# Local Null-Controllability for Some Quasi-Linear Phase-Field Systems with Neumann Boundary Conditions by one Control Force

Authors: Shengzhu Shi; Dazhi Zhang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this article, we study the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller in the frame of classical solutions. In order to minimize the number of control forces, we prove the Carleman inequality for the associated linear system. By constructing a sequence of optimal control problems and an iteration method based on the parabolic regularity, we find a qualified control in Hölder space for the linear system. Based on the theory of Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough.


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## Body

## 1. Introduction

In this paper, we are concerned with the local null-controllability for some quasi-linear phase-field systems with Neumann boundary conditions by a control force acting on an arbitrary small open setω⊂⊂Ω. Here Ω⊂ℝNN≥1 is a connected bounded domain with C2,θ boundary ∂Ω for some θ∈0,1. For a given T>0, we consider the cylindrical domain QT=Ω×0,T with lateral boundary ΣT=∂Ω×0,T, and by n=nx, we denote the outward unit normal vector to Ω at a point x∈∂Ω. Consider the following quasi-linear phase-field system:(1)wt+lψt−∑i,j=1N∂∂xjaijw∂w∂xi+fw,ψ=g,inQT,ψt−Δψ+hψ+rw=0,inQT,∂w∂nA=∑i,j=1Naijwnj∂w∂xi=0,∂ψ∂n=0,onΣT,wx,0=w0x,ψx,0=ψ0x,inΩ,where l∈ℝ, l≠0 is a constant, r∈Cθ,θ/2QT¯ and there exists a constant r0 such that r≥r0>0 in ω×0,T, w is the temperature function while ψ is the phase-field function, the initial datum w0,ψ0 is given in C2+θΩ¯2. And gx,t is a control function in the space(2)Uω=gx,t∈Cθ,θ/2QT¯:suppg⊂ω¯×0,T.Besides, we assume thatf⋅,⋅:ℝ×ℝ⟶ℝ is a C2 function with f0,0=0, h⋅:ℝ⟶ℝ is a C2 function with h0=0, aij⋅:ℝ⟶ℝ are C2 functions satisfying aij=aji and the uniform parabolic condition, i.e., there exist constants 0<Λ˜<Λ such that(3)Λ˜ξ2≤∑i,j=1Naijζξiξj≤Λξ2,∀ζ∈ℝ,ξ∈ℝN.In addition, we put(4)A=∑i,j=1Nsupζ≤1aijζ2+aij′ζ2+supη≤1h′η2+sup|ζ≤1,η≤1∂∂ζfζ,η2+∂∂ηfζ,η2+1.Phase-field systems model a large amount of physical phase transition phenomena. Usually, the phase-field functionψ describes the level of liquid solidification l>0 or solid crystallization l<0. In particular, the quasi-linear system (1) describes a kind of phenomenon where heat conduction coefficients aij depend on the temperature w in a manner as aij=aijw in the phase transition process. Different from Dirichlet boundary conditions which mean the phase on the boundary should be maintained, Neumann boundary conditions in system (1) describe that there is no flux of phases at the boundary. We refer to [1, 2] and the references therein for more detailed descriptions about the model.In recent years, lots of researchers have focused on addressing the controllability problems of linear and semilinear equations. For example, Barbu [3] investigated the local controllability of the phase-field system with two control forces and gave the observability estimate for the adjoint system with the global Carleman inequality established in [4]. The work of [3] illuminates us to expand the results of partial differential equations (a variety of results have been established, see for instance [5–10]) to coupled systems [11, 12] which model real physical or biological phenomena. In addition, controlling a system with a minimum number of forces is a common problem in the control theory [13, 14]. From this point of view, Ammar Khodja etc. gave the Carleman inequality for the adjoint system by constructing a suitable functional Λ with suitable weights (see Lemma 3.4 in [13]). It is worth noticing that their results are based on the Carleman inequality given in [15] which makes it possible to improve the regularity of the control function. This method has also been applied to solve the null-controllability for some general reaction-diffusion systems which arise in mathematical biology in [16]. Observing that [13] considered the system on a sub-domain of ℝN with 1≤N<6, González-Burgos and Pérez-García [17] considered a more general case with an arbitrary N≥1 and the nonlinearity f=fu,∇u,ϕ,∇ϕ. They used a method called strategy of fictitious control functions to construct a control function g∈LrQT, ∀r≥2. Furthermore, [18] studied the case of unbounded domains. In addition, [11] studied a phase-field system with Neumann boundary conditions and showed the relationship between the optimal control and the controllability.However, few work has been done to address the controllability of the quasi-linear form such as (1). The main problem is that the generalized solution of the quasi-linear system is not good enough, which makes the fixed point theorem not applicable. To overcome this difficulty, we would like to recall some work on the quasi-linear parabolic equation. To our knowledge, [19] proved the local null-controllability of a quasi-linear diffusion equation in one spacial dimension with the Sobolev embedding relation L∞0,T;H01Ω⊆L∞Ω×0,T which is only valid for one spacial dimension. And for the multidimensional case, [20] analyzed the controllability in the frame of classical solutions and gave a control in the Hölder space with given initial datum of high regularity and small enough. Enlightened by [19, 20], we also investigate the problem in the frame of classical solutions, while the coupled systems make the case more complex.As for boundary conditions, it is necessary to notice that different boundary conditions describe different physical phenomena, however, there is little difference on mathematic controllability discussion. Thanks to the study in [21], we can find that the Carlemen inequality with Neumann boundary conditions has an additional weight function compared to the one with Dirichlet boundary conditions. Since there are abundant discussions on the case with homogeneous Dirichlet boundary conditions, we pay attention to the system with homogeneous Neumann boundary conditions. Moreover, all of our results established in this paper are valid for homogeneous Dirichlt boundary conditions.As introduced in [22], in the general case, the observability for the adjoint system implies the controllability for the original system. To prove this, we pay attention to the variational method introduced in [4, 21]. We extend the method in [4] from the case of the parabolic equation to the case of coupled systems. Moreover, our adjoint system is different from the usual one because of the different optimal control problem.The paper is organized as follows. Section2 introduces a list of notations and presents preliminaries. In Section 3, we establish a Carleman inequality for the phase-field system which plays a key role in minimizing the number of control forces. In Section 4, we give an observability estimate for the adjoint system. In Section 5, we obtain a qualified control in L2QT by constructing a sequence of optimal control problems. Then we prove the control function belongs to Cθ,θ/2QT¯ by an iteration method based on the parabolic regularity and the embedding theorem. Finally, by utilizing the Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough in Section 6.


## 2. Preliminaries

In this section, we present some preliminaries. We first introduce some results on the existence, uniqueness and regularity of the solution for the associated linear phase-field system(5)L1w,ψ=wt+lψt−∑i,j=1N∂∂xjbijx,t∂w∂xi+qw+dψ=g,inQT,L2w,ψ=ψt−Δψ+eψ+rw=0,inQT,∂w∂nB=∑i,j=1Nbijx,tnj∂w∂xi=0,∂ψ∂n=0,onΣT,wx,0=w0x,ψx,0=ψ0x,inΩ,where g∈Uω is a control function to be determined, r≥r0>0 in ω×0,T, bij⋅,⋅∈C1+θ,θ/2QT¯ satisfy bij=bji and the uniform parabolic condition, i.e., there exist constants 0<Λ˜≤Λ such that(6)Λ˜ξ2≤∑i,j=1Nbijx,tξiξj≤Λξ2,∀x,t∈QT,ξ∈ℝN,and q,d,e,r∈Cθ,θ/2QT¯. In what follows, we denote(7)B=1+∑i,j=1NbijC1+θ,θ/2QT¯+qCθ,θ/2QT¯+dCθ,θ/2QT¯+eCθ,θ/2QT¯+rCθ,θ/2QT¯.Sincel is a constant, without loss of generality, we assume l>0 in the following demonstration.We use standard notationsLpQT, Wk,pQT, Wr2,1QT, H1QT in Sobolev space for the Lp estimate and Cθ,θ/2QT¯ in Hölder space for the Schauder estimate (for more information, one can see [23]). Moreover, we also need the following Hilbert spaces in [11].(8)V=f∈H2Ω:∂f∂n=0on∂Ω,VB=f∈H2Ω:∂f∂nB=0on∂Ω,and the compatibility condition in [24, 25].(9)∂w∂nBx,0=∂w0∂nBx=0,∂ψ∂nx,0=∂ψ0∂nx=0,on∂Ω.In the sequel, the symbolsc, C stand for various positive constants depending on different parameters and the values may change.By analogy with the proof of Theorem 3.1.2 and Theorem 3.4.2 in [23], we have the following lemma.Lemma 1.
Ifg∈L2QT,w0,ψ0∈H1Ω2,bi,j∈C1+θ,θ/2QT¯satisfybi,j=bj,i and the uniform parabolic condition (6),q,d,e,r∈Cθ,θ/2QT¯, there exists a unique weak solutionw,ψ∈W22,1QT2of the system (5) satisfying the estimate(10)wW22,1QT+ψW22,1QT≤expCBw0H1Ω+ψ0H1Ω+gL2QT,whereC=CΩ,N,l,T,Λ,Λ˜andBis given in (7).By recalling theLp theory of linear parabolic equations of second order (see, for instance, Theorem 6.4 in [26], Theorem 9.2.2 and Theorem 9.2.4 in [23]), we give the result on the Lp estimate.Lemma 2.
For anyp≥2,g∈LpQT,w0∈W2−2/p,pΩ∩VB,ψ0∈W2−2/p,pΩ∩V, bi,j∈C1+θ,θ/2QT¯ satisfy bi,j=bj,i and the uniform parabolic condition (6),q,d,e,r∈Cθ,θ/2QT¯, there exists a unique strong solutionw,ψ∈Wp2,1QT2 of the system (5) satisfying the estimate(11)wWp2,1QT+ψWp2,1QT≤expCBw0W2−2/p,pΩ+ψ0W2−2/p,pΩ+gLpQT,whereC=CΩ,N,l,p,T,Λ,Λ˜andBis given in (7).If we improve the regularity ofg and the initial datum w0,ψ0 in Lemma 2, we can obtain the following Schauder estimate.Lemma 3.
Letθ∈0,1be an arbitrary fixed constant,bi,j∈C1+θ,θ/2QT¯satisfybi,j=bj,iand the uniform parabolic condition (6),q,d,e,r∈Cθ,θ/2QT¯. Ifg∈Cθ,θ/2QT¯, w0,ψ0∈C2+θΩ¯2 satisfy the compatibility condition (9), there exists a unique classical solutionw,ψ∈C2+θ,1+θ/2QT¯2of the system (5) satisfying the estimate(12)wC2+θ,1+θ/2QT¯+ψC2+θ,1+θ/2QT¯≤expCBw0C2+θΩ¯+ψ0C2+θΩ¯+gCθ,θ/2QT¯,whereC=CΩ,N,l,θ,T,Λ,Λ˜andBis given in (7).The proof of this Lemma is a standard consequence of the classical Schauder estimate theory. For more information, one can see Theorem 4.2, Theorem 5.1 in [26], Theorem 7.2.24 in [23] and Theorem 3.4 in [24]. In addition, the usual Schauder estimate always includes a maximum norm of the solution on the right-hand side. We use the Moser iteration ([23]) and the result in Lemma 1 to estimate the maximum norm.Finally, we introduce the embedding theorem of the spaceWr2,1QTr>1 introduced in [20, 27].Lemma 4.
Let N be a positive integer andr>1, then the following continuous embedding holds:(1)
IfN+2>2r, then Wr2,1QT⟶Lr∗QT, where r∗=N+2r/N+2−2r.(2)
IfN+2=2r, then Wr2,1QT⟶LsQT for any s>1.(3)
Ifθ=2−N+2/r is not an integer, then Wr2,1QT⟶Cθ,θ/2QT¯.


## 3. Carleman Inequality

In this section, we give a Carleman inequality for the adjoint system associated to the linear phase-filed system (5) based on the strategy developed in [21]. We need to notice that, unlike the usual form, the adjoint system we construct is not homogeneous, which is necessary because of the different optimal control problem we give in Section 5.Let us consider the following adjoint system associated to (5)(13)L1∗v,φ=−vt−∑i,j=1N∂∂xibijx,t∂v∂xj+qv+rφ=g1,inQT,L2∗v,φ=−φt−lvt−Δφ+dv+eφ=g2,inQT,∂v∂nB=∑i,j=1Nbijx,tni∂v∂xj=0,∂φ∂n=0,onΣT,vx,T=vTx,φx,T=φTx,inΩ.Following [21], let us introduce some weight functions with parameter λ>0 as follows:(14)ρx,t=eλβxtT−t,ρ˜x,t=e−λβxtT−t,αx,t=mxtT−t,α˜x,t=m˜xtT−t,where mx=eλβx−e2λβ and m˜x=e−λβx−e2λβ, the function β⋅∈C2Ω¯ satisfies(15)β⋅>0inΩ,β⋅=0on∂Ωandmin∇β⋅,x∈Ω/ω0>0,with ω0⊂⊂ω and β  =  β⋅CΩ¯.For simplicity, we define a family of weight functionsuk=skρke2sα+e2sα˜k=0,1,2⋯ with parameter s>0 and α,α˜,ρ,ρ˜ defined in (14). With above weight functions, we have a modified Carleman inequality for Neumann boundary value problems such as(16)∂z∂t−∑i,j=1N∂∂xjbijx,t∂z∂xi=g˜x,t,inQT,∂z∂nB=∑i,j=1Nbijx,tnj∂z∂xi=0,onΣT,zx,0=z0x,inΩ,where g˜∈L2QT, z0∈H1Ω and the coefficients fulfill conditions (6) and (7).Lemma 5.
There exists a constantλ0=λ0B≥1such that for anyλ≥λ0there exists a constants0=s0λ≥1such that for everys≥s0, the solution of (16) satisfies the inequality(17)δk,z=∬QTuk−1zt2+Δz2+uk+1∇z2+uk+3z2dxdt≤CeCλB2∬QTukg˜2dxdt+∬ω0×0,Tuk+3z2dxdt,where k=0,1,2⋯, u−1=s−1ρ−1e2sα+e2sα˜, the constant λ0 depends continuously on B, the constant s0 depends continuously on λ and the positive constant C is independent of s.Remark 1.
In [21], Lemma 1.2 gives a detailed proof of the Carleman inequality (17) with k=0. However, it can not be applied to our case directly because of the term lvt (see the proof of Lemma 6). Replacing the auxiliary functions(18)wx,t=esαzx,t,w˜x,t=esα˜zx,t
introduced in [21] with(19)wx,t=ρk/2esαzx,t,w˜x,t=ρ˜k/2esα˜zx,t,and following the proof procedure of Lemma 1.2 in [21], one can prove Lemma 5. The method of constructing an auxiliary function is a usual technique in proving Carleman inequality (see, for instance, [15, Theorem 7.1, p.288]).
Applying Lemma5 to the adjoint system (13), we get the following lemma.Lemma 6.
Letλ0,s0be the constants given in Lemma 5. Then, for everyλ≥λ0and(20)s≥s1=CeCλs0+qCQT¯2/3+dCQT¯1/2+eCQT¯2/3+rCQT¯B2T2,the solution v,φ of the adjoint system (13) satisfies the estimate(21)δ1,v+δ0,φ≤CeCλB4∬QTu1g12+u0g22dxdt+∬ω0×0,Tu4v2+u3φ2dxdt.Proof.
We apply Lemma5 to the first equation of the system (13) with k=1, g˜=g1−rφ−qv. This implies that there exists(22)s≥s11=maxs0,CeCλB2/3qCQT¯2/3T24,such that(23)CeCλB2qCQT¯2u1≤12u4.
And as a consequence, we have(24)δ1,v≤CeCλB2∬QTu1g12dxdt+rCQT¯2∬QTu1φ2dxdt+∬ω0×0,Tu4v2dxdt.
Similarly, we apply Lemma5 to the second equation of the system (13) with k=0, g˜=g2+lvt−dv−eφ. Taking(25)s≥s12=maxs0,CeCλB2/3eCQT¯2/3T24,we get(26)δ0,φ≤CeCλB2l2∬QTu0vt2dxdt+∬QTu0g22+u0|dv|2dxdt+∬ω0×0,Tu3φ2dxdt≤CeCλB2l2δ1,v+∬QTu0g22dxdt+dCQT¯2∬QTu0v2dxdt+∬ω0×0,Tu3φ2dxdt.
Therefore, we deduce the result by choosings≥s1. □
To prove the main result of this section, we need the following estimates ofukk=0,1,2,…, which will play a crucial role in our following demonstrations.Lemma 7.
For anys≥1,λ≥1, by the definition ofρ, we haveρ≥4/T2, therefore we haveuk≤T2/4uk+1and(27)ukt≤CeCλ1+T3uk+2,∇uk≤Cλ1+T2uk+1,D2uk≤Cλ21+T4uk+2,where C=Ck,N,β, D2u=∑i,j=1N∂2u/∂xi∂xj.Proof.
Fors≥1,λ≥1 and β⋅ given in (15), we have(28)ρt=ρ2t−TtT−t≤Tρ2,∇ρ=λρ∇βx≤Cλρ,D2ρ=λρλ∑i=1N∂β∂xi2+D2β≤Cλ2ρ,
provided by(29)2t−T≤T,eλβx≥1,1tT−t≤eλβxtT−t=ρ.
Again, in the same way, we get(30)α+α˜t≤CeCλTρ2,∇α+α˜≤Cλρ,D2α+α˜≤Cλ2ρ,by observing ρx,t>ρ˜x,t,αx,t>α˜x,t and(31)α+α˜≤2e2λβ−1tT−t≤Ce2λβρ=CeCλρ.
Thus, combined with (28) and (30), we get(32)ukt≤CeCλTuk+1+uk+2,∇uk≤Cλuk+uk+1,D2uk≤Cλ2uk+2uk+1+1+T24uk+2,by virtue of(33)D2f⋅g=fD2g+gD2f+2∑i,j=1N∂f∂xi∂g∂xj≤fD2g+gD2f+2N2∇f∇g.
Finally, noting thatuk≤T2/4uk+1,∀k=0,1,2⋯, we finish the proof of Lemma 7. □
There are plenty of methods to get the null-controllability by one control force. The method developed in [13] is to estimate the term ∬ω0×0,Te−2αφ2dxdt by the term ∬ω×0,Te−rαv2dxdt for any r∈0,2, which is deduced by constructing a weight function Λt. And in [17], the authors used a different weight e−2sαt−7T−t7. Whichever the method is, the computation is complex but necessary. Enlightened by above two methods and the properties of weight functions in Lemma 7, we prove the Carleman inequality for the adjoint system (13) as the following proposition.Proposition 1.
Letλ0,s0be the constants given in Lemma 5. Then, for everyλ≥λ0,s≥s1given in Lemma 6, the solutionv,φof the adjoint system (13) satisfies the estimate(34)δ0,φ+δ1,v≤CeCB+λ∬QTu3g12+g22dxdt+∬ω×0,Tu7v2dxdt.Proof.
Let us begin with introducing a truncation functionξ∈C0∞Ω satisfying(35)0≤ξ≤1,inΩ,ξ=1,inω0,ξ=0,inΩω,∇ξ≤C,inΩ,∇ξξ2/3=6ξ∧∇ξ∧∈L∞ΩN,andD2ξξ2/3=30∇ξ∧2+6ξ∧D2ξ∧∈L∞Ω,where ξ=ξ∧6 with ξ∧∈C0∞Ω satisfying (35), ω0⊂⊂ω and C is a constant depending on ω0 and ω.
Observe thatu3ξvφ+lvx,0=u3ξvφ+lvx,T=0 and integrate u3ξvφ+lvt in QT. Integrating by parts, recalling the boundary conditions in (13) and replacing vt, φ+lvt by their expressions given in (13), we see that(36)∬QTru3ξφ2dxdt=−∬QTu3ξ∑i,j=1Nbij∂v∂xj⋅∂φ∂xidxdt−l∬QTu3ξ∑i,j=1Nbij∂v∂xj⋅∂v∂xidxdt−∬QTφ∑i,j=1Nbij∂v∂xj⋅∂u3ξ∂xidxdt−l∬QTv∑i,j=1Nbij∂v∂xj⋅∂u3ξ∂xidxdt−∬QTu3ξq+lr+eφv+d+lqv2dxdt−∬QT∇u3ξv⋅∇φdxdt−∬QTu3tξvφ+lvdxdt+∬QTu3ξφ+lvg1dxdt+∬QTu3ξvg2dxdt=I1+I2+I3+I4+I5+I6+I7+I8+I9.
In order to estimate the first term on the right-estimate the first term on the right-hand side, we need the Young inequality withγ>0, i.e.,(37)ab≤γa2+14γb2,∀γ>0,a>0,b>0.
We will use the Young inequality with different coefficientsγi>0 for many times in the following estimates, and for simplicity, we do not point it out each time. Since the coefficients bij satisfy the condition (7), we have(38)I1≤BN2∬QTu3ξ∇v∇φdxdt≤γ1∬QTu1∇φ2dxdt+CB2γ1∬QTu5ξ∇v2dxdt,by virtue of ∂v/∂xi≤∇v and ∂φ/∂xj≤∇φ,i,j=1,2,…,N. We deduce from the uniform parabolic condition (6) that(39)I2=−l∬QTu3ξ∑i,j=1Nbij∂v∂xj⋅∂v∂xidxdt≤−Λ˜l∬QTu3ξ∇v2dxdt≤0.
Sinceu3ξ=0 in Ω/ω, it appears that(40)∂u3ξ∂xi=0,i=1,2,…,Non∂Ω.
Integrating by parts again, we rewriteI3 as(41)I3=−∬QTφ∑i,j=1Nbij∂v∂xj⋅∂u3ξ∂xidxdt=∬QTv∑i,j=1N∂∂xjbijφ∂u3ξ∂xidxdt=∬QTv∑i,j=1Nbij∂φ∂xj⋅∂u3ξ∂xidxdt+∬QTvφ∑i,j=1N∂bij∂xj⋅∂u3ξ∂xidxdt+∬QTvφ∑i,j=1Nbij∂∂xj∂u3ξ∂xidxdt=I31+I32+I33.
It follows from the estimates in Lemma7 and (35) that(42)∇ukξ=ξ∇uk+uk∇ξ≤Cukξ2/3+λ1+T2uk+1ξ.
Thus the above inequality withk=3 yields(43)I31≤BN2∬QTv∇φ∇u3ξdxdt≤CB∬QTv∇φλ1+T2u4ξ+u3ξ2/3dxdt≤γ1∬QTu1ξ∇φ2dxdt+Cλ2B24γ11+T4∬QTu7ξv2dxdt+γ1∬QTu1ξ∇φ2dxdt+CB24γ1∬QTu5ξ1/3v2dxdt≤2γ1∬QTu1∇φ2dxdt+Cλ2B2γ11+T4∬ω×0,Tu7v2dxdt.
Using similar arguments, we can obtain(44)I32≤2γ2∬QTu3φ2dxdt+Cλ2B2γ21+T8∬ω×0,Tu7v2dxdt.
By analogy with (42), we get(45)D2ukξ=2∑i,j=1N∂uk∂xi⋅∂ξ∂xj+ξD2uk+ukD2ξ≤2N2∇ξ∇uk+ξD2uk+ukD2ξ≤Cukξ2/3+2λ1+T2uk+1ξ2/3+λ21+T4uk+2ξ.
We deduce from (45) that(46)I33≤CB∬QTvφu3ξ23+2λ1+T2u4ξ23+λ21+T4u5ξdxdt≤3γ2∬QTu3φ2dxdt+Cλ4B2γ21+T8∬ω×0,Tu7v2dxdt.
Hence, by virtue of above estimates, we conclude that(47)I3≤2γ1∬QTu1∇φ2dxdt+5γ2∬QTu3φ2dxdt+Cλ4B21+T81γ1+1γ2∬ω×0,Tu7v2dxdt.
Similar to the estimate ofI31, we can prove the estimate(48)I4≤2γ3∬QTu2∇v2dxdt+Cλ2B2γ31+T6∬ω×0,Tu7v2dxdt.
It is easy to check that(49)I5≤γ2∬QTu3φ2dxdt+CB2γ21+T8∬ω×0,Tu7v2dxdt.
Proceeding as previously, we have(50)I6=∬QTv∇u3ξ⋅∇φdxdt+∬QTu3ξ∇v⋅∇φdxdt≤3γ1∬QTu1∇φ2dxdt+14γ1∬QTu5ξ∇v2dxdt+Cλ2γ11+T4∬ω×0,Tu7v2dxdt.
We boundI7 with the above estimate for u3t in Lemma 7 as follows(51)I7≤C1+T3∬QTu5ξv⋅φ+lvdxdt≤γ2∬QTu3φ2dxdt+CeCλγ21+T8∬ω×0,Tu7v2dxdt.
It appears that(52)I8≤γ2∬QTu3φ2dxdt+CT8∬ω×0,Tu7v2dxdt+14+14γ2∬QTu3g12dxdt,and(53)I9≤CT8∬ω×0,Tu7v2dxdt+14∬QTu3g22dxdt.
By virtue of above estimates forI1∼I9 and the condition rx,t≥r0>0 in ω×0,T, we have the estimate for (36) as follows(54)∬ω0×0,Tu3φ2dxdt≤6γ1r0∬QTu1∇φ2dxdt+8γ2r0∬QTu3φ2dxdt+2γ3r0∬QTu2∇v2dxdt+CB2γ1r0∬QTu5ξ∇v2dxdt+CeCλB2r01γ1+1γ2+1γ3∬ω×0,Tu7v2dxdt+14r01+1γ2∬QTu3g12+g22dxdt.
Our following goal is to get rid of∬QTu5ξ∇v2dxdt on the right-hand side of (54). By analogy with above discussions, we integrate u5ξv2t in QT and integrate by parts, then it appears that(55)Λ˜∬QTu5ξ∇v2dxdt≤∬QTu5ξ∑i,j=1Nbij∂v∂xi⋅∂v∂xjdxdt=−∬QTv∑i,j=1Nbij∂u5ξ∂xi⋅∂v∂xjdxdt−∬QTqu5ξv2dxdt−∬QTru5ξφvdxdt−12∬QTu5ξtv2dxdt+∬QTu5ξvg1dxdt=J1+J2+J3+J4+J5.
Taking into account the estimate onI31 and observing the estimate for ∇ukξ in (42) with k=5, we obtain(56)J1≤2γ4∬QTu5ξ∇v2dxdt+Cλ2B2γ41+T4∬ω×0,Tu7v2dxdt.
It is obvious that(57)J2=−∬QTqu5ξv2dxdt≤CqCQT¯T4∬ω×0,Tu7v2dxdt.
And it follows that(58)J3=−∬QTru5ξφvdxdt≤γ5∬ω×0,Tu3φ2dxdt+rCQ¯T24γ5∬QTu7v2dxdt
Furthermore, by virtue of the estimate forI7 and Lemma 7, the following result holds(59)J4=−12∬QTu5ξtv2dxdt≤CeCλ1+T8∬ω×0,Tu7v2dxdt.
Utilizing Young inequality again withγ=1/2, we get(60)J5=∬QTu5ξvg1dxdt≤12∬QTu3g12dxdt+12∬ω×0,Tu7v2dxdt.
Thus, from the estimates forJ1∼J5, we can rewrite (55) as(61)∬QTu5ξ∇v2dxdt≤1Λ˜∬QTu3g12dxdt+2γ5Λ˜∬QTu3ξφ2dxdt+CeCλB2Λ˜4Λ˜+1γ51+T4∬ω×0,Tu7v2dxdt,by choosing γ4=Λ˜/4.
Combining (54) and (61) and Lemma 6, we finish the proof of this part by choosing γ1=r0/12CeCλB4, γ2=r0/16CeCλB4, γ3=r0/4CeCλB4 and γ5=r0Λ˜/4CeCλB4γ1. □


## 4. Observability Estimate

This section addresses the study of the observability estimate for (13). Notice that, unlike the general case, the adjoint system (13) is not homogeneous and the observability estimate with uk can not deduce the null controllability for the linear system (5). To obtain the qualified observability estimate, we need to construct a new sequence of weight functions(62)μk=skρ1ke2sα1+e2sα˜1,k=0,1,2,…,where(63)ρ1=eλβxTT−t,α1=mxTT−t,α˜1=m˜xTT−t,with the parameters given in Section 3.Then, we have the following lemma.Lemma 8.
For anys≥kT2/2e2λβ−eλβ, we have(64)Cske−c1s≤uk,∀x,t∈Ω¯×T4,3T4,Cske−c1s≤μk,∀x,t∈Ω¯×0,3T4,uk≤μk≤CeCλsk,∀x,t∈QT¯,where C=Ck,T,β and c1=CT−2e2λβ−e−λ‖β‖−1.Proof.
By analogy with the proof of Lemmas1 and 2 in [6], we denote the auxiliary function with the variable τ⩾0 as follows(65)fxkτ=AskesB/τ+esB1/ττk,τ∈0,τmax,where Ax>0, B1x<Bx<0. Then, it can be verified that limτ⟶0fxkτ=fxk0=0 and fxkτ is monotone increasing on 0,−sB/k. Therefore by choosing s⩾maxx∈Ω¯kτmax/−Bx, we can obtained that(66)fxkτ≤fxkτmax≤Csk.
In addition, we notice thattT−t≤TT−t≤T2=τmax for all t∈0,T, thus for s≥kT2/2e2λβ−eλβ, we have(67)uk=fxktT−t≤μk=fxkTT−t≤fxkT2,∀x,t∈QT¯,and the other results can be obtained in the same way. □
Thus, with above estimates, the following result holds.Proposition 2.
Letλ0be the constant given in Lemma 5ands1be the constant given in Lemma 6. Then, for everyλ≥λ0,s≥maxs1,3T2/2e2λβ−eλβ, the solutionv,φof the adjoint system (6) satisfies the observability estimate(68)∫Ωφx,02dx+∫Ωvx,02dx≤CeCB+λ+s∬QTμ3g12+g22dxdt+∬ω×0,Tu7v2dxdt.Proof.
Recalling (13) and integrating φ+lv2+δv2t in Ω, we can easily prove the estimate(69)−12ddt∫Ωφ+lv2+δv2dx≤−∫Ω∇φ2+l∇φ⋅∇v+δΛ˜∇v2dx+12dCQT¯+12eCQT¯+14∫Ωφ+lv2dx+δ2rCQT¯+12eCQT¯∫Ωφ2dx+qCQT¯+12δdCQT¯+12rCQT¯+δ4δ∫Ωv2dx+∫Ωg12dx+∫Ωg22dx,where δ is a constant to be fixed. By virtue of Young inequality with δ=l2/4Λ˜, we get(70)−∫Ω∇φ2+l∇φ⋅∇v+δΛ˜∇v2dx≤0.
And by virtue of(71)∫Ωφ2dx≤∫Ωφ2dx+∫Ωφ+2lv2dx=2∫Ωφ+lv2dx+2l2∫Ωv2dx,we rewrite (69) as(72)−12ddt∫Ωφ+lv2+δv2dx≤c2∫Ωφ+lv2+δv2dx+∫Ωg12dx+∫Ωg22dx,where c2=CB and B is given in (7). Hence we get(73)ddte2c2t∫Ωφ+lv2+δv2dx+2∫0t∫Ωg12x,τ+g22x,τdxdτ≥0,for all t≥0. For simplicity, we set φ+lv2+δv2=ϖx,t. By the second estimate in Lemma 8 and the standard energy estimate (see, for instance, [28]), we have(74)∫Ωϖx,0dx≤ec22T∫Ωϖx,T4dx+2∬Ω×0,T/4g12+g22dxdt≤2Te3c22T∬Ω×T/4,3T/4ϖx,tdxdt+2∬Ω×0,3T/4g12+g22dxdt≤2Te3c22T∬Ω×T/4,3T/4ϖx,tdxdt+Cs−kec1s∬QTμkg12+g22dxdt.
It follows from the first inequality in Lemma8 and Proposition 1 that(75)∬Ω×T/4,3T/4ϖx,tdxdt≤∬Ω×T/4,3T/42φ2+2l2+δv2dxdt≤Cec1s∬Ω×T/4,3T/4u3φ2+u4v2dxdt≤CeCB+λ+s∬QTu3g12+g22dxdt+∬ω×0,Tu7v2dxdt.
Thus, by virtue of (74) and (75) and uk≤μk, we obtain the following estimate(76)∫Ωφ+lvx,02dx+δ∫Ωvx,02dx≤CeCB+λ+s∬QTμ3g12+g22dxdt+∬ω×0,Tu7v2dxdt,which ends the proof of the proposition with (71).


## 5. Null-Controllability for the Linear System

In this section, we consider the null-controllability for the linear system (5). Firstly, we find a control function g∈L2QT with suppg⊂ω¯×0,T such that(77)wx,T=ψx,T=0,inΩ,by constructing a sequence of optimal control problems. Then, we prove that the control function belongs to LrQTr≥2 based on the parabolic regularity. Finally, by utilizing an iteration method and the embedding theorem, we find that the control function belongs to the space Uω.Proposition 3.
Letλ0be the constant given inLemma5ands1be the constant given inLemma6. For anyw0,ψ0∈H1Ω2,λ≥λ0,s≥maxs1,9⋅7⋅T2/8e2λ‖β‖−eλ‖β‖, there exists a controlg∈L2QTwithsuppg⊂ω¯×0,Tsuch that the corresponding solutionw,ψof (5) satisfies(78)ωx,T=ψx,T=0,inΩ,Moreover,(79)ωL2QT+ψL2QT+gL2QT≤CeCB+λ+sω0L2Ω+ψ0L2Ω.Proof.
To pave a way for further numerical approximation, we consider the following optimal control problem, which is similar to that introduced by Fursikov and Imanuvilov [4] and used in [21, 29, 30](80)MinJkg1,g2=12∬QTμ3k−1w2+ψ2dxdt+12∬QTm1ku7k−1g12+m2ku7k−1g22dxdt,subject to(81)L1ω,ψ=g1,L2ω,ψ=g2,inQT,∂ω∂nB=∂ψ∂n=0,onΣT,ωx,0=ω0x,ψx,0=ψ0x,inΩ,and the terminal constraints(82)ωx,T=ψx,T=0,inΩ,in Banach space W22,1QT2×L2QT2, where the parameters λ≥λ0, s≥maxs1,9⋅7⋅T2/8e2λ‖β‖−eλ‖β‖ are fixed, μ3k=s3eλβx/τ13e2smx/τ1+e2sm˜x/τ1,u7k=s7eλβx/τ27e2smx/τ2+e2sm˜x/τ2 with mx, m˜x defined in (14) and(83)τ1=TT−t+Tk,τ2=t+TkT−t+Tk,m1k=1,x∈ω¯,k,x∈Ωω¯,m2k=k,x∈Ω.
According to the classical theory about optimal control problem (see [31]), since the control function g1,g2 are imposed on the whole domain QT, it is easy to prove that, for any w0,ψ0∈H1Ω2, the optimal control problem (80)–(82) has a unique solution, which we denote by wk,ψk,g1k,g2k. Applying the Lagrange principle to the optimal control problem, we get the following necessary optimality condition (see [31, 32]), i.e., for each k, there exists co-state function vk,φk satisfying the homogeneous boundary conditions in (13) such that(84)L1wk,ψk=g1k,L2wk,ψk=g2k,inQT,L1∗vk,φk=μ3k−1wk,L2∗vk,φk=μ3k−1ψk,inQT,m1ku7k−1g1k+vk=0,m2ku7k−1g2k+φk=0,inQT,wkx,T=0,ψkx,T=0,inΩ.
Noting that(85)L1∗vk,φk,wkL2QT+L2∗vk,φk,ψkL2QT=L1wk,ψk,vkL2QT+L2wk,ψk,φkL2QT−∫Ωw0vkx,0+lψ0vkx,0+ψ0φkx,0dx,and taking into account (84), we obtain(86)Jkg1k,g2k=−12∫Ωw0vkx,0+lψ0vkx,0+ψ0φkx,0dx≤12w0L2Ω+lψ0L2Ω∫Ωvkx,02dx1/2+12ψ0L2QT∫Ωφkx,02dx1/2≤w0L2Ω+1+lψ0L2Ω∫Ωvkx,02dx+∫Ωφkx,02dx1/2.
Recalling Proposition2 and the definition of m1k, we get(87)∫Ωvkx,02dx+∫Ωφkx,02dx≤CeCB+λ+s∬QTμ3μ3k−2wk2+ψk2dxdt+∬ω×0,Tu7m1k2u7k−2g1k2dxdt≤CeCB+λ+sJkg1k,g2k,by virtue of μ3≤μ3k, u7≤u7k which can be proved by analogy with Lemma 8 by noting(88)TT−t≤TT−t+Tk≤2T2,tT−t≤t+TkT−t+Tk≤9T24.
Hence, it follows from (86) and (87) that(89)Jkg1k,g2k≤CeCB+λ+sw0L2QT+ψ0L2QT2.
By virtue of (89), we have a subsequence w∧k,ψ∧k,g∧1k,g∧2k,v∧k,φ∧kk=1∞ such that(90)w∧k,ψ∧k→w,ψweaklyinL2QT2,v∧k,φ∧k→v,φweakly,inL2QT2,g∧1k→g=0inL2Ω/ω×0,T,g∧2k→g2=0,inL2QT.
Moreover, by passing to the limit in (84), we obtain that the pair w,ψ is a solution of (5) satisfying(91)wx,T=ψx,T=0,inΩ,with the control function g1=g=−χωu7v∈L2QT and χω is the characteristic function of ω. Finally, passing limit in (89) and taking into account the third inequality in Lemma 8, we obtain the estimate (79). □
Our following goal is to proveg=−χωu7v∈Uω by Lemma 3 (embedding theorem) and the parabolic regularity.Proposition 4.
Letλ0be the constant given inLemma5ands1be the constant given in Lemma 6. For anyw0,ψ0∈C2+θΩ¯2satisfying the compatibility condition (9),λ≥λmax=maxλ0,ln2M∗+1−2/βands≥smax=maxs1,7⋅2M∗+1T2/eλβ−1(M∗is a fixed finite positive integer), there exists a controlg∈Uωsuch that the corresponding solution of (5) satisfies(92)wx,T=ψx,T=0,inΩ.Moreover,(93)gCθ,θ/2QT¯≤CeCB+λ+sw0L2Ω+ψ0L2Ω.Proof.
Letsm=1−1/2m+1s, s∧m=1/2m+1s, m=1,2⋯, and define the following weight functions(94)u∧k,m=sρ¯ke2smα¯+e2smα˜¯,μ∧k,m=sρ1¯ke2s∧mα1¯+e2s∧mα˜1¯,where ρ,α,α˜ are defined in (14), ρ1,α1,α˜1 are defined in (63) and f¯t=minx∈Ω¯fx,t,f¯t=maxx∈Ω¯fx,t. Then, with w,ψ,g,v,φ given in Proposition 3, we denote(95)g1∗m=L1∗u∧7,mv,u∧7,mφ,g2∗m=L2∗u∧7,mv,u∧7,mφ,g1m=L1μ∧3,m−1w,μ∧3,m−1ψ,g2m=L2μ∧3,m−1w,μ∧3,m−1ψ.
By analogy with Lemma8, it can be verified that(96)u∧7,12≤Cs11μ3,fors≥114T2e2λβ−1,u∧9,12≤Cs14u4,fors≥144T2e2λβ−1.
In addition, for eachk, vk,φk, wk,ψk given in Proposition 3, by virtue of Proposition 1, we have(97)∬QTu3φk2+u4vk2dxdt≤CeCB+λ∬QTμ3k−1wk2+ψk2dxdt+∬ω×0,Tu7vk2dxdt.
Thus, form=1, passing to the limit in (84) and (97), we see that(98)∬QTg1∗12dxdt=∬QTu∧7,1L1∗v,φ−u∧7,1tv2dxdt≤∬QTu∧7,1μ3−1w2+u∧9,1v2dxdt≤C∬QTu∧7,12μ3μ3−1w2dxdt+∬QTu∧9,12u4u4v2dxdt≤CeCB+λ+s∬QTμ3−1w2+ψ2dxdt+∬ω×0,Tu7v2dxdt.
Passing to the limit in (89) and estimating g2∗1 in the same way, we arrive at(99)g1∗1L2QT+g2∗1L2QT≤CeCB+λ+sw0L2Ω+ψ0L2Ω,for s≥maxs1,15/4T2/e2λ‖β‖−1. Moreover, similar to the result in Lemma 1, we get(100)u∧7,1vW22,1QT+u∧7,1φW22,1QT≤eCB+λ+sg1∗1L2QT+g2∗1L2QT.
By Lemma4 with r=2, we have W22,1QT⟶Lr1QT for (101)r1=2N+2/N−2,N>2,anyconstantc>1,N≤2,which implies(102)u∧7,1vLr1QT+u∧7,1φLr1QT≤CeCB+λ+sw0L2Ω+ψ0L2Ω.
On the other hand, we find that(103)∬QTg112dxdt≤C∬QTu7μ∧3,12χωu7v2dxdt+∬QTμ∧5,1μ3μ∧3,12μ3−1w2+l2ψ2dxdt≤CeCB+λ+sw0L2Ω+ψ0L2Ω,fors≥maxs1,43T2e2λβ−eλβ.
By analogy with above estimates, we can arrive at(104)μ∧3,1−1wLr1QT+μ∧3,1−1ψLr1QT≤CeCB+λ+sw0L2Ω+ψ0L2Ω.
Form≥2, It is easy to prove that(105)u∧7,mμ∧3,m−1≤Cs7μ3,fors≥7⋅2mT2eλβ−12,u∧9,m≤Cs2u∧7,m−1,fors≥2m−1T2e2λβ−1,u7≤Cu∧7,m−1μ∧3,m,forλ≥ln2m+1−2β,μ∧5,mμ∧3,m−1≤Cs2μ∧3,m2,fors≥2m+1T2e2λβ−eλβ.
As noted in the proof of Proposition 4.1 in [20], there exists M∗ such that WrM∗−12,1QT⟶LrM∗QT and WrM∗2,1QT⟶Cθ,θ/2QT¯. Thus, processing as previously and taking into account above estimates, we have g=−χωu7v∈Cθ,θ/2QT¯ with λ≥λmax=maxλ0,ln2M∗+1−2/β and s≥smax=maxs1,7⋅2M∗+1T2/eλβ−1 by the embedding theory and the parabolic regularity.


## 6. Local Null-Controllability for the Quasi-Linear System

In this section, we show the local null-controllability of the quasi-linear phase-field system (1) by above propositions and the fixed point theorem.As classical discussion shows (we refer to [33] for more detailed information), we can write(106)fw,ψ−f0,0=∫01ddσfσw,σψdσ=∫01∂∂ζfσw,σψdσ⋅w+∫01∂∂ηfσw,σψdσ⋅ψ.Denoteq˜w,ψ=∫01∂fσw,σψ/∂ζdσ,d˜w,ψ=∫01∂fσw,σψ/∂ηdσ,(107)e˜η=hηη,η≠0,h′η,η=0,and set the following nonempty convex(108)K=ζ,η∈C2+θ,1+θ/2QT¯2:ζC2+θ,1+θ/2+ηC2+θ,1+θ/2≤1,ζ0,η0=w0,ψ0.It is easy to verify thatK is a compact subset of L2QT2 with small initial datum. For any ζ,η∈K, consider the following linearized system of (1)(109)ωt+lψt−∑i,j=1N∂∂xjaijζ∂w∂xi+q˜ζ,ηω+d˜ζ,ηψ=g,inQT,ψt−Δψ+e˜ηψ+rω=0,inQT,∂ω∂nA=∑i,j=1Naijζnj∂ω∂xi=0,∂ψ∂n=0,onΣT,ωx,0=ω0x,ψx,0=ψ0x,inΩ.Then, by settingaijζ=bijx,t, q˜ζ,η=qx,t, d˜ζ,η=dx,t and e˜η=ex,t, bij,q,d,e,r satisfy the condition in Section 2 related to the linear system (5). By Proposition 4, we can prove that (109) is local null-controllable and has the cost estimate(110)gCθ,θ/2QT¯≤CeCA+λ+sw0L2Ω+ψ0L2Ω.Combining the Kakutani’s fixed point theorem, we have the following theorem.Theorem 1.
Letλmaxandsmaxbe the constants given inProposition4. Assume that for every given initial datumw0,ψ0∈C2+θΩ¯2satisfying the compatibility condition (9), there exists a positive constantγsuch thatw0C2+θΩ¯+ψ0C2+θΩ¯≤γ. Then, for everys≥smaxandλ≥λmax, one can find a controlg∈Uωsuch that the corresponding solutionw,ψof (1) satisfies(111)wx,T=ψx,T=0,inΩ.Moreover, we have the cost estimate(112)gCθ,θ/2QT¯≤CeCA+λ+sw0L2Ω+ψ0L2Ω.Proof.
For anyζ,η∈K, put(113)Φζ,η=w,ψ∈K:∃g∈Cθ,θ/2QT¯andaconstantCsuchthatw,ψ,gsatisfies109−112.
This defines a mapΦ:K⟶2K with w0,ψ0∈C2+θΩ¯2 small enough. Further, for any ζ,η∈K, Φζ,η is a nonempty convex and compact subset of L2QT2 provided by Lemma 3. Also, Φ is upper semi-continuous. Therefore, by the Kakutani’s fixed point theorem, there exists w,ψ∈K such that w,ψ∈Φw,ψ and this ends the proof of the theorem.


## 7. Conclusion

In this paper, we derive the the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller under the frame of classical solutions. We first develop the corresponding Carleman inequality and obtain the observability estimation. Then, we derive the null-controllability for the linear system and the get the desired control function by constructing a sequence of optimal control problems. Finally, by the Kakutani’s fixed point theorem, we have the local null-controllability for the quasi-linear system.



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*Source: 7645304-2022-07-07.xml*