CONTROL OF A NETWORK OF MAGNETIC ELLIPSOIDAL SAMPLES Shruti Agarwal Gilles

19 Pages·2011·0.41 MB·English

MATHEMATICALCONTROLAND doi:10.3934/mcrf.2011.1.129
RELATEDFIELDS
Volume1,Number2,June2011 pp. 129–147
CONTROL OF A NETWORK OF
MAGNETIC ELLIPSOIDAL SAMPLES
Shruti Agarwal
IndianInstituteofTechnologyMadras
DepartmentofMathematics,Chennai-600036,India
Gilles Carbou
IMB,Universit´eBordeaux
351courslaLib´eration
33405Talence,France
Ste´phane Labbe´
LaboratoireJeanKuntzmann,Universit´edeGrenoble
TourIRMA,51ruedesMath´ematiques,BP53
38041GrenobleCedex9,France
Christophe Prieur
DepartmentofAutomaticControl,Gipsa-lab
961ruedelaHouilleBlanche,BP46
38402GrenobleCedex,France
(Communicated by Emmanuel Tr´elat)
Abstract. Inthiswork,wepresentamathematicalstudyofstabilityandcon-
trollabilityofone-dimensionalnetworkofferromagneticparticles. Thecontrol
isthemagneticfieldgeneratedbyadipolewhosepositionandwhoseamplitude
canbeselected. Theevolutionofthemagneticfieldinthenetworkofparticles
is described by the Landau-Lifschitz equation. First, we model a network of
ellipsoidalshapeferromagneticparticles. Then,weprovethestabilityofrele-
vantconfigurationsanddiscussthecontrollabilitybythemeansoftheexternal
magneticfieldinducedbythemagneticdipole. Finallysomenumericalresults
illustratethestabilityandthecontrollabilityresults.
1. Introduction. Thestudyofferromagneticsystemsisofgreatimportance, spe-
cially for the development of modern technological devices for which there is a con-
tinuous demand of more memory. Ferromagnetic materials are used in numerous
technological devices such as hard-disks, cellular phones, magnetic sensors, record-
ingheadsetc. Theseapplicationsaskforthestudyofasystemwhichisanassembly
of magnetic domains. In order to make these objects efficient and more useful, it
becomes necessary to control their magnetic behavior and to guarantee some con-
trollability and stability properties.
Fordatarecordingapplications, thefirstcrucialproblemisthecorrectrecording
oftheinformation. Inourcontextofanassemblyofferromagneticmaterials,thisis
2000 Mathematics Subject Classification. Primary: 93B05,35Q93;Secondary: 93C10.
Key words and phrases. Ferromagneticmaterial,Landau-Lifschitzequation,Control.
129
130 S. AGARWAL, G. CARBOU, S. LABBE´ AND C. PRIEUR
acontrollabilityissuewithrespecttorelevantconfigurations. Thisproblemisstud-
ied and solved by the control of the external magnetic field generated by a dipole.
The second problem is the accurate conservation of the data. From the mathemat-
ical point of view, this latter problem corresponds to the stability of the relevant
solutions(withoutanyexternalmagneticfield). Existenceandcontrollabilityresults
have been already proven for ferromagnetic materials. Consider e.g. [7] where the
existence of strong solutions for the Landau-Lifschitz equation is proven for finite
local time, and [2] for weak solutions. See [9] for an existence result for global time
of the Landau-Lisfchitz equation with a suitable control. Numerical simulations of
ferromagnetic materials are studied in [4, 13, 15] among other references.
The geometry of the ferromagnetic materials considered in this paper is close to
the one of [1] since in both works, the ferromagnetic domains are assumed to be
ellipsoidal. In [1], only one ellipsoidal domain is considered whereas in the current
paper, a network of ferromagnetic materials is considered with a coupling between
them.
In this work, we deal with a one dimensional network of ferromagnetic particles.
Weaimtoestablishsufficientconditionsonthecellssizeandonthenetworkgeom-
etrytoobtainastabilityresultfortherelevantmagnetizationconfigurations. More
precisely it is derived a sufficient condition on the volume of the ellipsoidal samples
andonthedistancebetweenthesamplesforparticularmagneticconfigurationtobe
locally asymptotically stable. This is our first main result. Our second main result
is acontrollability theoremwhen thecontrolis definedas theamplitudeofadipole
which is moving along the network with a constant speed. More precisely, given
two relevant magnetization configurations (one is the initial magnetization configu-
ration, whereas the other one is the desired final magnetization configuration), we
prove a controllability result for the magnetization by the mean of an applied mag-
netic field generated by a magnetic dipole modeling a magnetized point. Finally
we perform some simulations to illustrate both main results. We also check on a
numerical simulation that the stability of relevant configurations may be violated
when the geometry of the network and of the ellipsoidal samples does not satisfied
our sufficient condition.
The paper is organized as follows. In Section 2, the model is derived and the
problems that are solved in this paper are introduced, namely the stability of rel-
evant solutions of the Landau-Lifschitz equation and the controllability to these
magnetizations. Then both main results are given in Section 3. The proof of the
stability result is given in Section 4, whereas the proof of the controllability result
is given in Section 5. Some numerical simulations are performed in Section 6 to
illustrate the main results and to check the accuracy of the sufficient condition for
a magnetization configuration to be stable. Section 7 contains some concluding
remarks and suggests further research lines.
2. Model and control problems under consideration.
2.1. Ferromagnetism model. The general setting of the ferromagnetism is the
following (see [5], [14] and [16] and the references therein). We consider a finite
homogeneous ferromagnetic medium denoted by Ω. We denote by m the magneti-
zation:
m: IR+×Ω −→ IR3
(t,x,y,z) (cid:55)→ m(t,x,y,z).
CONTROL OF A NETWORK OF MAGNETIC ELLIPSOIDAL SAMPLES 131
The magnetic moment m links the magnetic induction B and the magnetic field H
by the relation B = m¯ +H, where m¯ is the extension of m by zero outside Ω. In
addition, we assume that the material is saturated so that the magnitude of m is
constant. We denote by · the scalar product in IR3 and |·| the associated norm.
After renormalization we assume that
|m|=1 at any point. (1)
The evolution of m is described by the Landau-Lifschitz equation:
dm
=−m×H −m×(m×H ), (2)
dt eff eff
where we denote by × the cross product on IR3. See [5] for a complete description
ofthephysicalmodel. Theexistenceoflocalweaksolutionof(2)wasestablishedin
[18]. Global existence of weak solution is studied in [10, 11], whereas the existence
anduniquenessofregularsolutionsforLandau-Lisfchitzequationisprovenin[7]for
a bounded domain, and in [8] for the domain R3. The effective field H = −∇E
eff
is derived from the micromagnetism energy E given by
E =E +E +E ,
exch dem a
where
• By normalizing the exchange, the exchange energy E writes
exch
1(cid:90)
E = |∇m|2.
exch 2
Ω
• E is the demagnetizing energy:
dem
1(cid:90)
E = |H (m)|2.
dem 2 d
IR3
The demagnetizing field H (m) is characterized by
d

curlH (m)=0,
 d
(3)
 div(H (m)+m¯)=0.
d
• The applied energy E reflects the effects of an applied magnetic field H :
a a
(cid:90)
E =− H ·m.
a a
Ω
Therefore we obtain that
H =∆m+H (m)+H .
eff d a
2.2. Simplified network model. Let us describe now the network model.
We deal with a one dimensional network of magnetic ellipsoidal shape samples.
The ellipsoids are supposed to have the same geometry and to be laid on the axis
IRe(cid:126), where (e(cid:126),e(cid:126),e(cid:126)) is the canonical basis of IR3. We denote by x the position
1 1 2 3 j
ofthejthcell,andweassumethatx =jl,wherel>0isthedistancebetweentwo
j
consecutive cells. In this paper, we consider a finite network, that is the indexes i
are in the finite set I = {0,1,2,...,N}. The ith cell Ω is obtained from Ω by a
i 0
translation of vector ile(cid:126), so that
1
(cid:26) (x−il)2 y2 z2 (cid:27)
Ω = (x,y,z)∈IR3, + + ≤1 ,
i L2 L2 L2
x y z
132 S. AGARWAL, G. CARBOU, S. LABBE´ AND C. PRIEUR
where L , L and L are three positive values for the length of the axes of the
x y z
ellipsoid in the directions e(cid:126), e(cid:126), and e(cid:126) respectively. We assume that the longest
1 2 3
axisoftheellipsoidsisinthedirectione(cid:126),thatisweassumethatL >max{L ,L }.
2 y x z
We denote by V = 4πL L L the volume of each cell.
3 x y z
We assume that the characteristic lengths of the cells are small compared to l
and to V 13, so that we assume that in each cell Ωi, the magnetization is constant
in the space variable (cf. [17], see also [1]) and is denoted by m (t). We use the
i
following notations:
(cid:110) (cid:111)
• for each k in N, (IRk)I = u=(u ,...,u )∈IRk×...×IRk ,
0 N
• (cid:107)u(cid:107)=sup|u |, where |·| is the euclidean norm in IRk,
i
i∈I
• (S2)I =(cid:8)u=(u ,...,u )∈(IR3)I, such that ∀i∈I,|u |=1(cid:9).
0 N i
Sotheunknownm=(m ,...,m )isdefinedonIR+withvaluesin(S2)I. Under
0 N
thisassumption,theexchangefieldvanishesandtheeffectivefieldincludesonlythe
demagnetizing and the applied fields.
Let us consider a magnetization configuration m = (m ) . In a fixed cell Ω ,
i i∈I j0
we split the stray field induced by the distribution m in two parts: the stray field
generated on Ω by m itself, denoted by Hint(m)(j ), and the field generated by
j0 j0 d 0
the other cells, denoted by Hext(m)(j ):
d 0
H (m)(j )=Hint(m)(j )+Hext(m)(j ).
d 0 d 0 d 0
From classical results (see [1] and [17]), the stray field generated by a uniformly
magnetized ellipsoid on itself is given by
 
α 0 0
Hdint(m)(j0)=− 0 β 0 mj0,
0 0 γ
whereα,β andγ dependontheellipsoidgeometry. Inourcase,wehave0<β <α
and 0<β <γ.
The stray field generated by the cell i on the cell j is given by
0 0
1 (cid:90) m 3 (cid:90) x−y
H (m )(x)=− i0 dy+ m ·(x−y)dy.
i0,j0 i0 4π |x−y|3 4π |x−y|5 i0
y∈Ωi0 y∈Ωi0
WeassumethatΩ andΩ aresmallsothatwewritethatH (m )isalmost
i0 j0 i0,j0 i0
constant in Ω and we approximate it by
j0
V m 3V m1
H (m )=− i0 + i0 e(cid:126),
i0,j0 i0 4πl3|i −j |3 4π l3|i −j |3 1
0 0 0 0
that is
V 1
H (m )= Am , (4)
i0,j0 i0 4πl3|i −j |3 i0
0 0
where
 
2 0 0
A= 0 −1 0 .
0 0 −1
CONTROL OF A NETWORK OF MAGNETIC ELLIPSOIDAL SAMPLES 133
The network exterior stray field at the cell j is given by
0
 2hext(m1)(j ) 
d 0
 
Hdext(m)(j0)= (cid:88) Hi0,j0(m(i0))= −hedxt(m2)(j0) , (5)
i0(cid:54)=j0  
−hext(m3)(j )
d 0
where(m1,m2,m3)arethecoordinatesofm,andwheretheoperatorhext :(IR3)I →
d
(IR3)I is defined by, for all u=(u ) in (IR3)I,
i i∈I
V (cid:88) 1
hext(u)(j )= u(j). (6)
d 0 4πl3 |j−j |3
0
j(cid:54)=j0
In order to control the network, we apply on it a magnetic field generated by
a suitable dipole of magnetic moment Me(cid:126) situated in the plane Vect(e(cid:126),e(cid:126)) at
2 1 2
a fixed distance δl from the network. We denote by (X,δl,0) the coordinates of
the dipole. We assume that the dipole is moving with a constant speed v, so that
X(t)=x +vt. Our control is the variable M(t).
0
By standard results (see e.g. [3, Chap. 4.12, Eq. (4.17)] or [12, Chap. 3.4,
Eq. (3.103)]), assuming that the dipole is relatively far from the network, the field
induced on the cell Ω by this dipole is given by
i0
µ M 1
H (t,M)(i )= 0 (2cos(θ)u +sin(θ) u ), (7)
app 0 4π r3 r θ
where
• r is the distance between the dipole and the cell:
r =[(x0+vt−i0l)2+δ2l2]21,
• u and u are given by
r θ
   
x +vt−i l δl
1 0 0 1
ur = r  −δl , uθ = r  x0+vt−i0l 
0 0
(cid:92)
• θ is the angle (−e(cid:126),u ),
2 r
• µ is the dielectric permitivity of the vacuum.
0
It yields the following model:
 for i∈I, m :IR+ −→S2 ⊂IR3,
 dm i
i =−m ×H (i)−m ×(m ×H (i)) for i∈I and t∈IR+, (8)
 Hdt (i)=−iDme+ffHext(mi)(i)+iH (etf,fM)(i),
eff i d app
with
 
a 0 0
D = 0 0 0 , (9)
0 0 b
where a = α−β > 0 and b = γ −β > 0 (for simplicity, we have replaced D by
D−βI without changing the model, since it only appears in the term m×Dm in
d
the equations). Without loss of generality, we assume that 0<a<b.
134 S. AGARWAL, G. CARBOU, S. LABBE´ AND C. PRIEUR
We call relevant configurations the magnetization distributions of the form:
m0 =ε e(cid:126), with ε ∈{+1,−1} for i∈I. (10)
i i 2 i
Let us interpret these relevant configurations as a memory state in an electronic
device, where ε =1 corresponds to a bit 1, ε =−1 corresponds to a bit 0.
i i
In order to ensure a good conservation of the memory, the key point is the
stability of the relevant configurations for the system (8) without applied field.
Let us introduce the problems under consideration.
Problem 1. Exponential stability of any relevant position.
For any initial conditions sufficiently close to a given relevant configuration, the
solution of the Landau-Lischitz equation (8) converges exponentially fast to the rel-
evant configuration, without any external magnetic field.
Problem 2. Controllability under the action of a dipole.
Given any pair of relevant configurations, if the initial condition is in a neigh-
borhood of the first relevant configuration, then, with a suitable amplitude of the
magnetic field created by the dipole, the magnetic field of the network enters in
finite time a neighborhood of the second relevant configuration and converges expo-
nentially fast to it thereafter.
TheseproblemsaresolvedinTheorems1and2respectivelyunderconditionson
the geometry of the network.
3. Statement of the main results. To state the stability result, we need to
introduce the following notation. For ν > 0 small enough, we define V (ν) and
+1
V (ν) by
−1
V (ν)=(cid:8)ξ =(ξ ,ξ ,ξ )∈S2,ξ >0 and aξ2+bξ2 <ν2(cid:9),
+1 1 2 3 2 1 3
V (ν)=(cid:8)ξ =(ξ ,ξ ,ξ )∈S2,ξ <0 and aξ2+bξ2 <ν2(cid:9).
−1 1 2 3 2 1 3
We remark that for ξ ∈ S2, if ξ > 0 (resp. if ξ < 0), the quantity aξ2 +bξ2
2 2 1 3
measures the distance between ξ and +e(cid:126) (resp. −e(cid:126)) since in this case:
2 2
a
|ξ−e(cid:126)|2 ≤aξ2+bξ2 ≤b|ξ−e(cid:126)|2.
2 2 1 3 2
For ε=(ε ,...,ε ) with ε ∈{−1,+1}, we denote for ν >0:
1 N i
V¯ (ν)=(cid:8)m∈(S2)I,∀i∈I, m ∈V (ν)(cid:9). (11)
ε i εi
Our first main result is the following:
Theorem 1. There exists γ > 0 (depending on a and b, but independent to the
0
size of the network) such that if
V
≤γ , (12)
l3 0
then there exist ν > 0, and c > 0 such that for all relevant configurations m0
0
associated to ε (that is m0 = ε e(cid:126) for all i), for all minit ∈ V¯ (ν ), the solution m
i i 2 ε 0
of (8) with M ≡0 starting from the initial condition m(0)=minit satisfies:
∀t≥0,m(t)∈V¯ (ν e−ct).
ε 0
Remark 1. This theorem means that the relevant configurations are uniformly
asymptotically stable for the Landau-Lifschitz equation (8) with zero applied field
(M ≡0).
CONTROL OF A NETWORK OF MAGNETIC ELLIPSOIDAL SAMPLES 135
The second question under consideration is the controllability of our network by
the mean of a dipole generating applied field. We consider m(cid:91) and m(cid:93) two relevant
configurations. We assume that at t=0 the magnetization of our network is close
tom(cid:91). Inordertoalignthemagnetizationwithm(cid:93),wedefinethecontrolt(cid:55)→M(t)
by
 +M if m(cid:91) =−m(cid:93) =e(cid:126),
 i i 2
for t∈[ivl,ivl +δvl[, M(t)= 0 if m(cid:91)i =m(cid:93)i,
 −M if m(cid:91) =−m(cid:93) =−e(cid:126). (13)
i i 2
N
(cid:91) l l l
for t∈/ [i ,i +δ [, M(t)=0.
v v v
i=0
for a suitable M > 0 and 0 ≤ δ ≤ 1. We denote T = (N +δ)l/v. For t ≥ T ,
f f
M(t)=0, that is the dipole is switched off.
Our second main result is the following:
Theorem 2. Let v be a fixed positive value. Let γ and c be given by Theorem 1.
0
There exists γ >0 with γ <γ , there exist ν >0, M>0 and δ >0 such that if
1 1 0 1
V
≤γ ,
l3 1
then we have the following controllability result:
let m(cid:91) and m(cid:93) be two relevant configurations, and let t(cid:55)→M(t) be the control given
by (13). If minit ∈ V¯ (ν ), then the solution m of (8) starting from the initial
ε(cid:91) 1
condition minit satisfies:
∀t≥Tf, m(t)∈V¯ε(cid:93)(ν1e−c(t−Tf)).
Theremainingofthepaperisorganizedasfollows. WeproveTheorem1,namely
the stability of all relevant configurations, in Section 4. We remark that our sta-
bility criteria depends neither on the size of the network N nor on the considered
relevant configuration m(cid:91). Section 5 is devoted to the proof of Theorem 2, i.e.
the controllability of a finite network by the mean of a magnetic dipole generating
an applied field. The paper ends with some numerical simulations and concluding
remarks (see respectively Sections 6 and 7).
4. Proof of the stability for the relevant configurations. In this section we
tackle the stability of a relevant configuration for the Landau-Lifschitz equation
without applied field. More precisely we prove Theorem 1 and we consider the
following system with unknown m:IR+ →(S2)I:
dm
= −m×(−Dm+Hext(m))−m×(m×(−Dm+Hext(m)))
dt d d
= m×F(m) (14)
with F(m) = −(−Dm+Hext(m))−m×(−Dm+Hext(m)). The existence and
d d
uniqueness of a solution of (14) for any initial condition m0 ∈ (S2)I follows from
the classical Cauchy-Lipschitz theorem and the constraint m (t)∈S2 for all i in I
i
and for all t≥0.
136 S. AGARWAL, G. CARBOU, S. LABBE´ AND C. PRIEUR
Let m0 be a fixed relevant configuration: m0 ∈(S2)I such that
∀i∈I, m0 =ε e(cid:126), ε ∈{−1,+1}.
i i 2 i
Because of the saturation constraint (1), we only deal with perturbations m of
m0 satisfying:
∀i∈I,∀t,|m (t)|=1.
i
So we describe such a perturbation writing for all i∈I:
m =ρ1e(cid:126) +ρ3e(cid:126) +ε e(cid:126) +λ(ρ )ε e(cid:126) (15)
i i 1 i 3 i 2 i i 2
(cid:112)
where ρ =(ρ1,ρ3) and λ(ρ )= 1−|ρ |2−1.
i i i i i
In order to find an equivalent formulation of (14) in the variable ρ ∈ C1(IR+;
(S2)I), we plug (15) in (14) and we project the obtained expression on both e(cid:126) and
1
e(cid:126) axis.
3
We obtain that m, given by (15), satisfies (14) if and only if ρ satisfies the
following system:
(cid:18) (cid:19)
dρ −a εb
= ρ+L(ρ)+N(ρ) (16)
dt −εa −b
where
 εhext(ρ3)−ρ3hext(ε)+2hext(ρ1)+ερ1hext(ε) 
d d d d
L(ρ)=  .
2εhext(ρ1)+ρ1hext(ε)−hext(ρ3)+ερ3hext(ε)
d d d d
The nonlinear term N =(N ,N ) is given by
1 3
N (ρ)= −ελ(ρ)(−bρ3−hext(ρ3))−ρ3hext(ελ(ρ))+ελ(ρ)((−aρ1+2hext(ρ1))ε
1 d d d
+ρ1hext(ε))+(−aρ1+2hext(ρ1))λ(ρ)+ερ1hext(ελ(ρ)) ,
d d d
N (ρ)= ελ(ρ)(−aρ1+2hext(ρ1))+ρ1hext(ελ(ρ))
3 d d
−ελ(ρ)(−hext(ε)ρ +(bρ3+hext(ρ3))ε)
d 3 d
+λ(ρ)(−bρ3−hext(ρ3))+εhext(ελ(ρ))ρ3 .
d d
Remark 2. By projection, it is clear that if m satisfies (14) then ρ satisfies (16).
For the converse implication, we remark that if m∈C1(R ;(S2)I) and satisfies:
+
dm
∀i∈{1,3}, ·e(cid:126) =(m×F(m))·e(cid:126),
dt i i
because of the constraint |m|=1, then m satisfies
dm
=m×F(m).
dt
This argument is used in a partial differential equation framework in [6] and in [9].
In addition, m0 is asymptotically stable for (14) if and only if 0 is asymptotically
stable for (16).
V
Let us study now the stability of 0 for (16) under a smallness condition on .
l3
The linear operator hext is estimated in the following way: for all u=(u ) in
d i i∈I
(IR3)I, then
V (cid:88) 1
(cid:107)hext(u)(cid:107)≤ (cid:107)u(cid:107). (17)
d 4πl3 |j|3
j(cid:54)=0
So, we estimate the linear operator L with the following lemma:
CONTROL OF A NETWORK OF MAGNETIC ELLIPSOIDAL SAMPLES 137
Lemma 3. There exists a constant K such that we have, for all ρ∈(IR2)I,
L
V
(cid:107)L(ρ)(cid:107)≤K (cid:107)ρ(cid:107).
Ll3
This constant K depends neither on a, b, V, l nor on the size of the network.
1
The nonlinear right-hand side term of (16) is estimated with straightforward
arguments in the following lemma.
V
Lemma 4. We assume that ≤1. There exists a constant K such that, for all
l3 N
ρ∈(IR2)I such that (cid:107)ρ(cid:107)≤ 1, we have
2
(cid:107)N (ρ)(cid:107)+(cid:107)N (ρ)(cid:107)≤K (cid:107)ρ(cid:107)2.
1 2 N
We define ρ˜∈C0(IR+;(IR)I) by
ρ˜(t)=(cid:0)a(ρ1(t))2+b(ρ3(t))2(cid:1)21 .
i i i
Recalling 0<a<b, we remark that
√
|(aρ1,bρ2)|≤ b|ρ˜| and a|ρ |2 ≤|ρ˜|2 ≤b|ρ |2. (18)
i i i i i i
In addition, m∈V¯ (ν) if and only if ρ˜<ν (see (11)).
ε
Multiplying (16) by (aρ1,bρ3), we have, for all i in I,
i i
1 d
[a(ρ1)2+b(ρ3)2]+a2(ρ1)2+b2(ρ3)2 =(L(ρ) +N(ρ) )·(aρ1,bρ3),
2dt i i i i i i i i
so using that 0<a<b, (18) and Lemmas 3 and 4, we obtain that
1 d (cid:0)|ρ˜|2(cid:1)+a|ρ˜|2 ≤ K V (cid:107)ρ(cid:107)√bρ˜ +K (cid:107)ρ(cid:107)2√bρ˜ ,
2dt i i Ll3 i N i
√
(cid:114)
V b b
≤ K (cid:107)ρ˜(cid:107)2+K (cid:107)ρ˜(cid:107)3 . (19)
Ll3 a N a
We define γ by
0
a32
γ = √ . (20)
0
bK
L
V
We assume that V and l are fixed so that < γ . We aim to show that 0 is
l3 0
asymptotically stable for (16).
We multiply (19) by e2at and we integrate from 0 to t. We obtain that, for all i
in I,
√
V (cid:114)b (cid:90) t b(cid:90) t
(ρ˜(t))2e2at ≤(cid:107)ρ˜(0)(cid:107)2+2K (cid:107)ρ˜(s)(cid:107)2e2asds+2K (cid:107)ρ˜(s)(cid:107)3e2asds.
i Ll3 a N a
0 0
So taking the supremum for i in I, we obtain that, for all t≥0,
√
V (cid:114)b (cid:90) t b(cid:90) t
(cid:107)ρ˜(t)(cid:107)2e2at ≤(cid:107)ρ˜(0)(cid:107)2+2K (cid:107)ρ˜(s)(cid:107)2e2asds+2K (cid:107)ρ˜(s)(cid:107)3e2asds.
Ll3 a N a
0 0
138 S. AGARWAL, G. CARBOU, S. LABBE´ AND C. PRIEUR
(cid:113) V
We denote by c = a−K V b. From (20), since < γ , c is positive. Now,
Ll3 a l3 0
ca
while (cid:107)ρ˜(s)(cid:107)≤ √ , we have, for all t≥0,
2K b
N
(cid:90) t
(cid:107)ρ˜(t)(cid:107)2e2at ≤(cid:107)ρ˜(0)(cid:107)2+(2a−c) (cid:107)ρ˜(s)(cid:107)2e2asds,
0
ca
and by Gronwall lemma, we obtain that, while (cid:107)ρ˜(s)(cid:107)≤ √ ,
2K b
N
(cid:107)ρ˜(t)(cid:107)2 ≤(cid:107)ρ˜(0)(cid:107)2e−ct.
ca ca
So, if (cid:107)ρ˜(0)(cid:107) ≤ √ , then for all t ≥ 0, (cid:107)ρ˜(t)(cid:107)2 remains less than √
2K b 2K b
N N
and
∀t≥0,(cid:107)ρ˜(t)(cid:107)2 ≤(cid:107)ρ˜(0)(cid:107)2e−ct.
ca
We set ν = √ , and it concludes the proof of Theorem 1. (cid:3)
0
2K b
N
5. Proof of the controllability result. Let m(cid:91) and m(cid:93) be two relevant config-
urations associated to ε(cid:91) and ε(cid:93) respectively, that is m(cid:91) = ε(cid:91)e(cid:126) and m(cid:93) = ε(cid:93)e(cid:126) for
i i 2 i i 2
i in I. In this section we prove Theorem 2, that is the controllability result to m(cid:91)
with initial condition in a neighborhood of m(cid:93). Let us introduce ν > 0 given by
0
Theorem 1.
For a fixed ζ ∈S2, for ν ∈[−1,1], we define W(ζ,ν) by
W(ζ,ν)=(cid:8)x∈S2, x·ζ ≥ν(cid:9).
 
3
1
We define ξ(cid:126) by ξ(cid:126) = √  −1 . We introduce ν1 and µ1 > 0 such that
10
0
0<ν <ν so that
1 0
V (ν )⊂W(ξ(cid:126),−1+µ ) and W(−e(cid:126),1−µ )⊂V (ν ). (21)
+1 1 1 2 1 −1 1
We introduce the following sequence (t ) of time instants, defined by t =il,
i i∈I i v
for all i∈I. We assume that
m(0)∈V (ν ). (22)
ε(cid:91) 1
Werecallthatourcontrolisdefinedby(13)whereδandMaretwoconstantvalues.
These former values will be selected so that, for i ∈ {0,...,N +1}, the following
property (P ) holds:
i

V (ν ) for j ≥i,
 ε(cid:91)j 1
(P ) ∀j ∈{0,...,N}, m (t )∈
i j i
 Vε(cid:93)(ν1) for j <i.
j
Note that Theorem 2 follows from (P ).
N+1
TheselectionofδandMisdonebyusinganinductionargumentontheproperty
(P ).
i
The property (P ) is true (whatever of the value of δ and M).
0
Let i ∈I such that (P ) holds. The conditions on M and δ such that (P )
0 i0 i0+1
are derived as follows. First, denoting t = i l, we give a bound on the applied
i0 0v
fieldduringthetimeinterval[t ,t +δ l)inSection5.1. ThenweproveinSection
i0 i0 v