,𝑡)=𝑢_𝑥()𝑢1,𝑡=0,_𝑥𝑥(0,𝑡)=−𝑘𝑢(0,𝑡)+𝑢(0,𝑡)⁷−𝜂₀𝑢(0,𝑡)+𝑢(0,𝑡)³,𝑢_𝑥𝑥(1,𝑡)=𝑘𝑢(1,𝑡)+𝑢(1,𝑡)⁷+𝜂₁𝑢(1,𝑡)+𝑢(1,𝑡)³,𝜂̇₀=𝛾𝑢(0,𝑡)²+𝑢(0,𝑡)⁴,𝜂̇₁=𝛾𝑢(1,𝑡)²+𝑢(1,𝑡)⁴,𝑢(𝑥,0)=𝑢⁰(𝑥),𝜂₀(0)=𝜂⁰₀,𝜂₁(0)=𝜂⁰₁ (2.2) satisfies the following theorem.

Theorem 2.1. Suppose that 𝑘>0,𝛾>0, the initial condition 𝑢⁰∈𝐻²(0,1), and 𝜂⁰₀≥0, 𝜂⁰₁≥0. If the problem (2.2) has a global solution (𝑢,𝜂₀,𝜂₁), then one has the equilibrium 𝑢(𝑥)≡0,𝜂₀=𝜂₁=0 is globally 𝐿²-stable, that is: ‖𝑢(𝑡)‖²+𝛿𝛾𝜂₀(𝑡)²+𝛿𝛾𝜂₁(𝑡)²≤𝑢⁰²+𝛿𝛾𝜂₀(0)²+𝛿𝛾𝜂₁(0)²,∀𝑡≥0 (2.3) and 𝑢 is regulated to zero in 𝐿² sense: lim_𝑡→∞‖𝑢(𝑡)‖=0. (2.4)

3. The Alternative to Barbalat’s Lemma

Recently, the Barbalat’s lemma has more and more important applications in control theory, especially in the adaptive control theory. It is easy to connect with Lyapunov method to analyze the stability and convergence of the system. In this section, we establish the following alternative to Barbalat’s lemma [8].

Lemma 3.1. Suppose that the function 𝑓(𝑡) defined on [0,∞) satisfies the following conditions:(i)𝑓(𝑡)≥0 for all 𝑡∈[0,∞),(ii)𝑓(𝑡) is differentiable on [0,∞) and there exists a constant 𝑀 such that 𝑓^′(𝑡)≤𝑀,forall𝑡≥0,(iii)∫^∞₀𝑓(𝑡)𝑑𝑡<∞.
Then one has lim_𝑡→∞𝑓(𝑡)=0. (3.1)

Proof. Since𝑓^′(𝑡)≤𝑀,forall𝑡≥0, we have 𝑓(𝑡) is uniformly continuous, and 𝑓₂(𝑡)−𝑓₁(𝑡)≤𝑓₂(𝑡)−𝑓₁(𝑡), (3.2) therefore |𝑓(𝑡)| is uniformly continuous.
Setting 𝐹(𝑡)=^𝑡₀𝑓(𝜏)𝑑𝜏,𝑡≥0, (3.3) we have 𝐹̇(𝑡)=|𝑓(𝑡)|, such that 𝐹̇(𝑡) is also uniformly continuous.
By the standard Barbalat’s lemma, we have lim_𝑡→∞𝐹̇(𝑡)=0, then lim_𝑡→∞𝑓(𝑡)=0. (3.4)

4. Proof of Stabilization

In this section, we prove our main result by the Lyapunov method. Now we present the proof of Theorem 2.1; first we prove the stability of the system, and then we prove the exisetence and uniqueness of the solution.

Step 1. Stability (2.3). We follow the Lyapunov approach: to this end, we introduce the energy function 𝐸=¹₀𝑢²𝑑𝑥 (4.1) and the Lyapunov function 𝛿𝑉=𝐸+𝛾𝜂²₀+𝜂²₁, (4.2) where 𝛾 is a positive constant. Using (1.1) and integrating by parts, we obtain 𝑉̇=2¹₀𝑢𝜀𝑢_𝑥𝑥−𝑢𝑢_𝑥−𝛿𝑢_𝑥𝑥𝑥𝑑𝑥+2𝛿𝛾𝜂₀−1𝜂̇6𝛿⁰+2𝛿𝛾𝜂₁−1𝜂̇6𝛿¹=2𝜀𝑢𝑢_𝑥¹₀−2𝜀¹₀𝑢²_𝑥2𝑑𝑥−3𝑢³¹₀−2𝛿𝑢𝑢_𝑥𝑥¹₀+𝛿𝑢²_𝑥¹₀+2𝛿𝛾𝜂₀−1𝜂̇6𝛿⁰+2𝛿𝛾𝜂₁−1𝜂̇6𝛿¹=−2𝜀¹₀𝑢²_𝑥2𝑑𝑥−3𝑢(1,𝑡)³−𝑢(0,𝑡)³+2𝛿𝑢(0,𝑡)𝜑₀−2𝛿𝑢(1,𝑡)𝜑₁+2𝛿𝛾𝜂₀−1𝜂̇6𝛿⁰+2𝛿𝛾𝜂₁−1𝜂̇6𝛿¹≤2𝛿𝑢(0,𝑡)𝜑₀−2𝛿𝑢(1,𝑡)𝜑₁−2𝜀¹₀𝑢²_𝑥1𝑑𝑥+3𝑢(0,𝑡)⁴+𝑢(0,𝑡)²+𝑢(1,𝑡)⁴+𝑢(1,𝑡)²+2𝛿𝛾𝜂₀−1𝜂̇6𝛿⁰+2𝛿𝛾𝜂₁−1𝜂̇6𝛿¹=2𝛿𝑢(0,𝑡)𝜑₀−2𝛿𝑢(1,𝑡)𝜑₁−2𝜀¹₀𝑢²_𝑥1𝑑𝑥+2𝛿6𝛿−𝜂₀+𝜂₀𝑢(0,𝑡)⁴+𝑢(0,𝑡)²1+2𝛿6𝛿−𝜂₁+𝜂₁𝑢(1,𝑡)⁴+𝑢(1,𝑡)²+2𝛿𝛾𝜂₀−1𝜂̇6𝛿⁰+2𝛿𝛾𝜂₁−1𝜂̇6𝛿¹=−2𝜀¹₀𝑢²_𝑥𝑑𝑥+2𝛿𝑢(0,𝑡)𝜑₀+𝜂₀𝑢(0,𝑡)³()(𝜑+𝑢0,𝑡−2𝛿𝑢1,𝑡)₁−𝜂₁𝑢(1,𝑡)³()1+𝑢1,𝑡+2𝛿6𝛿−𝜂₀𝑢(0,𝑡)⁴+𝑢(0,𝑡)²−𝜂̇₀𝛾1+2𝛿6𝛿−𝜂₁𝑢(1,𝑡)⁴+𝑢(1,𝑡)²−𝜂̇₁𝛾. (4.3) This leads us to select the adaptive feedback control: 𝜂̇₀=𝛾𝑢(0,𝑡)²+𝑢(0,𝑡)⁴,𝜂̇₁=𝛾𝑢(1,𝑡)²+𝑢(1,𝑡)⁴,𝜑₀=−𝑘𝑢(0,𝑡)+𝑢(0,𝑡)⁷−𝜂₀𝑢(0,𝑡)+𝑢(0,𝑡)³,𝜑₁=𝑘𝑢(1,𝑡)+𝑢(1,𝑡)⁷+𝜂₁𝑢(1,𝑡)+𝑢(1,𝑡)³, (4.4) where 𝑘 is any positive constant. By this control, we obtain 𝑉̇≤−2𝜀¹₀𝑢²_𝑥𝑑𝑥−2𝛿𝑘𝑢(0,𝑡)²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸. (4.5) Thus ‖𝑢(𝑡)‖²+𝛿𝛾𝜂₀(𝑡)²+𝛿𝛾𝜂₁(𝑡)²≤𝑢⁰²+𝛿𝛾𝜂₀(0)²+𝛿𝛾𝜂₁(0)²,𝑡≥0. (4.6) This shows that (2.3) holds.
And we have: 2𝜀^∞₀¹₀𝑢²_𝑥𝑑𝑥𝑑𝑡+2𝛿𝑘^∞₀𝑢(0,𝑡)²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸≤𝑢𝑑𝑡₀₂+𝛿𝛾𝜂₀(0)²+𝛿𝛾𝜂₁(0)². (4.7)

Step 2. Regulation (2.4). To prove (2.4), it suffices to verify conditions (ii) and (iii) of Lemma 3.1.
By (1.2) and (4.7), we obtain ^∞₀𝑢(𝑡)²𝑑𝑡≤𝐶𝛿,𝛾,𝑘𝑢⁰²+𝜂₀(0)²+𝜂₁(0)². (4.8) Here and in the sequel, 𝐶=𝐶(𝛿,𝛾,𝑘) denotes a generic positive constant depending on 𝜀,𝛾,𝑘, which may vary from line to line. Thus condition (iii) of Lemma 3.1 is fulfilled. On the other hand, using Young inequality and noting that 𝑎⁴≤𝑎⁸+𝑎²,𝑎²≤𝑎⁶+1, we have 𝑑‖𝑢(𝑡𝑑𝑡)‖²=2¹₀𝑢𝜀𝑢_𝑥𝑥−𝑢𝑢_𝑥−𝛿𝑢_𝑥𝑥𝑥𝑑𝑥=2𝜀𝑢𝑢_𝑥¹₀−2𝜀¹₀𝑢²_𝑥2𝑑𝑥−3𝑢³¹₀−2𝛿𝑢𝑢_𝑥𝑥¹₀+𝛿𝑢²_𝑥¹₀=−2𝛿𝑢(0,𝑡)𝑘𝑢(0,𝑡)+𝑢(0,𝑡)⁷+𝜂₀𝑢(0,𝑡)+𝑢(0,𝑡)³−2𝛿𝑢(1,𝑡)𝑘𝑢(1,𝑡)+𝑢(1,𝑡)⁷+𝜂₁𝑢(1,𝑡)+𝑢(1,𝑡)³−23𝑢(1,𝑡)³−𝑢(0,𝑡)³−2𝜀¹₀𝑢²_𝑥𝑢()𝑑𝑥≤−2𝛿𝑘0,𝑡²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸−2𝛿𝜂₀𝑢(0,𝑡)²+𝑢(0,𝑡)⁴+𝜂₁𝑢(1,𝑡)²+𝑢(1,𝑡)⁴+13𝑢(0,𝑡)²+𝑢(0,𝑡)⁴+𝑢(1,𝑡)²+𝑢(1,𝑡)⁴≤−2𝛿𝑘𝑢(0,𝑡)²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸+𝛿𝑘𝑢(0,𝑡)²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸+𝐶(𝛿,𝑘)𝜂²₀+𝜂²₁+𝛿𝑘𝑢(0,𝑡)²+𝑢(0,𝑡)⁸+𝑢(1,𝑡)²+𝑢(1,𝑡)⁸()()+𝐶𝛿,𝑘≤𝐶𝛿,𝑘1+𝜂²₀+𝜂²₁ (4.9) which, combining with (4.6), implies condition (ii) of Lemma 3.1.
Hence lim_𝑡→∞‖𝑢(𝑡)‖=0. (4.10)

5. Well-Posedness

In this section, we use the Banach fixed point theorem to prove that the problem (2.2) is well posed. To this end, for any constant 𝑇>0, we first consider the following linear boundary value problem for any fixed 𝜆∈𝐶([0,1]×[0,𝑇]):𝑢_𝑡−𝜀𝑢_𝑥𝑥+𝛿𝑢_𝑥𝑥𝑥+𝜆𝑢_𝑥𝑢=0,_𝑥(0,𝑡)=𝑢_𝑥()𝑢1,𝑡=0,_𝑥𝑥(0,𝑡)=−𝑘𝜆(0,𝑡)+𝜆(0,𝑡)⁷−𝜂₀𝜆(0,𝑡)+𝜆(0,𝑡)³,𝑢_𝑥𝑥(1,𝑡)=𝑘𝜆(1,𝑡)+𝜆(1,𝑡)⁷+𝜂₁𝜆(1,𝑡)+𝜆(1,𝑡)³,𝜂̇₀=𝛾𝜆(0,𝑡)²+𝜆(0,𝑡)⁴,𝜂̇₁=𝛾𝜆(1,𝑡)²+𝜆(1,𝑡)⁴,𝑢(𝑥,0)=𝑢⁰(𝑥),𝜂₀(0)=𝜂⁰₀,𝜂₁(0)=𝜂⁰₁. (5.1) We introduce some notation as follows.

If 𝑢₁ and 𝑢₂ are two solutions of the problem (5.1) corresponding to 𝜆₁ and 𝜆₂ respectively, we set 𝑧=𝑢₁−𝑢₂,𝜔=𝜆₁−𝜆₂. (5.2) For a general function 𝜙=𝜙(𝑥,𝑡), we set 𝜙_∞=max_{0≤𝑥≤10≤𝑡≤𝑇}𝜙(𝐻𝑥,𝑡)₂₀(0,1)=𝑢∈𝐻²(0,1)∶𝑢_𝑥(0,𝑡)=𝑢_𝑥(1,𝑡)=0. (5.3)

Lemma 5.1. If 𝜆∈𝐶([0,𝑇];𝐶[0,1]) and the initial data 𝑢⁰(𝑥)∈𝐻²₀(0,1), then the problem (5.1) has a unique weak solution 𝑢 satisfying 𝑢∈𝐶([0,𝑇];𝐻²₀(0,1)). Moreover, one has 𝑧_𝑥(𝑡)²≤𝑇‖𝜔‖²_∞𝐹𝜆_1∞,𝜆_2∞,𝑧⁰_𝑥, (5.4) where 𝐹(𝑥₁,𝑥₂,𝑥₃) is a positive continuous function.

Proof. We use the standard Galerkin method. This method relies on a number of prior estimates which can be usually obtained by using Gronwall’s inequality.
Step  1. Transformation to a Homogeneous Problem. In order to use the Galerkin method, we transform the problem (5.1) into a homogenous boundary value problem. We first assume that 𝜆 is infinitely differentiable with respect to both 𝑥 and 𝑡. Set 1𝜓=6(1−𝑥)³−𝑘𝜆(0,𝑡)+𝜆(0,𝑡)⁷−𝜂₀𝜆(0,𝑡)+𝜆(0,𝑡)³+16𝑥³𝑘𝜆(1,𝑡)+𝜆(1,𝑡)⁷+𝜂₁𝜆(1,𝑡)+𝜆(1,𝑡)³,𝑣=𝑢−𝜓. (5.5) Then it is clear that 𝑣 satisfies the following equation: 𝑣_𝑡−𝜀𝑣_𝑥𝑥+𝛿𝑣_𝑥𝑥𝑥+𝜆𝑣_𝑥𝑣=𝑓,_𝑥(0,𝑡)=𝑣_𝑥(1,𝑡)=𝑣_𝑥𝑥(0,𝑡)=𝑣_𝑥𝑥()𝑣()1,𝑡=0,𝑥,0=𝑣⁰(𝑥), (5.6) where 𝑓(𝑥,𝑡)=−𝜓_𝑡+𝜀𝜓_𝑥𝑥−𝛿𝜓_𝑥𝑥𝑥−𝜆𝜓_𝑥,𝑣⁰(𝑥)=𝑢⁰(𝑥)−𝜓(0)∈𝐻²₀(0,1). (5.7)
Step  2. Approximate Problem. Set 𝐻³₀(0,1)=𝜑∈𝐻³(0,1)∶𝜑_𝑥(0)=𝜑_𝑥(1)=𝜑_𝑥𝑥(0)=𝜑_𝑥𝑥(1)=0. (5.8) Let {𝜑_𝑖}^∞_𝑖=1 be an orthonormal basis in 𝐻²₀(0,1) such that each function 𝜑_𝑖 is in 𝐻³₀(0,1). Since 𝑣⁰(𝑥)∈𝐻²₀(0,1), 𝑣⁰(𝑥) can be expanded as 𝑣⁰(𝑥)=^∞_𝑗=1𝑏_𝑗𝜑_𝑗(𝑥). (5.9) Set 𝑣_𝑛(𝑥,𝑡)=^𝑛_𝑗=1𝑎_𝑗(𝑡)𝜑_𝑗(𝑥),𝑣⁰_𝑛(𝑥)=^𝑛_𝑗=1𝑏_𝑗𝜑_𝑗(𝑥). (5.10) Then we have 𝑣⁰−𝑣⁰_𝑛²_𝐻2=^∞_𝑗=𝑛+1𝑏²_𝑗→0,as𝑛→∞. (5.11) Consider the following approximate problem: 𝑣_𝑛𝑡−𝜀𝑣_𝑛𝑥𝑥+𝛿𝑣_{𝑛𝑥𝑥𝑥}+𝜆𝑣_𝑛𝑥𝑣()=𝑓,𝑥,0=𝑣⁰(𝑥). (5.12)(5.13) Multiplying (5.12) by 𝜑_𝑖 and integrating from 0 to 1, we obtain ^𝑛_𝑗=1𝛼_𝑖𝑗𝑎̇_𝑗(𝑡)+𝛽_𝑖𝑗(𝑡)𝑎_𝑗(𝑡)=𝛾_𝑖(𝑡),𝑎_𝑖(0)=𝑏_𝑖,𝑖=1,…,𝑛, (5.14) where 𝛼_𝑖𝑗=𝜑_𝑗,𝜑_𝑖,𝛽_𝑖𝑗=−𝜀𝜑_𝑗𝑥𝑥,𝜑_𝑖+𝛿𝜑_{𝑗𝑥𝑥𝑥},𝜑_𝑖+𝜆𝜑_𝑗𝑥,𝜑_𝑖,𝛾_𝑖(𝑡)=𝑓,𝜑_𝑖. (5.15) Since the matrix 𝛼_𝑖𝑗 is nonsingular, by the classical theory of ordinary differential equations, the linear problem (5.14) has a unique continuously differential solution on [0,𝑇]. Therefore, the approximate problems (5.12) and (5.13) have a unique solution 𝑣_𝑛 with 𝑣_𝑛∈𝐶¹([0,𝑇],𝐻³₀).
Step  3. A Priori Estimate on 𝑣_𝑛. In what follows, we denote by 𝑐=𝑐(𝜀,𝛿,𝑇,𝜆,𝑣⁰) a positive generic constant, independent of 𝑛, which may vary from line to line. Multiplying (5.12) by 𝑣_𝑛 and integrating from 0 to 1, we have 𝑑𝑣𝑑𝑡^𝑛(𝑡)²=2¹₀𝑣_𝑛𝜀𝑣_𝑛𝑥𝑥−𝛿𝑣_{𝑛𝑥𝑥𝑥}−𝜆𝑣_𝑛𝑥+𝑓𝑑𝑥=2𝜀𝑣_𝑛𝑣_𝑛𝑥¹₀−2𝜀¹₀𝑣²_𝑛𝑥𝑑𝑥−2𝛿𝑣_𝑛𝑣_𝑛𝑥𝑥¹₀+2𝛿𝑣²_𝑛𝑥¹₀−2𝜆¹₀𝑣_𝑛𝑣_𝑛𝑥𝑑𝑥+2¹₀𝑓𝑣_𝑛𝑣𝑑𝑥=−2𝜀_𝑛𝑥(𝑡)²−2𝜆𝑣_𝑛𝑥,𝑣_𝑛+2𝑓,𝑣_𝑛≤−2𝜀𝑣_𝑛𝑥(𝑡)²+𝜀𝑣_𝑛𝑥(𝑡)²+𝜆𝑣_𝑛(𝑡)²𝜀+𝑓(𝑡)²+𝑣_𝑛(𝑡)²≤𝑓(𝑡)²+‖𝜆‖1+^2∞𝜀𝑣_𝑛(𝑡)². (5.16) By Gronwall-Bellman’s inequality and using inequality ‖𝑣⁰_𝑛−𝑣⁰‖≤‖𝑣⁰_𝑛−𝑣⁰‖_𝐻2, 𝑣_𝑛(𝑡)²≤𝑣⁰_𝑛²+^𝑇₀𝑓(𝑡)²‖𝜆‖𝑑𝑡exp1+^2∞𝜀𝑡≤2𝑣⁰_𝑛²+2𝑣⁰_𝑛−𝑣⁰²_𝐻2+^𝑇₀𝑓(𝑡)²‖𝜆‖𝑑𝑡exp1+^2∞𝜀𝑡≤𝑐. (5.17)
Step  4. A Priori Estimate on 𝑣_𝑛𝑥. Multiplying (5.12) by 𝑣_𝑛𝑥𝑥 and integrating from 0 to 1, we have 𝑑𝑣𝑑𝑡^𝑛𝑥(𝑡)²=2¹₀𝑣_𝑛𝑥𝑥−𝜀𝑣_𝑛𝑥𝑥+𝛿𝑣_{𝑛𝑥𝑥𝑥}+𝜆𝑣_𝑛𝑥𝑣−𝑓𝑑𝑥≤−2𝜀_𝑛𝑥𝑥(𝑡)²+𝜀𝑣_𝑛𝑥𝑥(𝑡)²+𝜆𝑣_𝑛𝑥(𝑡)²𝜀+𝑓(𝑡)²𝜀+𝜀𝑣_𝑛𝑥𝑥(𝑡)²≤𝑓(𝑡)²𝜀+‖𝜆‖²_∞𝑣_𝑛𝑥(𝑡)²𝜀, (5.18) which implies 𝑣_𝑛𝑥(𝑡)²≤2𝑣⁰_𝑛𝑥²+1𝜀^𝑇₀𝑓(𝑡)²‖𝜆‖𝑑𝑡exp^2∞𝜀≤𝑐. (5.19)
Step  5. Existence and Uniqueness. By (5.17) and (5.19), we deduce that 𝑣_𝑛 and 𝑣_𝑛𝑥 are bounded in 𝐿^∞([0,𝑇],𝐿²(0,1)). Consequently, there exists a subsequence of {𝑣_𝑛} denoted by {𝑣_𝑛𝑘} such that 𝑣_𝑛𝑘 converges to a function 𝑣 in the weak-star topology of 𝐿^∞([0,𝑇],𝐿²(0,1)).
It is easy to see that 𝑣 is the weak solution of the problem (5.6) satisfying 𝑣∈𝐶[0,𝑇];𝐻²₀(0,1). (5.20) Therefore, for any differentiable function 𝜆, the problem (5.1) has a unique weak solution: 𝑢=𝑣+𝜓∈𝐶[0,𝑇];𝐻²₀(0,1). (5.21) The estimate (5.4) can be proved in the same way as in the proof of (5.19). Finally, the continuous differentiability assumption on 𝜆 can be relaxed by using (5.4). This completes the proof.

Theorem 5.2. For the initial data 𝑢⁰(𝑥)∈𝐻²₀(0,1), the problem (2.2) has a unique solution 𝑢 satisfying 𝑢∈𝐶([0,∞];𝐻²₀(0,1)).

Proof. Let 𝑇>0 be any constant and 𝜆∈𝐶([0,1]×[0,𝑇]). By Lemma 5.1, the problem (5.1) has a unique solution 𝑢 with 𝑢∈𝐶([0,1]×[0,𝑇]). Hence we define the nonlinear mapping 𝐴 by 𝐴𝜆=𝑢. (5.22)
Set 𝑅=2‖𝑢⁰_𝑥‖. By (5.4), we deduce that if 𝑇 is small enough, then 𝐴 maps 𝐵(0,𝑅) into 𝐵(0,𝑅) and 𝐴 is a contractive mapping, where 𝐵(0,𝑅)=𝜆∈𝐶([0,1]×[0,𝑇])∶‖𝜆‖_∞≤𝑅. (5.23) Therefore, by the Banach fixed point theorem, 𝐴 has a unique fixed point 𝑢^∗. So the problem (2.2) has a unique solution 𝑢^∗ for 𝑇 small enough. Since 𝑢^∗ is also the solution of the linear problem (5.1), by Lemma 5.1, we deduce that 𝑢^∗∈𝐶([0,𝑇];𝐻²₀(0,1)). By Theorem 2.1, the solution 𝑢^∗ can be continued to the whole real line, that is, 𝑢^∗∈𝐶([0,∞];𝐻²₀(0,1)). This completes the proof.

6. Conclusion

We have shown the adaptive boundary stabilization of the KdV-Burgers’ equation by nonlinear boundary control. It seems yet not to be discussed. As for the adaptive case, we establish an extension to Barbalat’s lemma to show the 𝐿² regulation of the solution of the KdV-Burgers’ equation. Also, we prove that the solution of the KdV-Burgers equation is well posed. However, we want to get the 𝐻¹ stability which implies boundedness of the solution. We will solve this problem in the future.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (no. 11171135, 71073072) and the Nature Science Foundation of Jiangsu (no. BK2010329) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003).