SIAM J. CONTROL OPTIM.c
2005 Socie y o Indus ial and Applied Ma hema ics
Vol. 43, No. 5, pp. 1616–1639
CONTROLS INSENSITIZING THE OBSERVATION OF A
QUASI-GEOSTROPHIC OCEAN MODEL∗
ENRIQUE FERN´
ANDEZ-CARA†, GALINA C. GARCIA‡,AND AXEL OSSES§
Abs ac . We conside a linea quasi-geos ophic ocean model wi h pa ially known ini ial
condi ions. We sea ch o con ols ha make he obse a ion locally insensi i e o he pe u ba ions
o he ini ial da a. Thei exis ence is equi alen o he null con ollabili y p ope y o an associa ed
cascade S okes-like sys em. Thanks o he p esence o he Co iolis e m, we a e able o p o e he
exis ence o such con ols. Ou s a egy is he ollowing. Fi s , we p o e a unique con inua ion
p ope y o he adjoin o he s a e sys em ha leads o app oxima e con ollabili y; hen, unde
ce ain assump ions, an obse abili y inequali y is es ablished o he adjoin . The p oo is inspi ed
by he a gumen s leading o he unique con inua ion p ope y. This inequali y leads o he desi ed
null con ollabili y esul .
Key wo ds. insensi izing con ols, Ca leman inequali ies, unique con inua ion, null con olla-
bili y, ocean model
AMS subjec classifica ions. 93B05, 35B37, 35B60, 35Q30
DOI. 10.1137/S0363012903433607
1. In oduc ion and main esul s.
1.1. Incomple e ini ial da a ocean model. Le Ω be a nonemp y open
bounded and connec ed subse o R2, wi h bounda y Γ o class C2and ou wa ds uni
no mal ec o ν=ν(x). Le ωbe a nonemp y open subse o Ω, T>0, Q=Ω×(0,T),
and Σ = Γ ×(0,T). In his pape , we will conside a linea quasi-geos ophic ocean
model [1, 15, 16] desc ibed by he ollowing equa ions:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
u −A∆u+γu +( 0+βx2)k∧u+1
ρ0
∇p=T+h1ωin Q,
di u=0 inQ,
u= 0 on Σ,
u(0) = u0+τu0in Ω,
(1.1)
whe e u(x, ) and p(x, ), espec i ely, deno e he eloci y and he p essu e o he
fluid a (x, )=(x1,x
2, )∈R2×R+. In his model, A ep esen s he ho izon al eddy
iscosi y coefficien , γis he bo om ic ion coefficien , ρ0is he fluid densi y, and
( 0+βx2)k∧uis he Co iolis e m, wi h k∧u=(−u2,u
1). In he igh -hand side, 1ω
deno es he cha ac e is ic unc ion o ωand Tis a gi en sou ce in L2(Q)2. The e m
∗Recei ed by he edi o s Augus 22, 2003; accep ed o publica ion (in e ised o m) July 1, 2004;
published elec onically Ma ch 11, 2005.
h p://www.siam.o g/jou nals/sicon/43-5/43360.h ml
†Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do.
1160, 41080 Se illa, Spain (ca a@nume .us.es). This au ho ’s wo k was pa ially suppo ed by
D.G.E.S. (Spain) g an s BFM2000-1317 and BFM2003-06446.
‡Facul ad de Ingenie ´ıa, Uni e sidad Ca ´olica de la San ´ısima Concepci´on, Casilla 297, Con-
cepci´on, Chile ([email p o ec ed]). This au ho ’s wo k was suppo ed by FONDAP in Applied Ma he-
ma ics, CONICYT Ph.D. g an s, and CONICYT-INRIA coope a ion ag eemen s (Chile).
§Depa amen o de Ingenie ´ıa Ma em´a ica, Uni e sidad de Chile, Casilla 170/3 Co eo 3, San-
iago, Chile, and Cen o de Modelamien o Ma em´a ico, UMI 2807/Uni e sidad de Chile-CNRS,
San iago, Chile ([email p o ec ed]hile.cl). This au ho ’s wo k was pa ially suppo ed by FONDAP
in Applied Ma hema ics, FONDECYT-CONICYT 1030808-7030059, and ECOS-CONICYT C01E02
g an s (Chile).
1616
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CONTROLS INSENSITIZING AN OCEAN MODEL 1617
τu0, whe e τ∈R, ep esen s a small unknown pe u ba ion o he ini ial eloci y field
u0, and h=h(x, ) is a con ol unc ion o be de e mined.
No ice ha he Co iolis o ce is ep esen ed by a ze o o de coupling e m in he
equa ions. I in oduces a diffe en beha io o he sys em depending on he di ec ion
in space. To simpli y he p esen a ion o he esul s, we will assume ha A=1,γ=1,
0=1,β= 1, and ρ0=1.
We in oduce he ollowing spaces, which a e usual in he analysis o S okes sys-
ems:
H={ ∈L2(Ω)2: di = 0 in Ω, ·ν= 0 on Γ},
V={ ∈H1
0(Ω)2: di = 0 in Ω},W=H2(Ω)2∩V.
Recall ha
W→V→H≡H→V→W,
whe e he embeddings a e dense and compac .
Fo any gi en u0,τu0∈Hwi h u00,Ω=1,anyT∈L2(Q)2, and any h∈
L2(ω×(0,T))2, he linea sys em (1.1) possesses a unique solu ion (u, p), wi h u∈
L2(0,T;V)∩H1(0,T;V) and p∈W−1,∞(0,T;L2(Ω)). (pis unique up o an addi i e
dis ibu ion only depending on .) This is easily p o ed by adap ing he a gumen s o
[17] o he p esence o a skew-symme ic Co iolis e m in he equa ions. No ice ha i
we had u0+τu0∈V, hen he couple (u, p) would sa is y u∈L2(0,T;W)∩H1(0,T;H)
and p∈L2(0,T;H1(Ω)).
We will be conce ned wi h he sea ch o con ols such ha he eloci y measu e-
men s o e an obse a ion se a e ei he insensi i e o almos insensi i e o small
a ia ions o he ini ial condi ions. To do his, we will use insensi izing con ol heo y.
1.2. Insensi izing con ols and con ollabili y. Le Obe an open nonemp y
subse o Ω and le us in oduce he ollowing unc ional, defined on he amily o
solu ions o (1.1):
Φ(u)=1
2T
0O
|u(x, )|2dx d .(1.2)
The no ion o insensi izing con ols was in oduced by Lions [13]. In he con ex
o (1.1)–(1.2), i eads as ollows.
De ini ion 1.1. We say ha he con ol h∈L2(ω×(0,T))2is Φinsensi izing
i
d
dτ Φ(u)τ=0 =0 ∀u0∈Hwi h u00,Ω=1.(1.3)
On he o he hand, we say ha h∈L2(ω×(0,T))2is Φε-insensi izing i

d
dτ Φ(u)τ=0≤ε∀u0∈Hwi h u00,Ω=1.(1.4)
O cou se, in (1.3) and in (1.4) uis, oge he wi h p, he solu ion o (1.1).
The Φ insensi izing ( esp., Φ ε-insensi izing) con ols hmus be in e p e ed as
hose leading o an obse a ion Φ(u) ha is locally independen ( esp., almos inde-
penden ) a he ini ial pe u ba ion τu0. The exis ence o such con ols is a pe inen
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1618 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
ques ion, since i is ealis ic o assume ha he ue ini ial condi ions o (1.1) a e
unknown. In ac , as no iced in [13], i would be mo e con enien o sea ch o Ψ
insensi izing (o Ψ ε-insensi izing) con ols, whe e
Ψ(u)=1
2T
0O
|cu l u(x, )|2dx d ,
bu his is beyond he scope o his a icle and will be he subjec o u u e wo k.
I is easy o cha ac e ize he insensi i i y ( esp., ε-insensi i i y) p ope y in e ms
o exac null con ollabili y ( esp., app oxima e con ollabili y) o a ela ed cascade
sys em. Indeed, le (¯u, ¯p) and (q, ) be he solu ions o he ollowing sys ems:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
¯u −∆¯u+¯u+(1+x2)k∧¯u+∇¯p=T+h1ωin Q,
di ¯u=0 inQ,
¯u= 0 on Σ,
¯u(0) = u0in Ω,
(1.5)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−q −∆q+q−(1 + x2)k∧q+∇π=¯u1Oin Q,
di q=0 inQ,
q= 0 on Σ,
q(T)=0 inΩ.
(1.6)
Then he con ol his Φ insensi izing ( esp., Φ ε-insensi izing ) i and only i
q(0) = 0 ( esp., q(0)0,Ω≤ε).(1.7)
Indeed, in iew o (1.2), condi ion (1.3) is equi alen o
T
0O
¯u·uτdx d =0  esp., (1.4) is equi alen o T
0O
¯u·uτdx d ≤ε,
whe e ¯uis he solu ion o (1.5) and uτis he solu ion o (1.1) diffe en ia ed wi h
espec o τ. Using he defini ion o (q,π) and in eg a ing by pa s, we ob ain
Ω
q(0) ·u0dx =0  esp., Ω
q(0) ·u0dx≤ε∀u0∈Hwi h u00,Ω=1.
This is equi alen o (1.7). See [18] o mo e de ail.
No ice ha since ¯u∈L2(0,T;V), we also ha e q∈L2(0,T;W)∩H1(0,T;H) and
π∈L2(0,T;H1(Ω)).
We a e hus in he p esence o a null con ollabili y p oblem ( esp., an app oxima e
con ollabili y p oblem) o a cascade sys em, whe e he con ol his no ac ing di ec ly
in he sys em sa isfied by q( he unc ion we wan o d i e o ze o a e a ime in e al
o leng h T) bu indi ec ly, h ough ¯u1O.
1.3. Main esul s. The e ha e been se e al ecen esul s conce ning he exis-
ence o insensi izing and ε-insensi izing con ols o pa abolic p oblems.
Thus, in [2] he exis ence o ε-insensi izing con ols o linea hea equa ions
wi h pa ially known ini ial and bounda y condi ions was es ablished. The same
was also ob ained o semilinea hea equa ions wi h globally Lipschi z-con inuous
nonlinea i ies. Since hen, i has been p o ed in [18] ha insensi izing con ols exis
o he same equa ions comple ed wi h ze o ini ial da a, unde sui able assump ions
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CONTROLS INSENSITIZING AN OCEAN MODEL 1619
on he sou ce e m. In [3], he au ho s ex ended hese esul s o o he mo e gene al
(sligh ly supe linea ) nonlinea i ies.
In his pape , we deal wi h he insensi izing and ε-insensi izing p oblems o he
case o he S okes- ype equa ions (1.1). Ou esul s we e ske ched in [7]. These a e
he fi s insensi i i y esul s in he li e a u e o equa ions o his ype, as a as we
know.
We will assume ha he ollowing geome ical hypo hesis is sa isfied, as in he
p e ious e e ences:
ω∩O=∅.(1.8)
Ou main esul s a e he ollowing.
Theo em 1.2. Le T>0and assume ha (1.8) is sa isfied. Then, o each
ε>0 he e exis s a con ol h∈L2(ω×(0,T))2which is Φε-insensi izing.
Theo em 1.3. Unde he assump ions o Theo em 1.2, i we also ha e u0=0
and T
0Ω
exp M −4T2dx d < +∞(1.9)
o an app op ia e cons an Mdepending on Ω,ω,O, and T, hen he e exis s a
con ol h∈L2(ω×(0,T))2which is Φinsensi izing.
I was p o ed in [18] o he linea hea equa ion ha , in gene al, we canno
expec he exis ence o insensi izing con ols o non anishing ini ial da a in L2(Ω)
when Ω ω=∅. The p oo o his esul is based on a coun e example o which
he app op ia e obse abili y inequali y ails when he ini ial da a belong o L2(Ω).
Simila a gumen s could be used o S okes sys ems. In iew o his, i is easonable
o impose in Theo em 1.3 ha u0=0.
This pape is o ganized as ollows. In sec ion 2, we p o e Theo em 1.2, whe e
we ob ain a unique con inua ion esul o an adjoin cascade sys em hanks o he
p esence o he Co iolis e m. In sec ion 3, we p o e Theo em 1.3. In his sec ion,
we show ha insensi izing con ols do exis i an app op ia e obse abili y inequali y
holds. We deduce his obse abili y inequali y in sec ion 3.2 by means o an app o-
p ia e global Ca leman inequali y o he same adjoin cascade sys em. The p oo o
his global Ca leman inequali y is gi en in sec ion 3.1 and ollows a chain o es ima es
based on he s eps o he unique con inua ion p oo . A he end o his sec ion and
o be sel -con ained, we gi e he p oo o a s anda d global Ca leman es ima e o
S okes-like sys ems ha is needed in sec ion 3.1. Finally, in sec ion 4, we summa ize
he key poin s o his a icle in some final ema ks.
2. P oo o Theo em 1.2. We can assume wi hou loss o gene ali y ha T=0
and u0= 0 in (1.5)–(1.6). I is well known ha he exis ence o ε-insensi izing con ols
o (1.5)–(1.6) is equi alen o a unique con inua ion p ope y o he associa e adjoin
sys em ⎧
⎪
⎪
⎨
⎪
⎪
⎩
φ −∆φ+φ+(1+x2)k∧φ+∇θ=0 inQ,
di φ=0 inQ,
φ= 0 on Σ,
φ(0) = φ0in Ω,
(2.1)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−z −∆z+z−(1 + x2)k∧z+∇ =φ1Oin Q,
di z=0 inQ,
z= 0 on Σ,
z(T)=0 inΩ
(2.2)
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1620 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
o a gi en φ0∈H. This coupled sys em possesses a unique solu ion (φ, θ), (z, ),
wi h a leas φ, z ∈L2(0,T;V)∩H1(0,T;V) and θ, ∈W−1,∞(0,T;L2(Ω)). (Again,
θand a e unique up o a dis ibu ion depending only on .)
Using (1.5)–(1.6) and (2.1)–(2.2) he ollowing duali y iden i y is easily deduced:
T
0ω
h·zdxd =Ω
q(0) ·φ0dx ∀h∈L2(ω×(0,T))2.
I is clea om his las iden i y ha he se {q(0) : h∈L2(ω×(0,T))2}is dense in
Hi he ollowing unique con inua ion esul holds.
Lemma 2.1. Assume (1.8).Le (φ, θ),(z, )be he solu ion o (2.1)–(2.2) wi h
φ0∈H. Then, i z=0in ω×(0,T), we necessa ily ha e z≡φ≡0and ∇ ≡∇θ≡0
in Q.
P oo . This is a di ec consequence o a mo e gene al unique con inua ion esul .
To s a e his esul p ecisely, le ω=ω∩O=∅and le us se
C1(ω)={(x1,x
2)∈Ω:∃x0
1s. . (x0
1,x
2)∈ω},Σ1(ω)=(Γ∩C1)×(0,T).(2.3)
(C1(ω) is he ho izon al componen o ω.) We will p o e ha i φ=(φ1,φ
2)is
oge he wi h θ,z, and a solu ion o
⎧
⎨
⎩
φ −∆φ+φ+(1+x2)k∧φ+∇θ=0 inQ,
di φ=0 inQ,
φ1= 0 on Σ1,
(2.4)
−z −∆z+z−(1 + x2)k∧z+∇ =φ1ωin Q,
di z=0 inQ,
and z=0inω×(0,T), hen φ≡0.
To p o e his asse ion, we di ide he p oo in o wo s eps. Wi hou loss o
gene ali y, we can assume ha ωis connec ed; o he wise we would eplace ωby one
o i s connec ed componen s.
In a fi s s ep, we deduce om he ac ha z=0inω×(0,T) ha φ2= 0 and
φ1is cons an i hey a e es ic ed o ω×(0,T). Thus, since z=0inω×(0,T)we
no ice ha cu l φ=0inω×(0,T) by applying he cu l ope a o in he equa ion o
zin (2.4). Using his ac , i we now apply he cu l ope a o o he fi s equa ion in
(2.4), hanks o he p esence o he Co iolis e m we ob ain ha
cu l ((1 + x2)k∧φ)=φ2+ di φ=φ2=0
in ω×(0,T). Now, since di φ= 0 and cu l φ=0inω×(0,T), we ha e ∇φ1=0in
ω×(0,T). The e o e φ1is cons an in ω×(0,T) and we ce ainly ob ain φ= (Cons .,0)
in ω×(0,T).
In a second s ep, le us in oduce
ψ=∂φ
∂x1
,π=∂θ
∂x1
(2.5)
and he coefficien ma ix:
a=1−(1 + x2)
(1 + x2)1
∈L∞
loc(Q).(2.6)
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CONTROLS INSENSITIZING AN OCEAN MODEL 1621
Then we ha e ψ −∆ψ+aψ +∇π=0 inQ,
di ψ=0 inQ,
(2.7)
(ψ,π)∈L2
loc(Q)2×D
(Q)(2.8)
wi h ψ=0inω×(0,T). He e, we use a sha p uniqueness p ope y o he S okes
sys em (2.7) p o ed in [5] ha says ha , unde he egula i y de e mined by (2.6)
and (2.8), one has ψ≡0inQ. Now, om (2.5) we ob ain ∂φi/∂x1≡0 o i=1,2in
Q. Since di φ=0inQwe also ha e ∇φ2=0inQand, om he ac ha φ2=0in
ω×(0,T), we deduce ha φ2≡0inQ( ecall ha Ω is connec ed).
On he o he hand, since ∂φ1/∂x1=0inQand φ1= 0 on Σ1, we see ha φ1=0
in C1×(0,T). Finally we ha e φ=(φ1,φ
2)=0inC1×(0,T), which is an open
subse o Q. We can conclude ha φ≡0inQusing again he uniqueness p ope y
o [5]. (We can also use he e he weake esul p o ed in [4].)
Rema k 1. The me hod used in he second pa o he p oo o Lemma 2.1 leads
o he ollowing uniqueness p ope y in any dimension n. Le (φ, θ) be he solu ion o
⎧
⎨
⎩
φ −∆φ+aφ+∇θ=0 inQ,
di φ=0 inQ,
φ= 0 on Σ1(ω),
(2.9)
whe e Q=Ω×(0,T), Ω is a nonemp y open bounded connec ed subse o Rn,ωis
an open nonemp y subse o Ω, Σ1and C1a e as defined in (2.3), and a∈L∞(Q). I
ais a unc ion independen o x1in Qand φis independen o x1in ω×(0,T), hen
φ anishes in Q. Indeed, le us in oduce ψ=∂φ/∂x1,π=∂θ/∂x1, which sa is y
a S okes p oblem simila o (2.9), and his p oblem does no in ol e φexplici ly
since ais independen o x1. Now, om he uniqueness p ope y in [5], ψ≡0inQ.
Consequen ly ∂φ/∂x1=0inQand φ= 0 on Σ1,soweha eφ=0inC1×(0,T).
Using he unique con inua ion p ope y in [5] once again, we ob ain ha φ≡0inQ.
Rema k 2. The p e ious ema k shows ha he Co iolis e m plays a c ucial
ole only in he fi s pa o he p oo o Lemma 2.1. In ac , he p esence o he
Co iolis e m allows us o p o e ha cu l φ=0inω×(0,T) implies ha he second
componen o φ anishes in ω×(0,T). This will also be impo an in he deduc ion
o he Ca leman inequali y la e .
Rema k 3. In he p oo o he p e ious lemma, i is no possible o use he
esul s o [4] conce ning uniqueness p ope ies o he S okes sys em when one o he
componen s o φ anishes in ω×(0,T). This is because he esul s in [4] equi e ha
he coefficien a, in oduced in (2.6), sa is y a12 =0.
3. P oo o Theo em 1.3. The p oo o he exis ence o insensi izing con ols
o (1.1), i.e., he exac null con ollabili y o (1.5)–(1.6), elies on he ollowing
obse abili y esul o he cascade adjoin sys em (2.1)–(2.2).
P oposi ion 3.1. Assume ha ω∩O =∅. The e exis posi i e cons an s Mand
K, depending only on Ω,ω,O, and T, such ha he inequali y
T
0Ω
exp −M −4|z|2dx d ≤KT
0ω
|z|2dx d (3.1)
holds o e e y solu ion o (2.1)–(2.2) wi h φ0∈H.
The p oo o his esul is based on a global Ca leman inequali y (see Theo-
em 3.3), as will be seen in sec ion 3.2. This Ca leman inequali y will be p o ed in
sec ion 3.1.
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1622 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
Le us now gi e he p oo o Theo em 1.3 using P oposi ion 3.1. Thus, le us
assume ha (1.8) is sa isfied, u0= 0, and (1.9) holds wi h Mbeing he cons an
u nished by P oposi ion 3.1.
The app oxima e con ol ho minimal no m in L2(ω×(0,T))2co esponding o
u0= 0, a sou ce e m Tsa is ying (1.9), and ole ance ε>0 can be ob ained by
minimizing in L2(Ω)2 he ollowing con ex unc ional [6, 14]:
Jε(φ0)=1
2T
0ω
|z|2dx d +T
0Ω
T·zdxd +εφ00,Ω.(3.2)
Thus, he minimum o Jεis a ained a some 
φ0ε∈L2(Ω)2. We deno e by (
φε,
θε),
(zε, ε) he co esponding solu ion o (2.1)–(2.2) wi h φ0=
φ0ε; hen he con ol
unc ion defined as
hε=zε1ω
(3.3)
is such ha he associa ed solu ion (¯uε,¯pε), (qε,π
ε) o (1.5)–(1.6) wi h u0= 0 sa isfies
qε(0)0,Ω≤ε.
I is no difficul o see ha
lim in
φ00,Ω→∞
Jε(φ0)
φ00,Ω
≥ε.
The p oo o his inequali y is classical; see [6]. I is implied by he unique con inua ion
p ope y o he cascade adjoin sys em ha we p esen ed abo e (see Lemma 2.1).
Fu he mo e, he ollowing op imali y condi ion mus be sa isfied a 
φ0ε:
T
0ω
|zε|2dx d +T
0Ω
T·zεdx d +ε
φ0ε0,Ω=0.(3.4)
By eplacing (3.3) in (3.4), in oducing he weigh eM −4, and using (3.1) and Young’s
inequali y, we easily deduce ha
T
0ω
|hε|2dx d ≤K2T
0Ω
exp(M −4)|T |2dx d .
Since {hε}is uni o mly bounded in L2(ω×(0,T))2, hen up o a subsequence, s ill
deno ed {hε},weha e
hεh weakly in L2(ω×(0,T))2,
¯uε→¯us ongly in L2(Q)2,and
qε→qs ongly in L2(Q)2,
as ε→0. O cou se, we ha e deno ed he e by (¯uε,¯pε), (qε,π
ε) and (¯u, ¯p), (q,π)
he solu ions o (1.5)–(1.6) associa ed wi h hεand h, espec i ely. No ice ha
qε(0)0,Ω≤εand consequen ly we ha e q(0) = 0. This ends he p oo o Theo-
em 1.3.
3.1. A global Ca leman es ima e. The goal o his sec ion is o p esen an es-
ima e o he Ca leman kind o he solu ions o he adjoin cascade sys em (2.1)–(2.2).
As men ioned abo e, his es ima e will be c ucial o he p oo o P oposi ion 3.1.
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CONTROLS INSENSITIZING AN OCEAN MODEL 1623
Le us fi s in oduce an open ball B0such ha B0⊂⊂ ω∩O and an auxilia y
unc ion η0∈C
2(Ω) sa is ying
η0(x)>0∀x∈Ω,η
0=0 on∂Ω,|∇η0(x)|>0∀x∈Ω B0.(3.5)
The exis ence o such a unc ion is p o ed in [9].
Le us also in oduce he weigh unc ions
α(x, )=e2λη0∞−eλη0
4(T− )4,α( ) = min
Ω
α(x, ),α
∗( ) = max
Ω
α(x, ),
ϕ(x, )= eλη0
4(T− )4,ϕ( ) = max
Ω
ϕ(x, ),ϕ
∗( ) = min
Ω
ϕ(x, ).
The ollowing p ope y o he unc ions α∗and αwill be needed la e .
Lemma 3.2. Fo any a>1 he e exis s λa>0such ha
aα( )>α
∗( )∀λ>λ
a,∀ ∈(0,T).
P oo . The p oo is elemen a y. I suffices o no ice ha we ha e a(e2x−ex)>
e2x−1i a>1 and xis sufficien ly la ge.
The main esul in his sec ion is he ollowing.
Theo em 3.3. Assume ha ω∩O=∅and le he unc ions α,ϕ,α, and ϕbe
as abo e. Fo each γ∈(0,1), he e exis cons an s s,
λ, and 
Cdepending on Ω,ω,
O,T, and γsuch ha one has
T
0Ω
e−2sα 1
sϕ(|z |2+|∆z|2)+sλ2ϕ|∇z|2+s3λ4ϕ3|z|2dx d
+T
0Ω
e−2sα 1
sϕ(|φ |2+|∆φ|2)+sλ2ϕ|∇φ|2+s3λ4ϕ3|φ|2dx d
≤
CT
0ω
e−(1+γ)sαs63λ32 ϕ67|z|2dx d (3.6)
o any s>sand λ>
λand o e e y solu ion (φ, θ),(z, ) o (2.1)–(2.2) associa ed
wi h ini ial da a φ0∈H.
The p oo will be di ided in se e al s eps and will be gi en in he ollowing
subsec ions. Fi s , we will apply a global Ca leman es ima e o he S okes sys em o
(2.1) and (2.2). This will lead o he es ima e (3.10). Then, o deduce (3.6), we will
ha e o es ima e he in eg al in he igh -hand side o (3.10) con aining φin e ms
o z. To his end, we will ollow he s eps o he p oo o Lemma 2.1 in e e se o de .
3.1.1. S ep 1: A fi s di ec Ca leman es ima e. Le I(s, λ; ) s and o
he quan i y
I(s, λ; )=T
0Ω
e−2sα 1
sϕ(| |2+|∆ |2)+sλ2ϕ|∇ |2+s3λ4ϕ3| |2dx d (3.7)
o any posi i e sand λand any sufficien ly egula unc ion = (x, ). We hen
ha e he ollowing.
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1624 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
Lemma 3.4. Fo each γ1∈(0,1) he e exis posi i e cons an s s1,λ1, and C1,
depending on Ω,ω,O,T, and γ1, wi h he ollowing p ope ies:
I(s, λ;z)≤C1T
0B0
e−(1+γ1)sαs7λ4ϕ15/2|z|2dx d
+T
0B0
e−2sα(sλϕ)2|φ|2dx d (3.8)
+T
0O
e−2sα (sϕ)1/2|φ|2+1
s3ϕ7/2|φ |2dx d 
and
I(s, λ;φ)≤C1T
0B0
e−(1+γ1)sαs7λ4ϕ15/2|φ|2dx d (3.9)
o any s>s
1and λ>λ
1and o e e y solu ion o (2.1)–(2.2) wi h φ0∈H.
The p oo o Lemma 3.4 is simila o he p oo o o he ecen global Ca leman
inequali ies o he S okes sys em. The main ideas a e due o Imanu ilo [10, 11];
also see [8] o o he ela ed esul s. The p oo is p esen ed in he appendix.
Le us fix γ, wi h 0 <γ<1. We a e now going o deduce se e al es ima es ha
hold o “sufficien ly la ge sand λ.” By his we mean ha hey a e sa isfied o any
s>¯sand any λ>¯
λ, whe e ¯sand ¯
λa e (la ge) posi i e cons an s depending only on
Ω, ω,O,T, and γ.
In wha ollows, Cdeno es a gene ic cons an , no necessa ily he same a each
occu ence, depending on Ω, ω,O,T, and (possibly) γ.
Le γ1be gi en in (γ,1). In iew o Lemma 3.4 applied o γ1,wege
I(s, λ;z)+I(s, λ;φ)≤CT
0B0
e−(1+γ1)sαs7λ4ϕ15/2(|z|2+|φ|2)dx d (3.10)
o sand λla ge enough.
Indeed, he las wo in eg als in (3.8) can be abso bed by he le -hand side o
I(s, λ;φ), since
Cs−3ϕ−7/2≤1
2(sϕ)−1and C(sϕ)1/2≤1
2s3ϕ3
o sufficien ly la ge s.
3.1.2. S ep 2: An es ima e o φin e ms o cu l φ.To simpli y he no a-
ion, le us se a= 7 and b=15/2. Then
I(s, λ;z)+I(s, λ;φ)≤CT
0B0
e−(1+γ1)sαsaλ4ϕb(|z|2+|φ|2)dx d .(3.11)
We will deno e by B1,B
2,... a sequence o balls cen e ed a he same poin as
B0and sa is ying
B0⊂⊂ B1⊂⊂ · · · ⊂⊂ ω∩O.
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CONTROLS INSENSITIZING AN OCEAN MODEL 1631
Le us fi s p o e ha he e exis posi i e cons an s M,C1such ha
T/2
0Ω
exp(−M −4)|z|2dx d +T
T/2Ω
|φ|2dx d ≤C1T
0ω
|z|2dx d .(3.31)
Fo his es ima e, le us fi s no ice ha o some cons an s Mand C, one has
e−2sα(x, )ϕ(x, )3≥Ce−M −4∀(x, )∈Ω×(0,T/2);
his is easy o see, in iew o he defini ions o αand ϕ. Now, using (3.6), we ge
T/2
0Ω
exp(−M −4)|z|2dx d ≤1
s3λ4I(s, λ;z)
≤CK T
0ω
e−(1+γ)sαs63 ϕ67|z|2dx d ,
and since he weigh e−(1+γ)sαϕ67 is bounded, we can es ima e he fi s e m in he
le -hand side o (3.31). On he o he hand, o ob ain an es ima e o φin e ms o z,
le us ecall he inequali y (3.6). Since e−2sα −12(T− )−12 is bounded om below
a om = 0 and =T, in iew o (3.30), we ha e
T
T/2Ω
|φ|2dx d ≤3T/4
T/4Ω
|φ|2dx d
≤C3T/4
T/4Ω
e−2sαs3λ4ϕ3|φ|2dx d ≤CT
0ω
e−(1+γ)sαϕ67|z|2dx d .
As be o e, om he ac ha e−(1+γ)sαϕ67 is bounded, we a e able o es ima e he
second e m in he le -hand side o (3.31).
Finally, he desi ed obse abili y inequali y (3.1) is ob ained using he ene gy
es ima e (3.29) and (3.31):
T
0Ω
exp(−M −4)|z|2dx d ≤T/2
0Ω
exp(−M −4)|z|2dx d +T
T/2Ω
|z|2dx d
≤CT/2
0Ω
exp(−M −4)|z|2dx d +T
T/2Ω
|φ|2dx d

≤CT
0ω
|z|2dx d .
Appendix. P oo o Lemma 3.4. Le us ecall ha his p oo is gi en o
he sake o comple eness, bu i is essen ially an adap a ion o ou amewo k o he
a gumen s p esen ed in [8] and [11]. Le us conside he sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−z −∆z+z−(1 + x2)k∧z+∇ =φ1Oin Q,
di z=0 inQ,
z= 0 on Σ,
z(T)=0 inΩ,
(3.32)
whe e φ∈L2(0,T;W)∩H1(0,T;H).
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1632 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
Recall ha B0is an open ball sa is ying B0⊂⊂ ω∩O and he auxilia y unc ion
η0sa isfies η0∈C
2(Ω),
η0(x)>0∀x∈Ω,η
0=0 on∂Ω,|∇η0(x)|>0∀x∈Ω B0.
We will need an addi ional open ball B00 ⊂⊂ B0, such ha we s ill ha e
|∇η0(x)|>0∀x∈Ω B00 .
We will di ide he p oo o Lemma 3.4 in o se e al s eps.
S ep 1. Following [8], we apply some well-known Ca leman es ima es o he hea
equa ion o (3.32). Thus, he e exis cons an s s0,λ0, and C>0 depending on Ω,
ω, and Tsuch ha o e e y λ>λ
0and s>s
0, he ollowing es ima e holds:
I(s, λ;z)≤CT
0B00
e−2sαs3λ4ϕ3|z|2dx d
+T
0Ω
e−2sα |∇ |2+|(1 + x2)k∧z|2+|φ1O|2dx d .(3.33)
Recall ha he defini ions o I(s, λ;z) and he weigh s αand ϕa e gi en in sec ion 3.1.
O cou se, we can choose sla ge enough o abso b he p e ious e m |(1+x2)k∧z|2
wi h he le -hand side (3.33). We hen ha e
I(s, λ;z)≤CT
0B00
e−2sαs3λ4ϕ3|z|2dx d
+T
0Ω
e−2sα|∇ |2dx d +T
0O
e−2sα|φ|2dx d 
(3.34)
o any λ>λ
0and any s>s
01 .
S ep 2. To es ima e he p essu e g adien ∇ in (3.34), we fi s apply he di e -
gence ope a o o (3.32), i.e., we w i e
∆ ( ) = di ((1 + x2)k ∧z)( )inΩ, ∈(0,T),(3.35)
and hen we use he ollowing esul by Imanu ilo and Puel [12], which is sa isfied
by weak solu ions o second o de ellip ic equa ions.
Lemma 3.5. Le usse β(x)=eλη0(x)and le ∈H1(Ω) be a solu ion o
∆ = di hin Ω,(3.36)
whe e h∈L2(Ω)2. Then he e exis posi i e cons an s τ2,λ01, and Csuch ha
Ω
e2τβ|∇ |2dx ≤CτΩ
e2τββ|h|2dx +τ1/2e2τg2
1/2,∂Ω
+τ2λ2B00
e2τββ2| |2dx +B00
e2τβ|∇ |2dx
(3.37)
o any τ>τ
2and any λ>λ
01 , whe e g= |∂Ω.
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CONTROLS INSENSITIZING AN OCEAN MODEL 1633
In pa icula , we ha e he ollowing o ( ) and g( )= ( )|∂Ω:
Ω
e2τβ|∇ ( )|2dx ≤CτΩ
e2τββ|(1 + x2)k∧z( )|2dx
+τ1/2e2τg( )2
1/2,∂Ω+τ2λ2B00
e2τββ2| ( )|2dx
+B00
e2τβ|∇ ( )|2dx.(3.38)
To es ima e he las in eg al in (3.38), le us in oduce an open se B01 such ha
B00 ⊂⊂ B01 ⊂⊂ B0and a unc ion ξ01 ∈C
2
0(B01) such ha
0≤ξ01 ≤1 and ξ01 =1inB00 .
In eg a ing by pa s, i ollows om (3.35) ha
B00
e2τβ|∇ ( )|2dx ≤B01
e2τβξ01|∇ ( )|2dx
=−B01
e2τβξ01 di ((1 + x2)k ∧z)( ) ( )dx
−1
2B01
e2τβ∇ξ01 ·∇| ( )|2dx −B01
ξ01∇e2τβ ·∇| ( )|2dx.
In eg a ing again by pa s, applying Young’s inequali y, and aking in o accoun ha
|∆(e2τβξ01)|≤Cτ2λ2β2e2τβ o some posi i e cons an C, a e some s aigh o wa d
compu a ions we deduce ha
B00
e2τβ|∇ ( )|2dx ≤Cτ2λ2B01
e2τββ2| ( )|2dx +B01
e2τβ|z( )|2dx.
Replacing his inequali y in (3.38), we ob ain he ollowing o each ∈(0,T):
Ω
e2τβ|∇ ( )|2dx ≤CτΩ
e2τββ|z( )|2dx
+τ1/2e2τg( )2
1/2,∂Ω
+τ2λ2B01
e2τββ2| ( )|2dx.
Now, le us pu τ=s/( 4(T− )4) and le us choose s>s
02 = max(s01 ,τ
2(T/2)8).
Then τ>τ
2. Le us mul iply by exp(−2sexp(2λη0∞)/( 4(T− )4)) he p e ious
inequali y and le us in eg a e wi h espec o in (0,T). This leads o he es ima e
T
0Ω
e−2sα|∇ |2dx d ≤CT
0Ω
e−2sαsϕ|z|2dx d
+T
0
e−2sα∗(sϕ∗)1/2g( )2
1/2,∂Ωd
+T
0ω1
e−2sα(sλϕ)2| |2dx d ,(3.39)
whe e α∗and ϕ∗we e in oduced in sec ion 3.1.
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1634 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
The fi s e m in he igh -hand side o (3.39) can be abso bed by he le -hand
side I(s, λ;z) in (3.33) o sla ge enough. Hence, we ob ain
I(s, λ;z)≤CT
0ω0
e−2sαs3λ4ϕ3|z|2dx d +T
0
e−2sα∗(sϕ∗)1/2g( )2
1/2,∂Ωd
+T
0ω1
e−2sα(sλϕ)2| |2dx d +T
0O
e−2sα|φ|2dx d 
(3.40)
o any λ>λ
01 and any s>s
03 .
S ep 3. S ep 3 es ima es he no m o he ace o he p essu e on he bounda y.
To his end, we in oduce h ee new unc ions:
χ( )=e−sα∗( )(sϕ∗( ))1/4,˜z=χ( )z, ˜ =χ( ) .
F om (3.32), we see ha (˜z, ˜ ) sa isfies
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−˜z −∆˜z+˜z+∇˜ =−χz+χ(1 + x2)k∧z+χφ1Oin Q,
di ˜z=0 inQ,
˜z= 0 on Σ,
˜z(T)=0 inΩ.
Using he con inui y o he ace ope a o and s anda d a p io i es ima es o he
p essu e, we deduce ha
T
0
˜ ( )2
1/2,∂Ωd ≤T
0
˜ ( )2
1,Ωd
≤CT
0Ω
e−2sα∗s5/2(ϕ∗)3|z|2dx d
+T
0O
e−2sα∗(sϕ∗)1/2|φ|2dx d .
We ha e used he e ha |χ( )|2≤Ce−2sα∗s5/2(ϕ∗( ))3 o all ∈(0,T). We hus
ob ain a new es ima e om (3.40):
I(s, λ;z)≤CT
0B00
e−2sαs3λ4ϕ3|z|2dx d +T
0B01
e−2sα(sλϕ)2| |2dx d
+T
0O
e−2sα(sϕ)1/2|φ|2dx d 
(3.41)
o any λ>λ
01 and any s>s
04 .
S ep 4. I emains o es ima e he local e m in he igh -hand side o (3.41)
con aining | |2in e ms o zand φ.
Assume ha he p essu e has been no malized in such a way ha
B01
( )dx =0 ∀ ∈(0,T).
Then he e exis s C>0 such ha
B01
| ( )|2dx ≤CB01
|∇ ( )|2dx ∀ ∈(0,T)
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CONTROLS INSENSITIZING AN OCEAN MODEL 1635
and also
T
0B01
e−2sα(sλϕ)2| |2dx d ≤CT
0B01
e−2sα(sλϕ)2|∇ |2dx d ,
whe e he unc ions α=α( ) and ϕ=ϕ( ) we e in oduced in sec ion 3.1.
F om (3.32), we see ha
T
0B01
e−2sα(sλϕ)2|∇ |2dx d ≤CT
0B01
e−2sα(sλϕ)2(|z|2+|φ|2)dx d
+T
0B01
e−2sα(sλϕ)2(|z |2+|∆z|2)dx d .
The e o e, in iew o (3.41), we ob ain
I(s, λ;z)≤CT
0B01
e−2sαs3λ4ϕ3|z|2+(sλϕ)2|φ|2dx d
+T
0B01
e−2sα(sλϕ)2(|z |2+|∆z|2)dx d
+T
0O
e−2sα(sϕ)1/2|φ|2dx d .(3.42)
S ep 5. The es o he p oo deals wi h he es ima es o he local in eg als
con aining |∆z|2and |z |2. Fi s , we will be conce ned wi h |∆z|2.
Le us in oduce a unc ion ξ0∈C
4
0(B0) such ha
0≤ξ0≤1 and ξ0=1inB01 .
Le us se z(x, )=e−sαϕξ
0∆z(T− ). We wan o es ima e he no m zL2(B01×(0,T ))2.
Following he a gumen s in [8] (see S ep 4), we can deduce ha
T
0B01
e−2sα(sλϕ)2|∆z|2dx d =T
0B01
s2λ2|z|2dx d
≤CT
0B0
e−2sαs4λ2ϕ9/2|z|2dx d +T
0B0
e−2sα(sλϕ)2|φ|2dx d .
(3.43)
Thus, om (3.42) we ha e
I(s, λ;z)
≤CT
0B0
e−2sαs4λ4ϕ9/2|z|2dx d +T
0B0
e−2sα(sλϕ)2|φ|2dx d
+T
0B01
e−2sα(sλϕ)2|z |2dx d +T
0O
e−2sα(sϕ)1/2|φ|2dx d .(3.44)
S ep 6. Now we wan o es ima e |z |2. Due o he egula i y p ope ies o φ,we
can use he e a mo e s aigh o wa d a gumen han in [8], whe e he igh -hand side
belongs only o L2(Q)2.
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1636 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
Fi s , no ice ha
T
0B01
e−2sα(sλϕ)2|z |2dx d ≤δT
0B01
e−2sα 1
sϕ|z |2dx d
+δT
0B01
e−2sα∗1
s3(ϕ∗)7/2|z |2dx d
+CδT
0B01
e−4sα∗+2sα∗s7λ4ϕ15/2|z|2dx d .
This is easily ob ained by in eg a ing by pa s in ime. We will la e choose δ>0
small enough.
We ha e he ollowing auxilia y esul .
Lemma 3.6. Le (z, )be he solu ion o (3.32). Then he ollowing es ima e
holds:
T
0Ω
e−2sα∗1
s3(ϕ∗)7/2|z |2dx d
≤CI(s, λ;z)+T
0O
e−2sα∗1
sϕ∗|φ|2+1
s3(ϕ∗)7/2|φ |2dx d .(3.45)
P oo . Mul iply (3.32) by e−2sα∗s−2(ϕ∗)−9/4z and in eg a e in Q. No icing ha
(e−2sα∗(ϕ∗)−9/4) ≤Ce−2sα∗s(ϕ∗)−1,
a e some compu a ions we deduce ha
T
0Ω
e−2sα∗1
s2(ϕ∗)9/4|∇z |2dx d
≤CT
0Ω
e−2sα∗1
sϕ∗(|z|2+|z |2)dx d
+T
0Ω
e−2sα∗s(ϕ∗)1/4|∇z|2dx d +T
0O
e−2sα∗1
sϕ∗|φ|2dx d 
+1
2T
0Ω
e−2sα∗1
s3(ϕ∗)7/2|z |2dx d .(3.46)
On he o he hand, i we compu e he ime de i a i e o (3.32) and hen we
mul iply he esul by e−2sα∗s−3(ϕ∗)−7/2z , we find ha
T
0Ω
e−2sα∗1
s3(ϕ∗)7/2|z |2dx d
≤T
0Ω
e−2sα∗1
s2(ϕ∗)9/4|∇z |2dx d (3.47)
+CT
0Ω
e−2sα∗1
sϕ∗|z |2dx d +T
0O
e−2sα∗1
s3(ϕ∗)7/2|φ |2dx d .
F om (3.46) and (3.47), we see ha (3.45) holds.
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CONTROLS INSENSITIZING AN OCEAN MODEL 1637
In iew o his lemma, we ha e
T
0B01
e−2sα(sλϕ)2|z |2dx d
≤CδI(s, λ;z)+T
0O
e−2sα∗1
sϕ∗|φ|2+1
s3(ϕ∗)7/2|φ |2dx d 
+CδT
0B01
e−4sα∗+2sα∗s7λ4ϕ15/2|z|2dx d .
I we assume ha γ1<1, hen (3−γ1)/2>1, and om Lemma 3.2 we deduce ha
(3 −γ1)α/2>α
∗ o sufficien ly la ge λ,say,λ>λ
02 . Consequen ly, −4α+2α∗<
−(1 + γ1)αand
T
0B01
e−2sα(sλϕ)2|z |2dx d
≤CδI(s, λ;z)+T
0O
e−2sα∗1
sϕ∗|φ|2+1
s3(ϕ∗)7/2|φ |2dx d 
+CδT
0B01
e−(1+γ1)sαs7λ4ϕ15/2|z|2dx d
o any λ>λ
02 and any s>s
04 .
F om (3.44) and his es ima e, choosing δ>0 small enough, we find
I(s, λ;z)
≤CT
0B0
e−(1+γ1)sαs7λ4ϕ15/2|z|2dx d +T
0B0
e−2sα(sλϕ)2|φ|2dx d
+T
0O
e−2sα (sϕ)1/2|φ|2+1
s3(ϕ∗)7/2|φ |2dx d 
o all λ>λ
02 and s>s
04 .
Ob iously, his yields (3.8). The p oo o (3.9) is e y simila and in ac much
simple , since he le -hand side o (2.1) is ze o.
Thus, we ha e p o ed Lemma 3.4 o λ1=λ02 and s1=s04 ( wo pa ame e s
depending on Ω, ω,O, and T).
4. Some final ema ks. The geome ical hypo hesis ω∩O=∅is equi ed o
p o e he exis ence o bo h ε-insensi izing and insensi izing con ols. In he fi s case,
his assump ion is used o p o e a unique con inua ion p ope y (Lemma 2.1). In
he case o insensi izing con ols, i is used o p o e an obse abili y inequali y. The
p oblem is comple ely open when ω∩O=∅(see [18]).
The exis ence o insensi izing con ols is gua an eed by he null con ollabili y
p ope y o a cascade sys em o quasi-geos ophic equa ions (1.5)–(1.6). In his case,
he con ol ac s indi ec ly on one a iable h ough he o he one. O cou se, his
con ollabili y p ope y is s onge han he null con ollabili y o a single quasi-
geos ophic sys em.
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1638 E. FERN´
ANDEZ-CARA, G. C. GARCIA, AND A. OSSES
To p o e he null con ollabili y p ope y o he cascade sys em, he e a e wo
main difficul ies.
Fi s is he need o a unique con inua ion esul o he adjoin sys em
(2.2)–(2.1). The p esence o he Co iolis e m pe mi s us o ela e he second compo-
nen o he eloci y and i s associa ed o ici y, and his is a key poin in he p oo
o uniqueness (see Rema k 2).
The second p oblem is he need o an obse abili y inequali y o he adjoin .
This inequali y comes om an app op ia e (global) Ca leman es ima e. The main
idea is o es ima e φin e ms o cu l φin a ball con ained in ω∩Oin he igh -hand
side o (3.10). This is possible again due o he p esence o he Co iolis e m (in ac ,
ou me hod does no wo k in he case o he usual S okes equa ions). We ew i e he
sys em using he s eam unc ion and he o ici y and we see ha he Co iolis e m
leads o an exp ession o he ho izon al de i a i e o he s eam unc ion in e ms o
he o ici y. In his way, we a e able o a oid es ima es o p essu e e ms, which
a e in gene al e y ha d o deduce (see he appendix in sec ion 3). Mo eo e , he
weigh in he igh -hand side o (3.10) is la ge han he weigh in he le -hand side.
Acco dingly, he e ms in he igh canno be abso bed di ec ly as in he case o he
hea equa ion (see [18]) and his ac equi es some addi ional wo k.
Acknowledgmen s. The au ho s wish o hank J.-P. Puel o he Labo a oi e de
Ma h´ema iques Appliqu´ees, Uni e si ´e Ve sailles/Sain Quen in-en-Y elines (F ance)
and S. Gue e o o he Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa (Spain), o e y use ul discussions.
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