Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 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
1
CASY, DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
Advanced Electronics and Information Research Center, Department of
Electronics and Information Technology, Chonbuk National University, Republic of Korea
3
Department of Electrical and Electronics Engineering, Seonam University, Republic of Korea
2
Correspondence should be addressed to Kil To Chong, kitchong@jbnu.ac.kr
Received 12 August 2012; Accepted 7 November 2012
Academic Editor: Yongfu Su
Copyright q 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.
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 ω ⊂ 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.
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.
2
Abstract and Applied Analysis
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:
yt − yxx yyx yxxx t
kt, τyxx dτ χω ut, x ft, x,
t, x ∈ 0, T × Ω,
0
y0, x y0 x,
yt, x 0,
x ∈ Ω,
1.1
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.
Abstract and Applied Analysis
3
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:
yt t, x − yxx t, x at, xyt, x x bt, xyt, x yxxx t, x t
kt, τyxx τ, xdτ
0
χω ut, x ft, x, t, x ∈ 0, T × Ω,
1.2
y0, x y0 x, x ∈ Ω
yt, x 0, t, x ∈ 0, T × ∂Ω,
where ∂3 y/∂x3 yxxx , yt, x is a solitary wave of dispersion at the point x and time t, y0 x
is an initial temperature distribution, the integral kernel kt, τ is called conservation of mass
or volume and has support in t1 , t2 , where 0 < t1 < t2 < T , ft, x is a forcing term such as
pressure force, χω is a characteristic function of the subset ω ⊂ Ω, and ut, x is a control over
an arbitrary sub domain ω of the domain Ω.
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 m and p > 1 or p ∞, denote as usual by W m,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 W m,p Ω, we will write H m Ω. Besides, we need the space L2 0, T : H 1 Ω
of all equivalence classes of square integrable functions from 0, T to H 1 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 :
1 for x ∈ ω,
0 for x ∈ Ω/ω;
iii Q :
0, T × Ω, Qω :
0, T × ω, and Σ :
0, T × ∂Ω;
iv H m 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 ; H 1 Ω:
the space of all equivalence classes of square integrable functions
from 0, T to H 1 Ω;
vi W m,p Ω {yx : yW m,p |α|≤m Ω
|Dα y|p dx
1/p
< ∞}.
4
Abstract and Applied Analysis
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
yt t, x − yxx t, x at, xyx t, x bt, xyt, x yxxx t, x t
kt, τyxx τ, xdτ
0
χω ut, x ft, x, t, x ∈ 0, T × Ω,
2.1
y0, x y0 x, x ∈ Ω,
yt, x 0, t, x ∈ 0, T × ∂Ω,
where a ∈ W 1,∞ 0, T : L2 Ω, b ∈ L∞ Q. The kernel kt, · 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. Let Ω 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
fx, t ≤ f1 x, tesϕx,t ϕx x, tϕxx x, t,
a.e. x, t ∈ Q.
2.2
For each y0 ∈ L2 Ω, and u ∈ L2 Q such that
uL2 Q ≤ C y0 2 f1 L2 Q ,
2.3
and the space yu ∈ C1 0, T ; H 1 0, T : L2 Ω ∩ L2 0, T ; H 1 Ω is a solution of 1.2.
Proof. The proof is similar to Theorem 1.3.1 from Barbu 10 and hence it will be omitted.
Theorem 2.2. Let Ω, 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 ∈ C0, T ; L2 Ω ∩ L2 0, T ; H 1 Ω such that
yu T, x 0
a.e. x ∈ Ω,
2
uL2 Q ≤ C y0 2 f1 L2 Q .
2.4
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.
Abstract and Applied Analysis
5
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
ϕx, t ψx > 0, ∀x ∈ Ω, ψx 0, ∀x ∈ ∂Ω, ψx x > 0,
ψx
,
βt
ψx x < 0, ψxx x < 0, ψxx x · ψx x > 0,
ψx x · ψxxx x > 0,
3.1
x ∈ Ω \ ω,
where βt tT − t.
Suppose that w is an adjoint state variable of system 2.1, then it has some solutions.
Therefore,
wt t, x wxx t, x c1 wx t, x − c2 wt, x wxxx t, x −
T
kτ, twxx τ, xdτ
t
gt, x,
t, x ∈ Σ,
3.2
wT, x wT x,
wt, x 0,
x ∈ Ω,
t, x ∈ 0, T × ∂Ω,
where at, x ≤ c1 > 0, bt, x ≤ c2 > 0, g ∈ L2 Q and wT ∈ L2 Ω.
Now we will state the theorem to prove system 3.2 of solution w,which will 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 function ψ 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:
e2sϕ
Q
−1 sϕ
|wt |2 |wx |2 |wxx |2 |wxxx |2
s5 ϕ4x ϕxx |w|2 s3 ϕ2x ϕxx |wx |2 sϕ|wxx |2 dx dt
2
g dx dt ≤ Ce2sϕ
Q
Qω
s4 ϕ2x ϕ2xx |w|2 dxdt .
3.3
6
Abstract and Applied Analysis
Proof. We set w e−sϕ q where s, ϕ are positive parameters and q satisfies 3.2 for our
convenient c1 c2 1, then
qt sϕt q s2 ϕ2x q − sϕxx q − 2sϕx qx qxx − s3 ϕ3x q 3s2 ϕx ϕxx q 3s2 ϕ2x qx
− 3sϕxx qx − 3sϕx qxx − sϕxxx q qxxx − q − sϕx q qx
T
3.4
kτ, te−sϕτ qτ, x dτ gesϕ .
− esϕ
t
As based on Barbu 10 we can introduce the operators
L1 q qt − sϕt q − sϕxx q s2 ϕ2x q − s3 ϕ3x q 3s2 ϕx ϕxx q − sϕxxx q − q − sϕx q,
3.5
L2 q qxx qxxx − 2sϕx qx 3s2 ϕ2x qx − 3sϕxx qx − 3sϕx qxx qx .
3.6
It follows from 3.4, 3.5, and 3.6 that
L1 q L2 q hs
3.7
in Q,
where
hs ge
sϕ
e
sϕ
T
kτ, t e
t
−sϕ
q
xx
dτ .
3.8
Taking L2 -norm of both sides of 3.7, we obtain
2
2
hs 2L2 Q L1 q
L2 Q L2 q
L2 Q 2 L1 q, L2 q L2 Q .
3.9
Let us analyze the scalar product in 3.9 as
L1 q, L2 q
L2
Eij ,
1 ≤ i ≤ 9, 1 ≤ j ≤ 7,
where Eij is an integral product of ith term in L1 q and jth term in L2 q.
3.10
Abstract and Applied Analysis
7
Now we will simplify the estimate L1 q, L2 qL2 Q by using Green’s theorem with
integration by parts:
L1 q, L2 q
L2 Q
2
2
2
∂q ∂q 3
1
2
3 3 ∂q sϕxxx dΣ s ϕx ϕxx dΣ s ϕx dΣ
∂v
2
∂v
2
∂v
Σ
Σ
Σ
2
2
2
∂q ∂q 1
1
1
2 2 ∂q −
s ϕx dΣ sϕxx dΣ sϕt dΣ sϕt qx2 dxdt
2 Σ
∂v
2 Σ
∂v
2 Σ
∂v
Q
1
1
s ϕt xx q2 dxdt s ϕt xxx q2 dxdt −2
s ϕt x qx2 dxdt sϕt qx2 dxdt
−
2 Q
2 Q
Q
Q
3
s2 ϕt x ϕx q2 dxdt s2 ϕt ϕxx q2 dxdt s2 ϕt ϕxx q2 dxdt
2 Q
Q
Q
3
3
s2 ϕt ϕxx q2 dxdt 3
s3 ϕt ϕx ϕxxx q2 dxdt s2 ϕt x q2 dxdt
2 Q
2
Q
Q
3
s2 ϕt ϕx qx2 dxdt −
s2 ϕt xx ϕx q2 dxdt 2
sϕxx qx2 dxdt
3
2 Q
Q
Q
3
2 2
2
3 2
2
s ϕxx q dxdt s ϕx ϕxxx q dxdt 3
s3 ϕx ϕ2xx q2 dxdt
−
2
Q
Q
Q
s2 ϕx ϕxx qx2 dxdt −2
s2 ϕ2x qx2 dxdt s2 ϕ2xx q2 dxdt2
s2 ϕx ϕxx q2 dxdt
−3
Q
Q
s2 ϕxx ϕxxx q2 dxdt − 2
−
Q
−9
3
s
Q
ϕx ϕ2xx qx2 dxdt
3
−
2
Q
3
2
Q
s4 ϕ3x ϕxxx q2 dxdt − 3
18
3
2
Q
s4 ϕ2x ϕ2xx q2 dxdt
s3 ϕ2x ϕxx q2 dxdt
3
2
Q
−6
2
Q
Q
s3 ϕ3x q2 dxdt
2
s
Q
Q
s3 ϕ3x qx2 dxdt
s5 ϕ4x ϕxx q2 dxdt 9
s4 ϕ4x qx2 dxdt 6
s2 ϕxx ϕxxx q2 dxdt 6
Q
Q
Q
s3 ϕx ϕxx ϕxxx q2 dxdt 7
Q
s4 ϕ3x ϕxx q2 dxdt 3
Q
Q
ϕx ϕxx qx2 dxdt
Q
Q
s4 ϕ2x ϕ2xx q2 dxdt
s4 ϕ3x ϕxxx q2 dxdt
3
Q
s2 ϕxx ϕxxx qx2 dxdt 3
s2 ϕxx ϕ2xxx q2 dxdt
Q
s2 ϕxx ϕxxx qx2 dxdt
9
2 2
2
3
2
2
s ϕxx q dxdt 6
s ϕx ϕxx q dxdt −
s4 ϕ3x ϕxxx q2 dxdt
4
2 Q
Q
Q
9
27
s4 ϕ2x ϕxx q2 dxdt s3 ϕ3xx q2 dxdt 9
s3 ϕ2x ϕxx qx2 dxdt
−
2 Q
2 Q
Q
15
s3 ϕ3xx q2 dxdt sϕxxx qx2 dxdt 2
sϕxxx qx2 dxdt
−
2 Q
Q
Q
3 2
2
2
2
s ϕx ϕxx qx dxdt −
s ϕxx ϕxxx q dxdt − 3
s2 ϕx ϕxxx qx2 dxdt.
−6
Q
Q
Q
3.11
8
Abstract and Applied Analysis
Since q ϕ 0 on ∂Ω, some manipulations are dominated by the same parameters and also
observed by powers of s. Finally we obtain
7
9 3
2
Eij D1 D2 i
1 j
1
Q
s5 ϕ4x ϕxx q2 dxdt 3
Q
s3 ϕ2x ϕxx qx2 dxdt,
3.12
where D1 , D2 are boundary terms:
2
2
2
∂q ∂q 3
1
2
3 3 ∂q D1 sϕxxx dΣ s ϕx ϕxx dΣ s ϕx dΣ
∂v
2 Σ
∂v
2 Σ
∂v
Σ
2
2
2
∂q ∂q ∂q 1
1
1
−
s2 ϕ2x dΣ sϕxx dΣ sϕt dΣ,
2 Σ
∂v
2 Σ
∂v
2 Σ
∂v
1
1
D2 sϕt qx2 dxdt −
s ϕt xx q2 dxdt s ϕt xxx q2 dxdt
2
2
Q
Q
Q
s ϕt x qx2 dxdt sϕt qx2 dxdt s2 ϕt x ϕx q2 dxdt s2 ϕt ϕxx q2 dxdt
−2
Q
Q
Q
Q
3
s2 ϕt ϕxx q2 dxdt 3
s3 ϕt ϕx ϕxxx q2 dxdt
2
Q
Q
Q
3
3
s2 ϕt x q2 dxdt 3
s2 ϕt ϕx qx2 dxdt −
s2 ϕt xx ϕx q2 dxdt
2 Q
2 Q
Q
s2 ϕx ϕxx q2 dxdt −
s2 ϕxx ϕxxx q2 dxdt − 2
s4 ϕ3x ϕxx q2 dxdt
2
3
2
s2 ϕt ϕxx q2 dxdt Q
3
−
2
3
2
Q
Q
Q
3
Q
6
Q
Q
s3 ϕ2x ϕxx q2 dxdt
7
s4 ϕ3x ϕxxx q2 dxdt 6
s3 ϕx ϕ2xx q2 dxdt −
9
2
3
2
Q
Q
Q
s4 ϕ3x ϕxxx q2 dxdt − 6
Q
Q
Q
Q
s4 ϕ2x ϕ2xx q2 dxdt
s2 ϕx ϕxx qx2 dxdt
s2 ϕxx ϕxxx q2 dxdt 9
s2 ϕxx ϕxxx q2 dxdt − 3
−
2
Q
s2 ϕxx ϕ2xxx q2 dxdt s ϕx ϕxx ϕxxx q dxdt 27
3
3.13
s4 ϕ3x ϕxxx q2 dxdt −
27
2
Q
Q
s2 ϕxx ϕxxx qx2 dxdt
s4 ϕ2x ϕxx q2 dxdt
s2 ϕx ϕxxx qx2 dxdt.
By the time derivative definition ϕ we can write
ϕt ≤ Cϕ2 ,
ϕtt ≤ Cϕ3 ,
3.14
Abstract and Applied Analysis
9
where C does not depend on s, t, x. Using 3.14 for s is sufficiently large and observes the
same powers, then
|D2 | ≤ C
Q
2
s4 ϕ2x ϕ2xx q dxdt
2
s ϕx ϕxx qx dxdt .
2
3.15
Q
Now multiplying 3.7 by sϕq|∇ψ|2 and integrating over Q, we obtain
2
sϕq∇ψ L1 q dx dt Q
2
sϕq∇ψ L2 qdxdt Q
2
sϕq∇ψ hs dxdt.
Q
Therefore,
2
sϕq∇ψ hs dxdt Q
2
sϕq∇ψ L1 qdxdt Q
2
sϕ∇ψ qqxx dxdt
Q
2
sϕ∇ψ qqxxx dxdt − 2
Q
2
s2 ϕϕx ∇ψ qqx dxdt
Q
3
2
2 2
s ϕ∇x ψ qqx dxdt − 3
s2 ϕϕxx ∇ψ qqx dxdt
3
Q
Q
2
s2 ϕϕx ∇ψ qqxx dxdt,
−3
Q
2
4
1
L1 q2 dxdt,
sϕq∇ψ L1 qdxdt ≤
s2 ϕ2 ∇ψ q2 dxdt 2
2 Q
Q
Q
2
2
2
1
sϕ∇ψ qqxx dxdt −
sϕ∇ψ qx2 dxdt sϕxx ∇ψ q2 dxdt
2
Q
Q
Q
sϕx Δψ ∇ψ q2 dxdt,
Q
2
2
1
sϕx ∇ψ qx2 dxdt sϕxxx ∇ψ q2 dxdt
2 Q
Q
2
2
sϕxx ∇ψ Δψq dxdt 2
sϕ∇ψ qx2 dxdt
2
sϕ∇ψ qqxxx dxdt Q
Q
−
Q
2
sϕx ∇ψ q2 dxdt −
Q
2
sϕ∇ψ qx qxx dxdt,
Q
3.16
10
Abstract and Applied Analysis
2
s2 ϕx ϕ∇ψ qqx dxdt −2
Q
Q
2
s2 ϕ2x ∇ψ q2 dxdt 3
s
3
Q
−3
2
ϕϕ2x ∇ψ qqx dxdt
2
s2 ϕxx ϕ∇ψ q2 dxdt
Q
s2 ϕx ϕΔψ ∇ψ q2 dxdt,
2
Q
4
s3 ϕ2x ϕ∇ψ q2 dxdt
≤3
Q
Q
s3 ϕϕ2x qx2 dxdt,
2
2
3
s2 ϕx ϕxx ∇ψ q2 dxdt s2 ϕϕxxx ∇ψ q2 dxdt
2 Q
Q
s2 ϕϕxx ∇ψ Δψq2 dxdt
3
2
3
s2 ϕϕxx ∇ψ qqx dxdt 2
Q
2
s ϕx ϕ∇ψ qqxx dxdt 3
−3
3
4
Q
2
Q
2
s ϕϕx ∇ψ qx2 dxdt − 3
2
s2 ϕx ϕxx ∇ψ q2 dxdt
2
Q
Q
2
3
s2 ϕ2x ∇ψ Δψq2 dxdt −
s2 ϕx ϕxx ∇ψ q2 dxdt
2 Q
Q
2 2
3
2
s ϕϕxxx ∇ψ q dxdt − 3
s2 ϕϕxx ∇ψ Δψq2 dxdt
−
2 Q
Q
s2 ϕ2x ∇ψ Δψq2 dxdt.
−
−3
Q
3.17
By using the above calculation in 3.16, it becomes
3
−
4
Q
s3 ϕϕ2x qx2 dxdt
2
sϕ∇ψ qhs dxdt −
Q
4
s2 ϕ2 ∇ψ q2 dxdt
Q
2
1
L1 q2 dxdt
sϕ∇ψ qx2 dxdt 4
Q
Q
2
2
2
sϕΔψ ∇ψ qx dxdt −
sϕ ∇ψ qx qxx dxdt 3
s3 ϕϕx ϕxx ∇ψ q2 dxdt
2
Q
3
2
Q
2
s ϕx ϕxx ∇ψ q2 dxdt 2
s ϕϕxx ∇ψ q2 dxdt 2
2
Q
Q
s2 ϕϕx ∇ψ Δψq2 dxdt
2
Q
Q
2
4
3
3
3
2
s2 ϕ2x ∇ ∇ψ q2 dxdt s2 ϕx ϕxx ∇ψ q2 dxdt s3 ϕϕ2x ∇ψ q2 dxdt
2 Q
2 Q
2 Q
2
2
2
1
5
s2 ϕx ϕxx ∇ψ q2 dxdt sϕx ∇ψ q2 dxdt 3
s2 ϕϕxxx ∇ψ q2 dxdt
2 Q
2 Q
Q
2
2
2
3
s2 ϕϕx ∇ψ qx2 dxdt − 3
s2 ϕx ϕxx ∇ψ q2 dxdt −
s2 ϕx ϕxx ∇ψ q2 dxdt
3
2 Q
Q
Q
Abstract and Applied Analysis
2 2
3
2
−
s ϕϕxxx ∇ψ q dxdt − 3
s2 ϕϕxx ∇ψ Δψq2 dxdt
2 Q
Q
4
1
2
s2 ϕϕx ∇ ∇ψ qx2 dxdt sϕ∇ψ qx2 dxdt.
−3
4 Q
Q
11
3.18
From the above equation the right hand side, we can observe the powers of s:
−3
4
3
s
Q
ϕϕ2x qx2 dxdt
D3 3
3
s
Q
4
ϕϕ2x ∇ψ q2 dxdt
Q
2
sϕqxx
dxdt,
3.19
where
2
sϕq∇ψ hs dxdt D3 Q
3
2
2
sϕ∇ψ qL1 qdxdt 3
Q
2
s2 ϕx ϕxx ∇ψ q2 dxdt Q
2
3
s ϕx ϕxx ∇ψ q2 dxdt 2
Q
Q
s3 ϕϕ2x ∇ψ Δψq2 dxdt
2
s ϕϕxxx ∇ψ q2 dxdt
2
2
s2 ϕϕx ∇ψ qx2 dxdt
2
s2 ϕϕxx ∇ψ q2 dxdt
Q
Q
3
2
Q
s2 ϕϕx ∇ψ Δψq2 dxdt − 3
2
3.20
2
Q
2
2
3
s2 ϕx ϕxx ∇ψ q2 dxdt −
s2 ϕx ϕxx ∇ψ q2 dxdt
2 Q
Q
2
3
s2 ϕϕxxx ∇ψ q2 dxdt − 3
s2 ϕϕxx ∇ψ Δψq2 dxdt.
−
2 Q
Q
4
Applying Cauchy’s inequality with 1 for D3 ,
2 2
3
2
L1 q2 dxdt 1
s2 ϕx ϕxx ∇ψ q dxdt
|hs | dxdt 4 Q
2 Q
Q
2 2
3
s2 ϕϕ2x ∇ψ qx dxdt.
1
|D3 | ≤
4
3.21
Q
Therefore
L1 q, L2 q
L2 Q
3
3
5 4
2
D2 D3 s ϕx ϕxx q dxdt s3 ϕ2x ϕxx qx2 dxdt
2 Q
4 Q
4
2
s3 ϕϕ2x ∇ψ q2 dxdt sϕqxx
dxdt.
3
Q
Q
3.22
12
Abstract and Applied Analysis
By using 3.22 in 3.7, we get
L1 q
2 L2 q
2 3
Q
s5 ϕ4x ϕxx q2 dxdt ⎡
2
2sϕ ⎣
g dxdt ≤ Ce
Q
Q
6
Q
3
2
Q
s3 ϕ2x ϕxx qx2 dxdt 2
Q
2
sϕqxx
dxdt
2
T
−sϕ kτ, t e q xx dτ dxdt
Q t
2
s4 ϕ2x ϕ2xx q dxdt ⎤
2
1
s2 ϕx ϕxx qx dxdt⎦ 2
Q
4 2
s3 ϕϕ2x ∇ψ q dxdt 6
3.23
L1 q
2 dxdt
Q
2 2
s2 ϕϕx ∇ψ qx dxdt.
Q
Recalling that s is sufficiently large and also observing that powers of s and |∇ψ| > 0 in Ω \ ω,
1
L1 q
2 L2 q
2 3
2
Q
s5 ϕ4x ϕxx q2 dxdt 3
2
Q
s3 ϕ2x ϕxx qx2 dxdt Q
2
sϕqxx
dxdt
⎡
2
T
−sϕ 2sϕ 2
2sϕ ⎣
g dxdt ≤C
e
e kτ, t e q xx dτ dxdt
t
Q
Q
Qω
2
s4 ϕ2x ϕ2xx q dxdt 3.24
⎤
2
s2 ϕϕx qx dxdt⎦.
Qω
From the above inequality qx is in ω0 ,which will 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 ρx 1 in ω0 and
ρx 0 in Ω \ ω. Now we will multiply 3.7 by ρsϕq and integrating over Q, then we have
ρsϕqL1 q dx dt Q
ρsϕqL2 q dx dt Q
ρqsϕhs dxdt.
Q
Since
sϕqqxx dxdt ρ
Q
ρsϕxx q dxdt −
2
Q
Q
ρsϕqqxxx dxdt Q
Q
ρsϕx qx2 dxdt
−
2
∂q − sϕ dΣ ρsϕx qx2 dxdt
∂v
Σ
Q
ρsϕxxx q2 dxdt
Q
ρsϕx qx2 dxdt,
3.25
Abstract and Applied Analysis
2
2
2
−3
ρs ϕϕx qqxx dxdt 3
ρs ϕϕx qx dxdt − 3
ρs2 ϕx ϕxx q2 dxdt
Q
Q
ρs ϕx qqx dxdt −2
Q
ρs2 ϕϕxx qqx dxdt Q
Q
ρs3 ϕϕ2x qqx dxdt −
The boundary term −
Σ
2
Q
−3
3
Q
ρs ϕxx ϕq dxdt 2
ρs2 ϕϕxxx q2 dxdt
Q
3
2
3
2
2
ρs2 ϕ2x q2 dxdt
Q
3
2
ρs2 ϕx ϕxx q2 dxdt −
Q
3
2
ρs3 ϕ3x q2 dxdt − 3
Q
13
ρs2 ϕϕxxx q2 dxdt
Q
ρs3 ϕϕx ϕxx q2 dxdt.
3.26
Q
sϕ|∂q/∂v|2 dΣ ≥ 0, then
2
ρs2 ϕϕx qx dxdt
Q
1
≤
4
2
2
3
e
ρs ϕϕxx q dxdt ρs2 ϕϕxxx q dxdt
2 Q
Q
Q
2
2
2
ρs2 ϕϕt q dxdt 3
ρs2 ϕx ϕxx q dxdt ρs2 ϕϕxxx q dxdt
2
3
g dxdt −
2
2sϕ Q
2
ρsϕt q dxdt −
2
Q
Q
3
ρs
Q
2
ϕϕ2x q dxdt
Q
Q
3.27
2
ρs4 ϕϕ3x q dxdt
2
T
−sϕ 1
2 2 2
2sϕ 2
ρs ϕ q dxdt ρe kτ, t e q xx dτ dxdt.
4
t
Q
Q
Applying as ρx 1 in ω0 ⊂ ω,
⎛
2
2
s ϕϕx qx dxdt ≤ C⎝ e2sϕ g dxdt 2
Qω
Q
e
Qω
Qω
⎞
2
s4 ϕϕ3x q dxdt⎠.
2
T
−sϕ kτ, t e q xx dτ dxdt
t
2sϕ 3.28
14
Abstract and Applied Analysis
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:
−1 2 2 qx qxx qxxx 2 dxdt
sϕ
Q
2
−1 sϕ L2 q dxdt ≤C
Q
3.29
2
−1 2
sϕ s ϕx ϕxx q dxdt
Q
−1 2
sϕ qt dxdt
Q
2
−1 sϕ L1 q dxdt ≤C
Q
Q
s4 ϕ2x ϕ2xx q2 dxdt
Q
3.30
s2 ϕ2t q2 dxdt
In view of the estimates 3.22–3.30, we obtain
−1 2 2 2 qt qx qxx qxxx 2 dxdt
sϕ
Q
Q
2 2
2
s5 ϕ4x ϕxx q s3 ϕ2x ϕxx qx sϕqxx dxdt
⎛
≤ C⎝
e
2
g 2sϕ Q
Qω
2
s4 ϕ2x ϕ2xx ϕϕ3x q dxdt
3.31
⎧
⎞
2 ⎫
⎨ T
⎬
e2sϕ kτ, t e−sϕ q xx dτ dxdt⎠.
⎭
⎩ t
Qω
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,
⎧
2 ⎫
⎨ T
⎬
e2sϕ kτ, t e−sϕ q xx dτ dxdt
⎭
⎩
t
Qω
≤
e
Q
≤
e2sϕ
Q
2
T
−sϕ kτ, t e q xx dτ dxdt
0
2sϕ t
2
t1
≤ Ck2L∞ e2sϕ
|kτ, t|2 sϕdτ
Q
t1 ,t2 ×Ω
t
2
2
s−1 ϕ−1 e−sϕ qxx dτ dxdt
t1
2
e2sϕ sϕ e−sϕ q xx dxdt,
3.32
Abstract and Applied Analysis
15
where C depends on Ω, ω, t1 , t2 , and T since e2sϕ s−1 ϕ−1 ≤ C < ∞, and e−2sϕ sϕ ≤ 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 − sϕx q, then
−1 2sϕ
sϕ e |∇w|2 dxdt ≤ C
Q
Q
2
−1
sϕ sϕx q dxdt ,
Q
−1 2sϕ
sϕ e |Δw|2 dxdt ≤ C
−1 2
sϕ qx dxdt Q
−1 2
sϕ qxx dxdt Q
−1 2sϕ
sϕ e |∇ · Δw|2 dxdt ≤ C
Q
Q
−1 2 2 2
sϕ s ϕx q dxdt ,
2
−1 sϕ qxxx dxdt Q
Q
3.33
2
e2sϕ s4 ϕ2x ϕ2xx q dxdt
.
Similarly wt e−sϕ qt − sϕt q,
−1
sϕ e2sϕ |wt |2 dxdt ≤ C
Q
−1 2
sϕ qt dxdt Q
Q
−1 2 2 2
sϕ s ϕt q dxdt .
3.34
From 3.32, we can observe the kernel term as
e
2sϕ
Qω
⎧
2 ⎫
⎨ T
⎬
−sϕ kτ, t e q xx dτ dxdt
⎭
⎩ t
!
≤ Ck2L∞
e2sϕ
Q
3.35
"
t1 ,t2 ×Ω
|wxx |2 dxdt .
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 Theorem 3.1 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:
2
Ω
|w0, x| dx ≤ Ce
μs
e
2
g dxdt 2sϕ Q
Qω
e2sϕ ϕ2x ϕ2xx |w|2 dxdt
.
3.36
Proof. We multiply 3.2 by w and integrate on Ω using Cauchy’s inequality, when 1/2,
we obtain
1 d
−
2 dt
1
|w| dx 2
|∇w| dx 2
Ω
Ω
2
#
≤
Ω
2
2
Ω
|w| dxdt 2
Ω
|∇ · Δw|2 dxdt
2
T
$
2 3 2
2 g |w| dx 2
kτ, twxx dτ dx.
2
Ω t
3.37
16
Abstract and Applied Analysis
Define
&
%
−2
μs/βt
,
γt sup e−2sϕ ϕ−2
x x, tϕxx x, t : x ∈ Ω ≤ Ce
μ 2e2mψ∞ .
3.38
Therefore,
1 d
−
2 dt
⎛
|w| dx ≤ C⎝
2
Ω
Ω
2
T
|w|2 dx
kτ, twxx dτ dx Ω t
Ω
2
g dx Ω
|∇w|2 dx 3.39
⎞
Ω
|∇ · Δw|2 dx⎠.
Integrating 3.39 on 0, t,
−
t
0
d
dt
'
≤C
(
|w| dx ds
2
Ω
⎛
t
⎝
Ω
0
2
T
|∇w|2 dx
kτ, twxx dτ dx Ω t
Ω
2
g dx ⎞
Ω
e2sϕ γtϕ2x ϕ2xx |w|2 dx 3.40
Ω
|∇ · Δw|2 dx⎠ds,
for t ∈ 0, T .
Therefore,
⎛
|w0, x| dx ≤ C⎝
2
Ω
Ω
e2sϕ ϕ2x ϕ2xx |w|2 dx
Ω
|∇w|2 dx Ω
Ω
2
g dx 2
T
kτ, twxx dτ dx
Ω t
3.41
⎞
|∇ · Δw|2 dx⎠
for t ∈ 0, T .
Now we fix t1 and t2 such as 0 < t1 < t2 < T , then integrating the above inequality:
t2 t1
Ω
|w0, x|2 e−μs/tt−T dxdt
≤C
t2
t1
⎛
⎝
Ω
e2sϕ ϕ2x ϕ2xx |w|2 dx
Ω
2
g dx
Ω
|∇w|2 dx
⎞
2
T
kτ, twxx dτ dx |∇ · Δw|2 dx⎠dt.
t
Ω
Ω
3.42
Abstract and Applied Analysis
17
As based on 3.32, the kernel has been modified as
2
t2 T
kτ, twxx dτ dxdt
t1 Ω t
≤
T t2
Ω
0
2 −2sϕ
|kτ, t| e
t
2
sϕdτ
t1
≤ Ck2L∞
|wxx | e
s ϕ dτ dxdt
3.43
t1
t2 t1
2 2sϕ −1 −1
Ω
e2sϕ s−1 ϕ−1 |wxx |2 dxdt.
Using 3.43 in 3.42, then
t2 t1
Ω
|w0, x|2 e−μs/tt−T dxdt
2
g dxdt ≤C
Q
t2 t1
Ω
e
2sϕ
|∇ · Δw|2 dxdt
|∇w| dxdt Q
2
ϕ2x ϕ2xx |w|2
3.44
Q
−1 −1
2
s ϕ |wxx | dxdt .
Since
inf
t∈t1 ,t2 %
&
e−μs/βt ≥ C > 0
3.45
and the Carleman estimate 3.3 using in 3.44, we obtain
2
Ω
|w0, x| dx ≤ C
e
Q
2
g dxdt 2sϕ Qω
e2sϕ ϕ2x ϕ2xx |w|2 dxdt
3.46
.
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 T > 0, a ∈ W 1,∞ 0, T : L2 ω, b ∈ L∞ Q, and y0 ∈ H01 Ω.
For every > 0, the penalized formula
&
%
Minimize H
u; u ∈ L2 Q > 0,
4.1
18
Abstract and Applied Analysis
where the functional H
is see 9, 11
H
u 1
2
1
2
2
|u| dxdt Q
Ω
yT, x2 dx.
4.2
Here y is a solution of 1.2, which is associated with the control u. Since H
is a continuous
strictly convex functional in L2 Q, then H
has a unique solution u
, y
for any > 0.
If u
, y
as → 0 is a null controllability solution of the system 1.1, then due to
penalization property 1/
Ω |yT, x|2 dx, the limit exists in an appropriate norm. As based
on the pontryagin maximum principle, the maximal condition on the control u
is
u
χω w
4.3
a.e in Q,
and wu ∈ C0, T ; L2 Ω ∩ L2 0, T ; H01 Ω is a solution for
w
t Δw
w
x − w
w
xxx −
T
1
w
− y
T, x,
w
0,
kτ, tw
xx dτ 0
t
in Q,
4.4
4.5
in Ω,
in Σ.
4.6
Now we multiply 4.4 by y
where y y
, 1.2 by w
and integrate on Q,
2
y
dxdt 1
Qω
Ω
y
T, x2 dx −
Ω
fw
dxdt 0.
y0 xw
x, 0dx Q
4.7
Applying the observability inequality 3.36, we have
y0 xw
x, 0dx ≤ C
Ω
1/2
2
|w
x, 0| dxdt
Qω
y0 .
2
4.8
Using the Carleman estimate 3.3 and Lemma 2.1 condition,
1/2 1/2
2
2sϕ 2 2
2
e ϕx ϕxx |w
| dxdt
f1 dxdt
fw
dxdt ≤ C
Q
Q
Q
1/2
≤C
Qω
e2sϕ ϕ2x ϕ2xx |w
|2 dxdt
f1 2 ,
L Ω
4.9
where C is a positive constant that is independent of and y0 . By virtue of 4.7–4.9, we
obtain
2
y
dxdt 1
y
T, x2 dx ≤ C y0 2 f1 2 2
, ∀
> 0.
4.10
2
L
Q
Ω
Q
Abstract and Applied Analysis
19
By the condition 4.10, we can observe
|u
|2 dxdt Q
1
Ω
y
T, x2 dx ≤ C y0 2 f1 2 2
2
L Q
.
4.11
Since u
is bounded in L2 Q, there exists a subsequence denoted by such that
u
−→ u∗ weakly in L2 Q,
as −→ 0.
y
−→ y∗ weakly in L2 0, T ; H 1 Ω
4.12
∗
Clearly y∗ yu , letting → 0 in 4.11, then we have |y
T, x|2 0 a.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 in yxx . 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.
References
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