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

Authors: Rajagounder Ravi Kumar; Kil To Chong; Jong Ho Park
Journal: Abstract and Applied Analysis
 (2012)

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## 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 subdomainω⊂Ω 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.


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## Body

## 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:(1.1)yt-yxx+yyx+yxxx+∫0t‍k(t,τ)yxxdτ=χωu(t,x)+f(t,x),(t,x)∈(0,T)×Ω,y(0,x)=y0(x),x∈Ω,y(t,x)=0,(t,x)∈(0,T)×∂Ω,
where Ω⊂Rn is a bounded domain with smooth boundary ∂Ω. 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 L2 and L∞ norms with domain of [0,1]. But the domain of attraction always containing [0,1] 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]):
(1.2)yt(t,x)-yxx(t,x)+(a(t,x)y(t,x))x+b(t,x)y(t,x)+yxxx(t,x)+∫0t‍k(t,τ)yxx(τ,x)dτ=χωu(t,x)+f(t,x),(t,x)∈(0,T)×Ω,y(0,x)=y0(x),x∈Ωy(t,x)=0,(t,x)∈(0,T)×∂Ω,
where ∂3y/∂x3=yxxx,y(t,x) is a solitary wave of dispersion at the point x and time t, y0(x) is an initial temperature distribution, the integral kernel k(t,τ) is called conservation of mass or volume and has support in (t1,t2), where 0<t1<t2<T, f(t,x) is a forcing term such as pressure force, χω is a characteristic function of the subset ω⊂Ω, and u(t,x) is a control over an arbitrary sub domain ω of the domain Ω.The paper is organized as follows. Section2 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 m and p>1 or p=∞, denote as usual by Wm,p(Ω) the sobolev space of functions in Lp(Ω) whose weak derivatives are of order less than or equal to m. When p=2 instead of Wm,p(Ω), we will write Hm(Ω). Besides, we need the space L2(0,T:H1(Ω)) of all equivalence classes of square integrable functions from (0,T) to H1(Ω). The spaces L2(0,T:L2(Ω)) and L∞(0,T:L2(Ω)) are analogously defined. Moreover, we set (see [18]) (i)
Ω={(t,x):0<t<T,0<x<∞};(ii)
χω:={1forx∈ω,0forx∈Ω/ω;(iii)
Q:=(0,T)×Ω,Qω:=(0,T)×ω,andΣ:=(0,T)×∂Ω;(iv)
Hm(Ω):= the Sobolev spaces of functions in L2(Ω) whose weak derivatives are of order less than or equal to m, where m is a positive integer;(v)
L2(0,T;H1(Ω)):= the space of all equivalence classes of square integrable functions from (0,T) to H1(Ω);(vi)
Wm,p(Ω)={y(x):∥y∥Wm,p=(∑|α|≤m‍∫Ω‍|Dαy|pdx)1/p<∞}.


## 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
(2.1)yt(t,x)-yxx(t,x)+a(t,x)yx(t,x)+b(t,x)y(t,x)+yxxx(t,x)+∫0t‍k(t,τ)yxx(τ,x)dτ=χωu(t,x)+f(t,x),(t,x)∈(0,T)×Ω,y(0,x)=y0(x),x∈Ω,y(t,x)=0,(t,x)∈(0,T)×∂Ω,
where a∈W1,∞(0,T:L2(Ω)),b∈L∞(Q). The kernel k(t,·) is smooth and has support in (t0,t1) where 0<t0<t1<T. 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 yxx∈ω⊂Ω.The following lemma is a fundamental tool to proving controllability results.Lemma 2.1.
LetΩ be an open bounded and connected subset of the boundary ∂Ω in class C2, ω an arbitrary subsets of Ω, and f∈L2(Q) such that
(2.2)|f(x,t)|≤|f1(x,t)|esφ(x,t)φx(x,t)φxx(x,t),a.e.(x,t)∈Q.
For each y0∈L2(Ω),andu∈L2(Q) such that
(2.3)∥u∥L2(Q)≤C(|y0|2+∥f1∥L2(Q)),
and the space yu∈C1[0,T];H1([0,T]:L2(Ω))∩L2(0,T;H1(Ω)) is a solution of (1.2).Proof.
The proof is similar to Theorem1.3.1 from Barbu [10] and hence it will be omitted.Theorem 2.2.
LetΩ,ω be as in Lemma 2.1. System (1.2) is exactly null controllable for each T>0, if there exists u∈L2(Q) and yu∈C([0,T];L2(Ω))∩L2(0,T;H1(Ω)) such that
(2.4)yu(T,x)=0a.e.x∈Ω,∥u∥L2(Q)≤C(|y0|2+∥f1∥L2(Q)2).(The proof of the theorem is given in Section4).


## 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 φ and ψ are assumed as
(3.1)φ(x,t)=ψ(x)β(t),ψ(x)>0,∀x∈Ω,ψ(x)=0,∀x∈∂Ω,|ψx(x)|>0,ψx(x)<0,ψxx(x)<0,(ψxx(x)·ψx(x))>0,(ψx(x)·(ψxxx(x)))>0,x∈Ω∖ω-,
where β(t)=t(T-t).Suppose thatw is an adjoint state variable of system (2.1), then it has some solutions. Therefore,
(3.2)wt(t,x)+wxx(t,x)+c1wx(t,x)-c2w(t,x)+wxxx(t,x)-∫tT‍k(τ,t)wxx(τ,x)dτ=g(t,x),(t,x)∈Σ,w(T,x)=wT(x),x∈Ω,w(t,x)=0,(t,x)∈(0,T)×∂Ω,
where ∥a(t,x)∥≤c1>0,∥b(t,x)∥≤c2>0,g∈L2(Q) and wT∈L2(Ω).Now we will state the theorem to prove system (3.2) of solution w,whichwill be a solution for (2.1). The question is how the system (3.2) of solution w will be a solution for (2.1). As based on Carleman estimate (assume the weight functions φ,ψ as (3.1)), we can prove that w will be a solution for (2.1) (see [12]).Theorem 3.1 (Carleman estimate).
The functionψ defined as in (3.1), the kernel has support in (t1,t2) (where 0<t1<t2<T), there exist positive constants C, and s≥s0 such that the following inequalities hold when w a solution of (3.2):
(3.3)∫Q‍e2sφ((sφ)-1(|wt|2+|wx|2+|wxx|2+|wxxx|2)+s5φx4φxx|w|2+s3φx2φxx|wx|2+sφ|wxx|2)dxdt≤Ce2sφ(∫Q‍|g|2dxdt+∫Qω‍s4φx2φxx2|w|2dxdt).Proof.
We setw=e-sφq (where s,φ are positive parameters) and q satisfies (3.2) (for our convenient c1=c2=1), then
(3.4)qt+sφtq+s2φx2q-sφxxq-2sφxqx+qxx-s3φx3q+3s2φxφxxq+3s2φx2qx-3sφxxqx-3sφxqxx-sφxxxq+qxxx-q-sφxq+qx-esφ(∫tT‍k(τ,t)e-sφ(τ)q(τ,x))dτ=gesφ.
As based on Barbu [10] we can introduce the operators
(3.5)L1q=qt-sφtq-sφxxq+s2φx2q-s3φx3q+3s2φxφxxq-sφxxxq-q-sφxq,(3.6)L2q=qxx+qxxx-2sφxqx+3s2φx2qx-3sφxxqx-3sφxqxx+qx.
It follows from (3.4), (3.5), and (3.6) that
(3.7)L1q+L2q=hsinQ,
where
(3.8)hs=gesφ+esφ(∫tT‍k(τ,t)(e-sφq)xxdτ).
Taking L2-norm of both sides of (3.7), we obtain
(3.9)∥hs∥L2(Q)2=∥L1q∥L2(Q)2+∥L2q∥L2(Q)2+2〈L1q,L2q〉L2(Q).
Let us analyze the scalar product in (3.9) as
(3.10)〈L1q,L2q〉L2=Eij,1≤i≤9,1≤j≤7,
where Eij is an integral product of ith term in L1q and jth term in L2q.
Now we will simplify the estimate〈L1q,L2q〉L2(Q) by using Green’s theorem with integration by parts:
(3.11)〈L1q,L2q〉L2(Q)=∫Σ‍sφxxx|∂q∂v|2dΣ+32∫Σ‍s2φxφxx|∂q∂v|2dΣ+12∫Σ‍s3φx3|∂q∂v|2dΣ-12∫Σ‍s2φx2|∂q∂v|2dΣ+12∫Σ‍sφxx|∂q∂v|2dΣ+12∫Σ‍sφt|∂q∂v|2dΣ+∫Q‍sφtqx2dxdt-12∫Q‍s(φt)xxq2dxdt+12∫Q‍s(φt)xxxq2dxdt-2∫Q‍s(φt)xqx2dxdt+∫Q‍sφtqx2dxdt+∫Q‍s2(φt)xφxq2dxdt+∫Q‍s2φtφxxq2dxdt+32∫Q‍s2φtφxxq2dxdt+32∫Q‍s2φtφxxq2dxdt+3∫Q‍s3φtφxφxxxq2dxdt+32∫Q‍s2(φt)xq2dxdt+3∫Q‍s2φtφxqx2dxdt-32∫Q‍s2(φt)xxφxq2dxdt+2∫Q‍sφxxqx2dxdt-∫Q‍s2φxx2q2dxdt+32∫Q‍s3φx2φxxxq2dxdt+3∫Q‍s3(φx)φxx2q2dxdt-3∫Q‍s2φxφxxqx2dxdt-2∫Q‍s2φx2qx2dxdt+∫Q‍s2φxx2q2dxdt+2∫Q‍s2φxφxxq2dxdt-∫Q‍s2φxxφxxxq2dxdt-2∫Q‍s4φx3φxxq2dxdt+3∫Q‍s3φx3q2dxdt-9∫Q‍s3φxφxx2qx2dxdt-32∫Q‍s3φx2φxxq2dxdt+2∫Q‍s3φx3qx2dxdt+7∫Q‍s3φxφxxφxxxq2dxdt+32∫Q‍s5φx4φxxq2dxdt+9∫Q‍s4φx2φxx2q2dxdt+32∫Q‍s4φx3φxxxq2dxdt-3∫Q‍s4φx4qx2dxdt+6∫Q‍s4φx3φxxxq2dxdt+18∫Q‍s4φx2φxx2q2dxdt-6∫Q‍s2φxφxxqx2dxdt+3∫Q‍s2φxxφxxx2q2dxdt+32∫Q‍s2φxxφxxxq2dxdt+6∫Q‍s2φxxφxxxqx2dxdt+3∫Q‍s2φxxφxxxqx2dxdt+4∫Q‍s2φxx2q2dxdt+6∫Q‍s3φxφxx2q2dxdt-92∫Q‍s4φx3φxxxq2dxdt-272∫Q‍s4φx2φxxq2dxdt+92∫Q‍s3φxx3q2dxdt+9∫Q‍s3φx2φxxqx2dxdt-152∫Q‍s3φxx3q2dxdt+∫Q‍sφxxxqx2dxdt+2∫Q‍sφxxxqx2dxdt-6∫Q‍s3φx2φxxqx2dxdt-∫Q‍s2φxxφxxxq2dxdt-3∫Q‍s2φxφxxxqx2dxdt.
Since q=φ=0on∂Ω, some manipulations are dominated by the same parameters and also observed by powers of s. Finally we obtain
(3.12)∑i=19‍∑j=17‍Eij=D1+D2+32∫Q‍s5φx4φxxq2dxdt+3∫Q‍s3φx2φxxqx2dxdt,
where D1,D2 are boundary terms:
(3.13)D1=∫Σ‍sφxxx|∂q∂v|2dΣ+32∫Σ‍s2φxφxx|∂q∂v|2dΣ+12∫Σ‍s3φx3|∂q∂v|2dΣ-12∫Σ‍s2φx2|∂q∂v|2dΣ+12∫Σ‍sφxx|∂q∂v|2dΣ+12∫Σ‍sφt|∂q∂v|2dΣ,D2=∫Q‍sφtqx2dxdt-12∫Q‍s(φt)xxq2dxdt+12∫Q‍s(φt)xxxq2dxdt-2∫Q‍s(φt)xqx2dxdt+∫Q‍sφtqx2dxdt+∫Q‍s2(φt)xφxq2dxdt+∫Q‍s2φtφxxq2dxdt+32∫Q‍s2φtφxxq2dxdt+32∫Q‍s2φtφxxq2dxdt+3∫Q‍s3φtφxφxxxq2dxdt+32∫Q‍s2(φt)xq2dxdt+3∫Q‍s2φtφxqx2dxdt-32∫Q‍s2(φt)xxφxq2dxdt+2∫Q‍s2φxφxxq2dxdt-∫Q‍s2φxxφxxxq2dxdt-2∫Q‍s4φx3φxxq2dxdt-32∫Q‍s3φx2φxxq2dxdt+7∫Q‍s3φxφxxφxxxq2dxdt+27∫Q‍s4φx2φxx2q2dxdt+32∫Q‍s4φx3φxxxq2dxdt+6∫Q‍s4φx3φxxxq2dxdt-6∫Q‍s2φxφxxqx2dxdt+3∫Q‍s2φxxφxxx2q2dxdt+32∫Q‍s2φxxφxxxq2dxdt+9∫Q‍s2φxxφxxxqx2dxdt+6∫Q‍s3φxφxx2q2dxdt-92∫Q‍s4φx3φxxxq2dxdt-272∫Q‍s4φx2φxxq2dxdt-∫Q‍s2φxxφxxxq2dxdt-3∫Q‍s2φxφxxxqx2dxdt.
By the time derivative definition φ we can write
(3.14)|φt|≤Cφ2,|φtt|≤Cφ3,
(where C does not depend on s,t,x). Using (3.14) for s is sufficiently large and observes the same powers, then
(3.15)|D2|≤C[∫Q‍s4φx2φxx2|q|2dxdt+∫Q‍s2φxφxx|qx|2dxdt].
Now multiplying (3.7) by sφq|∇ψ|^2 and integrating over Q, we obtain
(3.16)∫Q‍sφq|∇ψ|2L1qdxdt+∫Q‍sφq|∇ψ|2L2qdxdt=∫Q‍sφq|∇ψ|2hsdxdt.
Therefore,
(3.17)∫Q‍sφq|∇ψ|2hsdxdt=∫Q‍sφq|∇ψ|2L1qdxdt+∫Q‍sφ|∇ψ|2qqxxdxdt+∫Q‍sφ|∇ψ|2qqxxxdxdt-2∫Q‍s2φφx|∇ψ|2qqxdxdt+3∫Q‍s3φ|∇x2ψ|2qqxdxdt-3∫Q‍s2φφxx|∇ψ|2qqxdxdt-3∫Q‍s2φφx|∇ψ|2qqxxdxdt,∫Q‍sφq|∇ψ|2L1qdxdt≤12∫Q‍s2φ2|∇ψ|4q2dxdt+2∫Q‍|L1q|2dxdt,∫Q‍sφ|∇ψ|2qqxxdxdt=-∫Q‍sφ|∇ψ|2qx2dxdt+12∫Q‍sφxx|∇ψ|2q2dxdt+∫Q‍sφxΔψ(|∇ψ|)q2dxdt,∫Q‍sφ|∇ψ|2qqxxxdxdt=∫Q‍sφx|∇ψ|2qx2dxdt+12∫Q‍sφxxx|∇ψ|2q2dxdt+∫Q‍sφxx(|∇ψ|)Δψq2dxdt+2∫Q‍sφ|∇ψ|2qx2dxdt-∫Q‍sφx|∇ψ|2q2dxdt-∫Q‍sφ|∇ψ|2qxqxxdxdt,-2∫Q‍s2φxφ|∇ψ|2qqxdxdt=∫Q‍s2φx2|∇ψ|2q2dxdt+∫Q‍s2φxxφ|∇ψ|2q2dxdt+2∫Q‍s2φxφΔψ(|∇ψ|)q2dxdt,3∫Q‍s3φφx2|∇ψ|2qqxdxdt≤3∫Q‍s3φx2φ|∇ψ|4q2dxdt+34∫Q‍s3φφx2qx2dxdt,-3∫Q‍s2φφxx|∇ψ|2qqxdxdt=32∫Q‍s2φxφxx|∇ψ|2q2dxdt+32∫Q‍s2φφxxx|∇ψ|2q2dxdt+3∫Q‍s2φφxx(|∇ψ|)Δψq2dxdt-3∫Q‍s2φxφ|∇ψ|2qqxxdxdt=3∫Q‍s2φφx|∇ψ|2qx2dxdt-3∫Q‍s2φxφxx|∇ψ|2q2dxdt-3∫Q‍s2φx2(|∇ψ|)Δψq2dxdt-32∫Q‍s2φxφxx|∇ψ|2q2dxdt-32∫Q‍s2φφxxx|∇ψ|2q2dxdt-3∫Q‍s2φφxx(|∇ψ|)Δψq2dxdt-∫Q‍s2φx2(|∇ψ|)Δψq2dxdt.
By using the above calculation in (3.16), it becomes
(3.18)-34∫Q‍s3φφx2qx2dxdt=-∫Q‍sφ|∇ψ|2qhsdxdt+∫Q‍s2φ2|∇ψ|4q2dxdt+∫Q‍sφ|∇ψ|2qx2dxdt+14∫Q‍|L1q|2dxdt+2∫Q‍sφΔψ(|∇ψ|)qx2dxdt-∫Q‍sφ|∇ψ|2qxqxxdxdt+3∫Q‍s3φφxφxx|∇ψ|2q2dxdt+32∫Q‍s2φxφxx|∇ψ|2q2dxdt+∫Q‍s2φφxx|∇ψ|2q2dxdt+2∫Q‍s2φφx(|∇ψ|)Δψq2dxdt+32∫Q‍s2φx2∇(|∇ψ|2)q2dxdt+32∫Q‍s2φxφxx|∇ψ|2q2dxdt+32∫Q‍s3φφx2|∇ψ|4q2dxdt+52∫Q‍s2φxφxx|∇ψ|2q2dxdt+12∫Q‍sφx|∇ψ|2q2dxdt+3∫Q‍s2φφxxx|∇ψ|2q2dxdt+3∫Q‍s2φφx|∇ψ|2qx2dxdt-3∫Q‍s2φxφxx|∇ψ|2q2dxdt-32∫Q‍s2φxφxx|∇ψ|2q2dxdt-32∫Q‍s2φφxxx|∇ψ|2q2dxdt-3∫Q‍s2φφxx(|∇ψ|)Δψq2dxdt-3∫Q‍s2φφx∇(|∇ψ|2)qx2dxdt+14∫Q‍sφ|∇ψ|4qx2dxdt.
From the above equation (the right hand side), we can observe the powers of s:
(3.19)-34∫Q‍s3φφx2qx2dxdt=D3+3∫Q‍s3φφx2|∇ψ|4q2dxdt+∫Q‍sφqxx2dxdt,
where
(3.20)D3=∫Q‍sφq|∇ψ|2hsdxdt+∫Q‍sφ|∇ψ|2qL1qdxdt+3∫Q‍s2φφx|∇ψ|2qx2dxdt+32∫Q‍s2φxφxx|∇ψ|2q2dxdt+∫Q‍s2φφxx|∇ψ|2q2dxdt+2∫Q‍s2φφx(|∇ψ|)Δψq2dxdt-3∫Q‍s3φφx2(|∇ψ|)Δψq2dxdt+32∫Q‍s2φxφxx|∇ψ|2q2dxdt+32∫Q‍s2φφxxx|∇ψ|2q2dxdt+4∫Q‍s2φxφxx|∇ψ|2q2dxdt-32∫Q‍s2φxφxx|∇ψ|2q2dxdt-32∫Q‍s2φφxxx|∇ψ|2q2dxdt-3∫Q‍s2φφxx(|∇ψ|)Δψq2dxdt.
Applying Cauchy’s inequality (with ϵ=1) for D3,
(3.21)|D3|≤14∫Q‍|L1q|2dxdt+14∫Q‍|hs|2dxdt+32∫Q‍s2φxφxx|∇ψ|2|q|2dxdt+3∫Q‍s2φφx2|∇ψ|2|qx|2dxdt.
Therefore
(3.22)〈L1q,L2q〉L2(Q)=D2+D3+32∫Q‍s5φx4φxxq2dxdt+34∫Q‍s3φx2φxxqx2dxdt+3∫Q‍s3φφx2|∇ψ|4q2dxdt+∫Q‍sφqxx2dxdt.
By using (3.22) in (3.7), we get
(3.23)∥L1q∥2+∥L2q∥2+3∫Q‍s5φx4φxxq2dxdt+32∫Q‍s3φx2φxxqx2dxdt+2∫Q‍sφqxx2dxdt≤Ce2sφ[∫Q‍|g|2dxdt+∫Q‍|∫tT‍k(τ,t)(e-sφq)xxdτ|2dxdt+∫Q‍s4φx2φxx2|q|2dxdt+∫Q‍s2φxφxx|qx|2dxdt|∫tT‍k(τ,t)(e-sφq)xxdτ|2]+12∫Q‍∥L1q∥2dxdt+6∫Q‍s3φφx2|∇ψ|4|q|2dxdt+6∫Q‍s2φφx|∇ψ|2|qx|2dxdt.
Recalling that s is sufficiently large and also observing that powers of s and |∇ψ|>0 in Ω∖ω,
(3.24)12∥L1q∥2+∥L2q∥2+3∫Q‍s5φx4φxxq2dxdt+32∫Q‍s3φx2φxxqx2dxdt+∫Q‍sφqxx2dxdt≤C[∫Q‍e2sφ|g|2dxdt+∫Q‍e2sφ|∫tT‍k(τ,t)(e-sφq)xxdτ|2dxdt+∫Qω‍s4φx2φxx2|q|2dxdt+∫Qω‍s2φφx|qx|2dxdt|∫tT‍k(τ,t)(e-sφq)xxdτ|2].
From the above inequality qx is in ω0¯,whichwill be eliminated (because ω0¯ is subset of ω0). To prove qx as q in ω0 (term |qx|2 on the right hand side of (3.24)), the truncating function is necessary, so let us define the truncating function ρ∈C0∞(Ω) with ρ(x)=1inω0¯ and ρ(x)=0inΩ∖ω. Now we will multiply (3.7) by ρsφq and integrating over Q, then we have
(3.25)∫Q‍ρsφqL1qdxdt+∫Q‍ρsφqL2qdxdt=∫Q‍ρqsφhsdxdt.
Since
(3.26)ρ∫Q‍sφqqxxdxdt=∫Q‍ρsφxxq2dxdt-∫Q‍ρsφxqx2dxdt,∫Q‍ρsφqqxxxdxdt=∫Q‍ρsφxqx2dxdt-∫Σ‍sφ|∂q∂v|2dΣ+∫Q‍ρsφxqx2dxdt-∫Q‍ρsφxxxq2dxdt-3∫Q‍ρs2φφxqqxxdxdt=3∫Q‍ρs2φφxqx2dxdt-3∫Q‍ρs2φxφxxq2dxdt+32∫Q‍ρs2φφxxxq2dxdt-2∫Q‍ρs2φxqqxdxdt=∫Q‍ρs2φxxφq2dxdt+∫Q‍ρs2φx2q2dxdt-3∫Q‍ρs2φφxxqqxdxdt=32∫Q‍ρs2φxφxxq2dxdt-32∫Q‍ρs2φφxxxq2dxdt3∫Q‍ρs3φφx2qqxdxdt=-32∫Q‍ρs3φx3q2dxdt-3∫Q‍ρs3φφxφxxq2dxdt.
The boundary term -∫Σ‍sφ|∂q/∂v|2dΣ≥0, then
(3.27)∫Q‍ρs2φφx|qx|2dxdt≤14∫Q‍e2sφ|g|2dxdt-32∫Q‍ρs2φφxx|q|2dxdt+32∫Q‍ρs2φφxxx|q|2dxdt+∫Q‍ρs2φφt|q|2dxdt+3∫Q‍ρs2φxφxx|q|2dxdt+∫Q‍ρs2φφxxx|q|2dxdt+∫Q‍ρsφt|q|2dxdt-∫Q‍ρs3φφx2|q|2dxdt+∫Q‍ρs4φφx3|q|2dxdt+2∫Q‍ρs2φ2|q|2dxdt+14∫Q‍ρe2sφ|∫tT‍k(τ,t)(e-sφq)xxdτ|2dxdt.
Applying as ρ(x)=1inω0¯⊂ω,
(3.28)∫Qωs2φφx|qx|2dxdt≤C(∫Qe2sφ|g|2dxdt+∫Qω‍e2sφ|∫tTk(τ,t)(e-sφq)xxdτ|2dxdtc21c21+∫Qωs4φφx3|q|2dxdt|∫tTk(τ,t)(e-sφq)xxdτ|2).
The term s,φ is not in ω, so it will be eliminated from the above inequalities. To eliminate s,φ, we will multiply (3.5) and (3.6) on (sφ)-1 (see [8, 12]):
(3.29)∫Q‍(sφ)-1(|qx|2+|qxx|2+|qxxx|2)dxdt≤C(∫Q‍(sφ)-1|L2q|2dxdt+∫Q‍(sφ)-1s2φxφxx|q|2dxdt)(3.30)∫Q‍(sφ)-1|qt|2dxdt≤C(∫Q‍(sφ)-1|L1q|2dxdt+∫Q‍s4φx2φxx2q2dxdt+∫Q‍s2φt2q2dxdt)
In view of the estimates (3.22)–(3.30), we obtain
(3.31)∫Q‍(sφ)-1(|qt|2+|qx|2+|qxx|2+|qxxx|2)dxdt+∫Q‍(s5φx4φxx|q|2+s3φx2φxx|qx|2+sφ|qxx|2)dxdt≤C({|∫tT‍k(τ,t)(e-sφq)xxdτ|2}{|∫tT‍k(τ,t)(e-sφq)xxdτ|2}∫Q‍e2sφ|g|2+∫Qω‍s4(φx2φxx2+φφx3)|q|2dxdt+∫Qω‍e2sφ{|∫tT‍k(τ,t)(e-sφq)xxdτ|2}dxdt).
From (3.31), we will eliminate the memory kernel term (that appears on the right hand side), because it may not contain in ω. By using Hölder’s inequality and changing the order of integration,
(3.32)∫Qω‍e2sφ{|∫tT‍k(τ,t)(e-sφq)xxdτ|2}dxdt≤∫Q‍e2sφ|∫0T‍k(τ,t)(e-sφq)xxdτ|2dxdt≤∫Q‍e2sφ(∫t1t2‍|k(τ,t)|2sφdτ)(∫t1t2‍s-1φ-1|e-sφqxx|2dτ)dxdt≤C∥k∥L∞2e2sφ∫Q‍∫(t1,t2)×Ω‍e2sφsφ|(e-sφq)xx|2dxdt,
where C depends on Ω,ω,t1,t2, and T (since e2sφs-1φ-1≤C<∞, and e-2sφsφ≤C<∞ for all (t,x)∈((t1,t2)×Ω)). Finally reverting to the original variable w to complete the proof, let us choose ∇w=e-sφ(qx-sφxq), then
(3.33)∫Q‍(sφ)-1e2sφ|∇w|2dxdt≤C(∫Q‍(sφ)-1|qx|2dxdt+∫Q‍(sφ)-1sφx|q|2dxdt),∫Q‍(sφ)-1e2sφ|Δw|2dxdt≤C(∫Q‍(sφ)-1|qxx|2dxdt+∫Q‍(sφ)-1s2φx2|q|2dxdt),∫Q‍(sφ)-1e2sφ|∇·Δw|2dxdt≤C(∫Q‍(sφ)-1|qxxx|2dxdt+∫Q‍e2sφs4φx2φxx2|q|2dxdt).
Similarly wt=e-sφ(qt-sφtq),
(3.34)∫Q‍(sφ)-1e2sφ|wt|2dxdt≤C(∫Q‍(sφ)-1|qt|2dxdt+∫Q‍(sφ)-1s2φt2|q|2dxdt).
From (3.32), we can observe the kernel term as
(3.35)∫Qω‍e2sφ{|∫tT‍k(τ,t)(e-sφq)xxdτ|2}dxdt≤C∥k∥L∞2{∫Q‍e2sφ∫(t1,t2)×Ω‍|wxx|2dxdt}.
If s≥s0 is sufficiently large, then there exists a constant C>0 such that the above integral terms have been absorbed on ω⊂Ω (the right hand side of (3.31)). From the identifier (3.33)–(3.35), we can conform all the terms |wxxx|2,|wt|2,|wxx|2,|w|2 involving in s≥s0,ω⊂Ω and we obtain (3.3).Lemma 3.2.
Suppose that Theorem3.1 is satisfied. If there exist positive constants C,μ (independent of s) and w will be a solution of (2.1), such that the following inequality holds:
(3.36)∫Ω‍|w(0,x)|2dx≤Ceμs(∫Q‍e2sφ|g|2dxdt+∫Qω‍e2sφφx2φxx2|w|2dxdt).Proof.
We multiply (3.2) by w and integrate on Ω (using Cauchy’s inequality, when ϵ=1/2), we obtain
(3.37)-12ddt∫Ω‍|w|2dx+2∫Ω‍|∇w|2dx+12∫Ω‍|w|2dxdt+2∫Ω‍|∇·(Δw)|2dxdt≤∫Ω‍(2|g|2+32|w|2)dx+2∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dx.
Define
(3.38)γ(t)=sup{e-2sφφx-2(x,t)φxx-2(x,t):x∈Ω}≤Ceμs/β(t),μ=2e2m∥ψ∥∞.
Therefore,
(3.39)-12ddt∫Ω‍|w|2dx≤C(∫Ω‍|g|2dx+∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dx+∫Ω‍|w|2dx+∫Ω‍|∇w|2dx+|∫tT‍k(τ,t)wxxdτ|2∫Ω‍|∇·(Δw)|2dx).
Integrating (3.39) on (0,t),
(3.40)-∫0t‍ddt[∫Ω‍|w|2dx]ds≤C∫0t‍(∫Ω‍|g|2dx+∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dx+∫Ω‍|∇w|2dxC∫0t‍s+∫Ω‍e2sφγ(t)φx2φxx2|w|2dx+∫Ω‍|∇·(Δw)|2dx|∫tT‍k(τ,t)wxxdτ|2)ds,fort∈(0,T).
Therefore,
(3.41)∫Ω‍|w(0,x)|2dx≤C(∫Ω‍e2sφφx2φxx2|w|2dx+∫Ω‍|g|2dx+∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dxC∫0t‍s+∫Ω‍|∇w|2dx+∫Ω‍|∇·(Δw)|2dx|∫tT‍k(τ,t)wxxdτ|2)fort∈(0,T).
Now we fix t1 and t2 such as 0<t1<t2<T, then integrating the above inequality:
(3.42)∫t1t2‍∫Ω‍|w(0,x)|2e-μs/(t(t-T))dxdt≤C∫t1t2‍(|∫tT‍k(τ,t)wxxdτ|2∫Ω‍e2sφφx2φxx2|w|2dx+∫Ω‍|g|2dx+∫Ω‍|∇w|2dx+∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dx+∫Ω‍|∇·(Δw)|2dx)dt.
As based on (3.32), the kernel has been modified as
(3.43)∫t1t2‍∫Ω‍|∫tT‍k(τ,t)wxxdτ|2dxdt≤∫0T‍∫Ω‍(∫t1t2‍|k(τ,t)|2e-2sφsφdτ)(∫t1t2‍|wxx|2e2sφs-1φ-1dτ)dxdt≤C∥k∥L∞2∫t1t2‍∫Ω‍e2sφs-1φ-1|wxx|2dxdt.
Using (3.43) in (3.42), then
(3.44)∫t1t2‍∫Ω‍|w(0,x)|2e-μs/(t(t-T))dxdt≤C(∫Q‍|g|2dxdt+∫Q‍|∇w|2dxdt+∫Q‍|∇·(Δw)|2dxdt+∫t1t2‍∫Ω‍e2sφ(φx2φxx2|w|2+s-1φ-1|wxx|2)dxdt).
Since
(3.45)inft∈(t1,t2){e-μs/β(t)}≥C>0
and the Carleman estimate (3.3) using in (3.44), we obtain
(3.46)∫Ω‍|w(0,x)|2dx≤C(∫Q‍e2sφ|g|2dxdt+∫Qω‍e2sφφx2φxx2|w|2dxdt).


## 4. Controllability Results

Now we are ready to give the proof of Theorem2.2 result, which will be a main part of our work.Proof of Theorem2.2.
Let us fixT>0,a∈W1,∞(0,T:L2(ω)),b∈L∞(Q),and y0∈H01(Ω). For every ϵ>0, the penalized formula
(4.1)Minimize{Hϵ(u);u∈L2(Q)}>0,
where the functional Hϵ is (see [9, 11])
(4.2)Hϵ(u)=12∫Q‍|u|2dxdt+12ϵ∫Ω‍|y(T,x)|2dx.
Here y is a solution of (1.2), which is associated with the control u. Since Hϵ is a continuous strictly convex functional in L2(Q), then Hϵ has a unique solution (uϵ,yϵ) (for any ϵ>0). If (uϵ,yϵ) (as ϵ→0) is a null controllability solution of the system (1.1), then due to penalization property (1/ϵ)∫Ω‍|y(T,x)|2dx, the limit exists in an appropriate norm. As based on the pontryagin maximum principle, the maximal condition on the control uϵ is
(4.3)uϵ=χωwϵa.einQ,
and wu∈C([0,T];L2(Ω))∩L2(0,T;H01(Ω)) is a solution for
(4.4)(wϵ)t+Δwϵ+(wϵ)x-wϵ+(wϵ)xxx-∫tT‍k(τ,t)(wϵ)xxdτ=0inQ,(4.5)wϵ=-1ϵyϵ(T,x),inΩ,(4.6)wϵ=0,inΣ.
Now we multiply (4.4) by yϵ (where y=yϵ), (1.2) by wϵ and integrate on Q,
(4.7)∫Qω‍|yϵ|2dxdt+1ϵ∫Ω‍|yϵ(T,x)|2dx-∫Ω‍y0(x)wϵ(x,0)dx+∫Q‍fwϵdxdt=0.
Applying the observability inequality (3.36), we have
(4.8)|∫Ω‍y0(x)wϵ(x,0)dx|≤C(∫Qω‍|wϵ(x,0)|2dxdt)1/2|y0|2.
Using the Carleman estimate (3.3) and Lemma 2.1 condition,
(4.9)|∫Q‍fwϵdxdt|≤C(∫Q‍e2sφφx2φxx2|wϵ|2dxdt)1/2(∫Q‍f12dxdt)1/2≤C(∫Qω‍e2sφφx2φxx2|wϵ|2dxdt)1/2∥f1∥L2(Ω),
where C is a positive constant that is independent of ϵ and y0. By virtue of (4.7)–(4.9), we obtain
(4.10)∫Q‍|yϵ|2dxdt+1ϵ∫Ω‍|yϵ(T,x)|2dx≤C(|y0|22+∥f1∥L2(Q)2),∀ϵ>0.
By the condition (4.10), we can observe
(4.11)∫Q‍|uϵ|2dxdt+1ϵ∫Ω‍|yϵ(T,x)|2dx≤C(|y0|22+∥f1∥L2(Q)2).
Since uϵ is bounded in L2(Q), there exists a subsequence denoted by ϵ such that
(4.12)uϵ→u*weaklyinL2(Q),yϵ→y*weaklyinL2(0,T;H1(Ω))asϵ→0.
Clearly y*=yu*, letting ϵ→0 in (4.11), then we have |yϵ(T,x)|2=0a.e.x∈Ω. 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 inyxx. 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].



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*Source: 531659-2012-12-20.xml*