UNIFORM NULL CONTROLLABILITY FOR A DEGENERATING ...
UNIFORM NULL CONTROLLABILITY FOR A DEGENERATING REACTION-DIFFUSION SYSTEM APPROXIMATING A SIMPLIFIED CARDIAC MODEL
arXiv:1106.1788v6 [math.OC] 25 Sep 2015
FELIPE WALLISON CHAVES-SILVA∗ AND MOSTAFA BENDAHMANE
Abstract. This paper is devoted to the analysis of the uniform null controllability for a family of nonlinear reaction-diffusion systems approximating a parabolic-elliptic system which models the electrical activity of the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by means of a single control is shown. The proof is based on the combination of Carleman estimates and weighted energy inequalities.
1. Introduction Let Ω ⊂ RN (N = 2, 3) be a bounded connected open set whose boundary, ∂Ω, is sufficiently regular. Let T > 0, and let ω and O be two (small) nonempty subsets of Ω, which we will refer to as control domains. We will use the notation Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). The main objective of this paper is to study the properties of controllability and observability for a family of nonlinear reaction-diffusion systems which degenerates into a nonlinear parabolic-elliptic system which models the electrical activity in the cardiac tissue. To state the model, we let ui = ui (t, x) and ue = ue (t, x) represent the intracellular and extracellular electric potentials, respectively. Their difference, v = ui − ue , is called the transmembrane potential. The anisotropic properties of the media are modeled by intracellular and extracellular conductivity tensors Mi (x) and Me (x). The widely accepted model (see [11, 22, 35]) describing the electrical activity in the cardiac tissue reads as follows: cm ∂t v − div (Mi (x)∇ui ) + h(v) = f 1ω in Q, (1.1) cm ∂t v + div (Me (x)∇ue ) + h(v) = g1O in Q, where cm > 0 is the surface capacitance of the membrane, the nonlinear function h : R → R is the transmembrane ionic current (the most interesting case being when h is a cubic polynomial), and f and g are stimulation currents applied, respectively, to ω and O. System (1.1) is known as the bidomain model and is completed with Dirichlet boundary conditions for the intra- and extracellular electric potentials ui = ue = 0 on Σ
(1.2)
2010 Mathematics Subject Classification. 35K57, 93B05, 93B07, 93C10. Key words and phrases. reaction-diffusion system, monodomain model, Carleman estimates, uniform null controllability, observability. ∗ F. W. Chaves-Silva has been supported by the ERC project Semi Classical Analysis of Partial Differential Equations, ERC-2012-ADG, project number 320845; the Grant BFI-2011-424 of the Basque Government and partially supported by the Grant MTM2011-29306-C02-00 of the MICINN, Spain, the ERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research Networking Programme OPTPDE and the Grant PI2010-04 of the Basque Government. 1
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F. W. CHAVES-SILVA AND M. BENDAHMANE
and initial data for the transmembrane potential v(0, x) = v0 (x),
x ∈ Ω.
(1.3)
We point out that realistic models describing electrical activities in the heart also include a system of ODE’s for computing the ionic current as a function of the transmembrane potential and a series of additional “gating variables” which aim to model the ionic transfer across the cell membrane (see [23, 25, 31, 32]). In the case where f 1ω = g1O and Mi = µMe , for some constant µ ∈ R, the bidomain model is simplified into the following parabolic-elliptic system: µ in Q, cm ∂t v − µ+1 div Me (x)∇v + h(v) = f 1ω −div M(x)∇u = div M (x)∇v in Q, e i (1.4) v = ue = 0 on Σ, v(0) = v0 in Ω, where M = Mi + Me . System (1.4) is known as monodomain model and is a very interesting model from the implementation point of view, since it conserves some of the essential features of the bidomain model as excitability phenomena (see [11, 27, 36]). The main difference between the bidomain model (1.1) and the monodomain model (1.4) is the fact that the first model is a system of two coupled parabolic equations, while the second one is a system of parabolic-elliptic type. Therefore, from the control point of view, one could expect these two systems to have, at least a priori, different control properties. In this work we show that the properties of controllability and observability for the monodomain model can be seen as a limit process of the controllability properties of a family of coupled parabolic systems. Indeed, given ε ∈ R such that 0 < ε ≤ 1, we approximate the monodomain model by the following family of parabolic systems: µ ε ε + h(v ε ) = f ε 1ω in Q, cm ∂t v − µ+1 div Me (x)∇v ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e (1.5) ε ε on Σ, v = ue = 0 ε in Ω. v (0) = v0 , uεe (0) = ue,0 In this paper we give a positive answer to the following question: Question 1.1. If, for each ε > 0, there exists a control f ε that drives the solution (v ε , uεe ) of (1.5) to zero at time t = T , i.e., v ε (T ) = uεe (T ) = 0, is it true that, when ε → 0+ , the control sequence {f ε }ε>0 converges to a function f which drives the associated solution (v, ue ) of (1.4) to zero at time t = T ? This question of approximating an equation by another having different physical properties has been used several times in the case of parabolic equations degenerating into hyperbolic ones (see, for example, [12, 16, 21]) and hyperbolic equations degenerating into parabolic ones (see, for example, [29, 30]). However, as far as we know, this is the first time that controllability of parabolic systems degenerating into parabolic-elliptic systems is studied. It is also important to mention that families of parabolic systems which degenerate into parabolic-elliptic ones arise in many areas, such as biology, chemistry and astrophysics (see [9, 10, 26]).
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
3
As usual, in control theory, when dealing with the controllability of a nonlinear problem, we first consider the linearized version of (1.5): µ div Me (x)∇v ε + a(t, x)v ε = f ε 1ω in Q, cm ∂t v ε − µ+1 ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e (1.6) ε ε v = u = 0 on Σ, e ε v (0) = v0 , uεe (0) = ue,0 in Ω, where a is a bounded function. Given ε > 0, the first obstacle to answering, positively, Question 1.1, will be to drive (v ε , uεe ), solution of (1.6), to zero at time T by means of a control f ε in such a way that the sequence of controls {f ε }ε>0 converges when ε → 0+ . Once it is shown that such a convergent sequence of control, {f ε }ε>0 , for the linear system (1.6), exists, we employ a fixed point argument and conclude that the same is true for the nonlinear system (1.5). Thus, we introduce the adjoint system of (1.6): µ div (Me (x)∇ϕε ) + a(t, x)ϕε = div (Mi (x)∇ϕεe ) in Q, −cm ∂t ϕε − µ+1 −ε∂ ϕε − div (M (x)∇ϕε ) = 0 in Q, t e e (1.7) ε ε ϕ = ϕ = 0 on Σ, e ε in Ω. ϕ (T ) = ϕT , ϕεe (T ) = ϕe,T Using duality arguments, it is very easy to prove that the task of building such a convergent sequence of controls, {f ε }ε>0 , for (1.6) is equivalent to prove the following (uniform) observability inequality for the solutions of (1.7): ZZ ||ϕε (0)||2L2 (Ω) + ε||ϕεe (0)||2L2 (Ω) ≤ C |ϕε |2 dxdt, Qω := ω × (0, T ), (1.8) Qω
where (ϕT , ϕe,T ) ∈ L2 (Ω)2 and the constant C = C(ε, Ω, ω, ||a||L∞ , T ) remains bounded when ε → 0+ . We prove inequality (1.8) as a consequence of an appropriate Carleman inequality for the solution (ϕε , ϕεe ) of (1.7) (see section 3). We notice that, due to the fact the control is acting on the first equation of (1.6), in our Carleman inequality, we need to bound global integrals of ϕε and ϕεe in terms of a local integral of ϕε , uniformly with respect to ε. Two main difficulties appear: first, the coupling in the first equation of (1.7) is in div (Mi (x)∇ϕεe ) and not in ϕεe ; second, we must show that the constant we get in our Carleman inequality does not blow up when ε → 0+ . The first difficulty is not so hard to overcome. Indeed, for each ε > 0 fixed, inequality (1.8) is known to be true for system (1.7) (see [20]). However, the main novelty here is the fact that we obtain the boundedness of the observability constant C with respect to ε. As we will see, Carleman inequalities alone are not enough for this task, and we need to combine sharp Carleman estimates, with respect to ε, and weighted energy inequalities. As far as the controllability of non degenerate coupled parabolic systems is concerned, the situation is, by now, fairly well understood. For instance, in [20], the controllability of a quite general linear coupled parabolic system is studied and a null controllability result is obtained by means of Carleman inequalities. In [2], using a different strategy, the controllability of a nonlinear reaction-diffusion system of two coupled parabolic equations is analyzed, and the authors prove the null controllability for the linear system and the local null controllability of the nonlinear one. Another relevant work concerning the controllability of coupled systems is [15], in which the authors analyze the null controllability of a cascade system of m (m > 1) coupled parabolic equations and the authors are able to obtain null controllability for the
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F. W. CHAVES-SILVA AND M. BENDAHMANE
cascade system, whenever they have a good coupling structure. It is also worth mentioning the works [3], [4] and [18], where local and global controllability results for phase field systems were studied. For a general discussion about the controllability of coupled parabolic systems, see the survey paper [1]. Concerning controllability results for the bidomain model, since in both equations the couplings are given by the time derivatives of the electrical potentials, it seems very difficult to study controllability properties for such a model. To the best of our knowledge, for the bidomain model (1.1), the problems of null and approximate controllability are still open (even with two controls). Regarding the null controllability of the monodomain model (1.4), since the solution of the parabolic equation enters as a source term in the elliptic one, the following controllability result holds. Theorem 1.2. (1) If h is C 1 (R), globally Lipschitz and h(0) = 0, then, for every v0 ∈ L2 (Ω), there exists a control f ∈ L2 (ω × (0, T )) such that the solution (v, ue ) of (1.4) satisfies: v(T ) = ue (T ) = 0. Moreover, the control f satisfies the following estimate: 2
2
kf 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) ,
(1.9)
for a constant C = C(Ω, ω, T ) > 0. (2) If h is C 1 (R) and h(0) = 0, there exists γ > 0 such that, for every v0 ∈ W 2/3,6 (Ω) ∩ H01 (Ω) with ||v0 ||W 2/3,6 (Ω)∩H01 (Ω) ≤ γ, there exists a control f ∈ L6 (ω × (0, T )) such that the solution (v, ue ) of (1.4) satisfies: v(T ) = ue (T ) = 0. Moreover, the control f satisfies the following estimate: kf 1ω kL6 (Q) ≤ C kv0 kL2 (Ω) ,
(1.10)
for a constant C = C(Ω, ω, T ) > 0. Theorem 1.2, case 1, follows from [14, Theorem 3.1] and case 2 follows from [19, Theorem 3.5] (see also [14, Theorem 4.2]). This paper is organized as follows. In section 2, we state our main results. In section 3, we prove a uniform Carleman inequality for the adjoint system (1.7). Next, we show, in section 4, the uniform null controllability of (1.6). In section 5, we deal with the uniform null controllability of the nonlinear system (1.5). 2. Main results Throughout this paper we will assume that the matrices Mj , j = i, e are C ∞ , bounded, symmetric and positive semidefinite. Our first main result is a uniform Carleman estimate for the adjoint system (1.7). Theorem 2.1. Given any 0 < ≤ 1, there exist positive constants C = C(Ω, ω), λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for any (ϕT , ϕe,T ) ∈ L2 (Ω)2 and any a ∈ L∞ (Q), the solution (ϕε , ϕεe ) of (1.7) satisfies: ZZ ZZ e3sα |ρε |2 dxdt + s3 λ4 φ3 e3sα |ϕε |2 dxdt Q Q ZZ ≤ Ce6λ||ψ|| s8 λ4 φ8 e2sα |ϕε |2 dxdt, (2.1) Qω
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
5
2/3
for every s ≥ (T + (1 + ||a||L∞ )T 2 )s0 and λ ≥ λ0 , where ρε (x, t) = div (M (x)∇ϕεe (x, t)) and the weight functions φ and α are defined in (3.3) and (3.4), respectively. The proof of Theorem 2.1 follows from a combination of Carleman inequalities, for the heat equation, with a precise dependence on the degenerating parameter, and an energy inequality for the adjoint system (1.7). We prove Theorem 2.1 in section 3. Remark 2.2. As a direct consequence of the Carleman inequality (2.1), we have the unique continuation property for the solutions (ϕε , ϕεe ) of (1.7): “Given ε > 0, if ϕε = 0 in ω × (0, T ), then (ϕε , ϕεe ) ≡ (0, 0) in Q”. This unique continuation property for the adjoint system (1.7) implies, for each ε > 0, the approximate controllability at time T of system (1.6), with a control acting only on the first equation. The second main result of this paper gives the global null controllability of the linear system (1.6). Theorem 2.3. For any 0 < ε ≤ 1 and any (v0 , ue,0 ) ∈ L2 (Ω)2 , there exists a control f ε ∈ L2 (ω × (0, T )) such that the associated solution, (v ε , uεe ), to (1.6) is driven to zero at time T . That is to say, the associated solution satisfies: v ε (T ) = 0, uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) , (2.2) for a constant C = C(Ω, ω, ||a||L∞ , T ) > 0. From Theorem 2.1, the proof of Theorem 2.3 is standard. However, for the sake of completeness, we prove Theorem 2.3 in section 4. The third main result of this paper is concerned with the uniform null controllability of the nonlinear parabolic system (1.5). Theorem 2.4. Given any 0 < ε ≤ 1, we have: (1) If h is C 1 (R), globally Lipschitz and h(0) = 0, then, for every (v0 , ue,0 ) ∈ L2 (Ω)2 , there exists a control f ε ∈ L2 (ω × (0, T )) such that the solution (v ε , uεe ) of (1.5) satisfies: v ε (T ) = uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) ,
(2.3)
for a constant C = C(Ω, ω, T ) > 0. (2) If h is C 1 (R) and h(0) = 0, there exists γ > 0, does not depending on ε, such that, for every 2 (v0 , ue,0 ) ∈ W 2/3,6 (Ω) ∩ H01 (Ω) with ||(v0 , ue,0 )||W 2/3,6 (Ω) ≤ γ, there exists a control f ε ∈ L6 (ω × (0, T )) such that the solution (v ε , uεe ) of (1.5) satisfies: v ε (T ) = uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL6 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) ,
(2.4)
for a constant C = C(Ω, ω, T ) > 0. The proof of Theorem 2.4 is achieved through fixed point arguments, and it will be done in section 5.
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F. W. CHAVES-SILVA AND M. BENDAHMANE
Remark 2.5. In this paper we restrict the dimension to N = 2, 3, because the bidomain model makes sense only in such dimensions. Nevertheless, from the mathematical point of view, systems (1.4), (1.5) and (1.6) make sense for any N ∈ N (the 1-d case corresponding to the cable equation) and, taking the initial data in the appropriate space, all the results of this paper can be extended to higher dimensions. 3. Carleman inequality In this section we prove Theorem 2.1. To simplify the notation, we neglect the index ε and, since the only constant which matters in the analysis is ε, we assume that all the other constants are normalized to be the unity. In this case, the adjoint system (1.7) reads: −∂t ϕ − div (Me (x)∇ϕ) + a(x, t)ϕ = div (Mi (x)∇ϕe ) −ε∂ ϕ − div (M (x)∇ϕ ) = 0 t e e ϕ = ϕ = 0 e ϕ(T ) = ϕT , ϕe (T ) = ϕe,T
in Q, in Q, on Σ, in Ω.
(3.1)
We notice that, if ϕT and ϕe,T are regular enough, taking ρ(x, t) = div (Mi (x)∇ϕe (x, t)), the pair (ϕ, ρ) satisfies: −∂t ϕ − div (Me (x)∇ϕ) + a(x, t)ϕ = ρ in Q, −ε∂ ρ − div (M (x)∇ρ) = 0 in Q, t (3.2) ϕ = ρ = 0 on Σ, ϕ(T ) = ϕT , ρ(T ) = ρT in Ω. We prove the Carleman inequality (2.1) using system (3.2). Before starting the proof of the Carleman inequality, let us first define several weight functions which will be usefull in the sequel. Lemma 3.1. Let ω0 be an arbitrary nonempty open set such that ω0 ⊂ ω ⊂ Ω. There exists a function ψ ∈ C 2 (Ω) such that ψ(x) > 0, ∀x ∈ Ω, ψ ≡ 0 on ∂Ω, |∇ψ(x)| > 0 ∀x ∈ Ω\ω0 . Proof. See [14].
Using Lemma 3.1, we introduce the weight functions φ(x, t) =
eλ(ψ(x)+m||ψ||) eλm||ψ|| ; φ∗ (t) = min φ(x, t) = ; t(T − t) t(T − t) x∈Ω
eλ(ψ(x)+m||ψ||) − e2λmkψk eλ(m+1)kψk − e2λmkψk ; α∗ (t) = max α(x, t) = , t(T − t) t(T − t) x∈Ω for a parameter λ > 0 and a constant m > 1. Here, α(x, t) =
(3.3) (3.4)
kψ(x)k = max |ψ(x)| . x∈Ω ∗
Remark 3.2. From the definition of α and α it follows that, for λ large enough, 3α∗ ≤ 2α. Moreover, φ∗ (t) ≤ φ(x, t) ≤ eλkψk φ∗ (x, t) and |∂t α∗ | ≤ e2λkψk T φ2 .
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
7
Proof of Theorem 2.1 . For a better comprehension, we divide the proof into several steps. Step 1. First estimate for the parabolic system. In this step we obtain a first Carleman estimate for the adjoint system (1.7). For that, we will apply sharp Carleman inequalities, with respect to ε, to the system and get a global estimate of ϕ and ρ in terms of a local integral of ϕ and another in ρ. We consider a set ω1 such that ω0 ⊂⊂ ω1 ⊂⊂ ω and apply the sharp Carleman inequality (6.2), with ε = 1, and (6.15) to ϕ and ρ, respectively. We get ZZ ZZ N 2 X 2 2 s−1 φ−1 e2sα |ϕt | dxdt + s−1 φ−1 e2sα ∂xi xj ϕ dxdt Q
Q
+ s3 λ4
ZZ
i,j=1
ZZ
2
2
φ3 e2sα |ϕ| dxdt + sλ2 φe2sα |∇ϕ| dxdt Q Q Z Z ZZ 2 2 2 2sα 3 4 3 2sα ≤C e (|ρ| + |ϕ| )dxdt + s λ φ e |ϕ| dxdt Q
(3.5)
Qω1
and ZZ
2
e2sα |∂t ρ| dxdt + ε−2
ZZ
Q
N 2 X 2 ∂xi xj ρ dxdt
e2sα Q
i,j=1
ZZ 2 2 φ4 e2sα |ρ| dxdt + s2 λ2 ε−2 φ2 e2sα |∇ρ| dxdt Q Q ZZ 2 λkψk 4 4 −2 4 2sα ≤ Ce s λ ε φ e |ρ| dxdt,
+ s4 λ4 ε−2
ZZ
(3.6)
Qω1 2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 . Adding (3.5) and (3.6), and absorbing the lower order ZZ ZZ N X 2 −1 2sα −1 2sα φ e |ϕt | dxdt + φ e Q
Q
4 4
ZZ
+s λ
3 2sα
φ e
2
+ ε2
ZZ Q
2
ZZ
Q
+ s4 λ4
e2sα Q
2
φ4 e2sα |ρ| dxdt + s2 λ2
Q
≤C e
2
φe2sα |∇ϕ| dxdt
|ϕ| dxdt + s λ
e2sα |∂t ρ| dxdt +
ZZ
2 2 ∂xi xj ϕ dxdt
i,j=1 2 2
Q
ZZ
terms in the right-hand side, we get
N 2 X 2 ∂xi xj ρ dxdt
(3.7)
i,j=1
ZZ
2
φ2 e2sα |∇ρ| dxdt
Q λ||ψ|| 4 4
ZZ
s λ
4 2sα
φ e Qω1
2
4 4
ZZ
3 2sα
|ρ| dxdt + s λ
φ e
|ϕ| dxdt , 2
Qω1
2/3
for s ≥ (T + (1 + kakL∞ )T 2 )s0 and λ ≥ λ0 . Remark 3.3. If we were trying to drive the solution of (1.6) to zero by means of controls on both equations, inequality (3.7) would be sufficient. Step 2. Estimate of the local integral of ρ.
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F. W. CHAVES-SILVA AND M. BENDAHMANE
In this step we estimate the local integral on ρ in the right-hand side of (3.7). This will be done using equation (3.2)1 . Indeed, we consider a function ξ satisfying ξ ∈ C0∞ (ω), 0 ≤ ξ ≤ 1, ξ(x) = 1 ∀x ∈ ω1 and write Ceλ||ψ|| s4 λ4
ZZ
ZZ
e2sα φ4 |ρ|2 ξdxdt = Ceλ||ψ|| s4 λ4 Qω
e2sα φ4 ρ(−ϕt − div (Me ∇ϕ) + aϕ)ξdxdt
Qω
(3.8)
:= E + F + G. In the sequel, we estimate each parcel in the expression above. First, we have ZZ E = Ceλ||ψ|| s4 λ4 s∂t αe2sα φ4 ρϕξdxdt Qω ZZ ZZ + Ceλ||ψ|| s4 λ4 e2sα φ3 φt ρϕξdxdt + Ceλ||ψ|| s4 λ4 e2sα φ4 ∂t ρϕξdxdt Qω
Qω
:= E1 + E2 + E3 , and it is not difficult to see that E1 + E2 ≤
1 4 4 s λ 10
ZZ
e2sα φ4 |ρ|2 dxdt + Ce2λ||ψ|| s8 λ4
ZZ
Qω
e2sα φ8 |ϕ|2 dxdt Qω
and ε2 E3 ≤ 2
ZZ e
2sα
2λ||ψ|| −2 8 8
2
|∂t ρ| dxdt + Ce
ε
Qω
ZZ
s λ
e2sα φ8 |ϕ|2 dxdt.
Qω
Next, integrating by parts, we get e−λ||ψ|| s−4 λ−4 F =
N ZZ X i,j=1
+
s∂xi α e2sα φ4 ρ(Meij ∂xj ϕ)ξdxdt
Qω N ZZ X
i,j=1
+
N ZZ X i,j=1
+
e2sα φ3 ∂xi φ ρ(Meij ∂xj ϕ)ξdxdt Qω
e2sα φ4 ∂xi ρ (Meij ∂xj ϕ)ξdxdt Qω
N ZZ X i,j=1
e2sα φ4 ρ(Meij ∂xj ϕ)∂xi ξdxdt Qω
and we can show that F ≤
ZZ 1 e2sα φ4 |ρ|2 dxdt + s2 λ2 e2sα φ2 |∇ρ|2 dxdt 6 Qω Qω ZZ ZZ N 2 X 1 2 e2sα + Ce2λ||ψ|| s8 λ8 e2sα φ8 |ϕ|2 dxdt + ∂xi xj ρ dxdt. 2 Qω Qω i,j=1
1 4 4 s λ 10
ZZ
(3.9)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
9
Finally, we have 1 G ≤ s4 λ4 10
ZZ
e2sα φ4 |ρ|2 dxdt Qω 2λ||ψ|| 4 4
+ Ce
s λ
||a||2L∞
ZZ
e2sα φ4 |ϕ|2 dxdt. Qω
Putting E, F and G together in (3.7), we obtain ZZ ZZ N 2 X 2 e2sα |ϕt |2 dxdt + e2sα ∂xi xj ϕ dxdt Q
Q
4 4
ZZ
4 2sα
+s λ
φ e ZZ
2
2 2
ZZ
2 2sα
|ϕ| dxdt + s λ
Q
φ e
2
|∇ϕ| dxdtε
Q
Q
2
ZZ
e2sα |∂t ρ|2 dxdt Q
ZZ N 2 X 2 φ4 e2sα |ρ|2 dxdt ∂xi xj ρ dxdt + s4 λ4
e2sα
+
i,j=1
i,j=1
(3.10)
Q
ZZ + s2 λ2 φ2 e2sα |∇ρ|2 dxdt Q ZZ e2sα φ8 |ϕ|2 dxdt, ≤ Ce2λ||ψ|| ε−2 s8 λ8 Qω 2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 . Using (3.10), we can prove that, for every ε > 0, system (1.6) is null controllable. However, the sequence of controls obtained in this way will not be bounded when ε → 0+ . Therefore, we need to go a step further and improve estimate (3.10). This is the goal of the next step. Step 3. Weighted energy inequality. The reason why we do not get a bounded sequence of controls out of step 2 is because of the term ε−2 in the right-hand side of (3.10). In this step we prove a weighted energy inequality for equation (3.2)2 , which will be used to compensate this ε−2 term. Let us introduce the function ∗ 3 y = e 2 sα ρ. This new function satisfies ∗ 3 ε∂t y − div (M (x)∇y) = ε 32 s∂t α∗ e 2 sα ρ in Q, (3.11) y=0 on Σ, y(0) = y(T ) = 0 in Ω. Multiplying (3.11) by y and integrating over Ω, we get Z ∗ 3 ε d 3 2 2 ||y(t)||L2 (Ω) + C||∇y(t)||L2 (Ω) ≤ ε s∂t α∗ (t)e 2 sα (t) ρ(t)y(t)dx. 2 dt 2 Ω Integrating this last inequality form 0 to T and using Poincar´e’s and Young’s inequalities, it is not difficult to see that ZZ ZZ 3sα∗ 2 2 4λ||ψ|| e |ρ| dxdt ≤ Cε e s4 φ4 e2sα |ρ|2 dxdt. (3.12) Q
Q
Finally, from (3.10) and (3.12), we obtain ZZ ZZ ∗ e3sα |ρ|2 dxdt ≤ Ce6λ||ψ|| s8 λ4 Q
Qω
φ8 e2sα |ϕ|2 dxdt.
(3.13)
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F. W. CHAVES-SILVA AND M. BENDAHMANE
This last estimate gives a global estimate of ρ in terms of a local integral of φ, with a constant C which is bounded with respect to ε. Step 4. Last estimates and conclusion. In order to finish the proof of Theorem 2.1, we combine inequality (3.13) and a slightly different Carleman inequality to the equation (3.2)1 . Indeed, the following Carleman inequality holds: ZZ ZZ N 2 X 2 s−1 φ−1 e3sα |ϕt |2 dxdt + s−1 φ−1 e3sα ∂xi xj ϕ dxdt Q
Q
i,j=1
ZZ
ZZ
+ s3 λ4 φ3 e3sα |ϕ|2 dxdt + sλ2 φe3sα |∇ϕ|2 dxdt Q Q Z Z ZZ 3sα 2 3 4 3 3sα 2 ≤C e |ρ| dxdt + s λ φ e |ϕ| dxdt , Q
(3.14)
Qω
2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 , where ϕ is, together with ρ, solution of (3.2). Notice that here we have just changed the weight e2sα by e3sα . The proof of (3.14) is exactly the same as the proof of Theorem 6.1, just taking the appropriate change of variable in (6.3). ∗ Next, since e3sα ≤ e3sα , we have ZZ ZZ ∗ e3sα |ρ|2 dxdt ≤ e3sα |ρ|2 dxdt and by (3.13), we have that ZZ e
3sα
Q
Q
2
6λ||ψ|| 8 4
|ρ| dxdt ≤ Ce
Q
ZZ
φ8 e2sα |ϕ|2 dxdt.
s λ
Qω
From (3.13) and (3.14), it follows that ZZ ZZ ZZ 3sα 2 3 4 3 3sα 2 6λ||ψ|| 8 4 e |ρ| dxdt + s λ φ e |ϕ| dxdt ≤ Ce s λ Q
Q
φ8 e2sα |ϕ|2 dxdt,
(3.15)
Qω
which is exactly (2.1). By density, we can show that (3.15) remains true when we consider initial data in L2 (Ω). Therefore, the proof of Theorem 2.1 is finished. 4. Null controllability for the linearized system This section is devoted to proving the null controllability of linearized equation (1.6). It will be done by showing the observability inequality (1.8) for the adjoint system (1.7), and solving a minimization problem. The arguments used here are classical in control theory for linear PDE’s. Hence, we just give a sketch of the proof. Proof of Theorem 2.3. Combining the standard energy inequalities for system (3.2) and the Carleman inequality given by Theorem 2.1, we can show the following observability inequality for the solutions of (3.2): ZZ 2/3
||ϕ(0)||2L2 (Ω) + ε||ρ(0)||2L2 (Ω) ≤ eC(1+1/T +||a||L∞ +||a||L∞ T )
|ϕ|2 dxdt,
Qω
where C = C(Ω, ω) is a positive constant. Next, since ρ(x, t) = div (M (x)∇ϕe (x, t)) and ϕe = 0 on ∂Ω, we have that ||ϕe (t)||H 2 (Ω) ≤ C||ρ(t)||L2 (Ω) ,
(4.1)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
11
for all t ∈ [0, T ]. Therefore, it follows from (4.1) that 2/3
||ϕ(0)||2L2 (Ω) + ε||ϕe (0)||2L2 (Ω) ≤ eC(1+1/T +||a||L∞ +||a||L∞ T )
ZZ
|ϕ|2 dxdt,
(4.2)
Qω
which is the observability inequality (1.8). From (4.2) and the density of smooth solutions in the space of solutions of (3.1) with initial data in L2 (Ω), we see that the above observability inequality is satisfied by all solutions of (1.7) with initial data in L2 (Ω). Now, in order to obtain the null controllability for linear system (1.6), we solve, for any δ > 0, the following minimization problem: Minimize Jδ (ϕT , ϕe,T ), with ( Z Z 1 T 2 |ϕε | dx dt + ε(ue,0 , ϕεe (0)) Jδ (ϕT , ϕe,T ) = 2 0 ω
(4.3) )
+ (v0 , ϕε (0)) + δ(kϕT kL2 (Ω) + ε1/2 kϕe,T kL2 (Ω) ) , where (ϕ, ϕe ) is the solution of the adjoint problem (1.7) with initital data (ϕT , ϕe,T ) ∈ L2 (Ω)2 . It is an easy matter to check that Jδ is strictly convex and continuous. So, in order to guarantee the existence of a minimizer, the only thing remaining to prove is the coercivity of Jδ . Using the observability inequality (1.8) for the adjoint system (1.7), the coercivity of Jδ is straightfoward. Therefore, for each δ > 0, there exists a unique minimizer (ϕδe,T , ϕδT ) of Jδ . Let us denote by ϕε,δ the corresponding solution to (1.7) associated to this minimizer. Taking f ε,δ = ϕε,δ 1ω as a control for (1.6), the duality between (1.6) and (1.7) gives ||v ε,δ (T )||L2 (Ω) + ε1/2 ||uε,δ e (T )||L2 (Ω) ≤ δ, ε,δ where (v ε,δ , uε,δ . It also gives e ) is the solution of (1.6) associated to the control f ||f ε,δ 1ω ||2L2 (Q) ≤ C ||v0 ||2L2 (Ω) + ε||ue,0 ||2L2 (Ω) .
(4.4)
(4.5)
From (4.4) and (4.5), we get a control f ε (the weak limit of a subsequence of f ε,δ 1ω in L2 (ω × (0, T ))) that drives the solution of (1.6) to zero at time T . From (4.5), we have the following estimate on the control f ε , ||f ε 1ω ||2L2 (Q) ≤ C ||v0ε ||2L2 (Ω) + ε||uεe,0 ||2L2 (Ω) . (4.6) This finishes the proof of Theorem 2.3.
5. The nonlinear system In this section we prove Theorem 2.4. The proof is achieved through fixed point arguments. Proof of Theorem 2.4 (case 1): We consider the following linearization of system (1.5): µ ε ε + g(z)v ε = f ε 1ω in Q, cm ∂t v − µ+1 div Me (x)∇v ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e ε ε v = ue = 0 on Σ, ε ε v (0) = v0 , ue (0) = ue,0 in Ω,
(5.1)
12
F. W. CHAVES-SILVA AND M. BENDAHMANE
where h(s) , if |s| > 0, g(s) = s h0 (0), if s = 0.
(5.2)
It follows from Theorem 2.3 that, for each (v0 , ue,0 ) ∈ L2 (Ω)2 and z ∈ L2 (Q), there exists a control function f ε ∈ L2 (Q) such that the solution of (5.1) satisfies: v ε (T ) = uεe (T ) = 0. As we said before, the idea is to use a fixed point argument. For that, we need the following generalized version of Kakutani’s fixed point Theorem, due to Glicksberg [17]. Theorem 5.1. Let B be a non-empty convex, compact subset of a locally convex topological vector space X. If Λ : B −→ B is a convex set-valued mapping with closed graph and Λ(B) is closed, then Λ has a fixed point. In order to apply Glicksberg‘s Theorem, we define a mapping Λ : B −→ X as follows Λ(z) = {v ε ; (v ε , uεe ) is a solution of (5.1), such that v ε (T ) = uεe (T ) = 0, for a control f ε satisfying (2.2)}. Here, X = L2 (Q) and B is the ball B = {z ∈ L2 (0, T, H01 (Ω)), ∂t z ∈ L2 (0, T, H −1 (Ω)); ||z||2L2 (0,T ;H 1 (Ω)) + ||∂t z||2L2 (0,T ;H −1 (Ω)) ≤ M }. 0
It is easy to see that Λ is well defined and that B is a convex and compact subset of L2 (Q). Let us now prove that Λ is convex, compact and has closed graph. • Λ(B) ⊂ B. Let z ∈ B and v ε ∈ Λ(z). Since v ε satisfies (5.1)1 , the following inequality holds ||v ε ||2L2 (0,T ;H 1 (Ω)) + ||∂t v ε ||2L2 (0,T ;H −1 (Ω)) ≤ K1 . 0
(5.3)
In this way, if z ∈ B then Λ(z) ⊂ B, if we take M = K1 . • Λ(z) is closed in L2 (Q). Let z ∈ B fixed, and vnε ∈ Λ(z), such that vnε → v ε . Let us prove that v ε ∈ Λ(z). In fact, by definition we have that vnε is, together with a function uεe,n , and a control fnε , the solution of (5.1), with ||fnε 1ω ||2L2 (Q) ≤ C ||v0 ||2L2 (Ω) + ε||ue,0 ||2L2 (Ω) . Therefore, we can extract a subsequence of fnε , denoted by the same index, such that fnε 1ω → f ε 1ω weakly in L2 (Q). Since fnε is bounded, we can argue as in the previous section and show that ||vnε ||2L2 (0,T ;H 1 (Ω)) + ||∂t vnε ||2L2 (0,T ;H −1 (Ω)) ≤ M. 0
Hence, ε vn → v ε weakly in L2 (0, T ; H01 (Ω)), ε vn → v ε strongly in L2 (Q), ∂t vnε → ∂t v ε weakly in L2 (0, T ; H −1 (Ω)).
(5.4)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
13
Using the converges above and (5.1)2 , we see that there exists a function uεe such that ε ue,n → uεe weakly in L2 (0, T ; H01 (Ω)), ε ue,n → uεe strongly in L2 (Q), ∂t uεe,n → ∂t uεe weakly in L2 (0, T ; H −1 (Ω)). It follows that (v ε , uεe ) is a controlled solution of (5.1) associated to the control f ε . Hence, v ε ∈ Λ(z) and Λ(z) is closed and compact in L2 (Q). • Λ has closed graph in L2 (Q) × L2 (Q). We need to prove that if zn → z, vnε → v ε strongly in L2 (Q) and vnε ∈ Λ(zn ), then v ε ∈ Λ(z). Using the two previous steps, it is easy to show that v ε ∈ Λ(z). Therefore, we can apply Glicksberg’s Theorem to conclude that Λ has a fixed point. This proves Theorem 2.4 in the case where the nonlinearity is a C 1 globally Lipschitz function. Proof of Theorem 2.4 (case 2): We consider the linear system: µ div Me (x)∇v ε + a(z)v ε = f ε 1ω cm ∂t v ε − µ+1 ε∂ uε − div M(x)∇uε = div M (x)∇v ε i t e e ε ε v = u = 0 e ε v (0) = v0 , uεe (0) = ue,0 2 with (v0 , ue,0 ) ∈ W 2/3,6 (Ω) ∩ H01 (Ω) , z ∈ L∞ (Q) and Z 1 dh a(z) = (sz)ds. 0 dz
in Q, in Q, on Σ, in Ω,
(5.5)
Arguing as in the proof Theorem 2.3, we can show the null controllability of (5.5) with controls in L2 (ω × (0, T )). However, these L2 controls are not sufficient to apply fixed point arguments and obtain the null controllability of the nonlinear system (1.5). For this reason, we modify a little the functional (4.3), obtaining controls which will allow us to employ Schauder’s fixed point Theorem. Indeed, for any δ > 0, we consider the problem: Minimize Jδ (ϕT , ϕeT ), with ( Z Z 1 T 2 Jδ (ϕT , ϕeT ) = e2sα φ8 |ϕε | dx dt + ε(ue,0 , ϕεe (0)) 2 0 ω
(5.6)
) ε 1/2 + (v0 , ϕ (0)) + δ kϕT kL2 (Ω) + kϕe,T kL2 (Ω) , where (ϕε , ϕεe ) is the solution of the adjoint system (1.7) with initital data (ϕT , ϕe,T ) ∈ L2 (Ω)2 . ε,δ As in section 4, we show that problem (5.6) has a unique minimizer (ϕε,δ , ϕε,δ = e ). Defining f ε,δ 2sα 8 ε,δ ε,δ ε,δ e φ ϕ and using the fact that ϕ is, together with a ϕe , the solution of (1.7), we see that f is a solution of a parabolic equation, with homogeneous Dirichlet boundary conditions, null initial data and a right-hand side in L2 (Q). Hence, f ε,δ ∈ L2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L2 (Ω)) and in particular we have that f ε,δ ∈ L6 (Q). Arguing as in [2, Lemma 5], we can easily show that 2 2 ||f ε,δ 1ω ||2L6 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) , (5.7) where C > 0 is independent of and δ.
14
F. W. CHAVES-SILVA AND M. BENDAHMANE ε,δ Moreover, the solution (v ε,δ , uε,δ , satisfies e ) of (5.5), associated to f
||v ε,δ (T )||L2 (Ω) + ε1/2 ||uε,δ e (T )||L2 (Ω) ≤ δ. +
ε
(5.8)
6
Taking the limit when δ → 0 , we get a control f ∈ L (Q) (the weak limit of a subsequence of f ε,δ ) such that the associated solution (v ε , uεe ) to (5.5) satisfies v ε (T ) = uεe (T ) = 0. Next, we define a map F : L∞ (Q) −→ L∞ (Q) which, to each z ∈ L∞ (Q), associates v ε the solution, together with uεe , of (5.5) corresponding to z and to the control f ε built above. Note that this application is well defined since, from the regularity theory for parabolic equations (see, for instance, [28]), we have that v ε ∈ X := L6 (0, T ; W 2,6 (Ω)) ∩ W 1,6 (0, T ; L6 (Ω)) ∩ C([0, T ]; W 2/3,6 (Ω) ∩ H01 (Ω)). Let us now consider the set A defined as A := z ∈ L∞ (Q); ||z||L∞ (Q) ≤ 1 . It is clear that A is a convex closed subset of L∞ (Q). From (5.7), which still holds for f ε , and the smallness assumption on the initial data, we can easily show that F is continuous and that F (A) ⊂ A. Finally, since the space X is compactly embedded in L∞ (Q), we have that F (A) is compact in L∞ (Q). Therefore, F has a fixed point and the the proof of case 2 of Theorem 2.4 is finished. 6. Appendix: Some technical results In this section we prove the two sharp Carleman inequalities used in the proof of Theorem 2.1. We consider the parabolic equation N X ∂xi (aij (x)∂xi v(t, x)) = g(x, t) in Q, −∂t v(x, t) − i,j=1
v=0 v(T ) = vT
(6.1)
on Σ, in Ω,
where vT ∈ L2 (Ω) and g ∈ L2 (Q). We assume that the matrix aij has the form aij =
Mij , ε
and (Mij )ij is an elliptic matrix, i.e., there exists β > 0 such that
PN
i,j
Mij ξj ξi ≥ β|ξ|2 for all ξ ∈ RN .
6.1. A degenerating Carleman inequality. The first sharp Carleman inequality we prove is the following. Theorem 6.1. For any 0 < ≤ 1, there exist λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for every λ ≥ λ0 and s ≥ s0 (T + T 2 ), the solution v of (6.1) satisfies ZZ ZZ N 2 X 2 s−1 φ−1 e2sα |∂t v|2 dxdt + s−1 ε−2 φ−1 e2sα ∂xi xj v dxdt Q
Q
i,j=1
ZZ
ZZ
+ s3 λ4 ε−2 φ3 e2sα |v|2 dxdt + sλ2 ε−2 φe2sα |∇v|2 dxdt Q Q Z Z ZZ 2sα 2 3 4 −2 3 2sα 2 ≤C e |g| dxdt + s λ ε φ e |v| dxdt , Q
with C > 0 depending only on Ω, ω0 , ψ and β.
Qω
(6.2)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
15
Proof. For s > 0 and λ > 0, we consider the change of variable w(t, w) = esα v(t, w),
(6.3)
which implies w(T, x) = w(0, x) = 0. Using the fact that v is the solution of (6.1), we write L1 w + L2 w = gs ,
(6.4)
where N X
L1 w = −∂t w + 2sλ
2
φaij ∂xj ψ ∂xi w + 2sλ
i,j=1
L2 w = −
N X
N X
φaij ∂xi ψ ∂xj ψw,
(6.5)
i,j=1
∂xi (aij ∂xj w) − s2 λ2
i,j=1
N X
φ2 aij ∂xi ψ ∂xj ψ w + s∂t α w
(6.6)
i,j=1
and gs = esα g + sλ2
N X
φaij ∂xi ψ ∂xj ψ w − sλ
i,j=1
N X
φ∂xi (aij ∂xj ψ) w.
(6.7)
i,j=1
From (6.4), we have that ||L1 w||2L2 (Q) + ||L2 w||2L2 (Q) + 2(L1 w, L2 w)L2 (Q) = ||gs ||2L2 (Q) .
(6.8)
The rest of the proof is devoted to analyze the terms appearing in (L1 w, L2 w)L2 (Q) . First, we write N X (L1 w, L2 w)L2 (Q) = Iij , i,j=1
where Iij is the inner product in L2 (Q) of the ith term in the expression of L1 w and the jth term in L2 w. After a long, but straightforward, calculation, we can show that the following estimate holds ZZ ZZ 2(L1 w, L2 w)L2 (Q) ≥ 2s3 λ4 β 2 ε−2 φ3 |∇ψ|4 |w|2 dxdt + 2sλ2 β 2 ε−2 φ|∇ψ|2 |∇w|2 dxdt Q Q Z Z −2 2 2 4 2 2 2 3 3 2 − Cε T s λ + Ts λ + T s + s λ + Ts λ φ3 |w|2 dxdt Q ZZ −2 2 2 − Cε (sλ + λ ) φ|∇w| dxdt.
(6.9)
Q
We take λ ≥ λ0 and s ≥ s0 (T + T 2 ), and it follows, from Remark 6.2 below, that 2(L1 w, L2 w)
L2 (Q)
3 4 2 −2
ZZ
φ3 |w|2 dxdt
+ 2s λ β ε
Qω0 2 2 −2
ZZ
φ|∇w|2 dxdt
+ 2sλ β ε
Qω0
≥ 2s3 λ4 β 2 ε−2
ZZ
φ3 |w|2 dxdt + 2sλ2 β 2 ε−2 Q
ZZ
φ|∇w|2 dxdt. Q
(6.10)
16
F. W. CHAVES-SILVA AND M. BENDAHMANE
Remark 6.2. Since Ω\ω0 is compact and |∇ψ| > 0 on Ω\ω0 , there exists δ > 0 such that β|∇ψ| ≥ δ on Ω\ω0 . Putting (6.10) in (6.8), we get ZZ φ3 |w|2 dxdt ||L1 w||2L2 (Q) + ||L2 w||2L2 (Q) + 2β −2 s3 λ4 δ 4 ε−2 Q ZZ + 2sλ2 δ 2 ε−2 φ|∇w|2 dxdt Q ZZ ≤ ||gs ||2L2 (Q) + 2β −2 s3 λ4 δ 4 ε−2 φ3 |w|2 dxdt
(6.11)
Qω0
+ 2sλ2 δ 2 ε−2
ZZ
φ|∇w|2 dxdt. Qω0
Now we deal with the local integral involving ∇w on the right-hand side of (6.11). To this end, we introduce a cutt-off function ξ such that ξ ∈ C0∞ (ω), 0 ≤ ξ ≤ 1, ξ(x) = 1 ∀x ∈ ω0 . Using the ellipticity condition on aij , we can prove that ZZ βε−1 φξ 2 |∇w|2 dxdt Qω Z Z ZZ ≤C L2 wφξ 2 wdxdt + (sT + ε−1 s2 λ2 ) φ3 |w|2 dxdt Q Qω ZZ φ1/2 |∇w|ξφ1/2 wdxdt . + λε−1 Qω
Therefore, by Young’s inequality, we have that ZZ sλ2 δ 2 ε−2 φξ 2 |∇w|2 dxdt Q ZZ ω ZZ 1 ≤ |L2 w|2 dxdt + Cβ −2 s3 λ4 (δ 4 + δ 2 )ε−2 φ3 |w|2 dxdt. 4 Q Qω Thus, inequality (6.11) gives ||L1 w||2L2 (Q)
−2 3 4 −2
ZZ
φ3 |w|2 dxdt ZZ 2 −2 + sλ ε φ|∇w|2 dxdt Q ZZ sα 2 −2 3 4 −2 3 2 ≤ C ||e g||L2 (Q) + β s λ ε φ |w| dxdt . +
||L2 w||2L2 (QT )
+β
s λ ε
Q
(6.12)
Qω
Let us now we use the first two terms in left-hand side of (6.12) in order to add the integrals of |∆w|2 and |wt |2 to the left-hand side of (6.12). This is done using the expressions of L1 w and L2 w. Indeed, from (6.5) and (6.6), we have
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
ZZ
s−1 φ−1 |∂t w|2 dxdt + ε−2 Q
ZZ
s−1 φ−1
Q
17
N X ∂xi (Mij ∂xj w) 2 dxdt i,j=1
ZZ
ZZ
(6.13)
+ s3 λ4 ε−2 φ3 |w|2 dxdt + sλ2 ε−2 φ|∇w|2 dxdt Q Q ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2 φ3 |w|2 dxdt . Qω
Using the term in |∂xi (Mij ∂xj w)|2 on the lef-hand side of (6.13) and elliptic regularity, it is easy to show that s−1 ε−2
ZZ
φ−1
Q
N 2 X 2 ∂xi xj w dxdt
ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2
φ3 |w|2 dxdt . Qω
i,j=1
Estimate (6.13) then gives
ZZ
s−1 φ−1 |∂t w|2 dxdt + s−1 ε−2
ZZ
Q
φ−1 Q
ZZ
N 2 X 2 ∂xi xj w dxdt i,j=1
ZZ
+s3 λ4 ε−2 φ3 |w|2 dxdt + sλ2 ε−2 φ|∇w|2 dxdt Q Q ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2 φ3 |w|2 dxdt .
(6.14)
Qω
From (6.14) and the fact that w = esα v, we finish the proof of Theorem 6.1.
6.2. A Slightly changed Carleman inequality. Our second sharp Carleman inequality is the following. Theorem 6.3. For any 0 < ≤ 1, there exist λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for every λ ≥ λ0 and s ≥ s0 (T + T 2 ), the solution v of (6.1) satisfies ZZ
e2sα |∂t v|2 dxdt + ε−2 Q
ZZ
e2sα Q
N 2 X 2 ∂xi xj v dxdt i,j=1
ZZ
ZZ + s4 λ4 ε−2 φ4 e2sα |v|2 dxdt + s2 λ2 ε−2 φ2 e2sα |∇ρ|2 dxdt Q Q ZZ ZZ ≤ Ceλ||ψ|| (s φe2sα |g|2 dxdt + s4 λ4 ε−2 φ4 e2sα |v|2 dxdt), Q
(6.15)
Qω
with C > 0 depending only on Ω, ω0 , ψ and β. Proof. The starting point is the application of the Carleman inequality given in Theorem 6.1 to the equation (6.1). Indeed, we have
18
F. W. CHAVES-SILVA AND M. BENDAHMANE
ε
2
ZZ
−1 −1 2sα
s
φ
e
ZZ
2
Q
Q
+ s3 λ4
N 2 X 2 ∂xi xj v dxdt
s−1 φ−1 e2sα
|vt | dxdt +
i,j=1
ZZ
ZZ
φ3 e2sα |v|2 dxdt + sλ2 φe2sα |∇v|2 dxdt Q ZZ ZZ 2sα 2 3 4 ≤ C( e |g| dxdt + s λ φ3 e2sα |v|2 dxdt).
(6.16)
Q
Q
Qω 1
Next, we introduce the function y(x, t) = v(x, t)(φ∗ (t)) 2 . This new function satisfies 1 ε∂t y − div (M (x)∇y) = −ε (T −2t) in Q, φ∗ y + ε(φ∗ (t)) 2 g 2 y=0 on Σ.
(6.17)
Applying again the Carleman inequality given by Theorem 6.1, this time for y, we obtain, for s large enough, that ZZ
s−1 φ−1 e2sα |∂t y|2 dxdt + ε−2
ZZ
Q
s−1 φ−1 e2sα Q
+ s3 λ4 ε−2
N 2 X 2 ∂xi xj y dxdt i,j=1
ZZ
ZZ
φ3 e2sα |y|2 dxdt + sλ2 ε−2 φe2sα |∇y|2 dxdt Q ZZ ZZ ∗ 2sα 2 3 4 −2 ≤ C( φ e |g| dxdt + s λ ε φ3 e2sα |y|2 dxdt).
(6.18)
Q
Q
Qω
From the definition of y, it is easy to show that ZZ ZZ ZZ 1 s−1 φ−1 e2sα |vt (φ∗ ) 2 |2 dxdt ≤ s−1 φ−1 e2sα |∂t y|2 dxdt + e2sα φ|y|2 dxdt. Q
Q
(6.19)
Q
Using (6.19), inequality (6.18) becomes ZZ
s−1 φ−1 φ∗ e2sα |vt |2 dxdt + ε−2 Q
ZZ
s−1 φ∗ φ−1 e2sα Q
+ s3 λ4 ε−2
N 2 X 2 ∂xi xj v dxdt i,j=1
ZZ
ZZ
φ3 φ∗ e2sα |v|2 dxdt + sλ2 ε−2 φφ∗ e2sα |∇v|2 dxdt Q Q ZZ ZZ ∗ 2sα 2 3 4 −2 3 ∗ 2sα 2 ≤ C( φ e |g| dxdt + s λ ε φ φ e |v| dxdt). Q
(6.20)
Qω
From Remark 3.2, the result follows.
Acknowledgments This paper has been partially established during the visit of M. Bendahmane to the Basque Center for Applied Mathematics. The authors thank professor Enrique Zuazua for various fruitful discussions about this work and Erich Foster for a careful reading of this paper.
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
19
References [1] F. Ammar-Khodja, A. Benabdallah, M. Gonz´ alez-Burgos, L. de Teresa, Recents results on the controllability of linear coupled parabolic problems: a survey, Mathematical Control and Related Fields, 1 (3)(2011), 267–306. [2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928–943. [3] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, I. Kostine, Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 42 (5)(2003), 1661–1680. [4] V. Barbu, Local controllability of the phase field system, Nonlinear Anal. Ser. A Theory Methods, 50 (2002), 363–372. [5] M. Bendahmane, R. B¨ urger, R. Ruiz-Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities, Math. Comp. Simul., 80 (2010), 1821–1840. [6] M. Bendahmane, F. W. Chaves-Silva, Null controllability of a degenerated reaction-diffusion system in cardiac electrophysiology, C. R. Math. Acad. Sci. Paris, 350 (11-12)(2012), 587–590. [7] M. Bendahmane, K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185–218. [8] M. Bendahmane, K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl. Numer. Math., 59 (2009), 2266–2284. [9] P. Biler, Existence and asymptotics of solutions for a parabolic-elliptic system nonlinear non-flux boundary conditions, Nonilinear Analysis T. M. A., 19 (1992), 1121–1136. [10] P. Biler, W. Hebisch, T. Nadzieja, The Debye system: Existence and long time behavior of solutions, Nonlinear Analysis T. M. A., 23 (1994), 1189–1209. [11] P. Colli Franzone, G. Savar´ e, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution equations, semigroups and functional analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl. 50, Birkh¨ auser, Basel, 2002, 49–78. [12] J.-M. Coron, S. Guerrero, A singular optimal control: A linear 1-D parabolic hyperbolic example, Asymptot. Analisys, 44 (2005), 237–257. [13] E. Fern´ andez-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincar´ e, Analise Non Lin´ eaire, 17 (2000), 583–616. [14] A. V. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Research Institute of Mathematics, Seoul National University, Seoul, 1996. [15] M. Gonz´ alez-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91–113. [16] O. Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J. Funct. Analysis, 358 (2010), 852–868. [17] I. L. Glicksberg, A further Generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170–174. [18] M. Gonz´ alez-Burgos, R. P´ erez-Garc´ıa, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., 46 (2)(2006), 123–162. [19] S. Guerrero, Introduction to the Controllability of Partial Differential Equations, Lectures at the Pan-American Advances Studies Institute, Santiago, Chile, 2012. [20] S. Guerrero, Null Controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379–394. [21] S. Guerrero, G. Lebeau, Singular Optimal Control for a Transport-Diffusion equation, Comm. Partial Diff. Equations, 32 (2007), 1813–1836. [22] C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the bidomain model, Crit. Rev. Biomed. Engr., 21 (1993), 1–77. [23] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544. [24] O. Yu. Imanuvilov, M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems (College Station, Tex, 1999), Lectures Notes in Pure and Appl. Math. 218, Marcel Dekker, New York, 2001, 113-137. [25] J. Keener, J. Sneyd, Mathematical physiology, Interdisciplinary Applied Mathematics 8, Springer-Verlag, New York, 1998. [26] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. [27] K. Kunisch, M. Wagner, Optimal control of the bidomain system (I): The monodomain approximation with the RogersMcCulloch model, Nonlinear Analysis: Real World applications, 13 (2012), 1525–1550.
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[28] O. A. Ladyzhenskaja, V. A. Solonnikov, N. N. Ural’ceva, Linear and Quasilinear equations of Parabolic type, 23 (Translated from the Russian by S. Smith), Translations of Mathematical Monographs, American Mathematical Society, Providence), 1967. [29] A. Lopez, X. Zhang, E. Zuazua, Null Controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl., 79 (2000), 741–808. [30] A. Lopez, E. Zuazua, Null controllability of the 1 − d heat equation as a singular limit of the controllability of damped wave equations, C. R. Acad. Sci. Paris, 327 (1998), 753–758. [31] C.-H. Luo, Y. Rudy, A model of the ventricular cardiac action potential: Depolarization, repolarization, and their interaction, Circ Res., 68 (1991), 1501–1526. [32] D. Noble, A modification of the Hodgkin-Huxley equation applicable to Purkinje fibre action and pacemaker potentials, J. Physiol., 160 (1962), 317–352. [33] J.-P. Puel, Controllability of Partial Differential Equations, Lectures in the Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil, 2002. [34] S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electro-cardiology, Numer. Methods Partial Differential Equations, 18 (2002), 218–240. [35] L. Tung, A bidomain model for describing ischemic myocardial D-C potentials, PhD thesis, MIT, 1978. [36] M.Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field, Math. Methods Appl.Sci., 29 (2006),1631–1661. [37] E. Zuazua, Exact boundary controllability of the semilinear wave equation, in Nonlinear Partial Differential Equations and their applications, Vol. 10, Pitman Res. Notes Math. Ser., 220, Longman, Harlow, UK, 357-391, 1991. (F. W. Chaves-Silva) ´ de Nice Sophia-Antipolis, Laboratoire Jean Dieudonne ´, UMR CNRS 6621, Parc Valrose, 06108 Nice Universite Cedex 02, France. E-mail address: [email protected] (M. Bendahmane) ´matiques de Bordeaux, Universite ´ Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Institut Mathe Bordeaux, France. E-mail address: [email protected]
arXiv:1106.1788v6 [math.OC] 25 Sep 2015
FELIPE WALLISON CHAVES-SILVA∗ AND MOSTAFA BENDAHMANE
Abstract. This paper is devoted to the analysis of the uniform null controllability for a family of nonlinear reaction-diffusion systems approximating a parabolic-elliptic system which models the electrical activity of the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by means of a single control is shown. The proof is based on the combination of Carleman estimates and weighted energy inequalities.
1. Introduction Let Ω ⊂ RN (N = 2, 3) be a bounded connected open set whose boundary, ∂Ω, is sufficiently regular. Let T > 0, and let ω and O be two (small) nonempty subsets of Ω, which we will refer to as control domains. We will use the notation Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). The main objective of this paper is to study the properties of controllability and observability for a family of nonlinear reaction-diffusion systems which degenerates into a nonlinear parabolic-elliptic system which models the electrical activity in the cardiac tissue. To state the model, we let ui = ui (t, x) and ue = ue (t, x) represent the intracellular and extracellular electric potentials, respectively. Their difference, v = ui − ue , is called the transmembrane potential. The anisotropic properties of the media are modeled by intracellular and extracellular conductivity tensors Mi (x) and Me (x). The widely accepted model (see [11, 22, 35]) describing the electrical activity in the cardiac tissue reads as follows: cm ∂t v − div (Mi (x)∇ui ) + h(v) = f 1ω in Q, (1.1) cm ∂t v + div (Me (x)∇ue ) + h(v) = g1O in Q, where cm > 0 is the surface capacitance of the membrane, the nonlinear function h : R → R is the transmembrane ionic current (the most interesting case being when h is a cubic polynomial), and f and g are stimulation currents applied, respectively, to ω and O. System (1.1) is known as the bidomain model and is completed with Dirichlet boundary conditions for the intra- and extracellular electric potentials ui = ue = 0 on Σ
(1.2)
2010 Mathematics Subject Classification. 35K57, 93B05, 93B07, 93C10. Key words and phrases. reaction-diffusion system, monodomain model, Carleman estimates, uniform null controllability, observability. ∗ F. W. Chaves-Silva has been supported by the ERC project Semi Classical Analysis of Partial Differential Equations, ERC-2012-ADG, project number 320845; the Grant BFI-2011-424 of the Basque Government and partially supported by the Grant MTM2011-29306-C02-00 of the MICINN, Spain, the ERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research Networking Programme OPTPDE and the Grant PI2010-04 of the Basque Government. 1
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F. W. CHAVES-SILVA AND M. BENDAHMANE
and initial data for the transmembrane potential v(0, x) = v0 (x),
x ∈ Ω.
(1.3)
We point out that realistic models describing electrical activities in the heart also include a system of ODE’s for computing the ionic current as a function of the transmembrane potential and a series of additional “gating variables” which aim to model the ionic transfer across the cell membrane (see [23, 25, 31, 32]). In the case where f 1ω = g1O and Mi = µMe , for some constant µ ∈ R, the bidomain model is simplified into the following parabolic-elliptic system: µ in Q, cm ∂t v − µ+1 div Me (x)∇v + h(v) = f 1ω −div M(x)∇u = div M (x)∇v in Q, e i (1.4) v = ue = 0 on Σ, v(0) = v0 in Ω, where M = Mi + Me . System (1.4) is known as monodomain model and is a very interesting model from the implementation point of view, since it conserves some of the essential features of the bidomain model as excitability phenomena (see [11, 27, 36]). The main difference between the bidomain model (1.1) and the monodomain model (1.4) is the fact that the first model is a system of two coupled parabolic equations, while the second one is a system of parabolic-elliptic type. Therefore, from the control point of view, one could expect these two systems to have, at least a priori, different control properties. In this work we show that the properties of controllability and observability for the monodomain model can be seen as a limit process of the controllability properties of a family of coupled parabolic systems. Indeed, given ε ∈ R such that 0 < ε ≤ 1, we approximate the monodomain model by the following family of parabolic systems: µ ε ε + h(v ε ) = f ε 1ω in Q, cm ∂t v − µ+1 div Me (x)∇v ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e (1.5) ε ε on Σ, v = ue = 0 ε in Ω. v (0) = v0 , uεe (0) = ue,0 In this paper we give a positive answer to the following question: Question 1.1. If, for each ε > 0, there exists a control f ε that drives the solution (v ε , uεe ) of (1.5) to zero at time t = T , i.e., v ε (T ) = uεe (T ) = 0, is it true that, when ε → 0+ , the control sequence {f ε }ε>0 converges to a function f which drives the associated solution (v, ue ) of (1.4) to zero at time t = T ? This question of approximating an equation by another having different physical properties has been used several times in the case of parabolic equations degenerating into hyperbolic ones (see, for example, [12, 16, 21]) and hyperbolic equations degenerating into parabolic ones (see, for example, [29, 30]). However, as far as we know, this is the first time that controllability of parabolic systems degenerating into parabolic-elliptic systems is studied. It is also important to mention that families of parabolic systems which degenerate into parabolic-elliptic ones arise in many areas, such as biology, chemistry and astrophysics (see [9, 10, 26]).
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
3
As usual, in control theory, when dealing with the controllability of a nonlinear problem, we first consider the linearized version of (1.5): µ div Me (x)∇v ε + a(t, x)v ε = f ε 1ω in Q, cm ∂t v ε − µ+1 ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e (1.6) ε ε v = u = 0 on Σ, e ε v (0) = v0 , uεe (0) = ue,0 in Ω, where a is a bounded function. Given ε > 0, the first obstacle to answering, positively, Question 1.1, will be to drive (v ε , uεe ), solution of (1.6), to zero at time T by means of a control f ε in such a way that the sequence of controls {f ε }ε>0 converges when ε → 0+ . Once it is shown that such a convergent sequence of control, {f ε }ε>0 , for the linear system (1.6), exists, we employ a fixed point argument and conclude that the same is true for the nonlinear system (1.5). Thus, we introduce the adjoint system of (1.6): µ div (Me (x)∇ϕε ) + a(t, x)ϕε = div (Mi (x)∇ϕεe ) in Q, −cm ∂t ϕε − µ+1 −ε∂ ϕε − div (M (x)∇ϕε ) = 0 in Q, t e e (1.7) ε ε ϕ = ϕ = 0 on Σ, e ε in Ω. ϕ (T ) = ϕT , ϕεe (T ) = ϕe,T Using duality arguments, it is very easy to prove that the task of building such a convergent sequence of controls, {f ε }ε>0 , for (1.6) is equivalent to prove the following (uniform) observability inequality for the solutions of (1.7): ZZ ||ϕε (0)||2L2 (Ω) + ε||ϕεe (0)||2L2 (Ω) ≤ C |ϕε |2 dxdt, Qω := ω × (0, T ), (1.8) Qω
where (ϕT , ϕe,T ) ∈ L2 (Ω)2 and the constant C = C(ε, Ω, ω, ||a||L∞ , T ) remains bounded when ε → 0+ . We prove inequality (1.8) as a consequence of an appropriate Carleman inequality for the solution (ϕε , ϕεe ) of (1.7) (see section 3). We notice that, due to the fact the control is acting on the first equation of (1.6), in our Carleman inequality, we need to bound global integrals of ϕε and ϕεe in terms of a local integral of ϕε , uniformly with respect to ε. Two main difficulties appear: first, the coupling in the first equation of (1.7) is in div (Mi (x)∇ϕεe ) and not in ϕεe ; second, we must show that the constant we get in our Carleman inequality does not blow up when ε → 0+ . The first difficulty is not so hard to overcome. Indeed, for each ε > 0 fixed, inequality (1.8) is known to be true for system (1.7) (see [20]). However, the main novelty here is the fact that we obtain the boundedness of the observability constant C with respect to ε. As we will see, Carleman inequalities alone are not enough for this task, and we need to combine sharp Carleman estimates, with respect to ε, and weighted energy inequalities. As far as the controllability of non degenerate coupled parabolic systems is concerned, the situation is, by now, fairly well understood. For instance, in [20], the controllability of a quite general linear coupled parabolic system is studied and a null controllability result is obtained by means of Carleman inequalities. In [2], using a different strategy, the controllability of a nonlinear reaction-diffusion system of two coupled parabolic equations is analyzed, and the authors prove the null controllability for the linear system and the local null controllability of the nonlinear one. Another relevant work concerning the controllability of coupled systems is [15], in which the authors analyze the null controllability of a cascade system of m (m > 1) coupled parabolic equations and the authors are able to obtain null controllability for the
4
F. W. CHAVES-SILVA AND M. BENDAHMANE
cascade system, whenever they have a good coupling structure. It is also worth mentioning the works [3], [4] and [18], where local and global controllability results for phase field systems were studied. For a general discussion about the controllability of coupled parabolic systems, see the survey paper [1]. Concerning controllability results for the bidomain model, since in both equations the couplings are given by the time derivatives of the electrical potentials, it seems very difficult to study controllability properties for such a model. To the best of our knowledge, for the bidomain model (1.1), the problems of null and approximate controllability are still open (even with two controls). Regarding the null controllability of the monodomain model (1.4), since the solution of the parabolic equation enters as a source term in the elliptic one, the following controllability result holds. Theorem 1.2. (1) If h is C 1 (R), globally Lipschitz and h(0) = 0, then, for every v0 ∈ L2 (Ω), there exists a control f ∈ L2 (ω × (0, T )) such that the solution (v, ue ) of (1.4) satisfies: v(T ) = ue (T ) = 0. Moreover, the control f satisfies the following estimate: 2
2
kf 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) ,
(1.9)
for a constant C = C(Ω, ω, T ) > 0. (2) If h is C 1 (R) and h(0) = 0, there exists γ > 0 such that, for every v0 ∈ W 2/3,6 (Ω) ∩ H01 (Ω) with ||v0 ||W 2/3,6 (Ω)∩H01 (Ω) ≤ γ, there exists a control f ∈ L6 (ω × (0, T )) such that the solution (v, ue ) of (1.4) satisfies: v(T ) = ue (T ) = 0. Moreover, the control f satisfies the following estimate: kf 1ω kL6 (Q) ≤ C kv0 kL2 (Ω) ,
(1.10)
for a constant C = C(Ω, ω, T ) > 0. Theorem 1.2, case 1, follows from [14, Theorem 3.1] and case 2 follows from [19, Theorem 3.5] (see also [14, Theorem 4.2]). This paper is organized as follows. In section 2, we state our main results. In section 3, we prove a uniform Carleman inequality for the adjoint system (1.7). Next, we show, in section 4, the uniform null controllability of (1.6). In section 5, we deal with the uniform null controllability of the nonlinear system (1.5). 2. Main results Throughout this paper we will assume that the matrices Mj , j = i, e are C ∞ , bounded, symmetric and positive semidefinite. Our first main result is a uniform Carleman estimate for the adjoint system (1.7). Theorem 2.1. Given any 0 < ≤ 1, there exist positive constants C = C(Ω, ω), λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for any (ϕT , ϕe,T ) ∈ L2 (Ω)2 and any a ∈ L∞ (Q), the solution (ϕε , ϕεe ) of (1.7) satisfies: ZZ ZZ e3sα |ρε |2 dxdt + s3 λ4 φ3 e3sα |ϕε |2 dxdt Q Q ZZ ≤ Ce6λ||ψ|| s8 λ4 φ8 e2sα |ϕε |2 dxdt, (2.1) Qω
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
5
2/3
for every s ≥ (T + (1 + ||a||L∞ )T 2 )s0 and λ ≥ λ0 , where ρε (x, t) = div (M (x)∇ϕεe (x, t)) and the weight functions φ and α are defined in (3.3) and (3.4), respectively. The proof of Theorem 2.1 follows from a combination of Carleman inequalities, for the heat equation, with a precise dependence on the degenerating parameter, and an energy inequality for the adjoint system (1.7). We prove Theorem 2.1 in section 3. Remark 2.2. As a direct consequence of the Carleman inequality (2.1), we have the unique continuation property for the solutions (ϕε , ϕεe ) of (1.7): “Given ε > 0, if ϕε = 0 in ω × (0, T ), then (ϕε , ϕεe ) ≡ (0, 0) in Q”. This unique continuation property for the adjoint system (1.7) implies, for each ε > 0, the approximate controllability at time T of system (1.6), with a control acting only on the first equation. The second main result of this paper gives the global null controllability of the linear system (1.6). Theorem 2.3. For any 0 < ε ≤ 1 and any (v0 , ue,0 ) ∈ L2 (Ω)2 , there exists a control f ε ∈ L2 (ω × (0, T )) such that the associated solution, (v ε , uεe ), to (1.6) is driven to zero at time T . That is to say, the associated solution satisfies: v ε (T ) = 0, uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) , (2.2) for a constant C = C(Ω, ω, ||a||L∞ , T ) > 0. From Theorem 2.1, the proof of Theorem 2.3 is standard. However, for the sake of completeness, we prove Theorem 2.3 in section 4. The third main result of this paper is concerned with the uniform null controllability of the nonlinear parabolic system (1.5). Theorem 2.4. Given any 0 < ε ≤ 1, we have: (1) If h is C 1 (R), globally Lipschitz and h(0) = 0, then, for every (v0 , ue,0 ) ∈ L2 (Ω)2 , there exists a control f ε ∈ L2 (ω × (0, T )) such that the solution (v ε , uεe ) of (1.5) satisfies: v ε (T ) = uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL2 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) ,
(2.3)
for a constant C = C(Ω, ω, T ) > 0. (2) If h is C 1 (R) and h(0) = 0, there exists γ > 0, does not depending on ε, such that, for every 2 (v0 , ue,0 ) ∈ W 2/3,6 (Ω) ∩ H01 (Ω) with ||(v0 , ue,0 )||W 2/3,6 (Ω) ≤ γ, there exists a control f ε ∈ L6 (ω × (0, T )) such that the solution (v ε , uεe ) of (1.5) satisfies: v ε (T ) = uεe (T ) = 0. Moreover, the control f ε satisfies the estimate: 2 2 2 kf ε 1ω kL6 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) ,
(2.4)
for a constant C = C(Ω, ω, T ) > 0. The proof of Theorem 2.4 is achieved through fixed point arguments, and it will be done in section 5.
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F. W. CHAVES-SILVA AND M. BENDAHMANE
Remark 2.5. In this paper we restrict the dimension to N = 2, 3, because the bidomain model makes sense only in such dimensions. Nevertheless, from the mathematical point of view, systems (1.4), (1.5) and (1.6) make sense for any N ∈ N (the 1-d case corresponding to the cable equation) and, taking the initial data in the appropriate space, all the results of this paper can be extended to higher dimensions. 3. Carleman inequality In this section we prove Theorem 2.1. To simplify the notation, we neglect the index ε and, since the only constant which matters in the analysis is ε, we assume that all the other constants are normalized to be the unity. In this case, the adjoint system (1.7) reads: −∂t ϕ − div (Me (x)∇ϕ) + a(x, t)ϕ = div (Mi (x)∇ϕe ) −ε∂ ϕ − div (M (x)∇ϕ ) = 0 t e e ϕ = ϕ = 0 e ϕ(T ) = ϕT , ϕe (T ) = ϕe,T
in Q, in Q, on Σ, in Ω.
(3.1)
We notice that, if ϕT and ϕe,T are regular enough, taking ρ(x, t) = div (Mi (x)∇ϕe (x, t)), the pair (ϕ, ρ) satisfies: −∂t ϕ − div (Me (x)∇ϕ) + a(x, t)ϕ = ρ in Q, −ε∂ ρ − div (M (x)∇ρ) = 0 in Q, t (3.2) ϕ = ρ = 0 on Σ, ϕ(T ) = ϕT , ρ(T ) = ρT in Ω. We prove the Carleman inequality (2.1) using system (3.2). Before starting the proof of the Carleman inequality, let us first define several weight functions which will be usefull in the sequel. Lemma 3.1. Let ω0 be an arbitrary nonempty open set such that ω0 ⊂ ω ⊂ Ω. There exists a function ψ ∈ C 2 (Ω) such that ψ(x) > 0, ∀x ∈ Ω, ψ ≡ 0 on ∂Ω, |∇ψ(x)| > 0 ∀x ∈ Ω\ω0 . Proof. See [14].
Using Lemma 3.1, we introduce the weight functions φ(x, t) =
eλ(ψ(x)+m||ψ||) eλm||ψ|| ; φ∗ (t) = min φ(x, t) = ; t(T − t) t(T − t) x∈Ω
eλ(ψ(x)+m||ψ||) − e2λmkψk eλ(m+1)kψk − e2λmkψk ; α∗ (t) = max α(x, t) = , t(T − t) t(T − t) x∈Ω for a parameter λ > 0 and a constant m > 1. Here, α(x, t) =
(3.3) (3.4)
kψ(x)k = max |ψ(x)| . x∈Ω ∗
Remark 3.2. From the definition of α and α it follows that, for λ large enough, 3α∗ ≤ 2α. Moreover, φ∗ (t) ≤ φ(x, t) ≤ eλkψk φ∗ (x, t) and |∂t α∗ | ≤ e2λkψk T φ2 .
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
7
Proof of Theorem 2.1 . For a better comprehension, we divide the proof into several steps. Step 1. First estimate for the parabolic system. In this step we obtain a first Carleman estimate for the adjoint system (1.7). For that, we will apply sharp Carleman inequalities, with respect to ε, to the system and get a global estimate of ϕ and ρ in terms of a local integral of ϕ and another in ρ. We consider a set ω1 such that ω0 ⊂⊂ ω1 ⊂⊂ ω and apply the sharp Carleman inequality (6.2), with ε = 1, and (6.15) to ϕ and ρ, respectively. We get ZZ ZZ N 2 X 2 2 s−1 φ−1 e2sα |ϕt | dxdt + s−1 φ−1 e2sα ∂xi xj ϕ dxdt Q
Q
+ s3 λ4
ZZ
i,j=1
ZZ
2
2
φ3 e2sα |ϕ| dxdt + sλ2 φe2sα |∇ϕ| dxdt Q Q Z Z ZZ 2 2 2 2sα 3 4 3 2sα ≤C e (|ρ| + |ϕ| )dxdt + s λ φ e |ϕ| dxdt Q
(3.5)
Qω1
and ZZ
2
e2sα |∂t ρ| dxdt + ε−2
ZZ
Q
N 2 X 2 ∂xi xj ρ dxdt
e2sα Q
i,j=1
ZZ 2 2 φ4 e2sα |ρ| dxdt + s2 λ2 ε−2 φ2 e2sα |∇ρ| dxdt Q Q ZZ 2 λkψk 4 4 −2 4 2sα ≤ Ce s λ ε φ e |ρ| dxdt,
+ s4 λ4 ε−2
ZZ
(3.6)
Qω1 2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 . Adding (3.5) and (3.6), and absorbing the lower order ZZ ZZ N X 2 −1 2sα −1 2sα φ e |ϕt | dxdt + φ e Q
Q
4 4
ZZ
+s λ
3 2sα
φ e
2
+ ε2
ZZ Q
2
ZZ
Q
+ s4 λ4
e2sα Q
2
φ4 e2sα |ρ| dxdt + s2 λ2
Q
≤C e
2
φe2sα |∇ϕ| dxdt
|ϕ| dxdt + s λ
e2sα |∂t ρ| dxdt +
ZZ
2 2 ∂xi xj ϕ dxdt
i,j=1 2 2
Q
ZZ
terms in the right-hand side, we get
N 2 X 2 ∂xi xj ρ dxdt
(3.7)
i,j=1
ZZ
2
φ2 e2sα |∇ρ| dxdt
Q λ||ψ|| 4 4
ZZ
s λ
4 2sα
φ e Qω1
2
4 4
ZZ
3 2sα
|ρ| dxdt + s λ
φ e
|ϕ| dxdt , 2
Qω1
2/3
for s ≥ (T + (1 + kakL∞ )T 2 )s0 and λ ≥ λ0 . Remark 3.3. If we were trying to drive the solution of (1.6) to zero by means of controls on both equations, inequality (3.7) would be sufficient. Step 2. Estimate of the local integral of ρ.
8
F. W. CHAVES-SILVA AND M. BENDAHMANE
In this step we estimate the local integral on ρ in the right-hand side of (3.7). This will be done using equation (3.2)1 . Indeed, we consider a function ξ satisfying ξ ∈ C0∞ (ω), 0 ≤ ξ ≤ 1, ξ(x) = 1 ∀x ∈ ω1 and write Ceλ||ψ|| s4 λ4
ZZ
ZZ
e2sα φ4 |ρ|2 ξdxdt = Ceλ||ψ|| s4 λ4 Qω
e2sα φ4 ρ(−ϕt − div (Me ∇ϕ) + aϕ)ξdxdt
Qω
(3.8)
:= E + F + G. In the sequel, we estimate each parcel in the expression above. First, we have ZZ E = Ceλ||ψ|| s4 λ4 s∂t αe2sα φ4 ρϕξdxdt Qω ZZ ZZ + Ceλ||ψ|| s4 λ4 e2sα φ3 φt ρϕξdxdt + Ceλ||ψ|| s4 λ4 e2sα φ4 ∂t ρϕξdxdt Qω
Qω
:= E1 + E2 + E3 , and it is not difficult to see that E1 + E2 ≤
1 4 4 s λ 10
ZZ
e2sα φ4 |ρ|2 dxdt + Ce2λ||ψ|| s8 λ4
ZZ
Qω
e2sα φ8 |ϕ|2 dxdt Qω
and ε2 E3 ≤ 2
ZZ e
2sα
2λ||ψ|| −2 8 8
2
|∂t ρ| dxdt + Ce
ε
Qω
ZZ
s λ
e2sα φ8 |ϕ|2 dxdt.
Qω
Next, integrating by parts, we get e−λ||ψ|| s−4 λ−4 F =
N ZZ X i,j=1
+
s∂xi α e2sα φ4 ρ(Meij ∂xj ϕ)ξdxdt
Qω N ZZ X
i,j=1
+
N ZZ X i,j=1
+
e2sα φ3 ∂xi φ ρ(Meij ∂xj ϕ)ξdxdt Qω
e2sα φ4 ∂xi ρ (Meij ∂xj ϕ)ξdxdt Qω
N ZZ X i,j=1
e2sα φ4 ρ(Meij ∂xj ϕ)∂xi ξdxdt Qω
and we can show that F ≤
ZZ 1 e2sα φ4 |ρ|2 dxdt + s2 λ2 e2sα φ2 |∇ρ|2 dxdt 6 Qω Qω ZZ ZZ N 2 X 1 2 e2sα + Ce2λ||ψ|| s8 λ8 e2sα φ8 |ϕ|2 dxdt + ∂xi xj ρ dxdt. 2 Qω Qω i,j=1
1 4 4 s λ 10
ZZ
(3.9)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
9
Finally, we have 1 G ≤ s4 λ4 10
ZZ
e2sα φ4 |ρ|2 dxdt Qω 2λ||ψ|| 4 4
+ Ce
s λ
||a||2L∞
ZZ
e2sα φ4 |ϕ|2 dxdt. Qω
Putting E, F and G together in (3.7), we obtain ZZ ZZ N 2 X 2 e2sα |ϕt |2 dxdt + e2sα ∂xi xj ϕ dxdt Q
Q
4 4
ZZ
4 2sα
+s λ
φ e ZZ
2
2 2
ZZ
2 2sα
|ϕ| dxdt + s λ
Q
φ e
2
|∇ϕ| dxdtε
Q
Q
2
ZZ
e2sα |∂t ρ|2 dxdt Q
ZZ N 2 X 2 φ4 e2sα |ρ|2 dxdt ∂xi xj ρ dxdt + s4 λ4
e2sα
+
i,j=1
i,j=1
(3.10)
Q
ZZ + s2 λ2 φ2 e2sα |∇ρ|2 dxdt Q ZZ e2sα φ8 |ϕ|2 dxdt, ≤ Ce2λ||ψ|| ε−2 s8 λ8 Qω 2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 . Using (3.10), we can prove that, for every ε > 0, system (1.6) is null controllable. However, the sequence of controls obtained in this way will not be bounded when ε → 0+ . Therefore, we need to go a step further and improve estimate (3.10). This is the goal of the next step. Step 3. Weighted energy inequality. The reason why we do not get a bounded sequence of controls out of step 2 is because of the term ε−2 in the right-hand side of (3.10). In this step we prove a weighted energy inequality for equation (3.2)2 , which will be used to compensate this ε−2 term. Let us introduce the function ∗ 3 y = e 2 sα ρ. This new function satisfies ∗ 3 ε∂t y − div (M (x)∇y) = ε 32 s∂t α∗ e 2 sα ρ in Q, (3.11) y=0 on Σ, y(0) = y(T ) = 0 in Ω. Multiplying (3.11) by y and integrating over Ω, we get Z ∗ 3 ε d 3 2 2 ||y(t)||L2 (Ω) + C||∇y(t)||L2 (Ω) ≤ ε s∂t α∗ (t)e 2 sα (t) ρ(t)y(t)dx. 2 dt 2 Ω Integrating this last inequality form 0 to T and using Poincar´e’s and Young’s inequalities, it is not difficult to see that ZZ ZZ 3sα∗ 2 2 4λ||ψ|| e |ρ| dxdt ≤ Cε e s4 φ4 e2sα |ρ|2 dxdt. (3.12) Q
Q
Finally, from (3.10) and (3.12), we obtain ZZ ZZ ∗ e3sα |ρ|2 dxdt ≤ Ce6λ||ψ|| s8 λ4 Q
Qω
φ8 e2sα |ϕ|2 dxdt.
(3.13)
10
F. W. CHAVES-SILVA AND M. BENDAHMANE
This last estimate gives a global estimate of ρ in terms of a local integral of φ, with a constant C which is bounded with respect to ε. Step 4. Last estimates and conclusion. In order to finish the proof of Theorem 2.1, we combine inequality (3.13) and a slightly different Carleman inequality to the equation (3.2)1 . Indeed, the following Carleman inequality holds: ZZ ZZ N 2 X 2 s−1 φ−1 e3sα |ϕt |2 dxdt + s−1 φ−1 e3sα ∂xi xj ϕ dxdt Q
Q
i,j=1
ZZ
ZZ
+ s3 λ4 φ3 e3sα |ϕ|2 dxdt + sλ2 φe3sα |∇ϕ|2 dxdt Q Q Z Z ZZ 3sα 2 3 4 3 3sα 2 ≤C e |ρ| dxdt + s λ φ e |ϕ| dxdt , Q
(3.14)
Qω
2/3 kakL∞ )T 2 )s0
for s ≥ (T + (1 + and λ ≥ λ0 , where ϕ is, together with ρ, solution of (3.2). Notice that here we have just changed the weight e2sα by e3sα . The proof of (3.14) is exactly the same as the proof of Theorem 6.1, just taking the appropriate change of variable in (6.3). ∗ Next, since e3sα ≤ e3sα , we have ZZ ZZ ∗ e3sα |ρ|2 dxdt ≤ e3sα |ρ|2 dxdt and by (3.13), we have that ZZ e
3sα
Q
Q
2
6λ||ψ|| 8 4
|ρ| dxdt ≤ Ce
Q
ZZ
φ8 e2sα |ϕ|2 dxdt.
s λ
Qω
From (3.13) and (3.14), it follows that ZZ ZZ ZZ 3sα 2 3 4 3 3sα 2 6λ||ψ|| 8 4 e |ρ| dxdt + s λ φ e |ϕ| dxdt ≤ Ce s λ Q
Q
φ8 e2sα |ϕ|2 dxdt,
(3.15)
Qω
which is exactly (2.1). By density, we can show that (3.15) remains true when we consider initial data in L2 (Ω). Therefore, the proof of Theorem 2.1 is finished. 4. Null controllability for the linearized system This section is devoted to proving the null controllability of linearized equation (1.6). It will be done by showing the observability inequality (1.8) for the adjoint system (1.7), and solving a minimization problem. The arguments used here are classical in control theory for linear PDE’s. Hence, we just give a sketch of the proof. Proof of Theorem 2.3. Combining the standard energy inequalities for system (3.2) and the Carleman inequality given by Theorem 2.1, we can show the following observability inequality for the solutions of (3.2): ZZ 2/3
||ϕ(0)||2L2 (Ω) + ε||ρ(0)||2L2 (Ω) ≤ eC(1+1/T +||a||L∞ +||a||L∞ T )
|ϕ|2 dxdt,
Qω
where C = C(Ω, ω) is a positive constant. Next, since ρ(x, t) = div (M (x)∇ϕe (x, t)) and ϕe = 0 on ∂Ω, we have that ||ϕe (t)||H 2 (Ω) ≤ C||ρ(t)||L2 (Ω) ,
(4.1)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
11
for all t ∈ [0, T ]. Therefore, it follows from (4.1) that 2/3
||ϕ(0)||2L2 (Ω) + ε||ϕe (0)||2L2 (Ω) ≤ eC(1+1/T +||a||L∞ +||a||L∞ T )
ZZ
|ϕ|2 dxdt,
(4.2)
Qω
which is the observability inequality (1.8). From (4.2) and the density of smooth solutions in the space of solutions of (3.1) with initial data in L2 (Ω), we see that the above observability inequality is satisfied by all solutions of (1.7) with initial data in L2 (Ω). Now, in order to obtain the null controllability for linear system (1.6), we solve, for any δ > 0, the following minimization problem: Minimize Jδ (ϕT , ϕe,T ), with ( Z Z 1 T 2 |ϕε | dx dt + ε(ue,0 , ϕεe (0)) Jδ (ϕT , ϕe,T ) = 2 0 ω
(4.3) )
+ (v0 , ϕε (0)) + δ(kϕT kL2 (Ω) + ε1/2 kϕe,T kL2 (Ω) ) , where (ϕ, ϕe ) is the solution of the adjoint problem (1.7) with initital data (ϕT , ϕe,T ) ∈ L2 (Ω)2 . It is an easy matter to check that Jδ is strictly convex and continuous. So, in order to guarantee the existence of a minimizer, the only thing remaining to prove is the coercivity of Jδ . Using the observability inequality (1.8) for the adjoint system (1.7), the coercivity of Jδ is straightfoward. Therefore, for each δ > 0, there exists a unique minimizer (ϕδe,T , ϕδT ) of Jδ . Let us denote by ϕε,δ the corresponding solution to (1.7) associated to this minimizer. Taking f ε,δ = ϕε,δ 1ω as a control for (1.6), the duality between (1.6) and (1.7) gives ||v ε,δ (T )||L2 (Ω) + ε1/2 ||uε,δ e (T )||L2 (Ω) ≤ δ, ε,δ where (v ε,δ , uε,δ . It also gives e ) is the solution of (1.6) associated to the control f ||f ε,δ 1ω ||2L2 (Q) ≤ C ||v0 ||2L2 (Ω) + ε||ue,0 ||2L2 (Ω) .
(4.4)
(4.5)
From (4.4) and (4.5), we get a control f ε (the weak limit of a subsequence of f ε,δ 1ω in L2 (ω × (0, T ))) that drives the solution of (1.6) to zero at time T . From (4.5), we have the following estimate on the control f ε , ||f ε 1ω ||2L2 (Q) ≤ C ||v0ε ||2L2 (Ω) + ε||uεe,0 ||2L2 (Ω) . (4.6) This finishes the proof of Theorem 2.3.
5. The nonlinear system In this section we prove Theorem 2.4. The proof is achieved through fixed point arguments. Proof of Theorem 2.4 (case 1): We consider the following linearization of system (1.5): µ ε ε + g(z)v ε = f ε 1ω in Q, cm ∂t v − µ+1 div Me (x)∇v ε∂ uε − div M(x)∇uε = div M (x)∇v ε in Q, t e i e ε ε v = ue = 0 on Σ, ε ε v (0) = v0 , ue (0) = ue,0 in Ω,
(5.1)
12
F. W. CHAVES-SILVA AND M. BENDAHMANE
where h(s) , if |s| > 0, g(s) = s h0 (0), if s = 0.
(5.2)
It follows from Theorem 2.3 that, for each (v0 , ue,0 ) ∈ L2 (Ω)2 and z ∈ L2 (Q), there exists a control function f ε ∈ L2 (Q) such that the solution of (5.1) satisfies: v ε (T ) = uεe (T ) = 0. As we said before, the idea is to use a fixed point argument. For that, we need the following generalized version of Kakutani’s fixed point Theorem, due to Glicksberg [17]. Theorem 5.1. Let B be a non-empty convex, compact subset of a locally convex topological vector space X. If Λ : B −→ B is a convex set-valued mapping with closed graph and Λ(B) is closed, then Λ has a fixed point. In order to apply Glicksberg‘s Theorem, we define a mapping Λ : B −→ X as follows Λ(z) = {v ε ; (v ε , uεe ) is a solution of (5.1), such that v ε (T ) = uεe (T ) = 0, for a control f ε satisfying (2.2)}. Here, X = L2 (Q) and B is the ball B = {z ∈ L2 (0, T, H01 (Ω)), ∂t z ∈ L2 (0, T, H −1 (Ω)); ||z||2L2 (0,T ;H 1 (Ω)) + ||∂t z||2L2 (0,T ;H −1 (Ω)) ≤ M }. 0
It is easy to see that Λ is well defined and that B is a convex and compact subset of L2 (Q). Let us now prove that Λ is convex, compact and has closed graph. • Λ(B) ⊂ B. Let z ∈ B and v ε ∈ Λ(z). Since v ε satisfies (5.1)1 , the following inequality holds ||v ε ||2L2 (0,T ;H 1 (Ω)) + ||∂t v ε ||2L2 (0,T ;H −1 (Ω)) ≤ K1 . 0
(5.3)
In this way, if z ∈ B then Λ(z) ⊂ B, if we take M = K1 . • Λ(z) is closed in L2 (Q). Let z ∈ B fixed, and vnε ∈ Λ(z), such that vnε → v ε . Let us prove that v ε ∈ Λ(z). In fact, by definition we have that vnε is, together with a function uεe,n , and a control fnε , the solution of (5.1), with ||fnε 1ω ||2L2 (Q) ≤ C ||v0 ||2L2 (Ω) + ε||ue,0 ||2L2 (Ω) . Therefore, we can extract a subsequence of fnε , denoted by the same index, such that fnε 1ω → f ε 1ω weakly in L2 (Q). Since fnε is bounded, we can argue as in the previous section and show that ||vnε ||2L2 (0,T ;H 1 (Ω)) + ||∂t vnε ||2L2 (0,T ;H −1 (Ω)) ≤ M. 0
Hence, ε vn → v ε weakly in L2 (0, T ; H01 (Ω)), ε vn → v ε strongly in L2 (Q), ∂t vnε → ∂t v ε weakly in L2 (0, T ; H −1 (Ω)).
(5.4)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
13
Using the converges above and (5.1)2 , we see that there exists a function uεe such that ε ue,n → uεe weakly in L2 (0, T ; H01 (Ω)), ε ue,n → uεe strongly in L2 (Q), ∂t uεe,n → ∂t uεe weakly in L2 (0, T ; H −1 (Ω)). It follows that (v ε , uεe ) is a controlled solution of (5.1) associated to the control f ε . Hence, v ε ∈ Λ(z) and Λ(z) is closed and compact in L2 (Q). • Λ has closed graph in L2 (Q) × L2 (Q). We need to prove that if zn → z, vnε → v ε strongly in L2 (Q) and vnε ∈ Λ(zn ), then v ε ∈ Λ(z). Using the two previous steps, it is easy to show that v ε ∈ Λ(z). Therefore, we can apply Glicksberg’s Theorem to conclude that Λ has a fixed point. This proves Theorem 2.4 in the case where the nonlinearity is a C 1 globally Lipschitz function. Proof of Theorem 2.4 (case 2): We consider the linear system: µ div Me (x)∇v ε + a(z)v ε = f ε 1ω cm ∂t v ε − µ+1 ε∂ uε − div M(x)∇uε = div M (x)∇v ε i t e e ε ε v = u = 0 e ε v (0) = v0 , uεe (0) = ue,0 2 with (v0 , ue,0 ) ∈ W 2/3,6 (Ω) ∩ H01 (Ω) , z ∈ L∞ (Q) and Z 1 dh a(z) = (sz)ds. 0 dz
in Q, in Q, on Σ, in Ω,
(5.5)
Arguing as in the proof Theorem 2.3, we can show the null controllability of (5.5) with controls in L2 (ω × (0, T )). However, these L2 controls are not sufficient to apply fixed point arguments and obtain the null controllability of the nonlinear system (1.5). For this reason, we modify a little the functional (4.3), obtaining controls which will allow us to employ Schauder’s fixed point Theorem. Indeed, for any δ > 0, we consider the problem: Minimize Jδ (ϕT , ϕeT ), with ( Z Z 1 T 2 Jδ (ϕT , ϕeT ) = e2sα φ8 |ϕε | dx dt + ε(ue,0 , ϕεe (0)) 2 0 ω
(5.6)
) ε 1/2 + (v0 , ϕ (0)) + δ kϕT kL2 (Ω) + kϕe,T kL2 (Ω) , where (ϕε , ϕεe ) is the solution of the adjoint system (1.7) with initital data (ϕT , ϕe,T ) ∈ L2 (Ω)2 . ε,δ As in section 4, we show that problem (5.6) has a unique minimizer (ϕε,δ , ϕε,δ = e ). Defining f ε,δ 2sα 8 ε,δ ε,δ ε,δ e φ ϕ and using the fact that ϕ is, together with a ϕe , the solution of (1.7), we see that f is a solution of a parabolic equation, with homogeneous Dirichlet boundary conditions, null initial data and a right-hand side in L2 (Q). Hence, f ε,δ ∈ L2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L2 (Ω)) and in particular we have that f ε,δ ∈ L6 (Q). Arguing as in [2, Lemma 5], we can easily show that 2 2 ||f ε,δ 1ω ||2L6 (Q) ≤ C kv0 kL2 (Ω) + ε kue,0 kL2 (Ω) , (5.7) where C > 0 is independent of and δ.
14
F. W. CHAVES-SILVA AND M. BENDAHMANE ε,δ Moreover, the solution (v ε,δ , uε,δ , satisfies e ) of (5.5), associated to f
||v ε,δ (T )||L2 (Ω) + ε1/2 ||uε,δ e (T )||L2 (Ω) ≤ δ. +
ε
(5.8)
6
Taking the limit when δ → 0 , we get a control f ∈ L (Q) (the weak limit of a subsequence of f ε,δ ) such that the associated solution (v ε , uεe ) to (5.5) satisfies v ε (T ) = uεe (T ) = 0. Next, we define a map F : L∞ (Q) −→ L∞ (Q) which, to each z ∈ L∞ (Q), associates v ε the solution, together with uεe , of (5.5) corresponding to z and to the control f ε built above. Note that this application is well defined since, from the regularity theory for parabolic equations (see, for instance, [28]), we have that v ε ∈ X := L6 (0, T ; W 2,6 (Ω)) ∩ W 1,6 (0, T ; L6 (Ω)) ∩ C([0, T ]; W 2/3,6 (Ω) ∩ H01 (Ω)). Let us now consider the set A defined as A := z ∈ L∞ (Q); ||z||L∞ (Q) ≤ 1 . It is clear that A is a convex closed subset of L∞ (Q). From (5.7), which still holds for f ε , and the smallness assumption on the initial data, we can easily show that F is continuous and that F (A) ⊂ A. Finally, since the space X is compactly embedded in L∞ (Q), we have that F (A) is compact in L∞ (Q). Therefore, F has a fixed point and the the proof of case 2 of Theorem 2.4 is finished. 6. Appendix: Some technical results In this section we prove the two sharp Carleman inequalities used in the proof of Theorem 2.1. We consider the parabolic equation N X ∂xi (aij (x)∂xi v(t, x)) = g(x, t) in Q, −∂t v(x, t) − i,j=1
v=0 v(T ) = vT
(6.1)
on Σ, in Ω,
where vT ∈ L2 (Ω) and g ∈ L2 (Q). We assume that the matrix aij has the form aij =
Mij , ε
and (Mij )ij is an elliptic matrix, i.e., there exists β > 0 such that
PN
i,j
Mij ξj ξi ≥ β|ξ|2 for all ξ ∈ RN .
6.1. A degenerating Carleman inequality. The first sharp Carleman inequality we prove is the following. Theorem 6.1. For any 0 < ≤ 1, there exist λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for every λ ≥ λ0 and s ≥ s0 (T + T 2 ), the solution v of (6.1) satisfies ZZ ZZ N 2 X 2 s−1 φ−1 e2sα |∂t v|2 dxdt + s−1 ε−2 φ−1 e2sα ∂xi xj v dxdt Q
Q
i,j=1
ZZ
ZZ
+ s3 λ4 ε−2 φ3 e2sα |v|2 dxdt + sλ2 ε−2 φe2sα |∇v|2 dxdt Q Q Z Z ZZ 2sα 2 3 4 −2 3 2sα 2 ≤C e |g| dxdt + s λ ε φ e |v| dxdt , Q
with C > 0 depending only on Ω, ω0 , ψ and β.
Qω
(6.2)
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
15
Proof. For s > 0 and λ > 0, we consider the change of variable w(t, w) = esα v(t, w),
(6.3)
which implies w(T, x) = w(0, x) = 0. Using the fact that v is the solution of (6.1), we write L1 w + L2 w = gs ,
(6.4)
where N X
L1 w = −∂t w + 2sλ
2
φaij ∂xj ψ ∂xi w + 2sλ
i,j=1
L2 w = −
N X
N X
φaij ∂xi ψ ∂xj ψw,
(6.5)
i,j=1
∂xi (aij ∂xj w) − s2 λ2
i,j=1
N X
φ2 aij ∂xi ψ ∂xj ψ w + s∂t α w
(6.6)
i,j=1
and gs = esα g + sλ2
N X
φaij ∂xi ψ ∂xj ψ w − sλ
i,j=1
N X
φ∂xi (aij ∂xj ψ) w.
(6.7)
i,j=1
From (6.4), we have that ||L1 w||2L2 (Q) + ||L2 w||2L2 (Q) + 2(L1 w, L2 w)L2 (Q) = ||gs ||2L2 (Q) .
(6.8)
The rest of the proof is devoted to analyze the terms appearing in (L1 w, L2 w)L2 (Q) . First, we write N X (L1 w, L2 w)L2 (Q) = Iij , i,j=1
where Iij is the inner product in L2 (Q) of the ith term in the expression of L1 w and the jth term in L2 w. After a long, but straightforward, calculation, we can show that the following estimate holds ZZ ZZ 2(L1 w, L2 w)L2 (Q) ≥ 2s3 λ4 β 2 ε−2 φ3 |∇ψ|4 |w|2 dxdt + 2sλ2 β 2 ε−2 φ|∇ψ|2 |∇w|2 dxdt Q Q Z Z −2 2 2 4 2 2 2 3 3 2 − Cε T s λ + Ts λ + T s + s λ + Ts λ φ3 |w|2 dxdt Q ZZ −2 2 2 − Cε (sλ + λ ) φ|∇w| dxdt.
(6.9)
Q
We take λ ≥ λ0 and s ≥ s0 (T + T 2 ), and it follows, from Remark 6.2 below, that 2(L1 w, L2 w)
L2 (Q)
3 4 2 −2
ZZ
φ3 |w|2 dxdt
+ 2s λ β ε
Qω0 2 2 −2
ZZ
φ|∇w|2 dxdt
+ 2sλ β ε
Qω0
≥ 2s3 λ4 β 2 ε−2
ZZ
φ3 |w|2 dxdt + 2sλ2 β 2 ε−2 Q
ZZ
φ|∇w|2 dxdt. Q
(6.10)
16
F. W. CHAVES-SILVA AND M. BENDAHMANE
Remark 6.2. Since Ω\ω0 is compact and |∇ψ| > 0 on Ω\ω0 , there exists δ > 0 such that β|∇ψ| ≥ δ on Ω\ω0 . Putting (6.10) in (6.8), we get ZZ φ3 |w|2 dxdt ||L1 w||2L2 (Q) + ||L2 w||2L2 (Q) + 2β −2 s3 λ4 δ 4 ε−2 Q ZZ + 2sλ2 δ 2 ε−2 φ|∇w|2 dxdt Q ZZ ≤ ||gs ||2L2 (Q) + 2β −2 s3 λ4 δ 4 ε−2 φ3 |w|2 dxdt
(6.11)
Qω0
+ 2sλ2 δ 2 ε−2
ZZ
φ|∇w|2 dxdt. Qω0
Now we deal with the local integral involving ∇w on the right-hand side of (6.11). To this end, we introduce a cutt-off function ξ such that ξ ∈ C0∞ (ω), 0 ≤ ξ ≤ 1, ξ(x) = 1 ∀x ∈ ω0 . Using the ellipticity condition on aij , we can prove that ZZ βε−1 φξ 2 |∇w|2 dxdt Qω Z Z ZZ ≤C L2 wφξ 2 wdxdt + (sT + ε−1 s2 λ2 ) φ3 |w|2 dxdt Q Qω ZZ φ1/2 |∇w|ξφ1/2 wdxdt . + λε−1 Qω
Therefore, by Young’s inequality, we have that ZZ sλ2 δ 2 ε−2 φξ 2 |∇w|2 dxdt Q ZZ ω ZZ 1 ≤ |L2 w|2 dxdt + Cβ −2 s3 λ4 (δ 4 + δ 2 )ε−2 φ3 |w|2 dxdt. 4 Q Qω Thus, inequality (6.11) gives ||L1 w||2L2 (Q)
−2 3 4 −2
ZZ
φ3 |w|2 dxdt ZZ 2 −2 + sλ ε φ|∇w|2 dxdt Q ZZ sα 2 −2 3 4 −2 3 2 ≤ C ||e g||L2 (Q) + β s λ ε φ |w| dxdt . +
||L2 w||2L2 (QT )
+β
s λ ε
Q
(6.12)
Qω
Let us now we use the first two terms in left-hand side of (6.12) in order to add the integrals of |∆w|2 and |wt |2 to the left-hand side of (6.12). This is done using the expressions of L1 w and L2 w. Indeed, from (6.5) and (6.6), we have
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
ZZ
s−1 φ−1 |∂t w|2 dxdt + ε−2 Q
ZZ
s−1 φ−1
Q
17
N X ∂xi (Mij ∂xj w) 2 dxdt i,j=1
ZZ
ZZ
(6.13)
+ s3 λ4 ε−2 φ3 |w|2 dxdt + sλ2 ε−2 φ|∇w|2 dxdt Q Q ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2 φ3 |w|2 dxdt . Qω
Using the term in |∂xi (Mij ∂xj w)|2 on the lef-hand side of (6.13) and elliptic regularity, it is easy to show that s−1 ε−2
ZZ
φ−1
Q
N 2 X 2 ∂xi xj w dxdt
ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2
φ3 |w|2 dxdt . Qω
i,j=1
Estimate (6.13) then gives
ZZ
s−1 φ−1 |∂t w|2 dxdt + s−1 ε−2
ZZ
Q
φ−1 Q
ZZ
N 2 X 2 ∂xi xj w dxdt i,j=1
ZZ
+s3 λ4 ε−2 φ3 |w|2 dxdt + sλ2 ε−2 φ|∇w|2 dxdt Q Q ZZ ≤ C ||esα g||2L2 (Q) + s3 λ4 ε−2 φ3 |w|2 dxdt .
(6.14)
Qω
From (6.14) and the fact that w = esα v, we finish the proof of Theorem 6.1.
6.2. A Slightly changed Carleman inequality. Our second sharp Carleman inequality is the following. Theorem 6.3. For any 0 < ≤ 1, there exist λ0 = λ0 (Ω, ω) ≥ 1 and s0 = s0 (Ω, ω) ≥ 1 such that, for every λ ≥ λ0 and s ≥ s0 (T + T 2 ), the solution v of (6.1) satisfies ZZ
e2sα |∂t v|2 dxdt + ε−2 Q
ZZ
e2sα Q
N 2 X 2 ∂xi xj v dxdt i,j=1
ZZ
ZZ + s4 λ4 ε−2 φ4 e2sα |v|2 dxdt + s2 λ2 ε−2 φ2 e2sα |∇ρ|2 dxdt Q Q ZZ ZZ ≤ Ceλ||ψ|| (s φe2sα |g|2 dxdt + s4 λ4 ε−2 φ4 e2sα |v|2 dxdt), Q
(6.15)
Qω
with C > 0 depending only on Ω, ω0 , ψ and β. Proof. The starting point is the application of the Carleman inequality given in Theorem 6.1 to the equation (6.1). Indeed, we have
18
F. W. CHAVES-SILVA AND M. BENDAHMANE
ε
2
ZZ
−1 −1 2sα
s
φ
e
ZZ
2
Q
Q
+ s3 λ4
N 2 X 2 ∂xi xj v dxdt
s−1 φ−1 e2sα
|vt | dxdt +
i,j=1
ZZ
ZZ
φ3 e2sα |v|2 dxdt + sλ2 φe2sα |∇v|2 dxdt Q ZZ ZZ 2sα 2 3 4 ≤ C( e |g| dxdt + s λ φ3 e2sα |v|2 dxdt).
(6.16)
Q
Q
Qω 1
Next, we introduce the function y(x, t) = v(x, t)(φ∗ (t)) 2 . This new function satisfies 1 ε∂t y − div (M (x)∇y) = −ε (T −2t) in Q, φ∗ y + ε(φ∗ (t)) 2 g 2 y=0 on Σ.
(6.17)
Applying again the Carleman inequality given by Theorem 6.1, this time for y, we obtain, for s large enough, that ZZ
s−1 φ−1 e2sα |∂t y|2 dxdt + ε−2
ZZ
Q
s−1 φ−1 e2sα Q
+ s3 λ4 ε−2
N 2 X 2 ∂xi xj y dxdt i,j=1
ZZ
ZZ
φ3 e2sα |y|2 dxdt + sλ2 ε−2 φe2sα |∇y|2 dxdt Q ZZ ZZ ∗ 2sα 2 3 4 −2 ≤ C( φ e |g| dxdt + s λ ε φ3 e2sα |y|2 dxdt).
(6.18)
Q
Q
Qω
From the definition of y, it is easy to show that ZZ ZZ ZZ 1 s−1 φ−1 e2sα |vt (φ∗ ) 2 |2 dxdt ≤ s−1 φ−1 e2sα |∂t y|2 dxdt + e2sα φ|y|2 dxdt. Q
Q
(6.19)
Q
Using (6.19), inequality (6.18) becomes ZZ
s−1 φ−1 φ∗ e2sα |vt |2 dxdt + ε−2 Q
ZZ
s−1 φ∗ φ−1 e2sα Q
+ s3 λ4 ε−2
N 2 X 2 ∂xi xj v dxdt i,j=1
ZZ
ZZ
φ3 φ∗ e2sα |v|2 dxdt + sλ2 ε−2 φφ∗ e2sα |∇v|2 dxdt Q Q ZZ ZZ ∗ 2sα 2 3 4 −2 3 ∗ 2sα 2 ≤ C( φ e |g| dxdt + s λ ε φ φ e |v| dxdt). Q
(6.20)
Qω
From Remark 3.2, the result follows.
Acknowledgments This paper has been partially established during the visit of M. Bendahmane to the Basque Center for Applied Mathematics. The authors thank professor Enrique Zuazua for various fruitful discussions about this work and Erich Foster for a careful reading of this paper.
UNIFORM CONTROLLABILITY FOR A DEGENERATING SYSTEM
19
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