New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence
Abstract
We consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllability. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0.
Full text
New phenomena o he null con ollabili y o pa abolic
sys ems: Minimal ime and geome ical dependence
Fa id Amma Khodja∗
Assia Benabdallah †
Manuel Gonz´
alez-Bu gos ‡
and Luz de Te esa§
June 12, 2015
Abs ac
We conside he null con ollabili y p oblem o wo coupled pa abolic equa ions wi h a
space-depending coupling e m. We analyze bo h bounda y and dis ibu ed null con ollabil-
i y. In each case, we exhibi a minimal ime o con ol, ha is o say, a ime T0∈[0,∞] such
ha he co esponding sys em is null con ollable a any ime T > T0and is no i T < T0. In
he dis ibu ed case, his minimal ime depends on he ela i e posi ion o he con ol in e al
and he suppo o he coupling e m. We also p o e ha , o a ixed con ol in e al and a
ime τ0∈[0,∞], he e exis coupling e ms such ha he associa ed minimal ime is τ0.
Con en s
1 In oduc ion and main esul s 2
2 Some p elimina y esul s 8
3 Bounda y con ollabili y p oblem 13
3.1 Bounda y app oxima e con ollabili y ......................... 14
3.2 Bounda y null con ollabili y .............................. 15
3.2.1 Posi i e bounda y con ollabili y esul .................... 16
3.2.2 Nega i e bounda y con ollabili y esul .................... 17
4 Dis ibu ed app oxima e con ollabili y 18
5 P oo o Theo em 1.3: The posi i e null con ollabili y esul 20
5.1 The momen p oblem .................................. 20
5.2 Cons uc ion o he unc ions 1and 2........................ 22
5.3 Sol ing he momen p oblem .............................. 24
5.3.1 The case k∈Λ1................................. 25
5.3.2 The case k∈Λ2................................. 26
5.3.3 The case k∈Λ3................................. 27
5.4 Conclusion ........................................ 27
∗Labo a oi e de Ma h´ema iques de Besan¸con, UMR 6623, Uni e si ´e de F anche-Com ´e, 16 ou e de G ay, 25030
Besan¸con cedex, F ance. E-mail: [email p o ec ed]
†Aix Ma seille Uni e si ´e, CNRS, Cen ale Ma seille, I2M, UMR 7373, 13453 Ma seille, F ance. E-mail:
[email p o ec ed]
‡Dp o. E.D.A.N., Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain. Suppo ed by g an MTM2013-
41286-P, Minis y o Economy and Compe i i eness (Spain). E-mail: [email p o ec ed]
§Ins i u o de Ma em´a icas, UNAM, Ci cui o Ex e io , C.U. 04510 D.F., M´exico. Suppo ed by p ojec IN102799
o D.G.A.P.A. (Mexico). E-mail: [email p o ec ed]
1
6 P oo o Theo em 1.3: The nega i e null con ollabili y esul 28
7 Complemen a y esul s. Some examples 29
A P oo o Lemma 7.1 35
1 In oduc ion and main esul s
This pape deals wi h he con ollabili y o non-scala pa abolic equa ions wi h a educed numbe
o con ols. The con ol o pa abolic sys ems is a challenging issue, which has a ac ed he in e es
o he con ol communi y in he las decade. These pa abolic sys ems a ise, o example, in he
s udy o chemical eac ions and in a wide a ie y o ma hema ical biology and physical si ua ions
(see e.g. [23], [33], [16], ...). Mo e p ecisely, he aim o his pape is o in es iga e he ela ionship
be ween he loca ion o he con ols and he ac ion o he coupling e ms. We will see ha in his
amewo k new phenomena a ise.
To his end, le us ix T > 0 and ω= (a, b)⊂(0, π) and conside he ollowing con ol p oblems:
y −yxx +q(x)A0y= 0 in QT:= (0, π)×(0, T),
y(0,·) = Bu, y(π, ·) = 0 on (0, T),
y(·,0) = y0in (0, π),
(1.1)
and
y −yxx +q(x)A0y=B 1ωin QT,
y(0,·)=0, y(π, ·) = 0 on (0, T),
y(·,0) = y0in (0, π),
(1.2)
whe e A0∈L(R2) and B∈R2a e espec i ely gi en by:
A0=0 1
0 0 and B=0
1.(1.3)
In sys ems (1.1) and (1.2), q∈L∞(0, π) is a gi en unc ion, y0is he ini ial da um and u∈L2(0, T)
and ∈L2(QT) a e he con ol unc ions.
Le us ema k ha o e e y u∈L2(0, T) ( esp., ∈L2(QT)) and y0∈H−1(0, π;R2) ( esp.,
y0∈L2(0, π;R2)), sys em (1.1) ( esp., sys em (1.2)) possesses a unique solu ion de ined by ans-
posi ion ( esp., a unique weak solu ion) which sa is ies
y∈L2(QT;R2)∩C0([0, T]; H−1(0, π;R2))
( esp., y∈L2(0, T;H1
0(0, π;R2)) ∩C0([0, T]; L2(0, π;R2)))
and depends con inuously on he da a uand y0, i.e., he e exis s a cons an C=C(T)>0 such
ha
kykL2(QT;R2)+kykC0([0,T ];H−1(0,π;R2)) ≤Cky0kH−1(0,π;R2)+kukL2(0,T )
( esp., kykL2(0,T ;H1
0(0,π;R2)) +kykC0([0,T ];L2(0,π;R2)) ≤Cky0kL2(0,π;R2)+k kL2(QT)).
Le us ecall ha he unc ion y∗∈L2(QT;R2)∩C0([0, T]; H−1(0, π;R2)) ( esp., he unc ion
y∗∈L2(0, T;H1
0(0, π;R2)) ∩C0([0, T]; L2(0, π;R2))) is a ajec o y o sys em (1.1) ( esp., o
sys em (1.2)) i y∗is he solu ion o (1.1) ( esp., o (1.2)) co esponding o he da a u∗∈L2(0, T)
and y∗
0∈H−1(0, π;R2) ( esp., ∗∈L2(QT) and y∗
0∈L2(0, π;R2)). Wi h he p e ious no a ions,
we de ine:
De ini ion 1.1. 1. I will be said ha sys em (1.1) ( esp., sys em (1.2)) is app oxima ely con-
ollable in H−1(0, π;R2) ( esp., in L2(0, π;R2)) a ime Ti o e e y y0, yd∈H−1(0, π;R2)
( esp., y0, yd∈L2(0, π;R2)) and o e e y ε > 0, he e exis s a con ol u∈L2(0, T) ( esp.,
∈L2(QT)) such ha he solu ion y o (1.1) ( esp., o (1.2)) sa is ies
ky(·, T)−ydkH−1(0,π;R2)≤ε( esp., ky(·, T )−ydkL2(0,π;R2)≤ε).
2
2. I will be said ha sys em (1.1) ( esp., sys em (1.2)) is null con ollable a ime Ti o
e e y y0∈H−1(0, π;R2) ( esp., y0∈L2(0, π;R2)), he e exis s a con ol u∈L2(0, T) ( esp.,
∈L2(QT)) such ha he solu ion y o (1.1) ( esp., o (1.2)) sa is ies
y(·, T) = 0 in H−1(0, π;R2) ( esp., in L2(0, π;R2)).
3. Finally, i will be said ha sys em (1.1) ( esp., sys em (1.2)) is exac ly con ollable o ajec-
o ies a ime T > 0 i o e e y y0∈H−1(0, π;R2) and e e y ajec o y y∗o sys em (1.1)
( esp., o e e y y0∈L2(0, π;R2) and e e y ajec o y y∗o sys em (1.2)), he e exis s a
con ol u∈L2(0, T) ( esp., ∈L2(QT)) such ha he solu ion y o (1.1) ( esp., o (1.2))
sa is ies
y(·, T) = y∗(·, T ) in H−1(0, π;R2) ( esp., in L2(0, π;R2)).
In his wo k we a e in e es ed in s udying he con ollabili y p ope ies o sys ems (1.1)
and (1.2). Le us obse e ha we a e exe ing only one con ol o ce on he sys ems (a bounda y o
dis ibu ed con ol) bu we wan o con ol he co esponding s a e ywhich has wo componen s.
In ac , he i s equa ion in (1.1) and (1.2) is indi ec ly con olled by means o he e m q(x)y2.
O cou se, his coupling e m mus be di e en om ze o, i.e., q6≡ 0.On he o he hand, using
he linea i y o sys ems (1.1) and (1.2), i is easy o see ha he null con ollabili y p ope y a
ime To he p e ious sys ems is equi alen o he exac con ollabili y o ajec o ies a ime T
o hese sys ems.
Sys ems (1.1) and (1.2) a e pa icula classes o mo e gene al n×npa abolic con ol sys ems
o he o m:
y −D∆y+A(x, )y=B 1ωin QT:= Ω ×(0, T),
y=Cu1Γ0,on ΣT:= ∂Ω×(0, T),
y(·,0) = y0in Ω,
(1.4)
whe e ωand Γ0a e, espec i ely, open subse s o he smoo h bounded domain Ω ⊂RNand o i s
bounda y ∂Ω, D= diag (d1,· · · , dn)∈L(Rn), wi h n≥1, is a posi i e ma ix, B, C ∈L(Rm,Rn),
wi h m≤n, a e gi en ma ices, and A= (aij )1≤i,j≤2∈L∞(QT;L(Rn)) is a ma ix- alued
unc ion. When m<n, he issue o his sys em is o con ol he whole componen s o he sys em
wi h a con ol unc ion ac ing, locally in space o on a pa o he bounda y, only on some o hem.
We e e o [5] o a e iew o esul s o he con ollabili y p oblem o sys em (1.4).
The i s esul s on con ollabili y o he scala case, n= 1, conce ns he one-dimensional case
N= 1. They ha e been es ablished by H.O. Fa o ini and D.L. Russell (see [19,20]) h ough
he momen me hod. The con ollabili y o he N-dimensional case, s ill o he scala equa ion
(n= 1), has been es ablished la e by G. Lebeau and L. Robbiano in [31] and by A. Fu siko
and O. Yu. Imanu ilo in [22] using Ca leman es ima es. I is in e es ing o poin ou ha he
bounda y and dis ibu ed null con ollabili y o scala pa abolic p oblems is alid o any posi i e
ime T, o any Γ0⊂∂Ω and o any ω⊂Ω.
Le us also unde line he e e ence [18], whe e he au ho p o es he exis ence o a minimal
con ol ime o he one-dimensional hea equa ion wi h con ols on he o m (x)u( ), wi h
∈H−1(0, π), a gi en ixed unc ion, and u∈L2(0, T).
The i s esul s on con ollabili y o coupled pa abolic equa ions (n > 1) ha e been es ablished
in [35,13,3,25]. They conce n mainly sys em (1.4) wi h n= 2, C= 0 (dis ibu ed con ol) and
B=0
1.
In all he p e ious wo ks he au ho s use Ca leman inequali ies o he co esponding adjoin
sys em o (1.4). The main assump ion on he ma ix- alued unc ion Ais ha he e exis an open
subse ω0⊂ωand a posi i e cons an σsuch ha
a12 ≥σ > 0 o a12 ≤ −σ < 0 in ω0×(0, T).(1.5)
3
I is in e es ing o poin ou ha in [25], unde he weake assump ion
|a12| ≥ σ > 0 in ω0×(0, T ),(1.6)
he au ho s p o e a null con ollabili y esul a ime T > 0 o some gene aliza ions o sys em (1.4).
The p e ious con ollabili y esul s ha e been ex ended in [26] o n≥2 when sys em (1.4)
has a pa icula s uc u e: cascade sys ems. To his end, he au ho s assume a gene aliza ion
o assump ion (1.5) on he coupling ma ix A(·,·) and, again, use Ca leman inequali ies o he
adjoin p oblem o p o ing he null con ollabili y esul .
In [4], a necessa y and su icien condi ion o he app oxima e and null con ollabili y a ime
T > 0 is es ablished when Ais a cons an ma ix. This condi ion does no depend on Tand
gene alizes he algeb aic Kalman condi ion (see [28]), well-known o he con ollabili y o ini e
dimensional sys ems. In he case n= 2, his necessa y and su icien condi ion educes o a12 6= 0.
Le us now desc ibe he exis ing esul s on bounda y con ollabili y o sys em (1.4) (B= 0).
The e a e ew esul s on his amewo k and mos o hem conce n he one-dimensional case
(N= 1), D=Id and Aa cons an ma ix. When D=Id and Ais a cons an ma ix, a necessa y
and su icien condi ion is exhibi ed in [21] and [6]. This condi ion is di e en om he one ha
cha ac e izes he dis ibu ed null con ollabili y o sys em (1.4) in he cons an case (see [4]).
As a consequence and unlike he scala case, we deduce ha he dis ibu ed and bounda y null
con ollabili y p ope ies o non-scala pa abolic sys ems a e in gene al no equi alen .
The bounda y null con ollabili y p oblem o sys em (1.4) in he cons an case is mo e in ica e
i D6=Id. When n= 2, he bounda y null con ollabili y p ope y holds i Tis g ea e han a
minimal ime T0∈[0,∞] which depends on he coe icien s o he cons an ma ices Dand A
(see [10]). Fo ins ance, i he di usion ma ix D, he coupling ma ix Aand he con ol ec o
Ca e gi en by
D= diag (1, d2), d 6= 0,1, A =0 1
0 0 , C =0
1,(1.7)
hen sys em (1.4) is app oxima ely con ollable a ime T > 0 i and only i dis an i a ional
numbe and he minimal ime T0o null con ollabili y depends on he diophan ine app oxima ion
o d. Le us also unde line ha his phenomenon (minimal ime o con ollabili y o pa abolic
equa ions) has been obse ed o he i s ime in he scala case in [18], bu conce ning poin wise
con ols. We would like o commen ha , o sys em (1.4) wi h he p e ious da a (1.7), i is
possible o selec posi i e numbe s d > 0 o which sys em (1.4) is app oxima ely con ollable a
any posi i e ime Tand ne e null con ollable (see [10]). Unlike he scala case, om he esul s
in [10] we in e ha he app oxima e and null bounda y con ollabili y p ope ies o non-scala
pa abolic sys ems a e, in gene al, no equi alen .
In [21], [6] and [10], he au ho s use he momen me hod (see [19,20]) o p o e he posi i e
null con ollabili y esul a ime T. They ca y ou a s udy on bounds o bio hogonal amilies o
exponen ials associa ed o complex sequences. In ac , he p e ious minimal ime T0is ela ed o
he index o condensa ion o he sequence o eigen alues o he ope a o associa ed o he sys em
(see [10]).
Finally, in [12] he au ho s ex end he one-dimensional bounda y null con ollabili y esul s
om [21] and [6] o he N-dimensional case when he domain Ω is a cylind ical domain.
Unlike he dis ibu ed con ollabili y p oblem o sys em (1.4), Ca leman es ima es o he
co esponding adjoin sys em seem no o be sui able when dealing wi h he bounda y null con-
ollabili y p oblem o sys em (1.4).
Le us come back o sys ems (1.1) and (1.2). Fi s , obse e ha om he null con ollabili y
esul s a ed in [25], i he unc ion qsa is ies (1.6), wi h σ > 0 and ω0⊂ωan open in e al, hen
sys em (1.2) is null con ollable a any posi i e ime T. The e o e, a na u al ques ion a ises: wha
happens i Supp q∩ω=∅? The i s and pa ial answe conce ns he app oxima e con ollabili y
o his sys em. Mo e p ecisely, in [29], he app oxima e con ollabili y o sys em (1.2) a e e y
ime T > 0 is p o ed when q= 1Owi h Oa nonemp y open subse o Ω. La e , o he pa ial
4
answe s a e gi en o he null con ollabili y o sys ems (1.1) and (1.2) unde sign condi ions on
he unc ion q(see [2], [34], [1] and [17]):
q6≡ 0 and q≥0 o q≤0 in (0, π).(1.8)
These esul s ha e been ob ained as a consequence o he co esponding hype bolic esul s by using
he ansmu a ion s a egy (see [32]). O cou se in he N-dimensional case (N≥2), hey assume
he Geome ic Con ol Condi ion (CGC) de ined in [11] on bo h se s ωand Supp q. Clea ly hese
assump ions a e no necessa y in he pa abolic se ing.
The i s sa is ying answe wi hou sign condi ions on qconce ns he null con ollabili y o
sys ems (1.1) and (1.2) when qsa is ies he condi ion
Zπ
0
q(x)dx 6= 0.(1.9)
Unde his condi ion, in [7] he au ho s gi e a necessa y and su icien condi ion o he app oxima e
and null con ollabili y a ime T > 0 o sys em (1.1). As a consequence, hey also ob ain he null
con ollabili y p ope y a any posi i e ime T o (1.2) unde he same condi ions.
The i s gene al esul o he dis ibu ed con ollabili y o sys em (1.2) conce ns he app oxi-
ma e con ollabili y and is p o ed in [14]. Fo gene al open se s ω, he au ho s p o ide a necessa y
and su icien condi ion o he app oxima e con ollabili y o sys em (1.2) in e ms o Supp qand
he connec ed componen s o Ω ω.
Some esul s p esen ed he e ha e been announced in [9]. In ac , in [9] he con ollabili y o
sys em (1.1) and he con ollabili y o sys em (1.2) when he unc ion qand he con ol in e al
ω= (a, b) sa is y he geome ical condi ion
Supp q⊂[0, a] o Supp q⊂[b, π] (1.10)
a e analyzed. Unde he p e ious condi ion (1.10), a minimal ime o bounda y and dis ibu ed
null con ollabili y, T0(q)∈[0,+∞], a ises in such a way ha hese sys ems a e null con ollable
a ime T i T∈(T0(q),∞) and a e no when T∈(0, T0(q)).
In his pape , we a e going o p o ide a comple e answe o he con ollabili y p oblem o
sys em (1.1) and sys em (1.2) wi hou imposing condi ion (1.10) and when he con ol domain is
an in e al, ω= (a, b). Mo e p ecisely, we will analyze he con ollabili y p ope ies o sys em (1.1),
o a gene al unc ion q∈L∞(0, π), and o sys em (1.2), when qsa is ies
Supp q∩ω=∅,(1.11)
i.e., when Supp q⊂[0, a]∪[b, π].
In he sequel, we se ϕk he no malized eigen ec o s o he Di ichle laplacian in (0, π), i.e.,
ϕk(x) = 2
πsin(kx),∀x∈(0, π), k ≥1.
On he o he hand, he co esponding eigen alues a e gi en by k2,k≥1.
Fo any k≥1, we associa e wi h he unc ion q∈L∞(0, π) sa is ying (1.11) he sequences
{Ik(q)}k≥1and {Ii,k(q)}k≥1,i= 1,2, gi en by
I1,k(q) := Za
0
q(x)|ϕk(x)|2dx, I2,k(q) := Zπ
b
q(x)|ϕk(x)|2dx,
Ik(q) := I1,k(q) + I2,k(q) = Zπ
0
q(x)|ϕk(x)|2dx.
(1.12)
Le us p esen ou bounda y con ol esul s, ha is, ou main esul ela ed o sys em (1.1).
Theo em 1.1. Le us conside A0and Bgi en by (1.3)and q∈L∞(0, π), a gi en unc ion.
Then, one has:
5
1. Sys em (1.1)is app oxima ely con ollable a ime T > 0i and only i
Ik(q)6= 0 ∀k≥1.(1.13)
2. Assume ha condi ion (1.13)holds and de ine
e
T0(q) := lim sup −log |Ik(q)|
k2∈[0,∞].(1.14)
Then, i T > e
T0(q)sys em (1.1)is null con ollable a ime T. On he o he hand, i
T < e
T0(q)sys em (1.1)is no null con ollable a ime T.
This esul has been announced in [9].
Rema k 1.2. The app oxima e con ollabili y esul s a ed in Theo em 1.1 does no depend on
he inal ime T: app oxima e con ollabili y o sys em (1.1) a a ime T0>0 is equi alen o he
app oxima e con ollabili y o sys em (1.1) a any ime T > 0. On he o he hand, condi ion (1.13)
cha ac e izes he app oxima e con ollabili y p ope y o sys em (1.1). Thus, (1.13) is a necessa y
condi ion o he null con ollabili y a ime T > 0 o his sys em.
Rema k 1.3. No e ha he sequences {Ii,k(q)}k≥1,i= 1,2, and {Ik(q)}k≥1a e con e gen and
om a simple compu a ion one has:
lim Ik(q) = 1
πZπ
0
q(x)dx, lim I1,k(q) = 1
πZa
0
q(x)dx, lim I2,k(q) = 1
πZπ
b
q(x)dx.
F om his, i eadily ollows ha he sequence {Ik(q)−1}k∈Λis bounded and e
T0(q) = 0 whene e
condi ion (1.9) holds ( o he exp ession o he se Λ, see (1.16)). Obse e ha , unde condi-
ion (1.8) on he unc ion q, (1.13) holds and e
T0(q) = 0. In pa icula , Theo em 1.1 gene alizes
he one-dimensional pa abolic bounda y con ollabili y esul s ob ained in [1] and [34].
Rema k 1.4. We will see in Sec ion 7 ha he e a e unc ions q∈L∞(0, π) such ha e
T0(q)>0
(in ac , e
T0(q) may ake any alue in [0,∞]). In pa icula , Theo em 1.1 implies ha , e en
in a pa abolic se ing, a posi i e ime o con ol may appea and ha , unlike he scala case,
bounda y app oxima e and null con ollabili y a e no equi alen p ope ies in he non-scala case
(see also [10] o a simila esul ). On he o he hand, Theo em 1.1 also in e s nega i e bounda y
con ollabili y esul s o hype bolic e sions o sys em (1.1). Indeed, i q∈L∞(0, π) is such ha
e
T0(q)>0, he ansmu a ion s a egy (see [32]) implies ha he co esponding hype bolic e sion
o (1.1) is no con ollable in he na u al space associa ed o he sys em (see Theo em 3.6 in [1])
a any ime T > 0.
Fo he dis ibu ed con ol p oblem (sys em (1.2)), le us i s ecall a ecen esul on app ox-
ima e con ollabili y:
Theo em 1.2 ([14]).Le us conside A0and Bgi en by (1.3)and q∈L∞(0, π), a unc ion
sa is ying (1.11). Then, sys em (1.2)is app oxima ely con ollable a ime T > 0i and only i
|Ik(q)|+|I1,k(q)| 6= 0 ∀k≥1.(1.15)
Fo he sake o comple eness his esul will be p o ed in Sec ion 4.
Rema k 1.5. As in he bounda y case, he app oxima e con ollabili y esul o sys em (1.2)
does no depend on he inal ime T: sys em (1.2) is app oxima ely con ollable a a ime T0>0
i and only i i is app oxima ely con ollable a any ime T > 0.
6
To s a e ou null con ollabili y esul o sys em (1.2) when q∈L∞(0, π) sa is ies (1.11), we
need some de ini ions and no a ions. Fi s , le us de ine he se s
Λ := {k≥1 : Ik(q)6= 0}= Λ1∪Λ2,
Λ1:= {k∈Λ : I1,k(q)6= 0},Λ2:= {k∈Λ : I1,k(q) = 0}and
Λ3:= {k≥1 : Ik(q) = 0},
(1.16)
whe e Ik(q) and I1,k(q) a e gi en in (1.12). Obse e ha Λ1, Λ2and Λ3a e disjoin se s and, o
cou se, Λ1∪Λ2∪Λ3= Λ ∪Λ3=N∗.
On he o he hand, le us assume ha he unc ion q∈L∞(0, π) is such ha condi ion (1.15)
holds. Thus, we can in oduce he quan i ies −log |I1,k(q)|and −log |Ik(q)|whe e we will use he
no a ion −log |x|=∞when x= 0. Wi h his no a ion and unde assump ion (1.15), we deduce
min{− log |I1,k(q)|,−log |Ik(q)|} ∈ R,∀k≥1.
One has:
Theo em 1.3. Le us conside A0∈L(R2)and B∈R2, gi en by (1.3), and q∈L∞(0, π), a
unc ion sa is ying (1.11). Le us also assume condi ion (1.15), and de ine
T0(q) := lim sup min{− log |I1,k(q)|,−log |Ik(q)|}
k2.(1.17)
Then, gi en T > 0, one has:
1. Assume ha T > T0(q). Then, sys em (1.2)is null con ollable a ime T.
2. I T < T0(q), hen sys em (1.2)is no null con ollable a ime T.
As in he bounda y case, condi ion (1.15) cha ac e izes he dis ibu ed app oxima e con ol-
labili y o sys em (1.2). This implies ha (1.15) is a necessa y condi ion o he dis ibu ed null
con ollabili y a ime T > 0 o (1.2).
We end he p esen a ion o ou main esul s wi h some ema ks.
Rema k 1.6. Unde condi ion (1.15), he minimal ime T0(q) is well-de ined and, aking in o
accoun Rema k 1.3, sa is ies T0(q)∈[0,∞]. We will check in Sec ion 7 ha , gi en he con ol
in e al ω= (a, b), he e a e unc ions q∈L∞(0, π), which ul ill condi ion (1.11), o which
T0(q)>0 and e en T0(q) = ∞. Fo such unc ions, sys em (1.2) is app oxima ely con ollable
a all posi i e ime Tbu i is no null con ollable a ime Ti T∈(0, T0(q)). Again and
unlike he scala case, he dis ibu ed app oxima e p ope y is no equi alen , in gene al, o he
dis ibu ed null con ollabili y p ope y in he non-scala case. Also, ollowing he easoning in
Rema k 1.4, om Theo em 1.3, we deduce ha when q∈L∞(0, π) sa is ies (1.11) and T0(q)>0,
he hype bolic e sion o sys em (1.2) is no con ollable in he na u al space associa ed o he
sys em (see Theo em 3.5 in [1] and De ini ion 1.1 in [17] o he de ini ion o his space) a any
ime T > 0.
Rema k 1.7. Le us ix a unc ion q∈L∞(0, π) and a con ol in e al ω ha sa is ies (1.11).
Taking in o accoun ha condi ion (1.13) implies (1.15) and he inequali y T0(q)≤e
T0(q), he
bounda y con ollabili y a ime T > 0 o sys em (1.1) implies he dis ibu ed con ollabili y a
ime T > 0 o sys em (1.2) p o ided he con ol in e al ωsa is ies (1.11). Bu , i is in e es ing
o no e ha he e exis unc ions q∈L∞(0, π) and con ol in e als ω ul illing condi ion (1.11)
o which T0(q)<e
T0(q) (see Example 7.3). This p o ides ano he di e ence wi h he scala case:
bounda y and dis ibu ed con ollabili y a e no equi alen in he non-scala pa abolic se ing.
Howe e , i ω= (a, b) and q∈L∞(0, π) a e such ha (1.10) holds, hen T0(q) = e
T0(q) and
sys em (1.1) is null con ollable a ime T > 0 i and only i sys em (1.2) is also null con ollable
a ime T.
7
Rema k 1.8. The minimal ime T0(q) (see (1.17)) depends on he unc ion qbu also on he
posi ion o he con ol in e al ω(sa is ying (1.11)). This ac p o ides a new phenomenon in he
amewo k o he dis ibu ed con ollabili y o non-scala pa abolic p oblems: he dependence o
he con ollabili y esul on he posi ion o he con ol se . Indeed, we will also see in Sec ion 7(see
Example 7.3) ha , gi en τ0∈(0,∞] (which could be τ0=∞), he e exis a unc ion q∈L∞(0, π)
and con ol in e als ω1, ω2⊂(0, π), sa is ying (1.11), such ha (1.15) holds, o ω1and ω2, and
T(1)
0(q) = 0 and T(2)
0(q) = τ0>0.
In he p e ious equali ies, T(i)
0(q) is he minimal ime associa ed o he unc ion qand o he
in e al ωi(see (1.17)). In conclusion, sys em (1.2) is null con ollable a e e y posi i e ime T, i
he con ol is exe ed on ω1, bu i is no null con ollable a ime Ti T∈(0, τ0) and he con ol
is exe ed on ω2. This is ano he big di e ence wi h he scala pa abolic case. This dependence
o he zone o con ol was highligh ed in [14], in he case o he app oxima e con ollabili y o
sys em (1.2).
Rema k 1.9. Unde assump ion (1.8) on he unc ion q, condi ions (1.13) and (1.15) hold o e e y
in e al ω⊂(0, π) sa is ying (1.11). This means ha sys ems (1.1) and (1.2) a e app oxima ely
con ollable a any posi i e ime T. In ac , aking in o accoun Rema k 1.3, we ge ha e
T0(q) =
T0(q) = 0 and sys ems (1.1) and (1.2) a e also null con ollable a any posi i e ime T. Thus, ou
esul s eco e he one-dimensional pa abolic e sion o he esul s in [29], [34], [1] and [17], wi h
less es ic i e assump ions on q.
The es o he pape is o ganized as ollows: In Sec ion 2we se and analyze some p elimina y
esul s ela ed o he spec um and he (gene alized) eigenspaces o he ope a o associa ed wi h
sys ems (1.1) and (1.2). Sec ion 3is de o ed o s udying he bounda y con ollabili y p oblem o
sys em (1.1), namely o he p oo o Theo em 1.1. Fo cla i y, his sec ion has been di ided in o wo
subsec ions; in he i s one i can be ound he p oo s conce ning he app oxima e con ollabili y
o sys em (1.1). In Subsec ion 3.2, he null-con ollabili y p ope y o his sys em is p o ed. The
dis ibu ed app oxima e con ollabili y p oblem is conside ed in Sec ion 4. Theo em 1.3 is p o ed
in Sec ions 5( he posi i e null-con ollabili y pa ) and 6(nega i e null-con ollabili y pa ). The
las sec ion con ains some complemen a y esul s and some examples ha illus a e he di e en
si ua ions.
2 Some p elimina y esul s
In his sec ion we will gi e some p ope ies which will be used below. Le us conside he ec o ial
ope a o
L:= −d2
dx2Id +q(x)A0:D(L)⊂L2(0, π;R2)−→ L2(0, π;R2) (2.1)
wi h domain D(L) = H2(0, π;R2)∩H1
0(0, π;R2) and also i s adjoin L∗. We will always deno e
by h·,·i he s anda d scala p oduc o ei he L2(0, π;R) o L2(0, π;R2), by h·,·iX0,X he duali y
pai ing be ween he Hile space Xand i s dual X0.
We a e in e es ed in s udying he spec um o he ope a o s Land L∗. To his end, gi en a
unc ion q∈L∞(0, π), we conside he quan i y Ik(q) gi en by (1.12), k≥1. Wi h his no a ion,
one has:
P oposi ion 2.1. Le A0be gi en by (1.3)and conside he ope a o Lgi en by (2.1)and i s
adjoin L∗. Then,
1. The spec a o Land L∗a e gi en by σ(L) = σ(L∗) = {k2:k≥1}.
2. Gi en k≥1, i
Φ1,k =ϕk
0,Φ2,k =ψk
ϕk,
8
( esp., i
Φ∗
1,k := ϕk
ψk,Φ∗
2,k := 0
ϕk),
whe e ψkis he unique solu ion o he non-homogeneous S u m-Liou ille p oblem:
−ψxx −k2ψ= [Ik(q)−q(x)] ϕkin (0, π),
ψ(0) = 0, ψ(π)=0,
Zπ
0
ψ(x)ϕk(x)dx = 0,
(2.2)
hen,
(L−k2Id)Φ1,k = 0 and (L−k2Id)Φ2,k =Ik(q)Φ1,k (2.3)
( esp., L∗−k2IdΦ∗
1,k =Ik(q)Φ∗
2,k and L∗−k2IdΦ∗
2,k = 0).(2.4)
In pa icula , i k∈Λ hen k2is a simple eigen alue and Φ1,k and Φ2,k ( esp., Φ∗
2,k and
Φ∗
1,k) a e, espec i ely, an eigen unc ion and a gene alized eigen unc ion o he ope a o L
( esp., L∗) associa ed o k2, while i k∈Λ3 hen Φ1,k and Φ2,k a e bo h eigen unc ions o L
( esp., L∗) associa ed o k2.
P oo . Fi s , Lcan be w i en
L=−∆q
0−∆
whe e ∆ = d2
dx2:L2(0, π)−→ L2(0, π) wi h domain D(∆) = H2(0, π)∩H1
0(0, π) is, as is well-
known, boundedly in e ible wi h compac in e se. We can check ha :
L−1=(−∆)−1−(−∆)−1◦q◦(−∆)−1
0 (−∆)−1
which eadily implies ha L−1is a compac ope a o on L2(0, π;R2).Thus, he spec um o L
educes o i s poin spec um.
We ha e now o sol e he eigen alue p oblem:
−y00
1+qy2=λy1in (0, π),
−y00
2=λy2in (0, π),
y1(0) = y2(0) = 0, y1(π) = y2(π) = 0.
I y2≡0, hen, λ=k2is an eigen alue o Land aking y1=ϕkwe ob ain Φ1,k as associa ed
eigen unc ion o L. I we now assume ha y26≡ 0, hen, again λ=k2and y2=ϕkis a (no malized)
solu ion o he second o.d.e. Obse e ha he i s equa ion admi s a solu ion i and only i k∈Λ3,
i.e., Ik(q) = 0. In his case, Φ2,k is a second associa ed eigen unc ion o L. In conclusion, i k∈Λ3,
hen k2is a double eigen alue o L.
F om he abo e conside a ions, i is clea ha i Ik(q)6= 0, hen he eigen alue k2o Lis
simple and Φ1,k is an associa ed eigen unc ion. Obse e ha , aking Φ2,k = (y1, y2), he equa ion
(L−k2Id)Φ2,k =cΦ1,k w i es:
−y00
1−k2y1=cϕk−qy2in (0, π),
−y00
2−k2y2= 0 in (0, π),
y1(0) = y2(0) = 0, y1(π) = y2(π) = 0.
Thus, again, choosing y2=ϕkand inse ing his exp ession in he i s equa ion, we ge o y1:
(−y00
1−k2y1= [c−q]ϕk,
y1(0) = y2(0) = 0
9
3.2.1 Posi i e bounda y con ollabili y esul
Le us assume ha T > e
T0(q)∈[0,∞) (see (1.14)). Ou objec i e is o p o e ha sys em (1.1)
is exac ly con ollable o ze o a ime T. To his end, o y0∈H−1(0, π;R2), we will e o mula e
he null con ollabili y p oblem as a momen p oblem.
Using P oposi ions 3.1 and 3.2, we deduce ha he con ol u∈L2(0, T) d i es he solu ion
o (1.1) o ze o a ime Ti and only i u∈L2(0, T) sa is ies
ZT
0
u( )B∗θx(0, )d =−hy0, θ(·,0)iH−1,H1
0,∀θ0∈H1
0(0, π;R2),
whe e θ∈C0([0, T]; H1
0(0, π;R2)) is he solu ion o he adjoin p oblem (3.1) associa ed wi h θ0.
Since B∗is a basis o H1
0(0, π;R2) (see Co olla y 2.5), he null con ollabili y p ope y a ime T
o sys em (1.1) is equi alen o ind u∈L2(0, T ) such ha
ZT
0
u( )B∗θi,k
x(0, )d =−hy0, θi,k(·,0)iH−1,H1
0,∀k≥1,∀i= 1,2,(3.5)
whe e θi,k is he solu ion o sys em (3.1) associa ed wi h θ0= Φ∗
i,k ( o he exp ession o he
unc ion Φ∗
i,k, see P oposi ion 2.1). Le us ake
u( ) = (T− ), ∈(0, T).
De eloping he equali y (3.5), one has:
1. I we ake θ0= Φ∗
2,k, he solu ion o he adjoin p oblem is θ2,k(·, ) = e−k2(T− )Φ∗
2,k and (3.5)
becomes,
ZT
0
e−k2 ( )d =−1
k π
2e−k2Thy0,Φ∗
2,kiH−1,H1
0:= e−k2T
M(k)
1(y0),∀k≥1.
I is easy o see ha
M(k)
1(y0)≤Cky0kH−1(0,π;R2),∀k≥1,(3.6)
o a posi i e cons an Cindependen o kand y0.
2. Le us now ake θ0= Φ∗
1,k. In his case he solu ion o he adjoin sys em (3.1) is
θ1,k(·, ) = e−k2(T− )Φ∗
1,k −(T− )Ik(q)e−k2(T− )Φ∗
2,k
and, hen, he equali y (3.5) ans o ms in o (u( ) = (T− ), ∈(0, T))
ψ0
k(0) ZT
0
e−k2 ( )d −Ik(q)ϕ0
k(0) ZT
0
e−k2 ( )d
=−e−k2Thhy0,Φ∗
1,kiH−1,H1
0−TIk(q)hy0,Φ∗
2,kiH−1,H1
0i.
Thus, he con ol u= (T− ·) mus also sa is y
ZT
0
e−k2 ( )d =e−k2T
Ik(q)
M(k)
2(y0),∀k≥1,
whe e
M(k)
2(y0) := 1
k π
2nψ0
k(0)
M(k)
1(y0) + hhy0,Φ∗
1,kiH−1,H1
0−TIk(q)hy0,Φ∗
2,kiH−1,H1
0io.
Using he p ope ies o he unc ion ψks a ed in P oposi ion 2.2 (see (2.6)), one has
M(k)
2(y0)≤Cky0kH−1(0,π;R2),∀k≥1.(3.7)
o a new posi i e cons an Cindependen o kand y0.
16
Summa izing, we ha e p o ed ha u∈L2(0, T) is such ha he solu ion yo sys em (1.1)
sa is ies y(·, T) = 0 in (0, π) i and only i =u(T− ·)∈L2(0, T) sa is ies
ZT
0
e−k2 ( )d =e−k2T
M(k)
1(y0),
ZT
0
e−k2 ( )d =e−k2T
Ik(q)
M(k)
2(y0),∀k≥1,
(3.8)
wi h
M(k)
1(y0) and
M(k)
2(y0) sa is ying (3.6) and (3.7).
F om he esul s in [21] (see also [6]), we can conclude ha he sequence
ne1,k := e−k2 , e2,k := e−k2 ok≥1
admi s a bio hogonal amily {q1,k, q2,k}k≥1in L2(0, T ), i.e., a amily {q1,k, q2,k}k≥1in L2(0, T)
sa is ying ZT
0
e ,kqs,j( )d =δkj δ s,∀k, j ≥1,1≤ , s ≤2,(3.9)
which mo eo e sa is ies ha o e e y ε > 0 he e exis s a cons an Cε,T >0 such ha
kqi,kkL2(0,T )≤Cε,T eεk2,∀k≥1, i = 1,2.(3.10)
Using he o mulas in (3.8) and he p ope y (3.9), we in e ha an explici o mal solu ion o
he momen p oblem (3.5) is gi en by
u(T− ) = ( ) = X
k≥1
e−k2T
M(k)
1(y0)q1,k( ) + 1
Ik(q)
M(k)
2(y0)q2,k( ).
Le us see ha his se ies de ines an elemen o L2(0, T) when T > e
T0(q), i.e., he p e ious
se ies con e ges in L2(0, T) i T > e
T0(q). Indeed, om he de ini ion o he minimal ime e
T0(q)
(see (1.14)) and o any ixed ε > 0, we can in e ha he e exis s a posi i e cons an Cεsuch ha
1
|Ik(q)|≤Cεek2(e
T0(q)+ε),∀k≥1.
On he o he hand, we can use he bound (3.10) and ge a new posi i e cons an Cε,T o which
e−k2T
M(k)
1(y0)q1,k +1
Ik(q)
M(k)
2(y0)q2,kL2(0,T )
≤Cε,T
e−k2Teεk2
|Ik(q)|
≤Cε,T e−k2(T−
e
T0(q)−2ε).
This las inequali y p o es he absolu e con e gence o he se ies which de ines he con ol usince
εmay be chosen a bi a ily small. This p o es he null con ollabili y o sys em (1.1) a ime T
when T > e
T0(q).
3.2.2 Nega i e bounda y con ollabili y esul
In o de o inish he p oo o Theo em 1.1, le us p o e ha i 0 < T < e
T0(q), hen sys em (1.1)
is no null con ollable a ime T. Recall ha condi ion (1.13) holds. We a gue by con adic ion.
Assume ha sys em (1.1) is null con ollable a ime T < e
T0(q). By means o P oposi ion 3.3,
his las ac is equi alen o he exis ence o a posi i e cons an Csuch ha he obse abili y
inequali y (3.3) holds o e e y solu ion θo he adjoin p oblem (3.1). Le us wo k wi h he
pa icula solu ions associa ed wi h ini ial da a θ0=akΦ∗
1,k +bkΦ∗
2,k, wi h ak, bk∈R, o be
17
de e mined, and Φ∗
1,k and Φ∗
2,k gi en in P oposi ion 2.1. Wi h his choice, he solu ion θko (3.1)
is gi en by
θk(·, ) = ake−k2(T− )Φ∗
1,k −(T− )Ik(q) Φ∗
2,k+bke−k2(T− )Φ∗
2,k,∀k≥1.
Thus, he obse abili y inequali y (3.3) becomes
A1,k ≤CA2,k,∀k≥1,
wi h
A1,k := e−2k2Tnk2|ak|2+h|ak|2kψ0
kk2
L2(0,π)+k2(bk−TIk(q)ak)2io≥e−2k2Tk2|ak|2,∀k≥1,
and
A2,k := ZT
0
e−2k2 |akψ0
k(0) + (bk− Ik(q)ak)ϕ0
k(0)|2d , ∀k≥1.
Taking ak= 1 and bk=−ψ0
k(0)/ϕ0
k(0) = −1
kpπ
2ψ0
k(0), he inequali y obse abili y ans o ms
in o
e−2k2Tk2≤A1,k ≤CA2,k =C2
π|Ik(q)|2k2ZT
0
2e−2k2 d , ∀k≥1,
ha is o say, o a new cons an C > 0 no depending on k, one has,
1≤Ce2k2T|Ik(q)|2,∀k≥1.(3.11)
F om he de ini ion o e
T0(q), we ob ain he exis ence o an inc easing unbounded subsequence
{kn}n≥1such ha
e
T0(q) = lim
n→∞
−log |Ikn(q)|
k2
n
∈(0,∞].
Assume ha 0 <e
T0(q)<∞( he case e
T0(q) = ∞is much simple and he de ails a e le o he
eade ). In his case, o e e y ε > 0, he e exi s a posi i e in ege nεsuch ha
e
T0(q)−ε≤−log |Ikn(q)|
k2
n
,∀n≥nε.
This las inequali y oge he wi h (3.11) p o ide he new inequali y
1≤Ce−2k2
n(e
T0(q)−T−ε),∀n≥nε.
The p e ious inequali y gi es a con adic ion i we ake 0 <ε<e
T0(q)−T/2. This ends he
p oo .
4 Dis ibu ed app oxima e con ollabili y
In his sec ion we will add ess he p oblem o he app oxima e con ollabili y a ime T > 0 o
sys em (1.2), i.e, we will p o e Theo em 1.2. As said abo e, Theo em 1.2 is a di ec consequence
o he esul s on app oxima e con ollabili y s a ed in [14]. Fo he sake o comple eness we will
p o ide a di ec p oo o he esul .
As in Sec ion 3, we will i s es ablish he ela ion be ween sys em (1.2) and (3.1). On he o he
hand, we will also gi e a gene al cha ac e iza ion o he con ollabili y p ope ies o sys em (1.2).
One has:
18
P oposi ion 4.1. Le us conside A0and Bgi en by (1.3)and q∈L∞(0, π), a gi en unc ion.
Then, o any y0∈L2(0, π;R2), ∈L2(QT)and θ0∈L2(0, π;R2), one has
ZZQT
(x, )1ωB∗θ(x, )dx d =hy(·, T), θ0i−hy0, θ(·,0)i,
whe e y, θ ∈L2(0, T;H1
0(0, π;R2))∩C0([0, T]; L2(0, π;R2)) a e, esp., he solu ions o (1.2)and (3.1)
associa ed o (y0, )and θ0.
Fo a p oo o he p e ious esul see o ins ance [15], [36] o [21].
P oposi ion 4.2. Unde assump ions o P oposi ion 4.1, one has:
1. Sys em (1.2)is app oxima ely con ollable a ime T > 0i and only i he ollowing unique
con inua ion p ope y holds:
“Le θ0∈L2(0, π;R2)be gi en and le θbe he co esponding solu ion o he adjoin p ob-
lem (3.1). Then, i B∗θ= 0 in ω×(0, T), one has θ0≡0in (0, π).”
2. Sys em (1.2)is null con ollable a ime T > 0i and only i he e exis s a posi i e cons an
Csuch ha he obse abili y inequali y
kθ(·,0)k2
L2(0,π;R2)≤CZZω×(0,T )
|B∗θ(x, )|2dx d (4.1)
holds o e e y θ0∈L2(0, π;R2). In (4.1),θis he adjoin s a e associa ed o θ0, i.e., he
solu ion o (3.1)associa ed o θ0.
Again, his esul is e y well known. Fo a p oo see, o ins ance, [37], [15] o [36].
We can al eady p o e Theo em 1.2. The a gumen s will be simila o hose used in Sec ion 3.
We ecall ha q∈L∞(0, π) is a unc ion sa is ying (1.11), whe e ω= (a, b).
Necessa y condi ion: Again, we a gue by con adic ion. Le us suppose ha condi ion (1.15)
does no hold, i.e., ha he e exis s k0≥1 such ha Ik0(q) = I1,k0(q) = 0. We will see ha he
dis ibu ed unique con inua ion p ope y o he adjoin sys em (3.1) ails o be ue.
Fi s , om P oposi ion 2.6, he unc ion ψk0is gi en by:
ψk0(x) = τk0ϕk0(x),∀x∈ω, (4.2)
(since Ik0(q) = I1,k0(q) = 0) whe e τk0is gi en in P oposi ion 2.6.
On he o he hand, le us ake θ0=aΦ∗
1,k0+bΦ∗
2,k0∈L2(0, π;R2), wi h a, b ∈R o be
de e mined. Again, he unc ions Φ∗
1,k0and Φ∗
2,k0a e eigen unc ions o he ope a o L∗(see
P oposi ion 2.1). Thus, he solu ion o he adjoin p oblem (3.1) is gi en by (3.4), so ha :
B∗θ(x, ) = e−k2
0(T− )(aψk0(x) + bϕk0(x)) = e−k2
0(T− )(aτk0+b)ϕk0(x),∀(x, )∈ω×(0, T),
hanks o (4.2). Jus aking a= 1 and b=−τk0we ob ain B∗θ≡0 in ω×(0, T) and θ6≡ 0. This
con adic s he dis ibu ed unique con inua ion p ope y o sys em (3.1). So, sys em (1.2) is no
app oxima ely con ollable a ime T > 0. This p o es he necessa y pa o Theo em 1.2.
Su icien condi ion: Le us assume ha condi ion (1.15) holds. The objec i e is o show
ha sys em (1.2) is app oxima ely con ollable a ime T, when q∈L∞(0, π) sa is ies (1.11).
This amoun s o p o e he dis ibu ed unique con inua ion p ope y o sys em (3.1) s a ed in
P oposi ion 4.2.
Le us ix θ0∈L2(0, π;R2) and assume ha he co esponding solu ion θo (3.1) sa is ies
B∗θ≡0 in ω×(0, T).
Since B∗is a basis o L2(0, π;R2) ( o he exp ession o B∗, see (2.7)), we can w i e
θ0=X
k≥1akΦ∗
1,k +bkΦ∗
2,k,
19
whe e he coe icien s a e gi en by ak=hθ0,Φ1,kiand bk=hθ0,Φ2,ki o any k≥1. As i has
been al eady obse ed, we ha e:
θ(·, ) = X
k≥1
e−k2(T− )akΦ∗
1,k −(T− )Ik(q) Φ∗
2,k+bkΦ∗
2,kin QT.
In ac , ollowing he ideas in Lemma 2.3, i is no di icul o p o e he con e gence o his
se ies in C0([0, T]; L2(0, π;R2)). Thus,
B∗θ(·, )|ω=X
k≥1
e−k2(T− )akB∗Φ∗
1,k|ω+bkB∗Φ∗
2,k|ω−X
k≥1
(T− )e−k2(T− )akIk(q)B∗Φ∗
2,k|ω
=X
k≥1
e−k2(T− )[akψk|ω+bkϕk|ω]−X
k≥1
(T− )e−k2(T− )akIk(q)ϕk|ω
o any ∈(0, T). Using again ha he amily {e−k2 , e−k2 }k≥1⊂L2(0, T) is minimal in L2(0, T)
and he assump ion B∗θ≡0 in ω×(0, T) we ge
akψk|ω+bkϕk|ω≡0 and akIk(q)ϕk|ω≡0∀k≥1
I is clea ha om he p e ious iden i ies ha ak=bk= 0 o all k∈Λ. On he o he hand,
aking in o accoun he exp ession o he ψkin ω(see P oposi ion 2.6), he las equali y becomes
(akτk+bk)ϕk(x)− π
2
I1,k(q)
kakcos(kx)=0 ∀x∈ω, ∀k∈Λ3.
Using he independence o ϕkand he unc ion cos(k·) in ω, we conclude ha ak=bk= 0 o e e y
k∈Λ3. This p o es ha θ0≡0. The e o e, we ha e p o ed he dis ibu ed con inua ion p ope y
o he solu ions o he adjoin p oblem (3.1) and he app oxima e con ollabili y o sys em (1.2)
a any posi i e ime T.
5 P oo o Theo em 1.3: The posi i e null con ollabili y
esul
This sec ion will be de o ed o p o ing he null con ollabili y o sys em (1.2) a ime T > 0, when
his ime sa is ies T > T0(q) (T0(q), gi en by (1.17), is assumed o be ini e in his sec ion). In
o de o make he p oo clea e , we will di ide i in o se e al s eps.
5.1 The momen p oblem
We s a he p oo o he i s poin o Theo em 1.3 by e o mula ing he null con ollabili y
p ope y o sys em (1.2) as a momen p oblem. To his end, le us conside T > T0(q) (T0(q)
is gi en by (1.17)). The aim is o p o e ha o any y0∈L2(0, π;R2) he e exis s a con ol
∈L2(QT) such ha he co esponding solu ion yo sys em (1.2) sa is ies y(·, T) = 0 in (0, π).
Le us ix an ini ial da um y0∈L2(0, π;R2). Thanks o P oposi ion 3.1 and 4.1, i is easy o
see ha he solu ion y∈C0([0, T]; L2(0, π;R2)) o sys em (1.2) associa ed wi h y0and a con ol
∈L2(QT) sa is ies y(·, T) = 0 in (0, π) i and only i he con ol ∈L2(QT) sa is ies
ZZQT
(x, )1ωB∗θ(x, )dx d =− hy0, θ(·,0)i,∀θ0∈L2(0, π;R2),
whe e θis he solu ion o he adjoin p oblem (3.1) co esponding o θ0. Using ha B∗is a basis
o L2(0, π;R2) (see Lemma 2.3) his las p ope y is equi alen o ∈L2(QT) and sa is ies
ZZQT
(x, )1ωB∗θi,k(x, )dx d =− hy0, θi,k(·,0)i,∀k≥1,∀i= 1,2,(5.1)
20
whe e θi,k deno es he solu ion o sys em (3.1) associa ed wi h θ0= Φ∗
i,k. By means o he
p e ious p oblem we ha e e o mula ed he null con ollabili y p ope y o sys em (1.2) as a
momen p oblem.
In o de o sol e he momen p oblem (5.1), he i s main idea is o sea ch con ols unde he
pa icula o m
(x, ) = 1(x) 1(T− ) + 2(x) 2(T− ),(x, )∈QT,(5.2)
whe e 1, 2∈L2(0, T) a e new con ols, only depending on , and 1, 2∈L2(0, π) a e app op ia e
unc ions sa is ying he condi ion Supp 1,Supp 2⊆ω= (a, b). This choice will be made clea e
a li le u he in he ex .
Fo k≥1 and θ0= Φ∗
2,k, he solu ion o (3.1) is gi en by θ2,k(·, ) = e−k2(T− )Φ∗
2,k. Thus, a e
a change o a iables, he momen p oblem (5.1) wi h con ols gi en by (5.2) eads as ollows:
1,k ZT
0
1( )e−k2 d + 2,k ZT
0
2( )e−k2 d =−e−k2Ty0,Φ∗
2,k,
whe e 1,k, 2,k a e, espec i ely, he Fou ie coe icien s wi h espec o ϕkco esponding o 1,
2:
i,k := Zπ
0
i(x)ϕk(x)dx, i = 1,2,∀k≥1.(5.3)
Fo θ0= Φ∗
1,k, he co esponding solu ion o (3.1) is gi en by
θ(·, ) = e−k2(T− )Φ∗
1,k −(T− )Ik(q) Φ∗
2,k.
F om he exp ession o unc ions Φ∗
i,k (see he s a emen o P oposi ion 2.1), o k≥1 and i= 1,
he equali y (5.1) wi h con ols gi en by (5.2) changes in o
e
1,k ZT
0
1( )e−k2 d +e
2,k ZT
0
2( )e−k2 d
−Ik(q) 1,k ZT
0
1( ) e−k2 d −Ik(q) 2,k ZT
0
2( ) e−k2 d
=−e−k2Ty0,Φ∗
1,k−TIk(q)y0,Φ∗
2,k,
whe e, o k≥1, e
1,k,e
2,k a e gi en by
e
i,k := Zπ
0
i(x)ψk(x)dx, i = 1,2.(5.4)
Le us poin ou ha , hanks o he p ope ies o he unc ion ψk(see (2.6)), one has
e
i,k≤C
k, i = 1,2,i k≥1,(5.5)
o some posi i e cons an C.
Summa izing, we ha e ans o med he null-con ollabili y p oblem a ime T > 0 o sys-
em (1.2) in o he ollowing momen p oblem: Find ∈L2(QT) unde he o m (5.2) such ha
1, 2∈L2(0, T) sa is y
1,k ZT
0
1( )e−k2 d + 2,k ZT
0
2( )e−k2 d =−e−k2Ty0,Φ∗
2,k
e
1,k ZT
0
1( )e−k2 d +e
2,k ZT
0
2( )e−k2 d
−Ik(q) 1,k ZT
0
1( ) e−k2 d −Ik(q) 2,k ZT
0
2( ) e−k2 d
=−e−k2Ty0,Φ∗
1,k−TIk(q)y0,Φ∗
2,k,
k≥1,(5.6)
21
wi h he no a ions in (5.3) and (5.4).
Ou objec i e is o sol e he p e ious momen p oblem unde he assump ion (1.15) and when
T > T0(q) (see (1.17)). To his end, we will cons uc app op ia e unc ions 1, 2∈L2(0, π)
sa is ying Supp 1,Supp 2⊆ω= (a, b). Le us ema k ha , i we ix k≥1, (5.6) is a linea
sys em o wo equa ions and ou unknown quan i ies:
ZT
0
1( )e−k2 d , ZT
0
2( )e−k2 d , ZT
0
1( ) e−k2 d and ZT
0
2( ) e−k2 d .
The momen p oblem (5.6) can be w i en as
AkVk+e
Ake
Vk=Fk∀k≥1,(5.7)
whi h o k≥1 :
Ak= 1,k 2,k
e
1,k e
2,k !,e
Ak= 0 0
−Ik(q) 1,k −Ik(q) 2,k !(5.8)
Vk:=
ZT
0
1( )e−k2 d
ZT
0
2( )e−k2 d
,e
Vk:=
ZT
0
1( ) e−k2 d
ZT
0
2( ) e−k2k2 d
,(5.9)
and
Fk=
−e−k2TDy0,Φ∗
2,kE
−e−k2TDy0,Φ∗
1,kE−TIk(q)Dy0,Φ∗
2,kE
.(5.10)
Remind ha i,k is he Fou ie coe icien o iwi h espec o ϕkand e
i,k is gi en by (5.4).
5.2 Cons uc ion o he unc ions 1and 2
In his subsec ion we will cons uc app op ia e unc ions 1, 2∈L2(0, T) sa is ying
Supp 1,Supp 2⊆ω,
which will allow us o sol e he momen p oblem (5.7) . One has:
Lemma 5.1. The e exis unc ions 1, 2∈L2(0, π)sa is ying Supp 1,Supp 2⊆ωand such ha
min {| 1,k|,| 2,k|} ≥ C
k3,∀k≥1,
|Bk|:= 1,k b
2,k − 2,k b
1,k≥C
k5,∀k≥1.
(5.11)
In (5.11)Cis a posi i e cons an only depending on 1and 2, i,k (i= 1,2) is he Fou ie
coe icien o he unc ion iwi h espec o ϕkand b
i,k is gi en by
b
i,k =Zπ
0
i(x) cos(kx)dx, k ≥1, i = 1,2.(5.12)
P oo . Le us conside he unc ions 1:= 1(a1,b1)and 2:= 1(a2,b2)wi h a1, b1, a2, b2∈ωand
ai< bi,i= 1,2. Then,
i,k =Zπ
0
i(x)ϕk(x)dx =2
k 2
πsin kai+bi
2sin kbi−ai
2,
b
i,k =Zπ
0
i(x) cos(kx)dx =2
kcos kai+bi
2sin kbi−ai
2.
22
Di ec compu a ions show ha
|Bk|=4
k2 2
πsin kb1−a1
2sin kb2−a2
2sin ka1+b1−a2−b2
2.
Le us now ake b1=a1+ 2`,a2=a1+`and b2=a1+ 3`, wi h a1∈(a, (3a+b)/4) and
`∈(0,(b−a)/4) such ha a1/π is a a ional numbe and `/π is an i a ional algeb aic numbe
o o de 2. In his case, we ha e ha (a1+`)/π and (a1+ 2`)/π a e also i a ional algeb aic
numbe s o o de 2. Thus, a1, b1, a2, b2∈ωand ai< bi,i= 1,2. On he o he hand, le us
admi he ollowing p ope y which will be p o ed below: i ξ/π ∈(0,∞) is an i a ional algeb aic
numbe o o de 2, hen
in
k≥1(k|sin(kξ)|)≥C, (5.13)
o a posi i e cons an Conly depending on ξ.
Coming back o he exp essions o 1,k, 2,k and |Bk|and aking in o accoun he p e ious
p ope y, one ob ains
| 1,k|=2
k 2
π|sin (k(a1+`))| |sin (k`)| ≥ C1
k3,
| 2,k|=2
k 2
π|sin (k(a1+ 2`))| |sin (k`)| ≥ C2
k3,
|Bk|=4
k2 2
π|sin (k`)|3≥C3
k5,∀k≥1,
wi h C1,C2and C3posi i e cons an s only depending on a1and `. This p o es (5.11).
Le us inalize he p oo showing inequali y (5.13). This inequali y is a consequence o Liou-
ille’s heo em on diophan ine app oxima ion:
Lemma 5.2 ([30]).Le νbe an i a ional algeb aic numbe o deg ee n≥2, i.e., νis an i a ional
numbe which is he oo o a polynomial o deg ee nwi h in ege coe icien s. Then, he e exis s
a posi i e numbe C, depending on ν, such ha
ν−p
q>C
qn,∀p, q ∈N∗, q > 0.
Le us conside ξ > 0 such ha ξ/π is an i a ional algeb aic numbe o deg ee 2 and le us
see inequali y (5.13). Fi s , o any k≥1 he e exis s hk∈N∗such ha
kξ
π−hk≤1
2,∀k≥1.
Indeed, we can ake hk=bkξ/πci kξ/π − bkξ/πc ≤ 1/2 o hk=bkξ/πc+ 1 o he wise (b·c is he
loo unc ion, i.e., o x∈R,bxcgi es he la ges in ege less han o equal o x).
I we now apply Lemma 5.2 wi h ν=ξ/π,n= 2, q=kand p=hkwe ge
Cπ
k≤ |kξ −hkπ| ≤ π
2,∀k≥1,
and
k|sin (kξ)|=k|sin (kξ −hkπ)|=ksin |kξ −hkπ| ≥ ksin Cπ
k≥2C, ∀k≥1.
In he las inequali y we ha e used
sin x
x≥2
π,∀x∈(0, π/2].
This p o es inequali y (5.13).
23
As a consequence o he p e ious esul , we also ha e:
Co olla y 5.3. Le us conside he unc ions 1and 2p o ided by Lemma 5.1 and he associa ed
ma ix Akgi en in (5.8). Then, he e exis s posi i e cons an s C1and C2(only depending on 1
and 2) such ha
|de Ak| ≥ C1
|I1,k(q)|
k6−C2
|Ik(q)|
k,∀k≥1.(5.14)
P oo . Le k≥1. We ha e (see (5.8))
de Ak= 1ke
2,k − 2,k e
1,k,
whe e 1kand 2ka e he Fou ie coe icien s o 1and 2and whe e e
1,k and e
2,k a e gi en
by (5.4). Using P oposi ion 2.6 and aking in o accoun ha Supp i⊂ω, one ge s
e
i,k =τk i,k +Zπ
0
i(x)gk(x)dx.
So
de Ak= 1,k Zπ
0
2(x)gk(x)dx − 2,k Zπ
0
1(x)gk(x)dx.
Using again P oposi ion 2.6,gkcan be w i en as
gk(x) = −Ik(q)
kZx
0
sin(k(x−ξ))ϕk(ξ)dξ − π
2
I1,k(q)
kcos(kx),∀x∈ω, ∀k≥1.
We deduce hen ha
de Ak=− π
2
I1,k(q)
k 1,k b
2,k − 2,k b
1,k−Ik(q)
k( 1,kG2,k − 2,kG1,k), k ≥1,
whe e b
i,k is gi en in (5.12) and
Gi,k =Zπ
0Zx
0
i(x) sin(k(x−ξ))ϕk(ξ)dξ dx,
o i= 1,2 and k≥1. Finally, om (5.11) and using ha he sequence {Gi,k}k≥1(i= 1,2) is
bounded, we deduce (5.14) o k≥1. This ends he p oo .
5.3 Sol ing he momen p oblem
We will de o e his subsec ion o sol ing he momen p oblem (5.7) when T > T0(q) (T0(q), gi en
by (1.17), is assumed o be ini e in his sec ion). To his end, we will wo k wi h he unc ions 1
and 2p o ided by Lemma 5.1 and Co olla y 5.3.
Theo em 5.4. Le y0∈L2(0, π;R2)be gi en and le us conside he momen p oblem (5.7).
Then, we can ind a solu ion o his p oblem unde he o m
ZT
0
i( )e−k2 d =e−k2TM(k)
1,i (y0),
ZT
0
i( ) e−k2 d =e−k2TM(k)
2,i (y0),
(5.15)
whe e he quan i ies M(k)
i,j (y0)∈R, wi h k≥1and 1≤i, j ≤2, sa is y he ollowing p ope y: o
any ε > 0 he e exis s a posi i e cons an Cε(only depending on ε) such ha
M(k)
i,j (y0)≤Cεek2(T0(q)+2ε)ky0kL2(0,π;R2),∀k≥1,1≤i, j ≤2.(5.16)
24
In he sequel, le us ix ε > 0. F om he de ini ion o he minimal ime T0(q), we can in e he
exis ence o a posi i e in ege kε o which
min {− log |I1,k(q)|,−log |Ik(q)|}
k2< T0(q) + ε, ∀k > kε.(5.17)
In o de o ind a solu ion o he momen p oblem (5.7) unde he o m (5.15), we a e going
o dis inguish i kbelongs o he se Λ1, he se Λ2o he se Λ3(see (1.16)).
5.3.1 The case k∈Λ1
Le us s a sol ing he momen p oblem (5.7) when k∈Λ1( o he de ini ion o Λ1, see (1.16)).
1. Le us i s conside k∈Λ1wi h k≤kε. Thanks o Lemma 5.1 (see (5.11)) we can deduce
ha 1,k 2,k 6= 0 o any k≥1. In his case, we sol e he momen p oblem (5.7) as ollows. Take
ZT
0
2( )e−k2 d =ZT
0
2( ) e−k2 d = 0,∀k∈Λ1, k ≤kε.
Wi h his choice, sys em (5.7) is equi alen o
1,k ZT
0
1( )e−k2 d =F(1)
k,
e
1,k ZT
0
1( )e−k2 d −Ik(q) 1,k ZT
0
1( ) e−k2 d =F(2)
k,
(5.18)
wi h k∈Λ1,k≤kεand whe e F(i)
k,i= 1,2, a e he componen s o Fk(see (5.10)). Obse e ha
in he se Λ1one has Ik(q)6= 0. The e o e, he p e ious p oblem can be sol ed as in he bounda y
case (see Sec ion 3.2) ob aining a solu ion unde he o m (5.15), o any k∈Λ1wi h k≤kε. In
pa icula , M(k)
1,2(y0) = M(k)
2,2(y0) = 0.
Using he p ope ies o e
i,k (see (5.5)) and aking in o accoun ha k∈Λ1and k≤kε, we
deduce he exis ence o a posi i e cons an Cεsuch ha
M(k)
i,j (y0)≤Cεky0kL2(0,π;R2),∀k∈Λ1, k ≤kε,1≤i, j ≤2.(5.19)
As a consequence, we ge inequali y (5.16) o any k∈Λ1, wi h k≤kε.
2. Le us now deal wi h he case k∈Λ1and k > kε. As be o e, ou objec i e is o sol e he
momen p oblem (5.7). To his end, o k > kε, le us spli he se Λ1in o wo subse s
Λ?
1,ε := k∈Λ1:k > kεand −1
k2log |Ik(q)| ≤ T0(q) + 3
2ε,
Λ1,ε := k∈Λ1:k > kεand −1
k2log |Ik(q)|> T0(q) + 3
2ε.
I k∈Λ?
1,ε, hen we eason as in he p e ious case. We ake
ZT
0
2( )e−k2 d =ZT
0
2( ) e−k2 d = 0,∀k∈Λ?
1,ε,
and he momen p oblem (5.7) is equi alen o (5.18), wi h k∈Λ?
1,ε. Again, we can compu e he
solu ion o his sys em, which is gi en by (5.15) (k∈Λ?
1,ε), whe e M(k)
1,2(y0) = M(k)
2,2(y0) = 0 and
M(k)
1,1(y0) = −1
1,k y0,Φ∗
2,k,
M(k)
2,1(y0) = −1
1,kIk(q) y0,Φ∗
1,k−TIk(q)y0,Φ∗
2,k+e
1,k
1,k y0,Φ∗
2,k!.
25
Le us also in oduce he unc ion q:
q(x) := (1 i x∈[a1, a1+`π],
−1 i x∈[a2+`π, a2+`π].
Wi h his unc ion q, he objec i e is o analyze he dependence o he minimal ime o null
con ollabili y o sys em (1.2) on he posi ion o he con ol open se ω= (a, b)⊂(0, π). To his
end, we will conside h ee di e en si ua ions:
1. Supp q∩ω6=∅: In his case, sys em (1.2) is a pa icula case o sys em (1.4) (C≡0) whe e
he coe icien a12 =qsa is ies condi ion (1.5) wi h σ= 1 and ω0could be a connec ed
componen o he in e io o he se Supp q∩ω6=∅. F om e y well-known esul s (see o
ins ance [35], [25] o [26]), we deduce ha sys em (1.2) is null con ollable a ime T o any
T > 0, ha is o say, he minimal ime o dis ibu ed null con ollabili y is ze o: T0(q) = 0.
2. a1+`≤a<b≤a2: In his case, condi ion (1.11) holds and i is easy o show (see (1.12))
I1,k(q) = 1
π`−1
ksin (k`π) cos (k(2a1+`π)),
I2,k(q) = −1
π`−1
ksin (k`) cos k2a1+3
2`π,
Ik(q) = I1,k(q) + I2,k(q) = −2
kπ sin (k`π) sin (k(a1+a2+`π)) sin (k(a2−a1))
=−2
kπ sin (k`π) sin (kα1π) sin (kα2π).
Thanks o he assump ion on α1,α2and `, we deduce ha Ik(q)6= 0 o any k≥1 and
q ul ills condi ion (1.15). Since ` > 0, we also ob ain he exis ence o k0≥1 such ha
|I1,k(q)|>|Ik(q)| o all k≥k0. The e o e (see (1.17)),
T0(q) = lim sup −log |I1,k(q)|
k2= 0.
In conclusion, unde he p e ious geome ical si ua ion, one ob ains ha sys em (1.2) is
app oxima ely and null con ollable a any posi i e ime T. Obse e ha he null con olla-
bili y p ope y o sys em (1.2) is independen o he diophan ine app oxima ion p ope ies
o he i a ional numbe `.
3. 0 ≤a < b ≤a1o a2+`≤a<b≤π: In his case, condi ion (1.11) also holds. Le us
analyze he case 0 ≤a<b≤a1. An analogous esul can be ob ained in he case a2+`≤
a < b ≤π. Wi h he p e ious choice, I1,k(q) = 0,
Ik(q) = −2
kπ sin (k`π) sin (kα1π) sin (kα2π),
and
T0(q) = lim sup −log |Ik(q)|
k2=e
T0(q).
Again, we will use he p ope ies o i a ional algeb aic numbe s p o ed be o e. To be p ecise,
as a consequence o inequali y (5.13) applied o α1πand α2π, we deduce he exis ence o
wo posi i e cons an s C1and C2such ha
−log 2
kπ −log |sin (k`π)|≤−log |Ik(q)|≤−log 2C1C2
k3π−log |sin (k`π)|,∀k≥1.
As a consequence,
T0(q) = lim sup −log |sin (k`π)|
k2,
32
and he minimal ime o null con ollabili y o sys em (1.2) depends on he diophan ine
app oxima ion p ope ies o he i a ional numbe `. Thanks o Lemma 7.1, gi en τ0∈
[0,∞], he e is `∈(0,1/5) sa is ying (7.1), ha is o say, he e is `∈(0,1/5) such ha
T0(q) = τ0. In con as wi h he geome ical si ua ion in i em 2, in he cu en case, he null
con ollabili y p ope y o sys em (1.2) s ongly depends on he diophan ine app oxima ion
p ope y o he i a ional numbe `.
Summa izing, wi h his example we ha e shown ha , gi en a unc ion q∈L∞(0, π), he null
con ollabili y p ope y o sys em (1.2) is di e en when he unc ion qand he con ol in e al
ωsa is y Supp q∩ω6=∅o Supp q∩ω=∅. Bu e en in his las case, i.e., in he case in which
condi ion (1.11) holds, he dis ibu ed null con ollabili y esul depends on he ela i e posi ion
o he se Supp qand he con ol in e al ω. Fo he same unc ion qand he same non-scala
pa abolic p oblem, we can ind con ol in e als sa is ying (1.11) o which he minimal ime o
null con ollabili y can be ze o and i we mo e he con ol in e al (s ill sa is ying (1.11)) he
minimal ime is posi i e o e en ∞. This phenomenon is e y well-known in he amewo k o he
con ollabili y o hype bolic p oblems bu , o ou knowledge, is new in he pa abolic amewo k.
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A P oo o Lemma 7.1
We will ob ain he p oo o Lemma 7.1 as a consequence o Lemma 5.2 and he esul :
Lemma A.1. 1. Le us ixed τ0∈(0,∞),x0∈[0,∞)and ε > 0. Then, he e exis an
i a ional numbe ν > 0and a sequence o a ional numbe s {pk/qk}k≥1such ha pkand qk
a e co-p ime posi i e in ege s, he sequences {pk}k≥1and {qk}k≥1a e s ic ly inc easing,
|ν−x0| ≤ εand lim eτ0q2
kν−pk
qk= 1.(A.1)
Mo eo e , o any k≥1one has
0<|qkν−pk|≤|qν −p|,∀p, q ∈N∗,wi h q < qk+1.(A.2)
2. Fo any σ∈(0,∞),x0∈[0,∞)and ε > 0, he e exis s an i a ional numbe ν > 0and a
sequence o a ional numbe s {pk/qk}k≥1such ha pkand qka e co-p ime posi i e in ege s,
he sequences {pk}k≥1and {qk}k≥1a e s ic ly inc easing and
|ν−x0| ≤ εand lim eq2+σ
kν−pk
qk= 0.(A.3)
The p e ious esul has been p o ed in [10] (see Lemma 6.22, Co olla y 6.25 and Appendix A).
Le us ix x0≥0 and ε > 0. In o de o p o e Lemma 7.1 we will use some ideas om [10].
We will di ide he p oo o Lemma 7.1 in o h ee di e en cases:
35
Case τ0= 0. Gi en x0≥0 and ε > 0, le us ake ν∈[x0−ε, x0+ε] a posi i e i a ional algeb aic
numbe o o de 2. F om (5.13) applied o ξ=νπ, we deduce he exis ence o a posi i e cons an
Csuch ha
|sin (kνπ)| ≥ C
k,∀k≥1.
Thus,
lim sup −log |sin (kνππ)|
k2≤lim sup −log (C/k)
k2= 0.
Taking in o accoun ha he p e ious limi supe io is always nonnega i e, we deduce (7.1). This
p o es he esul o τ0= 0.
Case τ0∈(0,∞). Gi en x0≥0, ε > 0 and τ0, we can apply he i s i em in Lemma A.1 and
conclude he exis ence o an i a ional numbe ν∈[x0−ε, x0+ε] sa is ying (A.1) and (A.2) o he
sequences o posi i e in ege s {pk}k≥1and {qk}k≥1. Wi h his choice we deduce ha νsa is ies
lim pk/qk=ν,
lim 1
qk
eτ0q2
k|νqk−pk|= 1 and
0<|νqk−pk|≤|νq −p|,∀p, q ∈N∗, q < qk+1.
(A.4)
Le us see ha he p e ious numbe νsa is ies (7.1).
F om he i s equali y in (A.4) we deduce lim |νqk−pk|= 0 and
e
T0(q) = lim sup −log |sin (νkπ)|
k2≥lim sup −log |sin (νqkπ)|
q2
k
= lim sup −log |sin [π(νqk−pk)]|
q2
k
= lim −log [π|νqk−pk|]
q2
k
= lim
−log πqke−τ0q2
k
q2
k
=τ0.
Then e
T0(q)≥τ0. Obse e ha he p e ious easoning also implies he exis ence o he ollowing
limi :
lim −log |sin [π(νqk−pk)]|
q2
k
=τ0.
Le us now p o e he inequali y e
T0(q)≤τ0. To his end, le us ix ε > 0. F om he p e ious
p ope y, he e exis s k0(ε)≥1 such ha
−log |sin [π(νqk−pk)]|
q2
k
≤τ0+ε, ∀k≥k0(ε).(A.5)
As in he p oo o Lemma 5.2, o e e y n≥1 he e is hn∈N∗ o which
|νn −hn| ≤ 1
2,∀n≥1.
Le us ake n0(ε) = qk0(ε)≥1. Thus, using ha he sequence {qk}k≥1is s ic ly inc easing,
i n≥n0(ε), we in e he exis ence o k≥k0(ε) such ha qk≤n<qk+1. These las p ope ies
oge he wi h he second o mula in (A.4) allow us o w i e
0<|νqk−pk| ≤ |νn −hn| ≤ 1
2,
and (|sin (νnπ)|=|sin (νnπ −hnπ)|= sin |νnπ −hnπ|
≥sin |νqkπ−pkπ|=|sin (νqkπ−pkπ)|,∀n≥n0(ε) : qk≤n<qk+1.
Since qk≤n<qk+1, he p e ious inequali y and (A.5) gi e
−log |sin (νnπ)|
n2≤−log |sin [π(νqk−pk)]|
q2
k
≤τ0+ε, ∀n≥n0(ε) : qk≤n<qk+1,
36
and
lim sup −log |sin (νnπ)|
n2≤τ0+ε∀ε > 0.
In conclusion, we ha e ob ained (7.1). This p o es Lemma A.1 when τ0∈(0,∞).
Case τ0=∞. Fo τ0=∞,x0≥0 and ε > 0, we apply he second i em in Lemma A.1 wi h,
o ins ance, σ= 1/2. We deduce he exis ence o a posi i e i a ional numbe νwhich ul ills
p ope y (A.2). Repea ing he a gumen s o he p e ious poin we deduce lim |νqk−pk|= 0 and
(σ= 1/2)
e
T0(q) = lim sup −log |sin (νkπ)|
k2≥lim sup −log |sin (νqkπ)|
q2
k
= lim sup −log |sin [π(νqk−pk)]|
q2
k
= lim −log [π|νqk−pk|]
q2
k
= lim
−log πqke−q2+σ
k
q2
k
=∞.
This shows ha e
T0(q) = ∞and inishes he hi d case and he p oo o Lemma 7.1.
37