Research Article | Open Access
Exact Null Controllability of KdV-Burgers Equation with Memory Effect Systems
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.
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  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 , 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.  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 ). 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 ). 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  discussed the approximate controllability for the similar type of equation with memory. Rosier  studied the exact boundary controllability of linear KdV equation with the half-line and Russell and Zhang also discussed the exact controllability and stabilizability of KdV equation. Sakthivel  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 , Kofman et al. , and Coron  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 ): 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 ) (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 ). 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  and hence it will be omitted.
(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).
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 ).
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  we can introduce the operators
It follows from (3.4), (3.5), and (3.6) that
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).
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