Abstract and Applied Analysis

Volume 2012 (2012), Article ID 531659, 20 pages

doi:10.1155/2012/531659

Research Article

Exact Null Controllability of KdV-Burgers Equation with Memory Effect Systems

Rajagounder Ravi Kumar,¹ Kil To Chong,² and Jong Ho Park³

¹CASY, DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

²Advanced Electronics and Information Research Center, Department of Electronics and Information Technology, Chonbuk National University, Republic of Korea

³Department of Electrical and Electronics Engineering, Seonam University, Republic of Korea

Received 12 August 2012; Accepted 7 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Rajagounder Ravi Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with exact null controllability analysis of nonlinear KdV-Burgers equation with memory. The proposed approach relies upon regression tool to prove controllability property of linearized KdV-Burgers equation via Carleman estimates. The control is distributed along with subdomain and the external control acts on the key role of observability inequality with memory. This description finally showed the exact null controllability guaranteeing the stability.

1. Introduction

In recent years there has been rapidly increasing interest in mathematical studies of dynamical and statistical property of nonlinear fields described by the Burgers equation (see [1–4]) and it has been motivated by several developments. As Burgers [5] noticed, the Burgers equation is a convenient analytical model for the physical turbulence, which is simultaneously taken into an account of two competing mechanisms: to determine properties of the strong hydrodynamic turbulence: the interior nonlinearity and viscosity.

Moreover, the quantitative description in many physical processes leads to the Burgers equation. One example here is an intense acoustical noise, such as the jet noise [6], where knowledge of dynamical and statistical properties of the Burgers turbulence can be directly applied to an analysis of nonlinear distortions. Another phenomenon is adequately for the nonlinear evolution of gravitational instability and the related characteristic of large-scale cellular structures (see [1, 7]).

Many problems have been modelled on nonlinear Burgers equations. There has been an enormous on-going research and it is directly investigating some nonlinear effects. For example, the simpler model equation with higher dimension, which encapsulates to essential features of the problem. But it is impossible to solve it directly in a higher dimension. After few years, these issues were carried out by Fernández-Cara et al. [8] and it has been discussed via Korteweg-de Vries (KdV). The KdV equation is a prototype of such a model also describing the competition between nonlinear and disperse effects in water waves.

The KdV equations are not directly related to the disperse waves, the density fields in Burgers turbulence, and it is rather difficult to do in higher dimensions. Although the KdV equation of analytical model is not a convenient and also nonefficient model of the strong turbulence, it holds at least one serious drawback from the viewpoint of physical applications (see [6]). Namely, it does not take into account pressure forces that could lead to smoothing of singularity, which appear in density fields driven by Burgerian velocity.

In this context we will consider the KdV-Burgers equations with memory, which is described by the density fields, pressure forces as well as the viscosity and its dispersion, and so forth. In various fields of physics and engineering, problems have been modelled by partial differential equation with memory (see [9]). It is essential to take an account of the effect of past history.

We consider a typical form of KdV-Burgers equation with memory; it is essential and suitable for the above physical situation:

where is a bounded domain with smooth boundary . The null controllability of linear parabolic equation without memory kernels has been extensively studied by several authors (see [8, 10–12], and references therein). Barbu and Iannelli [9] discussed the approximate controllability for the similar type of equation with memory. Rosier [13] studied the exact boundary controllability of linear KdV equation with the half-line and Russell and Zhang[14] also discussed the exact controllability and stabilizability of KdV equation. Sakthivel [15] has been proved the asymptotic stability of KdV-Burgers via Lyapunov function technique by using and norms with domain of . But the domain of attraction always containing is impossible; therefore, the problems of KdV-Burgers equations have not been fully investigated, and it is therefore still a challenging problem.

Some nonlinear control systems are modelled by partial differential equations; it will be a strong control. Unfortunately in some cases, even if the linearized control system around the equilibrium is not controllable. But the linear control system around the memory kernel is always controllable. This method has been introduced in Temam [16], Kofman et al. [7], and Coron [17] where they have discussed about the controllability depending on the relationship between the pressure forces and the magnitude of initial velocity in gas models. The distributed control is described by the behavior of water waves in a shallow channel, compressible gas and strong hydrodynamic turbulence, and so forth. It will be indicated as in (1.2) that appears as in control function.

We consider the linearized control KDV-Burgers equation with memory effects of dirichlet boundary conditions, which helps to develop (1.1) (as based on the existing result Fursikov and Imanuvilov [12]):

where is a solitary wave of dispersion at the point and time , is an initial temperature distribution, the integral kernel is called conservation of mass or volume and has support in , where , is a forcing term such as pressure force, is a characteristic function of the subset , and is a control over an arbitrary sub domain of the domain .

The paper is organized as follows. Section 2 gives some basic assumptions and formation of the problem. Section 3 gives the proof of the Carleman estimate and observability result. Section 4 gives exact null controllability result as based on a unique continuation result of adjoined problems and by using the observability inequality. A conclusion will be given in Section 5.

Notation. We describe some function spaces which will be useful to formulate our results. For each positive integer and or , denote as usual by the sobolev space of functions in whose weak derivatives are of order less than or equal to . When instead of , we will write . Besides, we need the space of all equivalence classes of square integrable functions from to . The spaces and are analogously defined. Moreover, we set (see [18])

(i); (ii)(iii); (iv):= the Sobolev spaces of functions in whose weak derivatives are of order less than or equal to , where is a positive integer; (v):= the space of all equivalence classes of square integrable functions from to ;(vi).

2. Assumptions and Main Results

A linearized control system (1.2) is a weak solution model of the control problem (1.1) (it is supporting to prove the stabilization), then system (1.2) can be written as

where . The kernel is smooth and has support in where . Moreover, there are plenty of works related to exact and approximate controllability properties of the parabolic system of type (1.2) without the memory effects (see [11]). In this paper we constructed a system with memory; finally by using Hölder’s inequality and changing the order of integration the memory term will be observed by .

The following lemma is a fundamental tool to proving controllability results.

Lemma 2.1. Let be an open bounded and connected subset of the boundary in class , an arbitrary subsets of , and such that

For each such that

and the space is a solution of (1.2).

Proof. The proof is similar to Theorem from Barbu [10] and hence it will be omitted.

Theorem 2.2. Let be as in Lemma 2.1. System (1.2) is exactly null controllable for each , if there exists and such that

(The proof of the theorem is given in Section 4).

3. Observability Results and Carleman Estimates

In this section we will derive the observability result via the Carleman estimate of adjoint system (2.1).

To prove the Carleman estimate, the weight functions are necessary. As based on Fernández-Cara et al. [8] and Rosier [13] the weight functions and are assumed as

where .

Suppose that is an adjoint state variable of system (2.1), then it has some solutions. Therefore,

where and .

Now we will state the theorem to prove system (3.2) of solution ,whichwill be a solution for (2.1). The question is how the system (3.2) of solution will be a solution for (2.1). As based on Carleman estimate (assume the weight functions as (3.1)), we can prove that will be a solution for (2.1) (see [12]).

Theorem 3.1 (Carleman estimate). The function defined as in (3.1), the kernel has support in (where ), there exist positive constants , and such that the following inequalities hold when a solution of (3.2):

Proof. We set (where are positive parameters) and satisfies (3.2) (for our convenient ), then

As based on Barbu [10] we can introduce the operators

It follows from (3.4), (3.5), and (3.6) that

where

Taking -norm of both sides of (3.7), we obtain

Let us analyze the scalar product in (3.9) as

where is an integral product of th term in and th term in .

Now we will simplify the estimate by using Green’s theorem with integration by parts:

Since , some manipulations are dominated by the same parameters and also observed by powers of . Finally we obtain

where are boundary terms:

By the time derivative definition we can write

(where does not depend on ). Using (3.14) for is sufficiently large and observes the same powers, then

Now multiplying (3.7) by and integrating over , we obtain

Therefore,

By using the above calculation in (3.16), it becomes

From the above equation (the right hand side), we can observe the powers of s:

where

Applying Cauchy’s inequality (with ) for ,

Therefore

By using (3.22) in (3.7), we get

Recalling that is sufficiently large and also observing that powers of and in ,

From the above inequality is in ,whichwill be eliminated (because is subset of . To prove as in (term on the right hand side of (3.24)), the truncating function is necessary, so let us define the truncating function with and . Now we will multiply (3.7) by and integrating over , then we have

Since

The boundary term , then

Applying as ,

The term is not in , so it will be eliminated from the above inequalities. To eliminate , we will multiply (3.5) and (3.6) on (see [8, 12]):

In view of the estimates (3.22)–(3.30), we obtain

From (3.31), we will eliminate the memory kernel term (that appears on the right hand side), because it may not contain in . By using Hölder’s inequality and changing the order of integration,

where depends on , and (since , and for all ). Finally reverting to the original variable to complete the proof, let us choose , then

Similarly ,

From (3.32), we can observe the kernel term as

If is sufficiently large, then there exists a constant such that the above integral terms have been absorbed on (the right hand side of (3.31)). From the identifier (3.33)–(3.35), we can conform all the terms involving in and we obtain (3.3).

Lemma 3.2. Suppose that Theorem 3.1 is satisfied. If there exist positive constants (independent of ) and will be a solution of (2.1), such that the following inequality holds:

Proof. We multiply (3.2) by and integrate on (using Cauchy’s inequality, when ), we obtain

Define

Therefore,

Integrating (3.39) on ,

Therefore,

Now we fix and such as , then integrating the above inequality:

As based on (3.32), the kernel has been modified as

Using (3.43) in (3.42), then

Since

and the Carleman estimate (3.3) using in (3.44), we obtain

4. Controllability Results

Now we are ready to give the proof of Theorem 2.2 result, which will be a main part of our work.

Proof of Theorem 2.2. Let us fix and . For every , the penalized formula

where the functional is (see [9, 11])

Here is a solution of (1.2), which is associated with the control . Since is a continuous strictly convex functional in , then has a unique solution (for any ). If (as ) is a null controllability solution of the system (1.1), then due to penalization property , the limit exists in an appropriate norm. As based on the pontryagin maximum principle, the maximal condition on the control is

and is a solution for

Now we multiply (4.4) by (where ), (1.2) by and integrate on ,

Applying the observability inequality (3.36), we have

Using the Carleman estimate (3.3) and Lemma 2.1 condition,

where is a positive constant that is independent of and . By virtue of (4.7)–(4.9), we obtain

By the condition (4.10), we can observe

Since is bounded in , there exists a subsequence denoted by such that

Clearly , letting in (4.11), then we have . This completes the proof of Theorem 2.2.

5. Conclusions

In this paper, we investigated the null controllability result for linearized KdV-Burgers equation with memory. It is suggested as a model which permits an analytical treatment of the Carleman estimate to prove the stability. The observability inequality has observed the null controllability, and the kernel term expression been observed in . It grants the stability, so our model could be heuristically useful in the studies of nonlinear water waves in a shallow channel, compressible gas, strong hydrodynamic turbulence as well as pressure force, and so forth, [1, 2].

Acknowledgments

This research was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (no. 2012-038978) and research funds of Chonbuk National University in 2011.

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