Applied Mathematics and Mechanics (English Edition), 2006, 27(1):109–116 c Editorial Committee of Appl. Math. Mech., ISSN 0253-4827
BOUNDARY CONTROL OF MKDV-BURGERS EQUATION TIAN Li-xin(
)1 , ZHAO Zhi-feng(
)
1,2
, WANG Jing-feng(
∗
)
1
(1.Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, P. R. China; 2 .Zhenjiang Watercraft College of PLA, Zhenjiang 212003, P. R. China) (Communicated by LIU Zeng-rong)
Abstract: The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given. Key words: boundary control; feedback control law; MKdV-Burgers equation Chinese Library Classification: O232 2000 Mathematics Subject Classification: 35Q53 Digital Object Identifier (DOI): 10.1007/s 10483-006-0114-z
Introduction Because of extensively applicable background of boundary control which is described by nonlinear PDE, the boundary control of nonlinear PDE attracts more attention of Physics, Mathematics and control engineers. With the periodic boundary conditions, some results were obtained by Russell and Zhang[1]. With the domain of the equation being whole real line, some results were obtained by Naumkin[2] . Weijiu-Liu and Krstic[3] attained some results of boundary control and numerical analysis of KdV equation by feedback control. On the above results, we will consider boundary control of MKdV-Burgers equation by feedback control. In this work, we will consider boundary control of the following MKdV-Burgers equation: ut − εuxx + uxxx + 6u2 ux = 0, 0 < x < 1, t > 0, u(0, t) = ux (1, t) = 0,
0
u(x, 0) = u (x), 0 < x < 1, t > 0,
3
uxx (1, t) = k1 u(1, t) + k2 u(1, t), t > 0. Definition 1
(1) (2) (3)
If a function u(x, t) ∈ C([0, T ], H01 (0, 1)) satisfies the following equation ut (t), ϕ − εux , ϕx − ux , ϕxx − uux , ϕ = [k1 u(1, t)3 + k2 u(1, t)], ϕ(1, t) − u0 , ϕ(0),
(4)
u(x, t) is called a weak solution of Eqs.(1)–(3). The rest part of this work is organized as follows. Section 1 is given the main theorems and main denotes. Section 2 is devoted to existence and stability of solution of system described by Eqs.(1)–(3). ∗ Received Feb.8,2004; Revised Sep.12,2005 Project supported by the National Natural Science Foundation of China (No.10071033) and the Natural Science Foundation of Jiangsu Province (No.BK2002003) Corresponding author TIAN Li-xin, Professor, Doctor, E-mail:
[email protected]
110
1
TIAN Li-xin, ZHAO Zhi-feng and WANG Jing-feng
Main Signals and Main Theorems
1 Suppose V (t) = 0 ux (x, t)2 dx is high order energy, H s (0, 1) is usual Sobolev space. For arbitrary s ∈ R, set H01 = {ϕ ∈ H 1 (0, 1) : ϕ(0) = 0}, H02 (0, 1) = {ϕ ∈ H 2 (0, 1) : ϕ(0) = ϕx (1) = 0}, H03 (0, 1) = {ϕ ∈ H 3 (0, 1) : ϕ(0) = ϕx (1) = ϕxx (1) = 0}. Let X is a Banach space, · and (·, ·) respectively denote norm and scalar product on L2 (0, 1). We define operator L, Lu = −uxxx +εuxx , whose domain is D(L) = H03 (0, 1). So we may easily prove adjoint operator L∗ of L given by L∗ u = uxxx + εuxx with the domain D(L∗ ) = {u ∈ H03 (0, 1)}. It is easa to be known that operators L and L∗ are close, dissipative, dense linear operator on the domain L2 (0, 1) and L is infinitesimal generator of C0 contraction semigroup on the domain L2 (0, 1). Suppose 2
W ={ω ∈ C 2 ([0, T ]; H01 (0, 1)) : ω(x, 0) = u0 (x), ωt (x, 0) = εu0xx (x) − u0xxx(x) − 6u0 (x)u0x (x)}, 2
2
2
1
ωW = [ max (ωx (t) + ωxt (t) + ωxtt (t) )] 2 . 0≤t≤T
W is a Banach space obviously. Theorem 1 (1) For initial value u0 (x) ∈ H01 (0, 1), there exists a unique weak solution u of Eqs.(1)–(3), which u satisfies the following L2 global-exponential stability estimate: 2 2 (5) u(t) ≤ u0 e−2εt , ∀t ≥ 0, and the H 1 global-asymptotic and semiglobal-exponential stability estimate is got as follows: max u(x, t)2 ≤ ux (t)2 2 2 ≤ M1 u0x exp(M2 u0 )e−εt , ∀t ≥ 0,
0≤x≤1
(6)
where M1 and M2 are positive real constants. (2) For the initial value u0 (x) ∈ H03 (0, 1) which satisfies compatibility condition u0xx (1) = k1 u0 (1)3 + k2 u0 (1) there exists a global classical solution u satisfying: u ∈ C([0, T ]; H03 (0, 1)) ∩ C 1 ([0, T ]; H01 (0, 1)) and
2 2 u2H 3 ≤ M3 u0 H 3 exp[M4 F (u0 H 3 )]e−εt . ω ε
(7)
4+ 4ω ε
12 2 Theorem 2 Set ω = ε+8 . 8 , k = max{2, ( ε+8 ) },F (r) = r + r 0 1 (1) For initial value u (x) ∈ H0 (0, 1), there exists a solution u of Eqs.(1)–(3), which u satisfies the following global-asymptotic and semiglobal-exponential stability estimate: 2 2+ 2ω 2 (8) u(t) ≤ K(u0 + u0 ε )e−2ωt , ∀t ≥ 0,
and 2
max u(x, t)2 ≤ ux (t) ≤ cF (u0
0≤x≤1
H1
) exp[cF (u0 )]e−ωt , ∀t ≥ 0.
(9)
(2) For the initial value u0 (x) ∈ H02 (0, 1) ∩ H 3 (0, 1) which satisfies compatibility condition u0xx (1) = k1 u0 (1)3 + k2 u0 (1), there exists a global classical solution u satisfying the following global-asymptotic and semiglobal-exponential stability estimate: 2 uH 3
≤c
3 i
F i (u0 H 3 ) exp[cF (u0 )]e−ωt , ∀t ≥ 0.
(10)
Boundary Control of MKdV-Burgers Equation
2
111
Proof of Theorems In order to prove Theorem 1 and Theorem 2, we first verify the following Lemma 1.
Lemma 1 Assume u0 (x) ∈ H02 (0, 1) ∩ H 7 (0, 1) and u0xx (1) = k1 u0 (1)3 + k2 u0 (1), then there exists a unique local classical solution u of Eqs.(1)–(3) when T > 0. Proof First, we verify that for an arbitrary fixed ω ∈ W , the following equations have a solution u ∈ W : ut − εuxx + uxxx + 6ω 2 ωx = 0, u(0, t) = ux (1, t) = 0,
0 < x < 1, 0
u(x, 0) = u ,
t > 0,
(11) 3
uxx (1, t) = k1 u(1, t) + k2 u(1, t),
t > 0.
(12)
Define nonlinear transform A such thatAω = u. If we can show that A has a unique fixed point u∗ by Banach fixed theorem, then we have that u∗ is the unique solution of Eqs.(1)–(3). Set 1 ψ = x(x − 1)2 [k1 ω 3 (1, t) + k2 ω(1, t)], υ = u − ψ. 2 From the definition of operator L, Eqs.(11) and (12) are transformed to an abstract Cauchy problem as follows: υt = Lυ + f, υ(0) = υ 0 . Assume ω, f are differential on both x and t and the linear operatorL on L2 (0, 1) is infinitesimal of C0 semigroup of contraction. According to semigroup theory, Eqs.(11) and (12) have a unique solution υ ∈ C 1 ([0, T ]; L2(0, 1)) ∩ C([0, T ]; H03 (0, 1)). Then it can be drawn that Eqs.(11) and (12) exist a unique solution u = υ + ψ ∈ C 1 ([0, T ]; L2 (0, 1)) ∩ C([0, T ]). Further suppose that if initial value u0 is sufficiently regular,u is also sufficiently regular. Assume that ω, u0 are sufficiently smooth and u1, u2 are respectively solutions of Eqs.(11) and (12) according to ω 1, ω2 and initial value u01 , u02 . Set z 0 = u01 − u02 ,
z = u1 − u2 , Then
η = ω1 − ω2 .
zt − εzxx + zxxx + 6ω12 ηx + 6η(ω1 + ω2 )ω2x = 0,
0 < x < 1,
0 < t < T.
(13)
Assume C(s1 , s2 )is a general continuous linear function. For the general function φ = φ(x, t), define φ∞ =
max
0≤x≤1,0≤t≤T
|φ(x, t)| ,
φ0,∞ = max φ(t) , 0≤t≤T
φ1,∞ = φt ∞ + max φx (t) . 0≤t≤T
For Eq.(13) integrating by part, it can be gotten: d dt
0
1
z(t)2 dx
=2 0
1
z[εzxx − zxxx − 6ω12 ηx − 6η(ω1 + ω2 )ω2x ]dx
2
2
≤ (η∞ + ηx 0,∞ )C(ω1 W , ω2 W ). It means that
2 z(t)2 ≤ T ηW C(ω1 W , ω2 W ) + z 0 .
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TIAN Li-xin, ZHAO Zhi-feng and WANG Jing-feng
Analogue to the above, it can be drawn: 2 zx 2 ≤ T ηW · C(ω1 W , ω2 W ) + zx0 . 2
2
2
(14) 2
So they can be also concluded the same results of zxt , zt , ztt and zxtt . For ω ∈ W, ϕ0 ∈ H02 (0, 1) ∩ H 7 (0, 1) and utilizing dense theorem, the solution of Eqs.(11) and (12) satisfies u ∈ C 2 ([0, T ], H01 (0, 1)). Moreover, since u(x, 0) = u0 (x), ut (x, 0) = εu0xx(x) − u0xxx(x) − 6(u0 (x))2 u0x (x), it can be drawn u ∈ W and A: W → W , which W satisfies Eq.(14). Supposing ω2 = 0, u02 = 0, we have 2 2 2 2 2 AωW ≤ u(0) + u0x (0) +ut (0) +uxt (0) 2
2
2
+ utt (0) +uxtt (0) +T ωw C(ωW ) 2
≤R2 + T ωW C(ωW ) where R = R(u0 H 7 ) is a positive real constant. Moreover , letting B(0, 2R) = {ω ∈ W : ωW 2 ≤ 2R } and T is sufficiently small. Then AωW ≤ R2 + T R2C(R) ≤ 4R2 holds . Then we have A : B(0, 2R) → B(0, 2R). Further, assuming T being sufficiently small to make T C(R) < 1, it can be drawn that A is a contraction injection. So utilizing Banach fixed point theorem, A has a unique fixed point u∗ ∈ W . Then for T being sufficiently small, there exists a unique solution u∗ of Eqs.(1)–(3). From the above, the proof of Lemma 1 is finished. The proof of Theorem 1 and Theorem 2 In order to prove the existence and uniqueness of the solution, we first give priori estimates for the solution of Eq.(1). (1) Stability estimate From Lemma 1, the systems (1)–(3) have a local solution u. Inequality (5) can easily be attained by energy inequality. In fact, we have d dt
0
1
u2 dx =2
0
1
uut dx
≤ − 2ε ux 2 1 ≤ − 2ε u2 dx. 0
2 So u ≤ u0 e−2εt holds. Therefore Eq.(5) holds. 1 In the following we will prove Eq.(8) by Lyapunov function 0 (x + 1)u(t)2 dt. From Eq.(1), we have 1 1 1 1 d 1 (x + 1)u2 dx ≤ ε (u2 + u2x )dx − 2ε u2x dx + 3 ux 2 u2 dx − 4 u2x dx dt 0 0 0 0 0 1 ε 2 u2x dx + 3 ux · u0 e−2εt ≤ −( + 4) 2 0 1 0 −2εt ε = −( + 4 − 3 u e ) u2x dx 2 0 2 ε + 8 − 6 u0 e−2εt 1 2 =− ux dx. (15) 2 0
Boundary Control of MKdV-Burgers Equation
Let T0 =
0 2 u ≤ 0 2 u >
0, 2
12u0 1 2ε ln( ε+8
),
1 12 (ε 1 12 (ε
113
+ 8), + 8).
It can be attained
1
u2 dx ≤
1
(x + 1)u2 dx 0 0 ≤ 2 μ0 e−2εT0 e−2ωt , 1 2 u2 dx ≤ u0 e−2εt 0
≤(
∀t ≥ T0 ,
2+ 2ω 12 ω ) ε u0 ε e−2ωt , ε+8
0 ≤ t ≤ T0 .
(16)
ω
12 ε Let K = max{2, ( ε+8 ) }. From Eq.(16), Eq.(8) can be attained. To prove Eq.(9), let us to estimate the following function:
B(t) = k2 u(1, t)2 + (k1 + 6)u(1, t)4 + (ε + 4)
1
0
u2x dx + u2x (0, t).
From Eq.(15), we attain d dt
1
0
2
(x + 1)u dx + B(t) ≤ 3
1
0
≤ 3(
0
4
u dx + ε 1
1
0
u2 dx)2 + ε
u2 dx
0
1
u2 dx.
(17)
To multiply Eq.(17) by eωt , we have d ωt (e dt
1
0
2 4+ 4ω (x + 1)u2 dx) + eωt B(t) ≤ C(u0 + u0 ε )e−ωt ,
(18)
where C = C(ε, ω) is general continuous linear function. Integrating Eq.(18) from 0 to ∞, so ∞ (19) eωs B(s)ds ≤ cF (u0 ). 0
x 1 From the definition of V (t), u(x, t)2 = ( 0 ux dx)2 ≤ x 0 u2x dx ≤ V (t) and 12 0
1
u2 ux uxx dx ≤ ≤
we attain
18 ε
1 0
u4 u2x dx + 2ε
18 V (t)3 + 2ε ε
0
1
0
1
u2xx dx
u2xx dx,
18 V˙ (t) ≤ 2k12 V (t)u(1, t)4 + V (t)3 + 2k22 V (t). ε
(20)
Multiplying Eq.(20) by eωt , we have d ωt 18 (e V (t)) ≤ ωeωt V (t) + [2k12 u(1, t)4 + V (t)2 + 2k22 ]V (t)eωt . dt ε
(21)
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TIAN Li-xin, ZHAO Zhi-feng and WANG Jing-feng
To integrate Eq.(21) from 0 to t and to utilize Eq.(19), it can be drawn eωt V (t) ≤ cF (u0 H 1 ) exp[cF (u0 )]. From the embedding theorem, it can be drawn ux (t)2 ≤ cF (u0 1 ) exp[cF (u0 )]e−ωt , H
∀t ≥ 0.
So Eq.(9) holds. Analogue to the above, we have t 2 2 4 ut ≤ 2 ut (0) exp( 162 ux (s) ds)e−ωt 0 ≤ cF (u0 H 3 ) exp(cF 2 (u0 ))e−ωt .
(22)
So we have uxx 2 ≤ c(ut 2 + ux 2 + ux 4 + ux 6 + ux 8 ) ≤c
4 i=1
F i (u0 H 3 ) exp[cF 2 (u0 )]e−ωt .
(23)
Analogue to Eq.(23) and utilizing Eq.(1), we can get that 2
2
2
6
uxxx ≤ c(uxx + ut + ux ) ≤c
4 i=1
Hence
F i (u0 H 3 ) exp[cF 2 (u0 )]e−ωt .
we have 0 2 u 3 = u2 + ux 2 + uxx 2 + uxxx 2 H 2
2
≤ 3 uxx + uxxx ≤c
4 i=1
F i (u0 H 3 ) exp[cF 2 (u0 )]e−ωt .
It means Eq.(10) holds. Analogue to the same procedure, we can obtain Eqs.(6) and (7). (2) Continuous dependence of solutions with respect to initial value From Eqs.(5), (6), (8), (9), we show that estimates (5)–(10) of the solution hold. In the following part we must establish the continuous dependence of solutions with respect to initial value. To replace ω1 , ω2 and η in Eq.(11) by u1 , u2 and z, we have that zt − εzxx + zxxx + 6u21 zx + 6z(u1 + u2 )u2x = 0,
0 < x < 1,
0 < t < T.
Then we have that 1 d 1 z(t)2 dx =2 z · zt dx dt 0 0 1 z[εzxx − zxxx − 6u21 zx − 6z(u1 + u2 )u2x ]dx =2 0
2 ≤ − 2z(1, t)zxx(1, t) − zx2 (0, t) − 2ε zx + 2 z (6 u21 zx + 6 z u1 + u2 u2x ).
Boundary Control of MKdV-Burgers Equation
115
From Eq.(19), we obtain that 2 2 u1 (t) − u2 (t) ≤ 6 u01 − u02 exp(c
0
t
2
2
u1x (s) + u2x (s) )ds)
2 ≤ 6C(u01 , u02 ) u01 − u02 ,
t ≥ 0.
(24)
Also we have
∞
0
2 2 (zx (0, t)2 + zx (t) )dt ≤ C(u01 , u02 ) u01 − u02 .
(25)
Moreover, we obtain d dt
1
0
zx2 (t)dx =2
1
0
zxx (−εzxx + zxxx + 6u21 zx + 6z(u1 + u2 )u2x )dx 2
2 2 (1, t) − zxx (0, t) − 2ε zxx ≤zxx 2 + 2 zxx ( u1 zx + u2x z (u1 + u2 )).
(26)
Utilizing Eqs.(19) and (25), we have that 2 2 u1x (t) − u2x (t) ≤ C(u01 , u02 ) u01 − u02 .
(27)
Further, we obtain that d dt
0
1
zt (t)2 dx =2
0
1
zt · ztt dx 2
2 (0, t) − 2ε zxt ≤ − 2zt (1, t)zxxt (1, t) − zxt 2 + 2 zt (6 u1t zxt + 6 zt u1t + u2t u2xt )
≤c z2
2
(uit 2 + uixt 2 ) + c zx 2
i=1 2
+ c zt
2
uit 2
i=1 2
2
2
(uix + uix ).
(28)
i=1
Further we show
∞ 0
2 ( uixt 2 )dt ≤ C(u01 H 3 , u02 H 3 ). i=1
Therefore from Eqs.(10), (22), (24), (25), (28), we obtain that 2 2 u1x (t) − u2x (t) ≤ C(u01 H 3 , u02 H 3 ) u01 − u02 H 3 ,
t ≥ 0.
(29)
t ≥ 0.
(30)
So we have that 2 2 u1 (t) − u2 (t)H 3 ≤ C(u01 H 3 , u02 H 3 ) u01 − u02 H 3 ,
(3) The proof of existence and uniqueness of local weak solution and classical solution For u0 (x) ∈ H01 (0, 1) utilizing Eqs.(27), (29), (30) and density argument, it can be proved that there exists a local weak and local classical solution of system described by Eqs.(1)–(3). Moreover for u0 (x) ∈ H02 (0, 1) ∩ H 3 (0, 1), it can be proved that there exists a local classical
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TIAN Li-xin, ZHAO Zhi-feng and WANG Jing-feng
solution of system described by Eqs.(1)–(3). The uniqueness of solution is directly drawn by Eq.(27). (4) Existence and uniqueness of global weak solution and classical solution From Eqs.(5), (8), (9), we obtain that the solution doesn’t blow up in finite time and the local solution can be continued to infinity. So we have the existence and uniqueness of global weak and classical solution. From now on, we have finished all proofs.
References [1] Russell D L and Zhang B Y. Exact controllability and stabilizability of the Korteweg-de Vries equation[J]. Trans Amer Math, 1996, 348 (9): 3643–3672. [2] Naumkin P I and Shishmarev I A. On the decay of step-like data for the Korteweg-de-Vries-Burgers equation[J]. Funktsional Anal Prilozhen, 1992, 26(2), 88–93. [3] Liu Weijiu and Miroslav Krstic. Global boundary stabilization of the Korteweg-de Vries Burgers equation[J]. Computational and Applied Mathematics, 2002, 21(1), 315–354.