CONTROLLABILITY OF A DEGENERATING

REACTION-DIFFUSION SYSTEM IN ELECTROCARDIOLOGY

MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Abstract. This paper is devoted to analyze the null controllability of a

nonlinear reaction-diffusion system approximating a parabolic-elliptic systemmodeling electrical activity in the heart. The uniform, with respect to the

degenerating parameter, null controllability of the approximating system by

a single control force acting on a subdomain is shown. The proof needs aprecisely estimate with respect to the degenerating parameter and it is done

combining Carleman estimates and energy inequalities.

1. Introduction

In this paper we study the controllability and observability properties for areaction-diffusion system which degenerates into a parabolic-elliptic system de-scribing the cardiac electric activity in a physical domain Ω ⊂ RN over a timespan (0, T ), T > 0 (Ω ⊂ R3 is the natural domain of the heart). To state our modelwe set ui = ui(t, x) and ue = ue(t, x) to represent, respectively, the spatial cellularand location x ∈ Ω of the intracellular and extracellular electric potentials, whosedifference v = v(t, x) = ui − ue is the transmembrane potential. The anisotropicproperties of the two media are modeled by intracellular and extracellular con-ductivity tensors Mi(x) and Me(x). The surface capacitance of the membraneis represented by the constant cm > 0. The transmembrane ionic current is rep-resented by a nonlinear function h(v) (the most interesting case being when thenonlinearity is cubic polynomial ) depending on the value of the potential v. Weassume that there is no stimulation currents applied to the intra- and extracellularspace.

The equations governing the cardiac model (known as the “bidomain model”)are given by the following coupled reaction-diffusion system:

cm∂tv − div (Mi(x)∇ui) + h(v) = 0,

cm∂tv + div (Me(x)∇ue) + h(v) = 0,(1.1)

for all (t, x) ∈ QT , where QT denotes the time-space cylinder (0, T )× Ω.We complete the system (1.1) with Dirichlet boundary conditions for the intra-

and extracellular electric potentials

(1.2) ui = 0 and ue = 0 on ΣT := (0, T )× ∂Ω,

and with initial data for the transmembrane potential

(1.3) v(0, x) = v0(x), x ∈ Ω.

Date: June 14, 2011.2000 Mathematics Subject Classification. 35K57, 93B05, 93B07, 93C10.Key words and phrases. reaction-diffusion system, controllability, observability, Carleman es-

timates, bidomain model, cardiac electric field.1

2 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

We emphasize that realistic models (bidomain model for e.g.) include a system ofODE’s for computing the ionic current as a function of the transmembrane potentialand a serie of additional “gating variables” aiming to model the ionic transfer acrossthe cell membrane (see, e.g., [19, 9, 20, 15]).

An equivalent formulation in the v, ue variables, which is more suitable from theimplementation viewpoint (see for e.g. [3]), is given by

cm∂tv + div(Me(x)∇ue

)+ h(v) = 0 in QT ,

−div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

(1.4)

where M = Mi + Me. If Mi ≡ µMe for some constant µ ∈ R, then the system(1.1) is equivalent to the following model (known as the “monodomain model”):

cm∂tv −µ

µ+ 1div(Me(x)∇v

)+ h(v) = 0 in QT ,

−div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT .

(1.5)

A first question we address is concerned with the null controllability to the system(1.5). In other words, given initial and boundary conditions, we want to knowwhether it is possible or not to find a control (our control might correspond topacemaker implantation) fχω such that v and ue are solutions of the system

(1.6)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ h(v) = fχω in QT ,

−div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0 in Ω.

Herein the function χω is the usual characteristic function acting on a subdomainω ⊂ Ω. Moreover, fχω = f(t, x)χω(x) is the control to be determined, which isassumed to be distributed over ω.

Another question we address is related to the controllability and the ε-observabilityproperties for the following approximated model of (1.6) with a forcing control term

(1.7)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ h(v) = fχω in QT ,

ε∂tue − div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0, ue(0) = ue,0 in Ω.

In the case where the control fχω remains bounded (with respect to ε) we see,formally, when ε → 0, that the solution (v := vε, ue := ue,ε) of (1.7) tends to thesolution (v, ue) to the problem (1.6). In this case we obtain the controllability of(1.6) as a limiting process. The ε-observability should be understood here as theway that the observability inequality “evolves” as ε→ 0.

Since we are using an approximated parabolic equation in the second equation ofthe system (1.7), it is natural decompose the initial condition v0 as v0 = ui,0−ue,0.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 3

In order to answer, at least partially, the questions we have made, we are led toconsider linearized versions of (1.6) and (1.7), respectively, by

(1.8)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ a(t, x)v = fχω in QT ,

−div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0 in Ω.

and

(1.9)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ a(t, x)v = fχω in QT ,

ε∂tue − div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0, ue(0) = ue,0 in Ω,

where a is a bounded function. We can address similar questions of controllabilityand ε-observability to this linearized versions as we already have done. At thispoint a remark has to be done. If we look for a moment to (1.8) (or (1.6)) we willsee that it is only necessary to control the function v on the first equation, and thecontrollability of ue will be for free. So it can be also easily seen that it is not thecase for (1.9) (or (1.7)).

To study the case of system (1.9), as in the case of one single parabolic equation,we are led to consider the adjoint system of (1.9):

(1.10)

−cm∂tϕ− µ

µ+1div (Me(x)∇ϕ) + a(t, x)ϕ = div (Mi(x)∇ϕe) in QT ,

−ε∂tϕe − div (M(x)∇ϕe) = 0 in QT ,

ϕ = 0, ϕe = 0 on ΣT ,

ϕ(T ) = ϕT , ϕe(T ) = ϕe,T in Ω.

It is well known that the null controllability for (1.9) is equivalent to the followingobservability inequality

(1.11) ||ϕe(0)||2L2(Ω) + ||ϕ(0)||L2(Ω) ≤ C∫∫

Qω

|ϕ|2dxdt, Qω := (0, T )× ω,

where C is an universal constant, i.e., inequality (1.11) holds for every solution ofthe adjoint system (1.10).

In this work, we assume that the domain Ω is assumed to be a bounded openset of RN with smooth boundary ∂Ω.It will be also assumed that the matrix Mj , j = i, e, are bounded, symmetric andpositive semidefinite. Our basic requirements are

(1.12) Mj ∈ C1(Ω;RN×N ), j = i, e,

(1.13) Mj(x)ξ · ξ ≥ CM |ξ|2 , for a.e. (t, x) ∈ QT and ∀ξ ∈ R3, j = i, e,

for a.e. x ∈ Ω, ∀ξ, ξ′ ∈ R3, with CM being a positive constant.Moreover, we assume that h = h(v), which is a function defined on R taking valuesin R and of class C1.We assume there exista a constant Ch > 0 such that

(1.14) h(0) = 0,h(v1)− h(v2)

v1 − v2≥ −Ch, ∀v1 6= v2,

4 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

and

(1.15) 0 < lim inf|v|→∞

h(v)

v3≤ lim sup|v|→∞

h(v)

v3<∞.

Remark. A consequence of conditions (1.14) and (1.15) is that ∀v ∈ R and fora.e. (t, x) ∈ QT

(1.16) |h(v)| ≤ C |v|(|v|2 + 1

),

for some constants C > 0.

Remark. It is can be easily seen that from (1.14), we get

(1.17) (h(v1)− h(v2)) (v1 − v2) + Ch (v1 − v2)2 ≥ 0,

∀v1, v2 ∈ R.

It is worth to mention some works that have been devoted to the theoreticaland numerical study of the bidomain model (1.1). For instance, Colli Franzoneand Savare [4] prove the existence of weak solutions to (1.1) using the theory ofevolution variational inequalities in Hilbert spaces. Applying the same approach,Sanfelici [22] proves the convergence of Galerkin approximations for this model.Recently, Bendahmane and Karlsen [1] proved the existence and uniqueness for anonlinear version of the bidomain equations (1.1) by using a uniformly parabolicregularization of the system and the Faedo–Galerkin method.

Regarding finite volume (FV) schemes for cardiac problems, Bendahmane andKarlsen [2] analyse a FV method for the bidomain model, supplying various ex-istence, uniqueness and convergence results. Bendahmane, Burger and Ruiz [3]analyse a parabolic-elliptic system with Neumann boundary conditions, adaptingthe approach in [2]; they also provide numerical experiments.

Let us mention some works in which the controllability of systems of coupledparabolic equations is analyzed. In fact, the study of the controllability of systemsof parabolic equations has obtained a lot of attention in the recent years. Forinstance, in [16] the authors analyze the controllability of a reaction-diffusion systemof a simple two coupled parabolic equations, obtaining the null controllability for thelinear system and the local null controllability of the nonlinear system. In [13] thecontrollability of a quite general two coupled linear parabolic system is studied andnull controllability is obtained combining Carleman inqualities and some energyinequalities. Another relevant work concerning to the controllability of coupledsystems is [8], in which the authors analyze the controllability of a cascade systemof m coupled parabolic equations, in that work the authors are able to obtain nullcontrollability for the cascade system whenever they have a good coupling structure.

But unlike the present work, the aforementioned works are devoted to systemsthat do not degenerate. So that a careful analysis is required in order to guar-antee uniform controllability with respect to the degenerating parameter ε. Thisapproach has been used several times in the case of parabolic equations degenerat-ing into hyperbolic ones, (e.g. [11], [14], [5]) and hyperbolic equations degeneratinginto parabolic ones (e.g. [17], [18]) but, as far as we know, this is the first timethat controllability of parabolic systems degenerating into parabolic-elliptic sys-tems is analyzed. It also seems to be, at least to our knowledge, the first time thatcontrollability of systems appearing in electrocardiology is studied.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 5

In passing, we want to mention that the systems of parabolic equations whichdegenerates into parabolic-elliptic ones arises in many areas such as biology ormodels describing gravitational interaction of particles.

In this paper we prove null controllability to (1.9) for each ε > 0 by establishingan observability estimate like (1.11) for its dual system (1.10). Moreover, theestimate on the control we obtain are uniform with respect to ε, that is, the constantC appearing in (1.11) does not depend on ε. We also analyze the controllabilityof the linear system (1.8), proving its null controllability. In addition we studythe controllability of the nonlinear systems (1.6) and (1.7) obtaining, under someassumptions on the nonlinearity, controllability results for these nonlinear systems.In the case of (1.7) we are able to obtain uniform estimates with respect to ε.

This paper is organized as follows: In Section 2, we state our main results.Section 3 is devoted to prove a Carleman inequality for a slightly changed versionof the adjoint system. In Section 4 we prove Theorem 2.3. Section 5 deals with thecontrollability for the nonlinear systems.

2. Main results

Before we state our main results, let us give a relevant definition of a weaksolution for the model (1.7).

Definition 2.1. A weak solution of (1.7) is a pair (v, ue) ∈ L2(0, T ;H10 (Ω)) ×

L2(0, T ;H10 (Ω)) such that ∂tv and ε∂tue belong to L2(0, T,H−1(Ω)) + L4/3(QT )

and L2(0, T,H−1(Ω)), respectively, and∫ T

0

cm 〈∂tv, ϕ〉 dt+λ

λ+ 1

∫∫QT

Me(x)∇v · ∇ϕdx dt

+

∫∫QT

h(v)ϕdx dt =

∫∫QT

fχωϕdx dt,

(2.1)

and

ε

∫ T

0

〈∂tue, ψ〉 dt+

∫∫QT

M(x)∇ue · ∇ψ dx dt+

∫∫QT

Mi(x)∇v · ∇ψ dx dt = 0,

(2.2)

for all ϕ ∈ L2(0, T ;H10 (Ω)) ∩ L4(QT ) and ψ ∈ L2(0, T ;H1

0 (Ω)).

We have the following result:

Theorem 2.1. Assume that conditions (1.12)-(1.15) hold. Let v0, u0,e be in L2(Ω),and fχω ∈ L2(QT ). Then, the system (1.7) with Dirichlet boundary conditions,possesses a unique weak solution.

The proof of Theorem 2.1 can be done exactly as in [1], so we omit it.

Remark. Note that the solution (v, ue) of the system (1.9) is also a weak solutionin the sense of Definition 2.1 with ∂tv ∈ L2(0, T,H−1(Ω)) and ϕ ∈ L2(0, T ;H1

0 (Ω)).

Now we state our main results. Our first main result give us the null controlla-bility of (1.8).

Theorem 2.2. Given v0 in L2(Ω). There exists a control f ∈ L2(ω×(0, T )) so thatthe associated solution to (1.8) is driven to zero at time T . That is, the associatedsolution satisfies

v(T ) = 0, ue(T ) = 0.

6 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Once the controllability of a single linear parabolic equation is nowadays quiteunderstood (see e.g. [7]), the proof of Theorem 2.2 will not be reproduced here.

Concerning the proof of Theorem 2.2, some words have to be said. The ideabehind is, as mentioned earlier, that equation (1.8)1 is somehow “independent” of(1.8)2, so that we can use the classical results of controllability for one single heatlike equation to drive v to zero at time T , and using (1.8)2 to see that ue has to bezero at time T .

Remark 1. From Theorem 2.2 follows that ui(T ) = 0, by definition.

Our second main result gives the null controllability of (1.9) (recall that ourfunctions v0, ue,0, f and the variables v, ue, depend on ε > 0).

Theorem 2.3. Given v0 and ue,0 in L2(Ω). For each ε > 0, there exists a controlf ∈ L2(ω × (0, T )) so that the associated solution to (1.9) is driven to zero at timeT . That is, the associated solution satisfies

v(T ) = 0, ue(T ) = 0.

Moreover, the control f is bounded by

(2.3) ‖fχω‖L2(QT ) ≤ C(‖v0‖L2(Ω) + ε ‖ue,0‖L2(Ω)

).

The proof of Theorem 2.3 will be done in Section 4. Note that from Theorem2.3, it is an easy task to show that there exists fχω := fεχω ∈ L2(ω × (0, T )) suchthat

fεχωε→0−→ fχω and (v := vε, ue := ue,ε)

ε→0−→ (v, ue) (weakly in L2(QT )),

where (vε, ue,ε) and (v, ue) are the solutions of (1.9) and (1.8), respectively. Inothers words, the null controllability of (1.8) can be seen as a limiting processof (1.9), when ε → 0. Observe that from (2.3) the estimate for the control fχωobtained by this limiting process follows

(2.4) ‖fχ‖L2(Qω) ≤ C ‖v0‖L2(Ω) .

In order to state the next results we need to define

(2.5) qN ∈ (2,∞) if N = 1, 2,N + 2

2< qN < 2

N + 2

N − 2if N ≥ 3.

The third main result of this paper gives the null controllability of the nonlinearparabolic-elliptic system (1.6).

Theorem 2.4. Given v0 in L2(Ω). We have• If h is C1(R), global lipschitz and satisfies h(0) = 0. Then there exists a controlfχω ∈ L2(ω × (0, T )) such that the solution (v, ue) of (1.6) satisfies

v(T ) = ue(T ) = 0.

• If h is of class C1 satisfying (1.14) and (1.15), v0 ∈ H10 (Ω)∩W 2(1− 1

qN),qN (Ω), with

||v0||L∞ ≤ γ, for sufficient small γ. There exists a control fχω ∈ LqN (ω × (0, T ))such that the solution v, ue of (1.6), with (v, ue) ∈W 2,1

qN (QT ), satisfies

v(T ) = ue(T ) = 0.

The last main result is concerned about the null controllability of the nonlinearparabolic system (1.7) (recall that our functions v0, ue,0, f and the variables v, ue,depend on ε > 0).

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 7

Theorem 2.5. Given v0 and ue,0 in L2(Ω). We have• If h is C1(R), global lipschitz and satisfies h(0) = 0. Then there exist a controlfχω ∈ L2(ω × (0, T )) such that the solution (v, ue) of (1.7) satisfies

v(T ) = ue(T ) = 0.

Besides, the control fχω has the estimate

(2.6) ‖fχω‖L2(QT ) ≤ C(‖v0‖L2(Ω) + ε ‖ue,0‖L2(Ω)

).

•Let h be a C1 function satisfying (1.14) and (1.15) and the initial data (v0, ue,0) ∈(H1

0 (Ω) ∩W 2(1− 1qN

),qN (Ω))2

, with ||(v0, ue,0)||L∞ ≤ γ, for sufficient small γ doesnot depending on ε. Then there exist a control fχω ∈ LqN (ω× (0, T )) such that thesolution (v, ue) of (1.7), with (v, ue) ∈ (W 2,1

qN (QT ))2, satisfies

v(T ) = ue(T ) = 0.

Theorem 2.5 is proved in section 5. The proof of Theorems 2.4 will be notreproduced here, once it is essentially done following the ideas of Theorems 2.2 and2.5.

3. One Carleman type inequality

In this section we prove a Carleman inequality for a slightly changed version ofthe adjoint system (1.10). This Carleman inequality will be used in the sequel toshow the observability inequality for the adjoint system (1.10), which will allow usto get controllability results for the linearized systems. The idea here is to keeptrack of the dependence with respect to ε in the Carleman inequality.

We consider the case where ϕT and ϕe,T are smooth enough and writing system(1.10) above in the concise way

(3.1)

−∂tϕ− div (Me(x)∇ϕ) + a(x, t)ϕ = div (Mi(x)∇ϕe) in QT ,

−ε∂tϕe − div (M(x)∇ϕe) = 0 in QT ,

ϕ = 0, ϕe = 0 on ΣT ,

ϕ(T ) = ϕT , ϕe(T ) = ϕe,T in Ω.

Putting ρ(x, t) = div (M(x)∇ϕe(x, t)) and using (3.1)2 we easily see that ρ = 0 onΣT . Thus the pair (ϕ, ρ) satisfies

(3.2)

−∂tϕ− div (Me(x)∇ϕ) + a(x, t)ϕ = Cρ in QT ,

−ε∂tρ− div (M(x)∇ρ) = 0 in QT ,

ϕ = 0, ρ = 0 on ΣT ,

ϕ(T ) = ϕT , ρ(T ) = ρT in Ω.

Before we state the Carleman inequality, let us define several weight functions whichwill be usefull in the sequel.

Lemma 3.1. Let ω0 be an arbitrary nonempty open set such that ω0 ⊂ ω ⊂ Ω.Then there exists a function ψ ∈ C2(Ω) such that

ψ(x) > 0,∀x ∈ Ω, ψ ≡ 0 on ∂Ω, |∇ψ(x)| > 0 ∀x ∈ Ω\ω0.

Proof. The proof of this lemma is given in [7].

8 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

From Lemma 3.1 we introduce the weight functions

(3.3) φ(x, t) =eλψ(x)

β(t); φ∗(t) = min

x∈Ωφ(x, t) =

1

β(t);

(3.4) α(x, t) =eλψ(x) − e2λ‖ψ‖

β(t); α∗(t) = max

x∈Ωα(x, t) =

eλ‖ψ‖ − e2λ‖ψ‖

β(t),

with β(t) = t(T − t), 0 ≤ t ≤ T , a parameter λ > 0 and

‖ψ(x)‖ = maxx∈Ω|ψ(x)| .

Remark 2. From the definition of α and α∗ we have 3α∗ ≤ 2α (for λ largeenough!). Moreover, from the definition it follows that

φ∗(t) ≤ φ(x, t) ≤ eλ‖ψ‖φ∗(x, t)

and

|∂tα∗| ≤ e2λ‖ψ‖Tφ2.

The main result of this section is given in the following theorem.

Theorem 3.2. There exist positive constants C = C(Ω, ω0), λ0 and s0 so that forevery ϕT , ϕe,T ∈ L2(Ω), a ∈ L∞(QT ), the corresponding solution to (3.2) satisfies(3.5)∫∫

QT

e3sα|ρ|2dxdt+s3λ4

∫∫QT

φ3e3sα|ϕ|2dxdt ≤ Ce6λ||ψ||s7λ4

∫Qω

φ8e2sα|ϕ|2dxdt,

for every s ≥ (T + (1 + ||a||2/3L∞ + ||a||2/5L∞)T 2 +T 4)s0, λ ≥ λ0 and appropriate weightfunctions φ and α defined in (3.3) and (3.4), respectively.

Remark 3. In this Carleman estimate ε does not appear. It is important becauseit will allow us take the limit when ε→ 0 in (1.9).

Proof. The proof will be divided in several steps:

• Step 1 First estimate for the parabolic systemIn this step we obtain a first Carleman estimate, which will depend on ε.We apply the following Carleman inequality for the solution ρ of (3.2)2

∫∫QT

s−1e2sα |∂tρ|2 dxdt+ ε−2

∫∫QT

s−1e2sα

∣∣∣∣∣∣N∑

i,j=1

∂2xixj

ρ

∣∣∣∣∣∣2

dxdt

+ s3λ4ε−2

∫∫QT

φ4e2sα |ρ|2 dxdt+ sλ2ε−2

∫∫QT

φ2e2sα |∇ρ|2 dxdt

≤ Ceλ‖ψ‖s3λ4ε−2

∫∫Qω

φ4e2sα |ρ|2 dxdt.

(3.6)

The proof of this inequality is given in appendix B, inequality (B.5).

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 9

Now we apply to (3.2)1 the Carleman inequality given in appendix A, inequality(A.2) with ε = 1, in order to obtain

∫∫QT

s−1φ−1e2sα |ϕt|2 dxdt+ s−1

∫∫QT

φ−1e2sα

∣∣∣∣∣∣N∑

i,j=1

∂2xixj

ϕ

∣∣∣∣∣∣2

dxdt

+ s3λ4

∫∫QT

φ3e2sα |ϕ|2 dxdt+ sλ2

∫∫QT

φe2sα |∇ϕ|2 dxdt

≤ C(s3

∫∫QT

e2sαφ4 |ρ|2 dxdt+ s3λ4

∫∫Qω

φ3e2sα |ϕ|2 dxdt),

(3.7)

for s ≥ (T + (1 + ‖a‖2/3L∞ + ‖a‖2/5L∞)T 2 + T 4)s0.Next we add (3.6) to (3.7), to obtain

∫∫QT

φ−1e2sα |ϕt|2 dxdt+

∫∫QT

φ−1e2sα

∣∣∣∣∣∣N∑

i,j=1

∂2xixj

ϕ

∣∣∣∣∣∣2

dxdt

+ s4λ4

∫∫QT

φ3e2sα |ϕ|2 dxdt+ s2λ2

∫∫QT

φe2sα |∇ϕ|2 dxdt

+ ε2

∫∫QT

e2sα |∂tρ|2 dxdt+

∫∫QT

e2sα

∣∣∣∣∣∣N∑

i,j=1

∂2xixj

ρ

∣∣∣∣∣∣2

dxdt

+ s4λ4

∫∫QT

φ4e2sα |ρ|2 dxdt+ s2λ2

∫∫QT

φ2e2sα |∇ρ|2 dxdt

≤ C(eλ||ψ||s4λ4

∫∫Qω

φ4e2sα |ρ|2 dxdt+ s4λ4

∫∫Qω

φ3e2sα |ϕ|2 dxdt).

(3.8)

At this point a remark has to be done. If we were trying to control (1.9) with twocontrol forces inequality (3.8) would be enough.

• Step 2 Estimation of the local integral of ρ.In this step we estimate the local integral involving ρ in the right-hand side of

(3.8). It will be done by using (3.2)1, to this end we write

Ceλ||ψ||s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt

= Ceλ||ψ||s4λ4

∫∫Qω

e2sαφ4ρ(−ϕt − div (Me∇ϕ) + aϕ)dxdt

:= A+B + C.

(3.9)

In the sequel we estimate each parcel in the expression above. We can write

A = Ceλ||ψ||s4λ4

∫∫Qω

s∂tαe2sαφ4ρϕdxdt

+ Ceλ||ψ||s4λ4

∫∫Qω

e2sαφ3φtρϕdxdt+ Ceλ||ψ||s4λ4

∫∫Qω

e2sαφ4∂tρϕdxdt

:= A1 +A2 +A3.

10 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

It is immediate to see that

A1 ≤1

10s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt+ Ce2λ||ψ||s8λ4

∫∫Qω

e2sαφ8|ϕ|2dxdt,(3.10)

A2 ≤1

10s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt+ Ce2λ||ψ||s6λ4

∫∫Qω

e2sαφ6|ϕ|2dxdt,(3.11)

and that

A3 ≤ε2

2

∫∫Qω

e2sα|∂tρ|2dxdt+ Ce2λ||ψ||ε−2s8λ8

∫∫Qω

e2sαφ8|ϕ|2dxdt.

Now we estimate B, to this we write

B = Ceλ||ψ||s4λ4N∑

i,j=1

∫∫Qω

s∂xiα e2sαφ4ρ(M ij

e ∂xjϕ)dxdt

+ Ceλ||ψ||s4λ4N∑

i,j=1

∫∫Qω

e2sαφ3∂xiφ ρ(M ij

e ∂xjϕ)dxdt

+ Ceλ||ψ||s4λ4N∑

i,j=1

∫∫Qω

e2sαφ4∂xiρ (M ij

e ∂xjϕ)dxdt

:= B1 +B2 +B3.

A simple calculation give us

B1 ≤1

10s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt+1

6s2λ2

∫∫Qω

e2sαφ2|∇ρ|2dxdt

+ Ce2λ||ψ||s8λ8

∫∫Qω

e2sαφ8|ϕ|2dxdt,

B2 ≤1

6s2λ2

∫∫Qω

e2sαφ2|∇ρ|2dxdt+1

10s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt

+ Ce2λ||ψ||s6λ8

∫∫Qω

e2sαφ6|ϕ|2dxdt,

and

B3 ≤1

2

∫∫Qω

e2sα|∂2xixj

ρ|2dxdt+1

6s2λ2

∫∫Qω

e2sαφ2|∇ρ|2dxdt

+ Ce2λ||ψ||s8λ8

∫∫Qω

e2sαφ8|ϕ|2dxdt.

We can also show that

C ≤ 1

10s4λ4

∫∫Qω

e2sαφ4|ρ|2dxdt

+ Ce2λ||ψ||s4λ4|a|2L∞∫∫

Qω

e2sαφ4|ϕ|2dxdt.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 11

Putting A, B, and C in (3.8), we get

∫∫QT

e2sα|ϕt|2dxdt+

∫∫QT

e2sα|N∑

i,j=1

∂2xixj

ϕ|2dxdt

+ s4λ4

∫∫QT

φ4e2sα|ϕ|2dxdt+ s2λ2

∫∫QT

φ2e2sα|∇ϕ|2dxdtε2

∫∫QT

e2sα|∂tρ|2dxdt

+

∫∫QT

e2sα|N∑

i,j=1

∂2xixj

ρ|2dxdt+ s4λ4

∫∫QT

φ4e2sα|ρ|2dxdt

+ s2λ2

∫∫QT

φ2e2sα|∇ρ|2dxdt

≤ Ce2λ||ψ||ε−2s8λ8

∫∫Qω

e2sαφ8|ϕ|2dxdt.

(3.12)

• Step 3. An energy Inequality.In this step we prove a weighted energy inequality for (3.2)2. This inequality willbe necessary in the sequel to obtain a Carleman type estimate in which ε does notplay any role .

We start with the change of variable y = e32 sα∗ρ and seeing that y solves the

system

(3.13)

ε∂ty − div (M(x)∇y) = ε 3

2s∂tα∗e

32 sα∗ρ in QT ,

y = 0 on ΣT ,

y(0) = y(T ) = 0 in Ω.

We multiply (3.13) by y and integrating on x, ellipticity of M give us

ε

2

d

dt||y||2L2(Ω) + C||∇y||2L2(Ω) ≤ ε

3

2

∫Ω

s∂tα∗e

32 sα∗ρydx.

An immediate consequence of this last inequality is∫∫QT

e3sα∗ |ρ|2dxdt ≤ Cε2e4λ||ψ||∫∫

QT

s3φ4e2sα|ρ|2dxdt.

From this and (3.12) we obtain the following Carleman type estimate

(3.14)

∫∫QT

e3sα∗ |ρ|2dxdt ≤ Ce6λ||ψ||s7λ4

∫∫Qω

φ8e2sα|ϕ|2dxdt.

• Step 4. Last estimates and conclusion.In this step we apply a slightly different Carleman inequality for (3.2)1. Combiningthis inequality with the results from previous steps we will be able to obtain aninequality which does not depend on ε.

12 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Indeed, we can show for (3.2)1 the inequality∫∫QT

s−1φ−1e3sα|ϕt|2dxdt+ s−1

∫∫QT

φ−1e3sα|N∑

i,j=1

∂2xixj

ϕ|2dxdt

+ s3λ4

∫∫QT

φ3e3sα|ϕ|2dxdt+ sλ2

∫∫QT

φe3sα|∇ϕ|2dxdt

≤ C(∫∫

QT

e3sα|ρ|2dxdt+ s3λ4

∫∫Qω

φ3e3sα|ϕ|2dxdt).

(3.15)

Herein we just changed the weight e2sα by e3sα. The proof is the same as in theCarleman inequality given by Theorem A.1, just taking a slightly different changeof variable in (A.3).

Next, since e3sα ≤ e3sα∗ , we obtain∫∫QT

e3sα|ρ|2dxdt ≤∫∫

QT

e3sα∗ |ρ|2dxdt,

and by (3.14) we have∫∫QT

e3sα|ρ|2dxdt ≤ Ce6λ||ψ||s7λ4

∫∫Qω

φ8e2sα|ϕ|2dxdt.

From (3.14) and (3.15) follows that(3.16)∫∫

QT

e3sα|ρ|2dxdt+s3λ4

∫∫QT

φ3e3sα|ϕ|2dxdt ≤ Ce6λ||ψ||s7λ4

∫∫Qω

φ8e2sα|ϕ|2dxdt,

which is exactly (3.5).It is not a difficulty task to show that (3.16) remains true when we consider L2

initial data. So that the Carleman inequality (3.5) holds for all initial data inL2(Ω).

4. Null controllability for the linearized system

This section is intended to prove the null controllability of linearized equation(1.9). It will be done by showing an observability inequality for the adjoint system(1.10) and solving a minimization problem. The arguments used here are classicalin control theory for linear PDE’s, so that we just give a sketch of the proof.

• Sketch of the proof of Theorem 2.3. Once the only constant which matter for usin (1.10) is ε, we can consider directly (3.1). As before, we will consider first thecase where we have solutions regular enough.

First we prove an observability inequality for the adjoint system (1.10). Bystandard energy inequality for (3.2) we see, using (3.16), that

(4.1) ||ρ(0)||2L2(Ω) + ||ϕ(0)||L2(Ω) ≤ C1eC2||a||∞T

∫∫Qω

|ϕ|2dxdt,

for some constant C1 > 0.Since ρ(x, t) = div (M(x)∇ϕe(x, t)) and ϕe = 0 on ∂Ω, we have

||ϕe(t)||H2(Ω) ≤ ||ρ(t)||L2(Ω),

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 13

for all t ∈ [0, T ]. Therefore, follows from (4.1) that

(4.2) ||ϕe(0)||2L2(Ω) + ||ϕ(0)||L2(Ω) ≤ C∫∫

Qω

|ϕ|2dxdt,

which is the observability inequality desired.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 Carleman inequality is satisfiedby all solutions of (3.1) with initial data in L2(Ω). Once system (3.1) is a variantof (1.10) we see that inequality (4.2) holds for all solutions of (1.10), with, possibly,a different constant.

Now in order to obtain the null controllability for linear system (1.9), we needto solve the following minimization problem. Given ϕT and ϕe,T in L2(Ω),

MinimizeJδ(ϕT , ϕe,T ), with

Jδ(ϕT , ϕe,T ) =

1

2

∫ T

0

∫ω

|ϕ(t, x)|2 dx dt+ ε(ue(0), ϕe(0))

+ (v(0), ϕ(0)) + δ(‖ϕT ‖L2(Ω) + ‖ϕe,T ‖L2(Ω))

,

(4.3)

where (ϕ,ϕe) is the solution of the adjoint problem (1.10) with initital data (ϕT , ϕe,T ).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 proveis the coercivity of Jδ.

But, once we have the observability inequality (4.2) for the adjoint problem(1.10), the coercivity of Jδ is immediate. Therefore, for each δ > 0 there existsan unique minimizer (ϕδe,T , ϕ

δT ) of Jδ. Let us denote by ϕε,δ the corresponding

solution to (1.10) associated to this minimizer.Taking fε,δχω = ϕε,δ as a control for (1.9). The duality between (1.9) and (1.10)

give us the approximated null controllability

(4.4) ||vε,δ(T )||L2(Ω) + ||uε,δe (T )||L2(Ω) ≤ δ,

where (vε,δ, uε,δe ) is the solution associated to the control fε,δχω. Also follows that

(4.5) ||fε,δχω||L2(Qω) ≤ C(||v0||L2(Ω) + ε||ue,0||L2(Ω)

).

Combining (4.4) and (4.5), we get a control fεχω (the weak limit of a subsequenceof fε,δχωin L2(ω× (0, T ))) that drives the solution of (1.9) to zero at time T . From(4.5) we have the following estimate on the control fεχω,

(4.6) ||fεχω||L2(Qω) ≤ C(||vε0||L2(Ω) + ε||uεe,0||L2(Ω)

).

This finishes the proof of Theorem 2.3.

5. The nonlinear system

In this section we prove Theorems 2.4 and 2.5. The proofs are done by applyingfixed point arguments.

14 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Proof of Theorem 2.5 (case 1): We consider the following linearization of system(1.7):

(5.1)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ g(z)v = fχω in QT ,

ε∂tue − div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0, ue(0) = ue,0 in Ω,

where

(5.2) g(s) =

h(s)

sif |s| > 0,

h′(s) if s = 0.

Follows from Theorem 2.3 that for each v0, ue,0 ∈ L2(Ω) and z ∈ L2(QT ), thereexist a control function fχω ∈ L2(QT ) such that the solution of (5.1) stisfies

v(T ) = ue(T ) = 0.

To continue we will apply a generalized version of Kakutani’s fixed point theoremdue to Glicksberg [12].

Theorem 5.1. Let B be a non-empty convex, compact subset of a locally convextopological vector space X. If Λ : B −→ X is a set-valued mapping convex, compactand with closed graphic. Then the set of fixed points of Λ is non-empty and compact.

Next, we define the mapping Λ : B −→ X as follows

Λ(z) = v; (v, ue) is a solution of (5.1), such that v(T ) = ue(T ) = 0,

for a control fχω satisfying (2.3).

where X = L2(QT ) and B is the ball

B = z ∈ L2(0, T,H10 (Ω), ∂tz ∈ L2(0, T,H−1(Ω));

||z||2L2(0,T ;H10 (Ω)) + ||∂tz||2L2(0,T ;H−1(Ω)) ≤M.

It’s easy to see that Λ is well defined and that B is convex and compact in L2(QT ).So we need to prove that Λ is convex, compact and has closed graphic. It will bedone into the next steps.• Λ(B) ⊂ B.Let z ∈ B and v ∈ Λ(z). Since v satisfies (5.1)1, the following inequality holds

(5.3) ||v||2L2(0,T ;H10 (Ω)) + ||∂tv||2L2(0,T ;H−1(Ω)) ≤ K1.

In this way if z ∈ B then Λ(z) ⊂ B, if we take M = K1.

• Λ(z) is closed in L2(QT ).Let z ∈ B fixed, and vn ∈ Λ(z), such that vn → v. Let’s prove that v ∈ Λ(z).

In fact, by definition we have that vn is, together with a function ue,n and acontrol fn, the solution of (5.1), with ||fnχω||L2(QT ) ≤ C

(||v0||L2(Ω)+ε||ue,0||L2(Ω)

).

Therefore we can extract a subsequence of fn, denoted by the same index, such that

fnχω → fχω weakly in L2(QT ).

Since fn is bounded, we can argue as in the previous section in order to obtain

(5.4) ||vn||2L2(0,T ;H10 (Ω)) + ||∂tvn||2L2(0,T ;H−1(Ω)) ≤ C.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 15

From this last inequality we can conclude that∣∣∣∣∣∣vn → v weakly in L2(0, T ;H1

0 (Ω)),vn → v strongly in L2(QT ),∂tvn → ∂tv weakly in L2(0, T ;H−1(Ω)).

Using the last converge above and (5.1)2, we see there exists a function ue suchthat ∣∣∣∣∣∣

ue,n → ue weakly in L2(0, T ;H10 (Ω)),

ue,n → ue strongly in L2(QT ),∂tue,n → ∂tue weakly in L2(0, T ;H−1(Ω)).

It’s immediate that (ue, v) is a controlled solution of (5.1) associated to the controlf . Hence v ∈ Λ(z) and Λ(z) is closed and compact of L2(QT ).

• Λ has closed graph in L2(QT )× L2(QT ).We need to prove that if zn → z and vn → v strongly in L2(QT ) and vn ∈ Λ(zn),

then v ∈ Λ(z). But it’s is an immediate consequence of the previous steps.Therefore by Glicksberg Theorem 5.1 we conclude that Λ has a fixed point. Whichproves Theorem 2.5 in the case which the nonlinearity is a C1 global Lipschitzfunction.

Proof of Theorem 2.5 (case 2): The proof of the local null controllability in thecase 2 is done exactly as in the equivalent one in [16].

We consider the linearization

(5.5)

cm∂tv − µ

µ+1div(Me(x)∇v

)+ a(z)v = fχω in QT ,

ε∂tue − div(M(x)∇ue

)= div

(Mi(x)∇v

)in QT ,

v = 0, ue = 0 on ΣT ,

v(0) = v0, ue(0) = ue,0 in Ω,

with (v0, ue,0) ∈(H1

0 (Ω) ∩W 2(1− 1qN

),qN (Ω))2

, z ∈ L∞(QT ) and

a(z) =

∫ 1

0

dh

dz(sz)ds.

Follows from the previous results the null controllability of (5.5) with a control inL2(ω × (0, T )). The idea is to get a control in LqN (QT ) in order to use a fixedpoint argument. This LqN (QT ) control is get by changing a bit the functional Jδin (4.3).In fact, we define the functional:

MinimizeJδ(ϕT , ϕeT ), with

Jδ(ϕT , ϕeT ) =

1

2

∫ T

0

∫ω

e2sαφ8 |ϕ(t, x)|2 dx dt+ ε(ue(0), ϕe(0))

+ (v(0), ϕ(0)) + δ(‖ϕT ‖L2(Ω) + ‖ϕe,T ‖L2(Ω)

),

(5.6)

where (ϕ,ϕe) is the solution of the adjoint problem (1.10) with initital data (ϕT , ϕe,T ).As before, we see that (5.6) has an unique minimizer (ϕδ, ϕδe). Defining fδ =

e2sαφ8ϕδ, we see, using the fact that ϕδ is, together with a ϕδe, the solution of (1.10),that fδ is a solution of a heat equation with null initial data and right-hand side

16 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

in L2(QT ). Using the regularizing effect of the heat equation and arguing exactlyas in [16] we can prove the inequality

(5.7) ||fδχω||LqN (QT ) ≤ C||(v0, ue,0)||2L2(QT ).

Taking the limit when δ → 0 we get a control fχω ∈ LqN (QT ) such that theassociated solution (v, ue) to (5.5) satisfies

v(T ) = ue(T ) = 0.

The proof is finished by applying the standard fixed point argument for the system(5.5), exactly as done in Theorem 6 in [16].

Appendix A. A Degenerating Carleman Inequality

In this appendix we show a Carleman estimate to the the following class ofparabolic systems

(A.1)

−∂tv(x, t)−

N∑i,j=1

∂xi(aij(x)∂xiv(t, x)) = g(x, t) in QT ,

v = 0 on ΣT ,

v(T ) = vT in Ω,

under the assumption that the matrix aij has the following form:

aij =Mij

ε,

where (Mij)ij is an elliptic matrix with∑Ni,jMijξjξi ≥ β|ξ|2.

The Carleman inequality we will prove is given by the the following Theorem.

Theorem A.1. Let ψ, φ, α the functions defined above. Then, there exists λ0 ≥ 1and s0 such that, for each, λ > λ0 and s ≥ s0(T + T 2 + T 4) the solution v(x, t) ofthe system (A.1) satisfies∫∫

QT

s−1φ−1e2sα|∂tv|2dxdt+ s−1ε−2

∫∫QT

φ−1e2sα|N∑

i,j=1

∂2xixj

v|2dxdt

+ s3λ4ε−2

∫∫QT

φ3e2sα|v|2dxdt+ sλ2ε−2

∫∫QT

φe2sα|∇v|2dxdt

≤ C(∫∫

QT

e2sα|g|2dxdt+ s3λ4ε−2

∫∫Qω

φ3e2sα|v|2dxdt),

(A.2)

with C > 0 depending on Ω, ω0, ψ and β.

This Carleman inequality is important to us because it gives explicitly the de-pendence with respect to ε.

Before to start the proof of Theorem A.1, we need the following two Remarks:

Remark 4. From the definition of the weight funtions φ and α we have

∇φ = λφ∇ψ = ∇α,

∇ψ = ν∂ψ

∂ν, ∇w = ν

∂w

∂νand

∂ψ

∂ν≤ 0 on ΣT .

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 17

It is also easy to see, since Ω\ω0 is compact and |∇ψ| > 0 on Ω\ω0, that thereexists δ > 0 such that

β|∇ψ| ≥ δ on Ω\ω0.

Proof. For s > 0 and λ > 0 we consider the change of variable

(A.3) w(t, w) = esαv(t, w),

which implies

w(T, x) = w(0, x) = 0.

Therefore we can re-write the state equation (A.1) in the form

(A.4) L1w + L2w = gs,

where

(A.5) L1w = −∂tw + 2sλ

N∑i,j=1

φaij∂xjψ ∂xi

w + 2sλ2N∑

i,j=1

φaij∂xiψ ∂xj

ψw,

(A.6) L2w = −N∑

i,j=1

∂xi(aij∂xj

w)− s2λ2N∑

i,j=1

φ2aij∂xiψ ∂xj

ψw + s∂tαw,

and

(A.7) gs = esαg + sλ2N∑

i,j=1

φaij∂xiψ ∂xj

ψw − sλN∑

i,j=1

φ∂xi(aij∂xj

ψ)w.

Taking the L2 norm in (A.4) we get

(A.8) ||L1w||2L2(QT ) + ||L2w||2L2(QT ) + 2(L1w,L2w)L2(QT ) = ||gs||2L2(QT ).

The idea is first to analyze the terms appearing in (L1w,L2w)L2(QT ). To this endwe write

(L1w,L2w)L2(QT ) =

N∑i,j=1

Iij .

Next, we calculate each component in a separated way as follows

I11 =

∫∫QT

∂tw

N∑i,j=1

∂xi(aij∂xjw)dxdt = 0,

18 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

I12 = s2λ2

∫∫QT

∂tw

N∑i,j=1

φ2 aij∂xiψ ∂xj

ψw dxdt

= −s2λ2N∑

i,j=1

∫∫QT

φ∂tφaij∂xiψ ∂xj

ψ |w|2 dxdt,

I13 = −s∫∫

QT

∂tw ∂tαw dxdt =s

2

∫∫QT

∂2ttα|w|2 dxdt,

I21 = −2sλ

∫∫QT

N∑i,j=1

φaij∂xiw ∂xjψ

N∑k,,l=1

∂xk(akl∂xl

w) dxdt

= −sλN∑

i,j,k,l=1

∫∫Σ

φ∂ψ

∂ν(∂w

∂ν)2aijνiνjaklνkνldΓdt

+ 2sλ2N∑

i,j,k,l=1

∫∫QT

φaij∂xiw ∂xj

ψakl∂xkψ ∂xl

w dxdt

− sλ2N∑

i,j,k,l=1

∫∫QT

φaij∂xjψ ∂xiψ akl∂xlw ∂xk

w dxdt

+ 2sλ

N∑i,j,k,l=1

∫∫QT

φ∂xk(aij∂xj

ψ)∂xiw akl∂xl

w dxdt

− sλN∑

i,j,k,l=1

∫∫QT

φ∂xi(aij∂xj

ψ)akl∂xlw ∂xk

w dxdt

− sλN∑

i,j,k,l=1

∫∫QT

φaij∂xjψ ∂xiakl ∂xlw ∂xk

w dxdt

:= I121 + . . .+ I6

21,

I22 = −2s3λ3N∑

i,j,k,l=1

∫∫QT

φ3aij∂xjψ ∂xi

w akl∂xkψ ∂xl

ψw dxdt

= 3s3λ4N∑

i,j,k,l=1

∫∫QT

φ3aij∂xjψ ∂xiψ |w|2akl∂xkψ ∂xl

ψ dxdt

+ s3λ3N∑

i,j,k,l=1

∫∫QT

φ3∂xi(aij∂xj

ψ)akl∂xkψ ∂xl

ψ |w|2 dxdt

+ s3λ3N∑

i,j,k,l=1

∫∫QT

φ3aij∂xjψ ∂xi

(akl∂xkψ ∂xl

ψ |w|2 dxdt

:= I122 + I2

22 + I322,

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 19

I23 = 2s2λ

N∑i,j=1

∫∫∫QT

φaij∂xjψ ∂xi

w ∂tαw dxdt

= −s2λ2N∑

i,j=1

∫∫QT

φaij∂xjψ ∂xi

ψ |w|2 ∂tαdxdt

− s2λ

N∑i,j=1

∫∫∫QT

φ∂xi(aij∂xj

ψ)|w|2∂tαdxdt

− s2λ2N∑

i,j=1

∫∫QT

φφtaij ∂xiψ ∂xj

ψ |w|2 dxdt

:= I123 + I2

23 + I323,

I31 = −2sλ2N∑

i,j,k,l=1

∫∫QT

φaij∂xiψ ∂xjψw ∂xk(akl∂xl

w) dxdt

= 2sλ3N∑

i,j,k,l=1

∫∫QT

φaij∂xiψ ∂xj

ψ ∂xkψwakl ∂xl

w dxdt

+ 2sλ2N∑

i,j,k,l=1

∫∫QT

φ∂xk(aij∂xi

ψ ∂xjψ)wakl ∂xl

w dxdt

+ 2sλ2N∑

i,j,k,l=1

∫∫QT

φaij∂xiψ ∂xjψ akl ∂xkw ∂xl

w dxdt

:= I131 + I2

31 + I331,

I32 = −2s3λ4N∑

i,j,k,l=1

∫∫QT

φ3aij∂xiψ ∂xj

ψ |w|2akl ∂xkψ ∂xl

ψ dxdt.,

I33 = 2s2λ2N∑

i,j=1

∫∫QT

φaij∂xiψ ∂xjψ |w|2 ∂tαdxdt.

Note that in this way can write

2(L1w,L2w)L2(QT ) = 2

(I12 + I13 + I1

21 + I221 + I3

21 + I421 + I5

21 + I621 + I1

22 + I222

+ I322 + I1

23 + I223 + I3

23 + I131 + I2

31 + I331 + I32 + I33

).

Now we use Remark 4 and the ellipticity of aij to obtain

I121 ≥ 0.

Moreover, we have

I12 = I323, I32 =

−2

3I122 and I3

21 = −I331

2.

20 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Therefore

2(L1w,L2w)L2(QT ) ≥2

(2I12 + I13 + I2

21 + I421 + I5

21 + I621 +

1

3I122 + I2

22

+ I322 + I1

23 + I223 + I1

31 + I231 +

1

2I331 + I33

).

(A.9)

Following the methodology of [7], we keep attention on the terms involving∫∫QT

φ|∇w|2dxdt and the terms involving

∫∫QT

φ3|w|2dxdt.

Now we denote by X all terms involving integration on |w|2 and by Y all termsinvolving integration on |∇w|2. So that

X = 2I12 + I13 + I222 + I3

22 + I123 + I2

23 + I33

and

Y = I421 + I5

21 + I621.

Therefore

2(L1w,L2w)L2(QT ) ≥ 2

(X + Y + I2

21 +1

3I122 + I1

31 + I231 +

1

2I331

).(A.10)

It is not difficult to see that

I221 +

2

3I122 + I3

31 ≥ 2s3λ4β2ε−2

∫∫QT

φ3|∇ψ|4|w|2dxdt

+ 2sλ2β2ε−2

∫∫QT

φ|∇ψ|2|∇w|2dxdt.(A.11)

Hence

2(L1w,L2w)L2(QT ) ≥ 2

(X + Y + I1

31 + I231

)+2s3λ4β2ε−2

∫∫QT

φ3|∇ψ|4|w|2dxdt

+ 2sλ2β2ε−2

∫∫QT

φ|∇ψ|2|∇w|2dxdt.

(A.12)

From Remark 4 follows that

2(L1w,L2w)L2(QT ) + 2β−2s3λ4δ4ε−2

∫∫Qω0

φ3|w|2dxdt

+ 2sλ2δ2ε−2

∫∫Qω0

φ|∇w|2dxdt

≥ 2

(X + Y + I1

31 + I231

)+2β−2s3λ4δ4ε−2

∫∫QT

φ3|w|2dxdt

+ 2sλ2δ2ε−2

∫∫QT

φ|∇w|2dxdt.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 21

Coming back to (A.8) we see that

||L1w||2L2(QT ) + ||L2w||2L2(QT ) + 2β−2s3λ4δ4ε−2

∫∫QT

φ3|w|2dxdt

+ 2sλ2δ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ ||gs||2L2(QT ) + 2β−2s3λ4δ4ε−2

∫∫Qω0

φ3|w|2dxdt

+ 2sλ2δ2ε−2

∫∫Qω0

φ|∇w|2dxdt− 2

(X + Y + I1

31 + I231

).

(A.13)

Now we analyze the term 2

(X + Y + I1

31 + I231

)in (A.13). It can be shown that

X ≤ Cε−2

(Ts2λ2 + T 2s+ s3λ3 + Ts2λ

)∫∫QT

φ3|w|2dxdt,(A.14)

for ε < 1. Similarly

(A.15) Y ≤ ε−2Csλ

∫∫QT

φ|∇w|2dxdt.

For the others two terms we have

I131 + I2

31 ≤ CT 4s2λ4ε−2

∫∫QT

φ3|w|2dxdt+ Cλ2ε−2

∫∫QT

φ|∇w|2dxdt,(A.16)

Puting (A.14), (A.15) and (A.16) in (A.13) we get, for s ≥ s0(T + T 2 + T 4) andλ ≥ λ0, with s0 and λ0 independents of T ,

||L1w||2L2(QT ) + ||L2w||2L2(QT ) + β−2s3λ4δ4ε−2

∫∫QT

φ3|w|2dxdt

+ sλ2δ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ ||esαg||2L2(QT ) + 2β−2s3λ4δ4ε−2

∫∫Qω0

φ3|w|2dxdt

+ 2sλ2δ2ε−2

∫∫Qω0

φ|∇w|2dxdt.

(A.17)

Now we want to deal with the local integral involving ∇w in the right-hand side of(A.17). To this end we introduce a cutt-off function ξ such that

ξ ∈ C∞0 (ω), 0 ≤ ξ ≤ 1, ξ(x) = 1 ∀x ∈ ω0.

22 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Multiplying L2w by φξ2w we get∫∫QT

L2wφξ2wdxdt

= s

∫∫QT

φξ2∂tα|w|2dxdt− s2λ2

∫∫QT

N∑i,j=1

φ3aij∂xiψ∂xjψ|w|2ξ2dxdt

+

∫∫QT

N∑i,j=1

aij∂xjw∂xiwξ2φdxdt+ 2

∫∫QT

N∑i,j=1

aij∂xjw∂xiξξφwdxdt

+ λ

∫∫QT

N∑i,j=1

aij∂xjw∂xiψξ2φwdxdt.

Using the ellipticity condition on aij we see that

βε−1

∫∫Qω

φξ2|∇w|2dxdt

≤ C(∫∫

QT

L2wφξ2wdxdt+ (sT + ε−1s2λ2)

∫∫Qω

φ3|w|2dxdt

+ λε−1

∫∫Qω

φ1/2|∇w|ξφ1/2wdxdt

).

An application of Young’s inequality we get

2sλ2δ2ε−2

∫∫Qω

φξ2|∇w|2dxdt

≤ 1

4

∫∫QT

|L2w|2dxdt+ Cβ−2T 2s2λ4δ4ε−2

∫∫Qω

φ3|w|2dxdt

+ 2β−1sλ2δ2ε−1(sT + ε−1s2λ2)

∫∫Qω

φ3|w|2dxdt

+ 2β−1sλ3δ2ε−2

∫∫Qω

φ1/2|∇w|ξφ1/2wdxdt

).

Assuming that β < 1, we obtain

2sλ2δ2ε−2

∫∫Qω

φξ2|∇w|2dxdt(A.18)

≤ 1

4

∫∫QT

|L2w|2dxdt+ Cβ−2s3λ4(δ4 + δ2)ε−2

∫∫Qω

φ3|w|2dxdt

+ 2β−1sλ3δ2ε−2

∫∫Qω

φ1/2|∇w|ξφ1/2wdxdt

).

Using Young’s inequality again

sλ2δ2ε−2

∫∫Qω

φξ2|∇w|2dxdt

≤ 1

4

∫∫QT

|L2w|2dxdt+ Cβ−2s3λ4(δ4 + δ2)ε−2

∫∫Qω

φ3|w|2dxdt.

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 23

Therefore we can improve (A.17),

||L1w||2L2(QT ) + ||L2w||2L2(QT ) + β−2s3λ4ε−2

∫∫QT

φ3|w|2dxdt

+ sλ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ C(||esαg||2L2(QT ) + β−2s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).

(A.19)

Now we need to use the first two terms in left-hand side of (A.19) in order toimprove more our Carleman inequality. Since s−1φ−1 ≤ C, it follows that∫∫

QT

s−1φ−1|L1w|2dxdt+ ||L2w||2L2(QT ) + s3λ4ε−2

∫∫QT

φ3|w|2dxdt

+ sλ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ C(||esαg||2L2(QT ) + s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).

(A.20)

Next, writing L1w = −∂tw +A, it is an easy task to show that∫∫QT

s−1φ−1|∂tw|2dxdt+ ||L2w||2L2(QT ) + s3λ4ε−2

∫∫QT

φ3|w|2dxdt

+ sλ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ C(||esαg||2L2(QT ) + s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).

(A.21)

Arguing in the same way with the norm involving L2w we obtain∫∫QT

s−1φ−1|∂tw|2dxdt+ ε−2

∫∫QT

s−1φ−1N∑

i,j=1

|∂xi(Mij∂xjw)|2dxdt

+ s3λ4ε−2

∫∫QT

φ3|w|2dxdt+ sλ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ C(||esαg||2L2(QT ) + s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).

(A.22)

The following inequality can be easily proved

s−1ε−2

∫∫QT

∣∣∣∣∣∣N∑

i,j=1

∂xi(Mij∂xj

(wφ−1/2))

∣∣∣∣∣∣2

dxdt

≤ ε−2

∫∫QT

s−1φ−1N∑

i,j=1

|∂xi(Mij∂xjw)|2dxdt+ Cε−2s−1λ2

∫∫QT

φ−1|∇w|2dxdt

+ Cε−2s−1(λ2 + λ4)

∫∫QT

φ−1|w|2dxdt.

(A.23)

24 MOSTAFA BENDAHMANE AND FELIPE WALLISON CHAVES-SILVA

Using this last inequality, inequality (A.22) and elliptic regularity one can showthat

s−1ε−2

∫∫QT

φ−1|N∑

i,j=1

∂2xixj

w|2dxdt ≤ C

(||esαg||2L2(QT ) + s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).

Putting this last estimate in (A.22) we obtain∫∫QT

s−1φ−1|∂tw|2dxdt+ s−1ε−2

∫∫QT

φ−1|N∑

i,j=1

∂2xixj

w|2dxdt

+s3λ4ε−2

∫∫QT

φ3|w|2dxdt+ sλ2ε−2

∫∫QT

φ|∇w|2dxdt

≤ C(||esαg||2L2(QT ) + s3λ4ε−2

∫∫Qω

φ3|w|2dxdt).(A.24)

From (A.24) and the fact that w = esαv, it is not difficult to get the Carlemaninequality (A.2).

Appendix B. A Slightly changed Carleman inequality

The aim of this appendix is to show the inequality (3.6), used in step 1. Wefollow essentially [10], just keeping track of the dependence with respect to ε.

The starting point is to use the Carleman inequality given by Theorem A.1 forthe solution ρ of (3.2)2, so that

ε2

∫∫QT

s−1φ−1e2sα|∂tρ|2dxdt+

∫∫QT

s−1φ−1e2sα|N∑

i,j=1

∂2xixj

ρ|2dxdt

+ s3λ4

∫∫QT

φ3e2sα|ρ|2dxdt+ sλ2

∫∫QT

φe2sα|∇ρ|2dxdt

≤ Cs3λ4

∫∫Qω

φ3e2sα|ρ|2dxdt.

(B.1)

Now we improve the above Carleman inequality with respect to the powers of φ,that is done by considering y(x, t) = ρ(x, t)(φ∗(t))

12 and seeing that y satisfies

(B.2)

∣∣∣∣ ε∂ty − div (M(x)∇y) = −ε (T−2t)2 φ∗y in QT ,

y = 0 on ΣT .

Applying again the Carleman inequality given by Theorem A.1, at this time for y,we obtain, for s large enough, that∫∫

QT

s−1φ−1e2sα|∂ty|2dxdt+ ε−2

∫∫QT

s−1φ−1e2sα|N∑

i,j=1

∂2xixj

y|2dxdt

+ s3λ4ε−2

∫∫QT

φ3e2sα|y|2dxdt+ sλ2ε−2

∫∫QT

φe2sα|∇y|2dxdt

≤ Cs3λ4ε−2

∫∫Qω

φ3e2sα|y|2dxdt.

(B.3)

CONTROLLABILITY OF A SYSTEM IN ELECTROCARDIOLOGY 25

From the definition of y it is easy to show that∫∫QT

s−1φ−1e2sα|∂tρ(φ∗)12 |2dxdt ≤

∫∫QT

s−1φ−1e2sα|∂ty|2dxdt+

∫∫QT

e2sαφ|y|2dxdt.

Therefore inequality (B.3) gives

∫∫QT

s−1φ−1φ∗e2sα|∂tρ|2dxdt+ ε−2

∫∫QT

s−1φ∗φ−1e2sα|N∑

i,j=1

∂2xixj

ρ|2dxdt

+ s3λ4ε−2

∫∫QT

φ3φ∗e2sα|ρ|2dxdt+ sλ2ε−2

∫∫QT

φφ∗e2sα|∇ρ|2dxdt

≤ Cs3λ4ε−2

∫∫Qω

φ3φ∗e2sα|ρ|2dxdt.

(B.4)

Finally from Remark 2 we get∫∫QT

s−1e2sα|∂tρ|2dxdt+ ε−2

∫∫QT

s−1e2sα|N∑

i,j=1

∂2xixj

ρ|2dxdt

+ s3λ4ε−2

∫∫QT

φ4e2sα|ρ|2dxdt+ sλ2ε−2

∫∫QT

φ2e2sα|∇ρ|2dxdt

≤ Ceλ||ψ||s3λ4ε−2

∫∫Qω

φ4e2sα|ρ|2dxdt.

(B.5)

Which is exactly inequality (3.6).

Acknowledgments

This paper has been partially established during the visit of M. Bendahmane atthe Basque Center for Applied Mathematics. The authors thank professor EnriqueZuazua for various fruitful discussions about this paper. F. W. Chaves-Silva hasbeen partially supported by Grant MTM2008- 03541 of the MICINN, Spain, theERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research NetworkingProgramme OPTPDE and the Grant PI2010-04 of the Basque Government.

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(Mostafa Bendahmane)

Institut Mathematiques de Bordeaux Universite Victor Segalen Bordeaux 2 Place dela Victoire 33076 Bordeaux, France.

E-mail address: mostafa.bendahmane@u-bordeaux2.fr

(Felipe Wallison Chaves-Silva)BCAM Basque Center for Applied Mathematics Bizkaia Technology Park, Building

500 E-48160 Derio - Basque Country - SpainE-mail address: felipewallison@gmail.com