3. Proof of Theorem 1.1

DOI:10.1051/cocv/2015030 www.esaim-cocv.org

THE LEBEAU–ROBBIANO INEQUALITY FOR THE ONE-DIMENSIONAL FOURTH ORDER ELLIPTIC OPERATOR AND ITS APPLICATION

Peng Gao

Abstract.In this paper, we establish the Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator by using a point-wise estimate. Based on this inequality, we obtain the null controllability of one-dimensional stochastic fractional order Cahn–Hilliard equation.

Mathematics Subject Classification. 93B05, 35K35, 60H15.

Received August 14, 2014. Revised April 25, 2015.

Published online May 27, 2016.

1. Introduction

In this paper, we investigate the one-dimensional fourth order elliptic operatorA onL²(I) as follows D(A) =H₀²(I)∩H⁴(I),

Ay=y_xxxx ∀y∈ D(A), whereI= (0,1).

Let{λi}^∞_i=1,0< λ₁≤λ₂≤. . . be the eigenvalues ofA and{ei}^∞_i=1 be the corresponding real eigenfunctions such thate_iL²_(I)= 1 (i= 1,2,3, . . .),which serves as an orthonormal basis ofL²(I) (see [13], Thm. 8.94).

The main result in this paper is an observability estimate on partial sums of eigenfunctions for the eigen- functions ofA,i.e. the Lebeau–Robbiano inequality:

Theorem 1.1. Let ω be a nonempty open subset ofI. There exist two positive constantsC₁, C₂ such that

λi≤r

|a_i|²≤C₁e^C2√r

ω

λi≤r

a_ie_i

2

dx

for every finiter >0 and every choice of{a_i}_λi≤r with a_i ∈C.

Keywords and phrases.Lebeau–Robbiano inequality, null controllability, stochastic fractional order Cahn–Hilliard equation.

1 School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P.R. China

2 Institute of Mathematics, Jilin University, Changchun 130012, P.R. China.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2016

The Lebeau–Robbiano inequality was first established in [5] for the Laplacian with homogeneous Dirichlet boundary condition, then the same result for more general boundary conditions and second order elliptic oper- ators were established in [7,11].

In the study on the controllability of PDE, the inequality of this type is the foundation of Lebeau–Robbiano iteration technology. By this inequality, [5] obtained the null controllability of a linear system of thermoelasticity in a compact,C^∞,n-dimensional connected Riemannian manifold. Then, this inequality was used to establish a certainL^∞-null controllability for the internal controlled heat equation when the control functions are restricted in arbitrary subset of positive measure in time variable in [14]. Later, in [10],viathe Lebeau–Robbiano inequality, the authors presented a time optimal control problem with control constraints of the rectangular type for internal controlled heat equations. Recently, the similar result as in [14] was extended to forward stochastic heat equations in [9] and second order parabolic equation with equivalued surface boundary conditions in [11].

In [3], the Lebeau–Robbiano inequality can also be used to depict nodal sets of sums of functions.

However, to the best of our knowledge, the Lebeau–Robbiano inequality for fourth order elliptic operator has not been established. It should be noted that Theorem1.1in this paper is not a straightforward generalization of the Lebeau–Robbiano inequality in [4], although the inequality has the same formal. To prove Theorem1.1, a point-wise estimate for the operator ∂_t²−∂⁴_x in Proposition 2.1 plays a crucial role. In order to obtain the point-wise estimate, we need to choose suitable grouping for the terms ofθ(y_tt−y_xxxx) (see I₁ andI₂ in the proof of Prop.2.1) since the high order of the operator. Due to the special properties of the operator∂_t²−∂_x⁴ and the boundary terms in the point-wise estimate, we need to construct several weight functions.

Based on Theorem1.1, we can obtain the second result in this paper: the null controllability of one-dimensional stochastic fractional order Cahn–Hilliard equation.

First, we give some assumptions.

(H1) Let (Ω,F,{Ft}t≥0, P) be a complete filtered probability space on which a one-dimensional standard Brownian motion{B(t)}_t≥0is defined such that {F_t}_t≥0is the natural filtration generated byB(·), aug- mented by all the P-null sets in F.Let H be a Banach space, and letC([0, T];H) be the Banach space of allH-valued strongly continuous functionals defined on [0, T].We denote byL²_F(0, T;H) the Banach space consisting of all H-valued {Ft}t≥0-adapted processes X(·) such that E(X(·)²_L2(0,T;H)) < ∞, with the canonical norm; by L^∞_F(0, T;H) the Banach space consisting of all H-valued{Ft}t≥0-adapted bounded processes; and by L²_F(Ω;C[0, T];H)) the Banach space consisting of all H-valued {F_t}_t≥0- adapted continuous processes X(·) such that E(X(·)²_C([0,T];H)) < ∞, with the canonical norm; by L^r_F(0, T;L²(Ω;H)) the Banach space consisting of allH-valued{F_t}_t≥0-adapted processesX such that EX²_HL^r_(0,T)<∞(1≤r≤ ∞),with the canonical norm.

(H2) E⊂[0, T] with positive measure.χ_ω andχ_E denote the characteristic function ofω andE,respectly.

(H3) We define the operatorA^αas follows

⎧⎪

⎪⎨

⎪⎪

⎩

D(A^α) =

u∈L²(I)|u=

i≥1a_ie_i and

i≥1|ai|²λ^2α_i <+∞

, A^αu=

i≥1a_iλ^α_ie_i whereu=

i≥1a_ie_i. (H4) a(t)∈L^∞_F(0, T;R), ξ=a²_L∞

F(0,T;R).

(H5) For eachr >0,we setX_r = span{ei(x)}λi≤r and denote byP_r the orthogonal projection fromL²(I) to X_r.

(H6) Throughout this paper, C(and sometimes C) denotes various positive constants. C(. . .) stands for a positive constant that depends on what are enclosed in the brackets.

Our second main result in this paper is the following theorem.

Theorem 1.2. Let α > ¹₂. For any y₀ ∈ L²(Ω,F₀, P;L²(I)), there is a control f ∈ L^∞_F(0, T;L²(Ω;L²(I))) such that the solutiony of system

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

dy+A^αy=a(t)ydB+χ_ωχ_Efdt y(0, t) = 0 =y(1, t)

y_x(0, t) = 0 =y_x(1, t) y(x,0) =y₀(x)

inI×(0, T), in(0, T), in(0, T), inI,

(1.1)

satisfiesy(T) = 0in I, P-a.s. Moreover, the controlf satisfies the following estimate f²_L^∞

F(0,T;L²(Ω;L²(I)))≤LEy0²_L2(I), whereL is a constant independent ofy₀.

Remark 1.3. We refer to ([1], Chap. 6) for the well-posedness of system (1.1) in the class L²_F(Ω;C([0, T];L²(I))).

The rest of this paper is organized as follows. In Section 2, we establish a point-wise estimate. Theorem1.1 is proved in Section 3 according to two important lemmas. Section 4 is devoted to the proof of Theorem1.2.

2. A point-wise estimate

In this section, we give a point-wise estimate which will be used in the proof of Theorem1.1and the estimate itself has independent interest.

Proposition 2.1. Setθ= e^l, l=λϕ, ϕ= e^μψ andu=θyinQ=I×(0, T),whereψ∈C^∞(Q)andy∈C⁴(Q).

Assume thatμ≥1andλ≥λ₀,whereλ₀=λ₀(μ, ψ)such thatλμ⁻¹ϕ≥1. Then we have the following point-wise estimate

λ⁷μ⁸ϕ⁷ψ_x⁸u²+λ⁵μ⁶ϕ⁵ψ⁶_xu²_x+λ³μ⁴ϕ³ψ_x⁴u²_xx+λμ²ϕψ²_xu²_xxx

+λ³μ⁴ϕ³ψ_x⁴u²_t+λμ²ϕψ_x²u²_xt+{. . .}_x+{. . .}_xx+{. . .}_xxx+{. . .}_xxxx+{. . .}_t+{. . .}_tt

≤C(ψ)

λ⁷μ⁷ϕ⁷u²+λ⁵μ⁵ϕ⁵u²_x+λ³μ³ϕ³u²_xx+λμϕu²_xxx+λ³μ³ϕ³u²_t+λμϕu²_xt+θ²|ytt−y_xxxx|²

, (2.1) where

{. . .}_x=

−3

2B_2xxu²_x+3

2B₂u²_xx−1

2B₄u²_xxx−4a_xu²_x+ 2a_xxxu²+b_xu²_xx−B₀u_tu_xxx+1

2B₁B₂u² +3

2(B₁B₄)_xxu²−3

2B₁B₄u²_x−(bB₁)_xu²+1

2B₂B₃u²_x+1

2B₃B₄u²_xx−(aB₃)_xu²+B₀B₃u_tu_x

−1

2B₂u²_t−3

2B_4xxu²_t+3

2B₄u²_xt−b_tu_tu_x+b_xu²_t−(B₄c)_xu²_x+1

2acu²+1 2bcu²_x

x

, {. . .}xx=

3

2B_2xu²_x−3axxu²+2au²_x−b 2u²_xx−3

2(B₁B₄)_xu²+1

2bB₁u²+1

2aB₃u²+3

2B_4xu²_t−1 2bu²_t+1

2B₄cu²_x

xx

, {. . .}_xxx=

−1

2B₂u²_x+ 2a_xu²+1

2B₁B₄u²−1 2B₄u²_t

xxx

, {. . .}_xxxx=

−1 2au²

xxxx

, {. . .}t=

1

2B₀B₁u²−1

2B₀B₃u²_x+B₂u_tu_x+1

2B₀u²_t−a_tu²+B₄u_tu_xxx+bu_tu_xx+1 2b_tu²_x

t

, {. . .}_tt=

1 2au²

tt

,

a= 4l_x²l_xx, b= 8l_xx, c=−10l_xl_xx, B₀=−2l_t, B₁=−l⁴_x, B₂= 4l³_x, B₃=−6l_x², B₄= 4l_x.

Remark 2.2. The key points in the proof of Proposition2.1are suitable grouping and the choice ofa, bandc.

Proof. From the assumptions ofλ, μ, ϕandl,we can obtain that

⎧⎪

⎨

⎪⎩

λⁿμ^mϕⁿ≤λⁿ⁺ⁱμ^m−iϕⁿ⁺ⁱ≤λⁿ⁺ⁱμ^m+iϕⁿ⁺ⁱ, ∀m, n∈Z and i∈N,

|lx . . . x

k

t . . . t j

| ≤C(ψ)λμ^k+jϕ, ∀k, j∈N, (2.2)

wherelx . . . x

k

t . . . t j

= _∂^∂_k^k+j_x∂j^l_t· Direct computation shows that

θ(y_tt−y_xxxx) =u_tt+A₀u_t+A₁u+A₂u_x+A₃u_xx+A₄u_xxx−u_xxxx where

A₀=−2l_t,

A₁=l_t²−l_tt−l⁴_x−4l_xl_xxx+l_xxxx+ 6l_x²l_xx−3l²_xx, A₂=−12lxl_xx+ 4l³_x+ 4l_xxx,

A₃=−6l²_x+ 6l_xx, A₄= 4l_x.

Set I₁=−u_xxxx+B₁u+B₃u_xx+u_tt+cu_x, I₂=B₂u_x+B₄u_xxx+B₀u_t+au+bu_xx,

R=θ(y_tt−y_xxxx)−I₁−I₂=S₀u+S₁u_x+S₂u_xx, where

S₀=l_t²−l_tt−4l_xl_xxx+l_xxxx−3l²_xx+ 2l²_xl_xx, S₁= 4l_xxx−2l_xl_xx, S₂=−2l_xx.

Step 1.We shall prove the following equality

I₁·I₂=u²{. . .}+u²_x{. . .}+u²_xx{. . .}+u²_xxx{. . .}+u²_t{. . .}+u²_xt{. . .}+u_xtu_xxx{. . .}

+u_tu_x{. . .}+u_tu_xxx{. . .}+{. . .}_x+{. . .}_xx+{. . .}_xxx+{. . .}_xxxx+{. . .}_t+{. . .}_tt, (2.3) where

u²{. . .}=u²

−1

2axxxx−1

2(B1B2)_x−1

2(B1B4)_xxx+1

2(bB1)_xx−1

2(B0B1)_t+1

2(aB3)_xx+1

2att+aB1−1 2(ac)_x

, u²_x{. . .}=u²_x1

2B2xxx+2axx+3

2(B1B4)_x−bB1−1

2(B2B3)_x−aB3+1

2(B0B3)_t−1

2btt+B2c+1

2(B4c)_xx−1 2(bc)_x

,

u²_xx{. . .}=u²_xx

−3

2B2x−a−1 2bxx−1

2(B3B4)_x+bB3−B4c

,

u²_xxx{. . .}=u²_xxx 1

2B4x+b

, u²_t{. . .}=u²_t1

2B2x−1

2B0t−a+1

2B4xxx−1 2bxx

,

u²_xt{. . .}=u²_xt

−3 2B4x+b

, uxtuxxx{. . .}=uxtuxxx{B0},

utux{. . .}=utux{−(B0B3)_x−B2t+bxt+B0c}, utuxxx{. . .}=utuxxx{B0x−B4t},

and{. . .}_x,{. . .}_xx,{. . .}_xxx,{. . .}_xxxx,{. . .}_t,{. . .}_ttare the same as in Proposition2.1.

Indeed, (2.3) can be obtained from the following equations u_xxxxB₂u_x= 1

2

B₂u²_x

xxx−3 2

B_2xu²_x

xx+3 2

B_2xxu²_x−B₂u²_xx

x+3

2B_2xu²_xx−1

2B_2xxxu²_x, u_xxxxB₄u_xxx= 1

2

B₄u²_xxx

x−B_4xu²_xxx , u_xxxxau= 1

2 au²

xxxx−2 a_xu²

xxx+

3a_xxu²−2au²_x

xx

+

4a_xu²_x−2a_xxxu²

x+au²_xx−2a_xxu²_x+1

2a_xxxxu², u_xxxxbu_xx= 1

2 bu²_xx

xx− b_xu²_xx

x−bu²_xxx+1 2b_xxu²_xx, u_xxxxB₀u_t= (B₀u_tu_xxx)_x−B_0xu_tu_xxx−B₀u_xtu_xxx,

B₁uB₂u_x= 1 2

B₁B₂u²

x−(B₁B₂)_xu² , B₁uB₄u_xxx= 1

2

B₁B₄u²

xxx−3 2

(B₁B₄)_xu²

xx+3

2(B₁B₄)_xu²_x +3

2

(B₁B₄)_xxu²−B₁B₄u²_x

x−1

2(B₁B₄)_xxxu², B₁ubu_xx= 1

2

bB₁u²

xx−

(bB₁)_xu²

x−bB₁u²_x+1

2(bB₁)_xxu², B₁uB₀u_t= 1

2

B₀B₁u²

t−(B₀B₁)_tu² , B₃u_xxB₂u_x= 1

2

B₂B₃u²_x

x−(B₂B₃)_xu²_x , B₃u_xxB₄u_xxx= 1

2

B₃B₄u²_xx

x−(B₃B₄)_xu²_xx , B₃u_xxau= 1

2

aB₃u²

xx−

(aB₃)_xu²

x−aB₃u²_x+1

2(aB₃)_xxu², B₃u_xxB₀u_t= 1

2(B₀B₃)_tu²_x−(B₀B₃)_xu_tu_x−1 2

B₀B₃u²_x

t+ (B₀B₃u_xu_t)_x, u_ttB₂u_x= (B₂u_tu_x)_t−B_2tu_tu_x−1

2

B₂u²_t

x−B_2xu²_t , u_ttB₀u_t= 1

2

B₀u²_t

t−B_0tu²_t , u_ttau= 1

2 au²

tt− a_tu²

t−au²_t+1 2a_ttu², u_ttB₄u_xxx= (B₄u_tu_xxx)_t−B_4tu_tu_xxx−1

2

B₄u²_t

xxx+3 2

B_4xu²_t

xx

−3 2

B_4xxu²_t−B₄u²_xt

x−3

2B_4xu²_xt+1

2B_4xxxu²_t, u_ttbu_xx= (bu_tu_xx)_t+b_xtu_tu_x−1

2b_ttu²_x+1 2

b_tu²_x

t−(b_tu_tu_x)_x−1 2

bu²_t

xx+ b_xu²_t

x+bu²_xt−1 2b_xxu²_t, B₄u_xxxcu_x= 1

2

B₄cu²_x

xx−

(B₄c)_xu²_x

x−B₄cu²_xx+1

2(B₄c)_xxu²_x, aucu_x= 1

2

acu²

x−(ac)_xu² , bu_xxcu_x= 1

2

bcu²_x

x−(bc)_xu²_x .

Step 2.We shall prove estimate (2.1).

Indeed,

u²{. . .}=u²{10λ⁷μ⁸ϕ⁷ψ⁸_x+R₀}, u²_x{. . .}=u²_x{22λ⁵μ⁶ϕ⁵ψ⁶_x+R₁}, u²_xx{. . .}=u²_xx{6λ³μ⁴ϕ³ψ_x⁴+R₂}, u²_xxx{. . .}=u²_xxx{10λμ²ϕψ²_x+R₃}, u²_t{. . .}=u²_t{2λ³μ⁴ϕ³ψ_x⁴+R₄}, u²_xt{. . .}=u²_xt{2l_xx},

u_xtu_xxx{. . .}=u_xtu_xxx{2l_t},

u_tu_x{. . .}=u_tu_x{−24lxtl²_x−4l_tl_xl_xx+ 8l_xxxt},

u_tu_xxx{. . .}=u_tu_xxx{−6l_xt}, (2.4) where

|R0| ≤Cλ⁷μ⁷ϕ⁷, |R1| ≤Cλ⁵μ⁵ϕ⁵, |R2| ≤Cλ³μ³ϕ³,

|R₃| ≤Cλμϕ, |R₄| ≤Cλ³μ³ϕ³. (2.5)

Here we have used (2.2). Similarly, we have

|u_xtu_xxx{. . .}| ≤C

λμϕu²_xxx+λμϕu²_xt ,

|u_tu_x{. . .}| ≤C

λ⁵μ⁵ϕ⁵u²_x+λ³μ³ϕ³u²_t ,

|utu_xxx{. . .}| ≤C

λμϕu²_xxx+λ³μ³ϕ³u²_t , S₀²u²≤Cλ⁷μ⁷ϕ⁷u²,

S₁²u²_x≤Cλ⁵μ⁵ϕ⁵u²_x,

S₂²u²_xx≤Cλ³μ³ϕ³u²_xx, (2.6)

this implies

I₁I₂≤ 1

2(I₁+I₂)²

= 1

2[θ(y_tt−y_xxxx)−R]²

≤θ²|ytt−y_xxxx|²+|R|²

≤C

θ²|ytt−y_xxxx|²+S₀²u²+S₁²u²_x+S₂²u²_xx

≤C

θ²|y_tt−y_xxxx|²+λ⁷μ⁷ϕ⁷u²+λ⁵μ⁵ϕ⁵u²_x+λ³μ³ϕ³u²_xx

. (2.7)

Thanks to (2.4)–(2.7) all together with (2.3), we can obtain (2.1).

3. Proof of Theorem 1.1

Through this section, we make the following assumptions:

(A1) LetT₀, T, T, γ, b₀ satisfy

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

0< γ < T< ^T₂⁰ < T< T₀−γ, ^T0

2 −T=T−^T₂⁰, max

T−^T₂⁰,

1 +_T0

2 −T₂¹₂

< b₀< ^T₂⁰ −γ.

(A2) For a measurableF ⊂Q₀I×(0, T₀) and a functiongsatisfiesg, g_x, g_xx, g_xxx, g_t, g_xt∈L²(F),we set g,F =

g²_L2(F)+gx²_L2(F)+gxx²_L2(F)+gxxx²_L2(F)+gt²_L2(F)+gxt²_L2(F)

¹₂ . (A3) DefineG{g∈L²(Q₀)|g_x, g_xx, g_xxx, g_xxxx, g_t, g_xt∈L²(Q₀)}.

Remark 3.1. The existence ofT₀, T, T, γ andb₀ is easy to obtain, for example,T₀= 4, T = 1, T= 3, γ= 0.5, b₀= 1.42 satisfies (A1).

We borrow some idea from [11].

In order to prove Theorem1.1, we investigate the following system

⎧⎪

⎨

⎪⎩

y_tt−y_xxxx= 0 y(0, t) = 0 =y(1, t) y_x(0, t) = 0 =y_x(1, t)

inQ₀, in (0, T₀), in (0, T₀).

(3.1)

First, we establish two important lemmas.

Lemma 3.2. There exists a constant α₁∈(0,1) such that we have the following estimate

y_L²_(I×(T,T))≤Cy^α_,ω×(γ,T¹ _0−γ)y^1−α_,Q0¹, (3.2) wherey∈ G solves (3.1).

Proof. Introduceφ∈C₀^∞(γ, T₀−γ) andψ₀∈C^∞(I) which enjoy the following properties

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

0≤φ(t)≤1 t∈(γ, T₀−γ), φ(t) = 1 t∈(^T₂⁰ −b₀,^T₂⁰ +b₀), ψ₀(x)>0 x∈I,

|ψ_0x|>0 x∈I\ω,

ψ₀(0) =ψ₀(1) = 0, ψ_0x(0)>0, ψ_0x(1)<0, ψ0_C(I)= 1.

(3.3)

The existence ofψ₀ can be found in [15].

Setψ₁(x, t) =ψ₀(x)−(t−^T₂⁰)²+C₁,whereC₁ is chosen to be so large to makeψ₁>0.According to (A1), it is obvious that (T, T)⊂(^T₂⁰ −b₀,^T₂⁰ +b₀).

We here letψ(x, t) =ψ₁(x, t) in Proposition2.1and apply (2.1) toy=φyandw=θy,some straightforward calculation gives that

Q0

λ⁷μ⁸ϕ⁷ψ⁸_1xw²+λ⁵μ⁶ϕ⁵ψ_1x⁶ w²_x+λ³μ⁴ϕ³ψ⁴_1xw²_xx+λμ²ϕψ_1x² w²_xxx

+λ³μ⁴ϕ³ψ⁴_1xw²_t+λμ²ϕψ_1x² w_xt² +{. . .}x+{. . .}xx+{. . .}xxx+{. . .}xxxx+{. . .}t+{. . .}tt

dxdt

≤C(ψ)

Q0

(λ⁷μ⁷ϕ⁷w²+λ⁵μ⁵ϕ⁵w²_x+λ³μ³ϕ³w²_xx+λμϕw_xxx²

+λ³μ³ϕ³w²_t +λμϕw_xt² +θ²|y_tt−y_xxxx|²)dxdt. (3.4) It follows from (3.3) that

w(0, t) =w(1, t) =w_x(0, t) =w_x(1, t) = 0 ∀t∈(0, T₀)