Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

Acta Mathematica Scientia 2010,30B(5):1593–1604 http://actams.wipm.ac.cn

CONTROLLABILITY FOR A PARABOLIC EQUATION WITH A NONLINEAR TERM INVOLVING THE STATE AND THE GRADIENT∗


)1

Xu Youjun (



Liu Zhenhai (

)2

1.School of Mathematics and Physics, University of South China, Hengyang 421001, China 2.School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410075, China E-mail: [email protected]; [email protected]

Abstract In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of RN . The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem. Key words controllability; Kakutani fixed-point theorem; nonlinear gradient term 2000 MR Subject Classification

1

35K55; 35K05; 93B05

Introduction and Main Results

Let Ω ⊂ RN , N ≥ 1, be an open and bounded set with boundary ∂Ω ∈ C 2 . Let ω be an open and non-empty subset of Ω. For T > 0, we denote Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). We consider the following quasilinear parabolic system: ⎧ ⎪ ⎪ ⎨ yt − y + f (y, ∇y) = ξ + v1ω in Q, ⎪ ⎪ ⎩

y=0

on Σ,

y(x, 0) = y0 (x)

in Ω,

(1)

r 2 where yt = ∂y ∂t , ξ ∈ L (Q), y0 ∈ L (Ω) are given and v is a control function to be determined in Lr (Q), 1ω denotes the characteristic function of the set w. The function f : R × RN is a C 1 -locally Lipschitz-continuous function, we can write

f (s, p) = f (0, 0) + g(s, p)s + G(s, p) · p, ∀ (s, p) ∈ R × RN for some L∞ loc functions g and G. These are respectively given by  1  1 d d f (σs, σp)dσ, G(s, p) = f (σs, σp)dσ, g(s, p) = 0 ds 0 dpi ∗ Received

(2)

1 ≤ i ≤ N.

May 19, 2008. revised January 15, 2009. The authors were supported financially by the National Natural Science Foundation of China (10971019), The author (Y. Xu) was supported financially by the Scientific Research Fund of Hunan Provincial Educational Department (09C852).

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lim

|(s,p)|→∞

|g(s, p)| log

3/2

(1 + |s| + |p|)

= 0,

lim

|(s,p)|→∞

Vol.30 Ser.B

|G(s, p)| log

1/2

(1 + |s| + |p|)

= 0,

N , if N ≥ 2; r = 2, if N = 1. 2 Let us consider an ideal trajectory y ∗ , solution of the problem without control: r >1+

⎧ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎨ yt − y + f (y , ∇y ) = ξ y∗ = 0 ⎪ ⎪ ⎩ ∗ y (x, 0) = y0∗ (x)

(3) (4)

in Q, on Σ,

(5)

in Ω,

where y0∗ ∈ L2 (Ω) and ξ ∈ Lr (Q) with r as in (4). Then, we know that under conditions (2) and (3), (5) has a local solution in time (see [9] and [10]). Moreover, there exists T ∗ > 0, such that for T < T ∗ , the solution y ∗ of (5) satisfies y ∗ ∈ C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)). The main goal of this article is to analyze the controllability properties of system (1). Definition 1.1 We say that system (1) is exactly controllable to the trajectories if, for any trajectory y ∗ solution of (5) and for any y0 ∈ L2 (Ω), for every T < T ∗ , there exists a control v ∈ Lr (0, T ; Lr (ω)) such that system (1) has a solution on (0,T) satisfying y(x, T ) = y ∗ (x, T ) in Ω.

(6)

Definition 1.2 We say that system (1) is null controllable at time T if, for each y0 ∈ L (Ω), there exists v ∈ Lr (0, T ; Lr (ω)), such that the corresponding initial boundary problem system (1) admits a solution y ∈ C 0 ([0, T ]; L2(Ω)) satisfying 2

y(x, T ) = 0 in Ω.

(7)

Definition 1.3 We say that system (1) is approximately controllable in L2 (Ω) at time T if, for any y0 ∈ L2 (Ω), for any yd ∈ L2 (Ω) and any ε > 0, there exists a control v ∈ Lr (0, T ; Lr (ω)), such that the corresponding initial boundary problem system (1) admits a solution y ∈ C 0 ([0, T ]; L2(Ω)) satisfying y(·, T ) − yd L2 (Ω) ≤ .

(8)

In system (1) y = y(x, t) is the state and v = v(x, t) is the control function with a support localized in ω. We aim at changing the dynamics of the system by acting on the subset ω of the domain Ω. The heat equation is a model for many diffusion phenomena such as heat conduction, properties of elastic plastic material, diffusion-reaction processes, etc. For instance system (1) provides a good description of the temperature distribution and evolution in a body occupying the region Ω. In system (1) ξ is a given heat source, then the control v represents a localized source of heat. The interest on analyzing the heat equation above relies not only in the fact that it is a model for a large class of physical phenomena, but also one of the most significant partial differential equations of parabolic type. Methods such as the variational methods, the related duality theory, etc., are presented to study the controllability of nonlinear control systems (1). The controllability analysis of nonlinear parabolic systems was thoroughly developed (see, e.g [1, 2–8, 11, 12]). However, in the study of the controllability of quasi-linear

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parabolic systems with superlinear nonlinearities, additional technical difficulties arise. In [1], Barbu obtained null Lr(N ) -controls v such that vLr(N ) (Q) ≤ Cy0 L2 (Ω) +2) with r(N ) ∈ [2, 2(N N −2 ] if N ≥ 3, r(N ) ∈ [2, ∞) if N = 2, and r(N ) ∈ [2, ∞) if N = 1. However, this technique can only be applied to the null controllability of the superlinear heat equation when N < 6 (which does not seem to be a natural restriction on N ). Again, Lr(N ) -estimates of the controls are needed for control cost in industrial applications, etc. We are motivated by [5], overcome such difficulty and generalize some previous results under the Dirichlet boundary conditions, in particular, those in [1] and [6].

The main result in this article is the following one: Theorem 1.1 Assume that f satisfies (2), (3) and ξ ∈ Lr (Q) with r satisfying (4). Then, system (1) is exactly controllable to the trajectory at time T . The proof of Theorem 1.1 is based on the null controllability of a linear problem (see Theorem 3.1), which is obtained from observability inequalities (see Theorem 2.1). The idea of combining the controllability of a linearized system and a fixed point argument in the proof will be applied in this article. The rest of this article is organized as follows: in Section 2, we present an observability inequality; in Section 3, we prove Theorem 1.1.

2

The Observability Inequality

In this section, we present an observability inequality that is a generalization of that given in [4] to the case of linear systems with Dirichlet boundary conditions, which is essential in the proof of Theorem 1.1. Before giving the proof of Theorem 1.1, we have to present some technical results (see [4]). Let us consider following problem ⎧ N ⎪  ∂Fi ⎪ ⎪ ⎪ −p − p = F + in Q, 0 ⎪ ⎨ t ∂xi i=1 (9) ⎪ p=0 on Σ, ⎪ ⎪ ⎪ ⎪ ⎩ p(x, T ) = p (x) in Ω, T where F0 , Fi ∈ L2 (Q)(1 ≤ i ≤ N ) and pT ∈ L2 (Ω). Lemma 2.1 [4] Let ω be a nonempty open subset of Ω and ω0 be an open subset of ω such that ω0 ⊂⊂ ω. Then, there exists a function Φ(x) ∈ C 2 (Ω), such that Φ(x) > 0 in ¯ 0. Ω, Φ = 0 on ∂Ω and |∇Φ| > 0 in Ω\ω For λ > 0, and any (x, t) ∈ Q, we set α(x, t) =

exp(C ∗ Φ(x)) − exp(2C ∗ ΦC(Ω) ) exp(C ∗ Φ(x)) , ϕ(x, t) = , t(T − t) t(T − t)

where C ∗ is an appropriate constant depending only on Ω and ω.

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Lemma 2.2 [4] Assume p is a solution of (9) associated to pT ∈ L2 (Ω) and f ∈ L2 (Q). Let ω be a nonempty open subset of Ω. Then, there exist positive constants C0 , σ0 (depending only on Ω and ω), such that   −1 −1 2 3 s exp(2sα)t (T − t) |∇p| + s exp(2sα)t−3 (T − t)−3 |p|2 dxdt Q Q   3 −3 −3 2 exp(2sα)t (T − t) |p| dxdt + exp(2sα)|f0 |2 dxdt ≤ C0 (s ω×(0,T )

+s2

N   i=1

Q

exp(2sα)t−2 (T − t)−2 |fi |2 dxdt), Q

for s ≥ s0 = σ0 (Ω, ω)(T + T 2 ), with α as in Lemma 2.1. In the sequel, unless otherwise specified, C will stand for a generic positive constant only depending on Ω and ω, whose value can change from line to line. Let us introduce the following (adjoint) system ⎧ ⎪ ⎪ ⎨ −qt − q − ∇ · (Bq) + aq = 0 in Q, (10) q=0 on Σ, ⎪ ⎪ ⎩ q(x, T ) = qT (x) in Ω. Theorem 2.1 For any a ∈ L∞ (Q), B ∈ L∞ (Q)N , and qT ∈ L2 (Ω), one has   2/r  2 |q|r dxdt , q(x, 0)L2 (Ω) ≤ C exp(K(T, a∞ , B∞ )

(11)

ω×(0,T )

2/3

where q is the solution of system (10), K(T, a∞, B∞ ) = 1 + a∞ T + a∞ T 2 + (T + T 2 )B2∞ , 1r + r1 = 1. C depends on Ω, ω, r , N . Proof Let ω  ⊂⊂ ω, applying Lemma 2.2, we get   −1 −1 2 3 s exp(2sα)t (T − t) |∇q| + s exp(2sα)t−3 (T − t)−3 |q|2 dxdt Q Q   3 −3 −3 2 ≤ C0 (s exp(2sα)t (T − t) |q| dxdt + exp(2sα)|aq|2 ω  ×(0,T ) Q  2 +s exp(2sα)t−2 (T − t)−2 |Bq|2 dxdt, (12) Q

for all s ≥ s0 . We can estimate the second term and the third term on the right side as follows:   2 2 exp(2sα)|aq| + s exp(2sα)t−2 (T − t)−2 |Bq|2 dxdt Q Q  2 −6 6 −2 2 2 2 exp(2sα)t−3 (T − t)−3 |q|2 dxdt. (13) ≤ (2 T a∞ + 2 T s B∞ ) Q

Together with (12), we have   −3 −3 2 exp(2sα)t (T − t) |q| dxdt ≤ C

ω  ×(0,T )

Q 1/3

2/3

for all s ≥ s1 = max{s0 , C0 T 2 a∞ , C0 T 2 B2∞ }.

exp(2sα)t−3 (T − t)−3 |q|2 dxdt,

(14)

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In contrast, we easily know that exp(2sα)t−3 (T − t)−3 ≤ 26 T −6 exp(−CsT −2 ), exp(2sα)t−3 (T − t)−3 ≥

 16 3 3

T −6 exp(−CsT −2 ),

¯ ∀ (x, t) ∈ Q,

(15)

¯ × [T /4, 3T /4], ∀ (x, t) ∈ Ω

(16)

2/3

for all s ≥ s2 = max{s1 , σ1 (T + T 2 + T 2 a∞ + T 2 B2∞ )}, σ1 only depends on Ω and ω. Together with (14), set Ms = s · max{exp(2λΦC(Ω) ) − exp(λΦ(x))}, then 

Ms 2 |q| dxdt ≤ T 6 exp − t(T − t) Q 

 ω

exp(2sα)t−3 (T − t)−3 |q|2 dxdt.

(17)

Let a function ξ(x) ∈ D(ω), such that ξ = 1 in ω  . We set p = ξv(t)q, where q is the solution of (10) and v(t) = exp(sα)t−3/2 (T − t)−3/2 . Note that p(T ) = p(0) = 0. Then, we have ⎧ ⎪ ⎪ ⎨ −pt − p = ξv∇ · (Bq) − aξvq − ξvt q − 2v∇ξ∇q − ξqv ⎪ ⎪ ⎩

in Q,

p=0

on Σ,

p(x, T ) = 0

in Ω.

(18)

For simplicity of the computations, we set p˜(x, t) = p(x, T − t) for any (x, t) ∈ Q. In a similar ˜ Thus, we have ˜ v˜, q˜, h. way, we introduce the functions a ˜, B, ⎧ ˜ q˜) − a ⎪ p = ξ˜ v ∇ · (B ˜ξ˜ v q˜ − ξ˜ vt q˜ − 2˜ v∇ξ∇˜ q − ξ q˜v˜ in Q, ⎪ ⎨ p˜t − ˜ ⎪ ⎪ ⎩

p˜ = 0

on Σ,

p˜(x, 0) = 0

in Ω.

(19)

Thanks to the regularizing effect of the heat equation, we have, for any t > 0, 1 ≤ r1 , r2 ≤ ∞, N

1

1

S(t)uLr1 (ω) ≤ Ct− 2 ( r2 − r1 ) uLr2 (ω) , ∀ u ∈ Lr2 (ω), N

1

1

1

S(t)uW 1,r1 (ω) ≤ Ct− 2 ( r2 − r1 )− 2 uLr2 (ω) , ∀ u ∈ Lr2 (ω),

(20) (21)

where {S(t)}t≥0 is the semigroup generated by the heat equation with Dirichlet boundary  conditions. Applying the L2 − Lr regularing effect of the heat equation, we obtain 

t

N

1

1

|v|(t − τ )− 2 ( 2 − r ) ˜ q (·, τ )Lr (ω) dτ 0  t N 1 1 1 |v|(t − τ )− 2 ( 2 − r )− 2 ˜ q (·, τ )Lr (ω) dτ +C(1 + B∞ ) 0  t N 1 1 |vt |(t − τ )− 2 ( 2 − r ) ˜ q (·, τ )Lr (ω) dτ. +C

(22)

|˜ v | ≤ CT −3 exp(−sCT −2 ), ∀(x, t) ∈ Q,

(23)

|˜ vt | ≤ CT −6 (s + T 2 ) exp(−sCT −2 ),

(24)

˜ p(·, t)L2 (ω ) ≤ C(1 + a∞ )

0

Note that

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For s ≥ s2 . Together with (22), we have ˜ p(·, t)L2 (ω ) ≤ CT −3 (1 + T 1/2 + T 1/2 a∞ + B∞ ) exp(−sCT −2 )  t N 1 1 1 (t − τ )− 2 ( 2 − r )− 2 ˜ q (·, τ )Lr (ω) dτ + CT 1/2 T −6 (s + T 2 ) exp(−sCT −2 ) · 0  t N 1 1 1 (t − τ )− 2 ( 2 − r )− 2 ˜ q (·, τ )Lr (ω) dτ, (25) · 0

for s ≥ s2 . If r satisfies

N 1 1 1 3 1 − +  + < , 2 2 r r 2 2

that is to say, r >

2(N + 2) . N +4

(26)

Thus, by Young’s inequality and estimating the L2 (0, T ; L2 (ω  ))-norm of p˜, we obtain ˜ p(·, t)L2 (0,T ;L2 (ω )) ≤ CT α T −3 (1 + s + T 2 + T 1/2 a∞ + B∞ )   2/r  · exp(−sCT −2 ) |˜ q (·, τ )|r dτ ,

(27)

ω×(0,T )

where C depends on Ω, ω  , ω, r , N ; α depends on r, N ; s ≥ σ(Ω, ω)T 2 . Note that for N < 4 and r as in (4), and condition (26) is satisfied. For N ≥ 4, we continue this process and obtain (27) for sufficiently small r . Combining with (17), we get    Ms 2 |q| dxdt ≤ T 6 exp − exp(2sα)t−3 (T − t)−3 |q|2 dxdt t(T − t)  Q ω ×(0,T ) ≤ CT α T −3 (1 + s + T 2 + T 1/2 a∞ + B∞ )   2/r  −2 |˜ q (·, τ )|r dτ . · exp(−sCT )

(28)

ω×(0,T )

Let θ0 ∈ C 1 [0, 1] be a function such that θ0 ∈ [0, 1] with θ0 = 1 on [0, 14 ] and θ0 = 0 on [ 34 , 1]. Now, we define a function θ(t) = θ0 ( Tt ) and rewrite (10) for θ(t)q, then, ⎧ ⎪ ⎪ −(θq)t − (θq) − ∇ · (Bθq) + aθq = qθt ⎪ ⎪ ⎪ ⎪ ⎨ θq = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ θq(x, 3T ) = 0 4

3T ), 4 3T on ∂Ω × (0, ), 4

in Ω × (0,

in Ω.

Multiplying (29) by θq and integrating on Ω, we have    1 d |θq|22 + |∇(θq)|22 + − a(θq)2 dx − ∇ · (Bθq)θqdx = qθt θqdx 2 dt Ω Ω Ω Thus, −

d |θq|22 + 2|∇(θq)|22 ≤ 2a∞ dt

 Ω

|θq|2 dx + 2

 Ω

 ∇ · (Bθq)θqdx + 2

Ω

qθt θqdx.

(29)

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By using H¨ older and Young inequalities and the trace theory, we have −

d (exp(2(a∞ + B2∞ )t)|θq|22 ) ≤ exp(2(a∞ + B2∞ )t) dt

 Ω

|θt ||θq|2 dx.

(30)

For any t ≥ 0, integrating this inequality with respect to time on [0, t] with t ∈ [ 3T 4 , T ], and noting that θ ≤ 1, |θt | ≤ C/T , we obtain  1 |q|2 dxdt. (31) q(x, 0)2L2 (Ω) ≤ exp(3(a∞ + B2∞ )T ) T 3T T Ω×( 4 , 4 ) Together with (16) and (14), we obtain 2 2 −2 ) q(x, 0)2L2 (Ω) ≤ CT 5 exp((a∞ T + a2/3 ∞ T + B∞ )T ) exp(CsT  · exp(2sα)t−3 (T − t)−3 |q|2 dxdt.

(32)

ω  ×(0,T )

By (27), we have   q(x, 0)2L2 (Ω) ≤ C exp(K(T, a∞ , B∞ ))



ω×(0,T )

|q(·, τ )|r dτ

2/r

,

(33)

for s ≥ s2 , where 2 2 2 K(T, a∞, B∞ ) = 1 + a∞ T + a2/3 ∞ T + (T + T )B∞ .

(34)

This completes the proof.

3

Proof of Theorem 1.1

In this section, we will prove Theorem 1.1. First, we consider the controllability of a linear heat equation involving gradient terms with Dirichlet boundary conditions. For given a ∈ L∞ (Q), B ∈ L∞ (Q)N and ξ ∈ Lr (Q), we analyze the linear system ⎧ ⎪ ⎪ ⎨ yt − y + ay + B · ∇y = ξ + v1ω in Q, (35) y=0 on Σ, ⎪ ⎪ ⎩ y(x, 0) = y0 in Ω, Theorem 3.1 Assume that T > 0, a ∈ L∞ (Q), B ∈ L∞ (Q)N . Then, there exists a positive constant M (depending on Ω, ω, T ), such that for any ξ ∈ Lr (Q), verifying   M |ξ|2 dxdt < ∞, exp (36) t(T − t) Q one can find a control function vˆ ∈ Lr (ω × (0, T )), such that the corresponding solution yˆ of (35) satisfying yˆ(x, T ) = 0 in Ω. (37) Furthermore, vˆ can be chosen in such a way that |ˆ v |Lr (ω×(0,T )) ≤ C exp(K(T, a∞ , B∞ ))(y0 L2 (Ω) + ξLr (Q) ),

(38)

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where K(T, a∞, B∞ ) is given in Theorem 2.1. Proof For every  > 0, we consider the functional J with  2/r  1 |q|r dxdt + qT 2L2 (Ω) J (qT ) = 2 ω×(0,T )   + q(x, 0)y0 (x)dx + ξqdxdt, ∀ qT ∈ L2 (Ω), Ω

(39)

Q

where q is the solution of (10) associated to qT ∈ L2 (Ω). It is easy to see that J is a continuous and strictly convex functional. Moreover, by (14), it verifies the unique continuation property, i.e., q = 0 in ω × (0, T ), then q ≡ 0. (40) By arguing as in [2], J is coercive. In fact, we have J (qT ) ≥ . qT L2 →∞ qT L2

(41)

lim inf

Therefore, J achieves its minimum at a unique point qˆT, ∈ L2 (Ω). Let qˆ be the solution of  (10) associated to qˆT, . Arguing as in [2], we take in (35) v = v where vˆ = sgn(ˆ q )|ˆ q |r −1 |ω , then, we find a solution yˆ satisfies ˆ y (·, T )L2 ≤ .

(42)

qT, ) ≤ J (0) = 0. By (39), we obtain In contrast, at the minimum qˆT, , we have J (ˆ     2/r 1 r |q| dxdt ≤− qˆ (x, 0)y0 (x)dx − ξ qˆ dxdt 2 ω×(0,T ) Ω Q    1/2 M )|ξ|2 dxdt exp( ≤ ˆ q (x, 0)L2 (Ω) y0 (x)L2 (Ω) + t(T − t) Q   1/2 M )|q|2 dxdt · exp(− . (43) t(T − t) Q By (17), (28), and (33), we obtain |ˆ v |Lr (ω×(0,T )) ≤ C exp(K(T, a∞ , B∞ ))(y0 L2 (Ω) + ξL2 (Q) ).

(44)

Since vˆ is uniformly bounded in Lr (ω × (0, T )), we choose a subsequence, still denoted by itself, and deduce that vˆ → vˆ weakly in Lr (ω × (0, T )), (45) where vˆ ∈ Lr (ω × (0, T )) and satisfies (38). Accordingly, yˆ (·, T ) → yˆ(·, T ), where yˆ is the solution of (35) associated to vˆ. Since we have (42) for all  > 0, then we obtain yˆ(·, T ) = 0 in Ω. This ends the proof. Remark 3.1 Applying the same argument of [6], it is also possible to obtain the control in L∞ (ω × (0, T )). Proof of Theorem 1.1 Let us consider a trajectory y ∗ , solution of (5). Let u = y − y ∗ , where y is the solution of system (1). Then, we have that ⎧ ⎪ ⎪ ⎨ ut − u + F (x, t; u, ∇u) = v1ω in Q, ⎪ ⎪ ⎩

u=0

on Σ,

u(x, 0) = u0 (x)

in Ω,

(46)

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where u0 = y0 − y0∗ and F (x, t; s, p) = f (y ∗ (x, t) + s, ∇y ∗ (x, t) + p)) − f (y ∗ (x, t), ∇y ∗ (x, t)), for ˜ t; s, p) · p, where all (x, t) ∈ Q, (s, p) ∈ R × RN . Note that F (x, t; s, p) = g˜(x, t; s, p)s + G(x, 

1

∂f ∗ (y (x, t) + λs, ∇y ∗ (x, t) + λp)dλ, ∂s

(47)

∂f ∗ (y (x, t) + λs, ∇y ∗ (x, t) + λp)dλ, for 1 ≤ i ≤ N. ∂pi

(48)

g˜(x, t; s, p) = 0



1

˜ t; s, p) = G(x, 0

By (3), we easily know that lim

|(s,p)|→∞

lim

|(s,p)|→∞

1 log

3/2

(1 + |s| + |p|) 1

log1/2 (1 + |s| + |p|)





1

0





0

1

∂f

(s0 + λs, p0 + λp)dλ = 0, ∂s

(49)

∂f

(s0 + λs, p0 + λp)dλ = 0 ∂pi

(50)

uniformly in (s0 , p0 ) ∈ K, for every compact set K ⊂ R × RN . Theorem 1.1 would be proved after we show that, if for each u0 ∈ L2 (Ω), there exists v ∈ Lr (ω × (0, T )), such that u(x, T ) = 0.

(51)

˜ ∈ C 0 (R × RN )N . For We first consider the case in which u0 ∈ L∞ (Ω) and g˜ ∈ C 0 (R × RN ), G each  > 0, there exists C > 0, such that ˜ t; s, p)|2 |˜ g(x, t; s, p)|2/3 + |G(x, ≤ C +  log(1 + |s| + |p|), ∀ (x, t) ∈ Q, ∀ (s, p) ∈ R × RN .

(52)

¯ ∩ Lr (0, T ; W 1,r (Ω)) and R > 0 be a constant whose value will be Let us set Z = C 0 (Q) determined below. We consider the truncation function TR : R → R and TR : RN → RN which is given respectively by ⎧ ⎨ s, if |s| ≤ R; TR (s) = ⎩ Rsgn(s) otherwise. and TR (p) = (TR (pi ))1≤i≤N , ∀ p ∈ RN .

(53)

For each z ∈ Z, we consider the linear system ⎧ ˜ ⎪ ⎪ ⎨ ut − u + G(x, t; TR (z), TR (∇z)) · ∇u + g˜(x, t; TR (z), TR (∇z))u = v1ω u=0 ⎪ ⎪ ⎩ u(x, 0) = u0 (x) We have that (54) is of form (35), with ⎧ ⎨ a = a = g˜(x, t; T (z), T (∇z)) ∈ L∞ (Q), z R R ⎩ B = Bz = G(x, ˜ t; TR (z), TR (∇z)) ∈ L∞ (Q)N .

in Q, on Σ,

(54)

in Ω.

(55)

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Consequently, we can apply Theorem 3.1 to (54). In fact, we will apply this result in a time interval [0, Tz ], where Tz = min{T, ˜ g(x, t; TR (z), TR (∇z))−2/3 , ˜ g(x, t; TR (z), TR (∇z))−1/3 },

(56)

here the subindex z denotes that it depends on z. This is a key point in this proof that will derive to approximate estimates (the idea is taken from [4]). By Theorem 3.1, we obtain the existence of the control vˆz ∈ Lr (ω × (0, Tz )), such that the solution uˆz of (54) in Ω × (0, Tz ) with v = vˆz satisfies u ˆz (x, Tz ) = 0 in Ω. (57) Moreover, ˆ vz Lr (ω×(0,Tz )) ≤ C exp[K(T, a∞, B∞ )]u0 L2 (Ω) .

(58)

Let us extend by zero uˆz and vˆz to the whole cylinder Q = Ω × (0, T ), which we still call uˆz and vˆz . It is clear that u ˆz is the corresponding solution of (54) associated to vˆz and u ˆz (x, T ) = 0

in Ω.

(59)

Note that the null controllability problem of (54) is equivalent to ⎧  ˜ ⎪ ⎪ ⎨ pt − p + G(x, t; TR (z), TR (∇z)) · ∇p + g˜(x, t; TR (z), TR (∇z))p = −η (t)U + v1ω ⎪ ⎪ ⎩

in Q,

p=0

on Σ,

p(x, 0) = 0

in Ω, (60)

which verifies p(x, T ) = 0 in Ω,

(61)  where p = u − η(t)U, η(t) ∈ C0∞ ([0, T ]) satisfies η ≡ 1 on 0, T3 , η ≡ 0 on 2T 3 , T , and U 

solves (54) with v = 0. Suppose that there exists a control v˜ ∈ L2 (ω × (0, T )) solving (60) with supp˜ v ⊂ B0 × [0, T ], B0 ⊂⊂ ω, is a nonempty open set, and let p˜ be the corresponding state, then p = (1 − θ(x))˜ p together with v = θ(x)η  (t)U + 2∇θ · ∇˜ p + θp˜ − (B · ∇θ)˜ p

(62)

solves the null controllability problem (60) where θ ∈ D(ω) verifies θ ≡ 1 in B0 . By classical energy estimation, we have p˜ ∈ Y = L2 (0, T ; H 2(Ω) ∩ H01 (Ω)) ∩ C([0, T ]; H01 (Ω)), ˜ t; TR (z), TR (∇z))2 ](u0 2 + ˜ ˜ pY ≤ exp[C(1 + ˜ g (x, t; TR (z), TR (∇z))∞ + G(x, v 2 ). ∞ Applying Proposition 2.1 in [5] and letting B0 ⊂⊂ B1 ⊂ ω, we have p˜ ∈ X r (0, T ; Ω\B¯1), ˜ pX r (0,T ;Ω\B¯1 ) ˜ t; TR (z), TR (∇z))2∞ ](u0 2 + ˜ ≤ exp[C(1 + ˜ g(x, t; TR (z), TR (∇z))∞ + G(x, v 2 ). In contrast, arguing as in [8], we know that ˜ t; TR (z), TR (∇z))2∞ ]u0 2 . ˜ v 2 ≤ C exp[(1 + ˜ g(x, t; TR (z), TR (∇z))∞ + G(x,

No.5

Y.J. Xu & Z.H. Liu: CONTROLLABILITY FOR A PARABOLIC EQUATION

1603

¯ B¯1 , we have p ∈ X r and Then, since supp(1 − θ(x)) ⊂ Ω\ ˜ t; TR (z), TR (∇z))2 ]u0 2 . g(x, t; TR (z), TR (∇z))∞ + G(x, pX r ≤ exp[C(1 + ˜ ∞ By applying Lemma 2.2 in [5], the space X r , r > C 0 (Q). Then, we have p ∈ Z and

N 2

+ 1, might be compactly embedded into

˜ t; TR (z), TR (∇z))2 ]u0 2 . g(x, t; TR (z), TR (∇z))∞ + G(x, pZ ≤ exp[C(1 + ˜ ∞

(63)

By (62), we obtain v ∈ Lr and ˜ t; TR (z), TR (∇z))2∞ ]u0 2 . g(x, t; TR (z), TR (∇z))∞ + G(x, vLr ≤ exp[C(1 + ˜

(64)

For any given v ∈ Lr (ω × (0, Tz ), pv ∈ Z is the solution of (60) in Q. Define A : Z → A(z) ∈ Lr (Q) with A(z) = {v ∈ Lr (ω × (0, Tz )) : p satisfies (60), p(·, T ) = 0, v verifies (64)},

(65)

and let Λ be the set-valued mapping defined on Z by Λ(z) = {pv : pv satisfies (60), v ∈ A(z), pv verifies (63)}.

(66)

Let us prove that Λ fulfills the assumptions of Kakutani’s fixed point theorem. First, we can check that Λ(z) is a nonempty set; moreover, ∀z ∈ Z, Λ(z) is a uniformly bounded closed β convex subset of X r . By Lemma 2.2 in [5], X r might be compactly embedded into C β, 2 (Q), β = ¯ R). 2 − Nr+2 . Then, there exists a compact set K ⊂ Z such that Λ(z) ⊂ K, ∀z ∈ B(0, Let us now prove that Λ is an upper semicontinuous multivalued mapping, that is to say, ¯ R) → sup μ, p is for any bounded linear form μ ∈ Z  , the real-valued function z ∈ B(0, u∈Λ(z)

upper semicontinuous. Correspondingly, let us show that

 ¯ R) : sup μ, p ≥ λ Bλ,μ = z ∈ B(0, p∈Λ(z)

is a closed subset of Z, for any λ ∈ R, μ ∈ Z  . To this end, we consider a sequence {zn } ⊂ Bλ,μ , such that zn → z ∈ Z. Our aim is to prove that z ∈ Bλ,μ . Since all sets Λ(zn ) are compact, then for any n ≥ 1, there exists pn ∈ Λ(zn ), such that μ, pn  = sup μ, p ≥ λ. From the p∈Λ(z)

definitions of Λ(zn ) and A(z), let vn ∈ A(z), pn ∈ Λ(zn ) solves (60) with control vn , such that pn → p¯ strongly in Z,

vn → v¯ weakly in Lr (Q).

˜ t; TR (z), TR (∇z)) are continuous functions, we have If g˜(x, t; TR (z), TR (∇z)), G(x, g˜(x, t; TR (zn ), TR (∇zn )) → g˜(x, t; TR (z), TR (∇z)) in C 0 (Q),

(67)

˜ t; TR (z), TR (∇z)) in C 0 (Q). ˜ t; TR (zn ), TR (∇zn )) → G(x, G(x,

(68)

Let n → ∞, then we obtain p¯ solves (60) with control function v¯. Moreover, p¯ and v¯ satisfy (63), (58), respectively, that is, v¯ ∈ A(z), p¯ ∈ Λ(z). Then, let n → ∞, we have, sup μ, p ≥ μ, p¯ ≥ λ. p∈Λ(z)

(69)

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ACTA MATHEMATICA SCIENTIA

Vol.30 Ser.B

So z ∈ Bλ,μ and hence, Λ(z) is upper semicontinuous. Finally, we have that there exists R > 0, such that ¯ R)) ⊂ B(0, ¯ R). Λ(B(0, (70) ¯ R) ⊂ Z, from (63) and (52), we obtain For any z ∈ B(0, pZ ≤ exp[C(1 + C + 2 log(1 + 2R))]u0 2 ≤ exp[C(1 + C )](1 + R)C u0 ∞ .

(71)

1 Taking  = 2C , we get pZ ≤ C(1 + R)1/2 u0 ∞ . Thus, we get (70) if R is large enough. Applying Kakutani fixed-point theorem, we obtain that there exists p ∈ Z, such that p ∈ Λ(z). Note that the definition of p and (54), we conclude that there exists u ∈ Z, such that

u ∈ {uv : vLr (ω×(0,T )) ≤ exp[C(1 + ˜ g(x, t; TR (z), TR (∇z))∞ ˜ t; TR (z), TR (∇z))2∞ ]u0 2 , uv Z ≤ exp[C(1 + ˜ +G(x, g(x, t; TR (z), TR (∇z))2/3 ∞ ˜ t; TR (z), TR (∇z))2 ]u0 2 }. +G(x, ∞

(72)

˜ t; TR (z), TR (∇z)) are not continuous functions, arguing as in [5], If g˜(x, t; TR (z), TR (∇z)), G(x, there exists a control v ∈ Lr (ω × (0, T )), such that (46) possesses a solution u satisfying (51). In the following, if u0 ∈ L2 (Ω), for δ > 0 small enough. Set v ≡ 0 for t ∈ (0, δ). Applying the regularizing effect of the heat equation, we conclude that the corresponding solution u of (54) satisfies u(·, δ) ∈ L∞ (Ω). Then, we argue as above for p(·, δ) in [δ, T ] and we get control v ∈ Lr (0, T ; Lr (ω)), such that (64) holds (see [9], [10]). This ends the proof of Theorem 1.1. References [1] Barbu V. Exact controllability of the superlinear heat equation. Appl Math Optim, 2000, 42: 73–89 [2] Fabre C, Puel J P, Zuazua E. Approximate controllability of the semilinear heat equation. Proc Roy Soc Edinburgh Sect A, 1995, 125: 31–61 [3] Doubova A, Fern´ andez-Cara E, Gonza´ aez-Burgos M, Zuazua E. On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM J Control Optim, 2003, 41(3): 798–819 [4] Bodart O, Gonza´ aez-Burgos M, Pe´eez-Garc a R. Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient. Nonlinear Anal, 2004, 57(5/6): 687–711 [5] Bodart O, Gonza´ aez-Burgos M, Pe´eez-Garc A R. Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. Comm Partial Differ Equs, 2004, 29(7/8): 1017–1050 [6] Fern´ andez-Cara E, Zuazua E. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann Inst Henri Poincar´e, Analyse non lin´eaire, 2000, 17(5): 583–616 [7] Barbu V. Controllability of parabolic and Navier-Stokes equations. Scientia Mathematica Japonica, 2002, 6: 143–211 [8] Doubova A, Osses A, Puel J -P. Exact controllability to trajectories for semilinear heat equations with discontinuous coefficients. ESAIM: COCV, 2002, 8: 621–661 [9] Weissler F B. Local existence and nonexistence for semilinear parabolic equations in Lp . Indiana Univ Math J, 1980, 29(1): 79–102 [10] Weissler F B. Semilinear evolution equations in Banach spaces. J Funct Anal, 1979, 32(3): 277–296 [11] Xu Y J, Liu Z H, Exact Controllability to trajectories for a semilinear heat equation with a superlinear nonlinearity. Acta Appl Math, 2010, 110(1): 57–71 [12] Li Tatsien, Rao Bopeng. Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics. Acta Math Sci, 2009, 29B(4): 980–990