C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280
h p:// ance.else ie .com/di ec /CRASS1/
Pa ial Di e en ial Equa ions/Op imal Con ol
On he con ollabili y o he N-dimensional Na ie –S okes
and Boussinesq sys ems wi h N−1 scala con ols
En ique Fe nández-Ca aa, Se gio Gue e oa, Oleg Yu ie ich Imanu ilo b,
Jean-Pie e Puelc
aDepa amen o E.D.A.N., Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain
bDepa men o Ma hema ics, Iowa S a e Uni e si y, 400 Ca e Hall, Ames, IA 50011-2064, USA
cLabo a oi e de ma héma iques appliquées, uni e si é de Ve sailles – S Quen in, 45, a enue des E a s Unis, 78035 Ve sailles, F ance
Recei ed 4 No embe 2004; accep ed 3 Decembe 2004
A ailable online 18 Janua y 2005
P esen ed by Pie e-Louis Lions
Abs ac
In his No e we p esen se e al con ollabili y esul s o nonlinea sys ems o he Na ie –S okes and Boussinesq kind. We
discuss he exis ence o pa icula con ols wi h a small numbe o deg ees o eedom. To ci e his a icle: E. Fe nández-Ca a
e al., C. R. Acad. Sci. Pa is, Se . I 340 (2005).
2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
Résumé
Su la con ôlabili é des sys èmes de Na ie –S okes e Boussinesq N-dimensionnels a ec N−1 con ôles scalai es.
Dans ce e No e on p ésen e quelques ésul a s de con ôlabili é pou des sys èmes non linéai es du ype Na ie –S okes e
Boussinesq. On analyse l’exis ence de con ôles pa iculie s a ec un nomb e pe i de deg és de libe é. Pou ci e ce a -
icle:E. Fe nández-Ca a e al., C. R. Acad. Sci. Pa is, Se . I 340 (2005).
2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
Ve sion ançaise ab égée
Ce e No e con ien quelques ésul a s de con ôlabili é pou des sys èmes non linéai es du ype Na ie –S okes
e Boussinesq. On che che des con ôles pa iculie s, a ec un nomb e pe i de deg és de libe é.
E-mail add esses: [email p o ec ed] (E. Fe nández-Ca a), [email p o ec ed] (S. Gue e o), [email p o ec ed] (O.Y. Imanu ilo ),
[email p o ec ed] (J.-P. Puel).
1631-073X/$ – see on ma e 2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
doi:10.1016/j.c ma.2004.12.013
276 E. Fe nández-Ca a e al. / C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280
On se donne un domaine bo né e égulie Ω⊂RN(N=2ouN=3), un ou e non ide (e pe i ) O⊂Ωe
un nomb e T>0. On considè e d’abo d le sys ème
y −y +(y ·∇)y +∇p= 1O,∇·y=0 dans Q=Ω×(0,T),
y=0su Σ=∂Ω ×(0,T),
y(0)=y0dans Ω,(1)
où 1Oes la onc ion ca ac é is ique de O. Considé ons l’espace de Banach E=H∩L4(Ω)N,où
H={w∈L2(Ω)N;∇·w=0 dans Ω, w ·n=0su ∂Ω}.(2)
On di a que (1) es localemen exac emen con ôlable aux ajec oi es en emps Tsi, pou ou e solu ion su i-
sammen éguliè e (¯y, ¯p) du sys ème non con ôlé
¯y −¯y+(¯y·∇)¯y+∇¯p=0,∇·¯y=0 dans Q,
¯y=0su Σ,
il exis e δ>0 el que, si y0−¯y(0)E⩽δ, alo s on peu ou e des con ôles ∈L2(O×(0,T))
Ne des é a s
associés (y, p) solu ion de (1) sa is aisan y(T) =¯y(T).
On considè e main enan le sys ème de Boussinesq





y −y +(y ·∇)y +∇p= 1O+θe
N,∇·y=0 dans Q,
θ −θ +y·∇θ=h1Odans Q,
y=0,θ=0su Σ,
y(0)=y0,θ(0)=θ0dans Ω,
(3)
où les con ôles son ∈L2(O×(0,T))
Ne h∈L2(O×(0,T)).Ondi aque(3)es localemen exac emen
con ôlable aux ajec oi es en emps Tsi, pou ou e solu ion su isammen éguliè e (¯y, ¯p, ¯
θ) du sys ème



¯y −¯y+(¯y·∇)¯y+∇¯p=¯
θeN,∇·¯y=0 dans Q,
¯
θ −¯
θ+¯y·∇¯
θ=0 dans Q,
¯y=0,¯
θ=0su Σ,
il exis e δ>0 el que, si (y0,θ0)−(¯y(0), ¯
θ(0))E×L2⩽δ, alo s on peu ou e des con ôles ∈L2(O×
(0,T))
Ne h∈L2(O×(0,T)) e des é a s associés (y,p,θ)sa is aisan y(T) =¯y(T) e θ(T)=¯
θ(T).
Finalemen , lo sque N=2, on considè e a les sys èmes a ec e me non linéai e onqué
y −y +(y ·∇)TM(y) +∇p= 1O,∇·y=0 dans Q,
y·n=0,∇×y=0su Σ,
y(0)=y0dans Ω,(4)
où M>0, TM(y) =(TM(y1), TM(y2)) e
TM(s) =−Msi s⩽−M,
ssi −M⩽s⩽M,
Msi s⩾M.
On di a que (4) es (globalemen ) con ôlable à zé o en emps Tsi, pou ou y0∈H,ilexis e ∈L2(O×
(0,T))
2 el que l’é a associé sa is ai y(T ) =0.
On ai les hypo hèses sui an es su O,¯ye ¯
θ:
∃x0∈∂Ω, ∃ε>0 els que O∩∂Ω ⊃B(x0;ε) ∩∂Ω, (5)
¯y∈L∞(Q)N,¯y ∈L20,T;Lσ(Ω)Nσ>1siN=2
σ>6/5siN=3(6)
e
¯
θ∈L∞(Q), ¯
θ ∈L20,T;Lσ(Ω)σ>1siN=2
σ>6/5siN=3.(7)
E. Fe nández-Ca a e al. / C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280 277
On a alo s les ésul a s sui an s :
Théo ème 0.1. Supposons que Osa is ai (5). Alo s, pou ou T>0,(1) es localemen exac emen con ôlable
en emps Taux ajec oi es (y,p) qui sa is on (6) a ec des con ôles ∈L2(O×(0,T))
Nqui posséden une
composan e nulle.
Théo ème 0.2. Supposons que Osa is ai (5) a ec ni(x0)= 0, pou un indice i<N. Alo s, pou ou T>0,(3)
es localemen exac emen con ôlable en emps Taux ajec oi es (¯y, ¯p, ¯
θ)qui sa is on (6),(7) a ec des con ôles
e h els que i≡ N≡0. En pa iculie , si N=2, on a la con ôlabili é exac e locale aux ajec oi es a ec des
con ôles ≡0e h∈L2(O×(0,T)).
Pou no e oisième ésul a , on in odui l’espace
W=∇×z=(∂2z, −∂1z);z∈L20,T;H1(Ω).
Alo sona:
Théo ème 0.3. Soi N=2. Alo s, pou ou T>0e ou M>0,lesys ème(4)es con ôlable à zé o en emps
Ta ec des con ôles 1O,où ∈W.
Les démons a ions dé aillées de ces ésul a s se on données dans un a ail à pa aî e.
1. In oduc ion and main esul s
Le Ω⊂RNbe a bounded and egula domain (N=2o N=3), le O⊂Ωbe a nonemp y (small) open subse
and le T>0 be gi en. In his No e, we p esen se e al con ollabili y p ope ies o some nonlinea sys ems o he
Na ie –S okes and Boussinesq kind.
We will i s be conce ned wi h he Na ie –S okes sys em
y −y +(y ·∇)y +∇p= 1O,∇·y=0inQ=Ω×(0,T),
y=0onΣ=∂Ω ×(0,T),
y(0)=y0in Ω,(8)
whe e 1Ois he cha ac e is ic unc ion o O. Le us in oduce he Banach space E=H∩L4(Ω)N, whe e
H=w∈L2(Ω)N;∇·w=0inΩ, w ·n=0on∂Ω.(9)
I will be said ha (8) is locally exac ly con ollable o he ajec o ies a ime Ti , o each su icien ly egula
solu ion (¯y, ¯p) o he uncon olled sys em
¯y −¯y+(¯y·∇)¯y+∇¯p=0,∇·¯y=0inQ,
¯y=0onΣ,
he e exis s δ>0 such ha , whene e y0−¯y(0)E⩽δ, we can ind con ols ∈L2(O×(0,T))
Nand associa ed
s a es (y, p) sa is ying y(T) =¯y(T).
We will also conside he Boussinesq sys em





y −y +(y ·∇)y +∇p= 1O+θe
N,∇·y=0inQ,
θ −θ +y·∇θ=h1Oin Q,
y=0,θ=0onΣ,
y(0)=y0,θ(0)=θ0in Ω,
(10)
278 E. Fe nández-Ca a e al. / C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280
whe e he con ols a e now ∈L2(O×(0,T))
Nand h∈L2(O×(0,T)). I will be said ha (10) is locally exac ly
con ollable o he ajec o ies a ime Ti , o any su icien ly egula solu ion (¯y, ¯p, ¯
θ) o he sys em



¯y −¯y+(¯y·∇)¯y+∇¯p=¯
θeN,∇·¯y=0inQ,
¯
θ −¯
θ+¯y·∇¯
θ=0inQ,
¯y=0,¯
θ=0onΣ,
he e exis s δ>0 such ha , i (y0,θ0)−(¯y(0), ¯
θ(0))E×L2⩽δ, hen we can ind con ols ∈L2(O×(0,T))
N
and h∈L2(O×(0,T)) and associa ed s a es (y,p,θ) sa is ying y(T ) =¯y(T ) and θ(T)=¯
θ(T).
Finally, o N=2 we will also conside he sys ems wi h unca ed nonlinea i y
y −y +(y ·∇)TM(y) +∇p= 1O,∇·y=0inQ,
y·n=0,∇×y=0onΣ,
y(0)=y0in Ω,(11)
whe e he bounda y condi ions a e o he Na ie kind. He e, M>0, TM(y) =(TM(y1), TM(y2)) and
TM(s) =−Mi s⩽−M,
si −M⩽s⩽M,
Mi s⩾M.
I will be said ha (11) is (globally) null con ollable a ime Ti , o each y0∈H, he e exis s ∈L2(O×
(0,T))
2such ha he associa ed s a e sa is ies y(T ) =0.
In he p e ious sys ems, = (x, ) and h=h(x, ) a e con ol unc ions. The goal o his No e is o p o e he
exis ence o con ols wi h a educed numbe o deg ees o eedom such ha he p e ious con ollabili y p ope ies
hold.
Some hypo heses will be imposed on he con ol domain and he ajec o ies. Mo e p ecisely, we will equen ly
assume ha
∃x0∈∂Ω, ∃ε>0 such ha O∩∂Ω ⊃B(x0;ε) ∩∂Ω, (12)
(B(x0;ε) is he ball cen e ed a x0o adius ε),
¯y∈L∞(Q)N,¯y ∈L20,T;Lσ(Ω)Nσ>1i N=2
σ>6/5i N=3(13)
and
¯
θ∈L∞(Q), ¯
θ ∈L20,T;Lσ(Ω)σ>1i N=2
σ>6/5i N=3.(14)
Ou i s wo esul s a e he ollowing:
Theo em 1.1. Assume ha Osa is ies (12). Then, o any T>0,(8) is locally exac ly con ollable a ime T o he
ajec o ies (¯y, ¯p) sa is ying (13) wi h con ols ∈L2(O×(0,T))
Nsuch ha i≡0 o some i.
Theo em 1.2. Assume ha Osa is ies (12) wi h nk(x0)= 0 o some k<N. Then, o each T>0,(10) is locally
exac ly con ollable a ime T o he ajec o ies (¯y, ¯p, ¯
θ) sa is ying (13), (14) wi h con ols and hsuch ha
k≡ N≡0. In pa icula , i N=2, we ha e local exac con ollabili y o he ajec o ies wi h con ols ≡0and
h∈L2(O×(0,T)).
Fo ou las esul , le us in oduce he space
W=∇×z=(∂2z, −∂1z);z∈L20,T;H1(Ω).
We ha e he ollowing:
E. Fe nández-Ca a e al. / C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280 279
Theo em 1.3. Le N=2. Then, o any T>0and any M>0,(11)is null con ollable a ime Twi h con ols o
he o m 1O, whe e ∈W.
In he ollowing sec ions, we will indica e he main ideas used in he p oo s o he p e ious esul s ( o simplici y,
we will only e e o Theo ems 1.1 and 1.3). The de ailed p oo s will be gi en in a o hcoming pape .
2. The null con ollabili y o some simila linea sys ems
Following well known a gumen s, we will i s deduce null con ollabili y esul s o linea ized e sions o (8)
and (11), namely:
y −y +(¯y·∇)y +(y ·∇)¯y+∇p= + 1O,∇·y=0inQ,
y=0onΣ,
y(0)=y0in Ω
(15)
(whe e ¯ysa is ies (13) and = (x, )sa is ies app op ia e decay assump ions nea =T) and
y −y +(y ·∇)¯y+∇p= 1O,∇·y=0inQ,
y·n=0,∇×y=0onΣ,
y(0)=y0in Ω
(16)
(whe e we assume ha N=2 and ¯y∈L∞(Q)2).
Fo he null con ollabili y o (15) wi h k≡0, he main ool is a global Ca leman es ima e o he solu ions o
he associa ed adjoin sys em
−ϕ −ϕ −(Dϕ) ¯y+∇π=g, ∇·ϕ=0inQ,
ϕ=0onΣ,
ϕ(T ) =ϕ0in Ω,(17)
whe e Dϕ =∇ϕ+ ∇ϕand g∈L2(Q)N. Indeed, assume ha ( o ins ance) N=3 and n1(x0)= 0. Then he ask
is o p o e ha he solu ions o (17) sa is y

Q
ρ2
1|ϕ|2dxd ⩽C(Ω,O,T, ¯y)
Q
ρ2
2|g|2dxd +
O×(0,T )
ρ2
3|ϕ2|2+|ϕ3|2dxd (18)
o some app op ia e weigh s ρi=ρi(x, ) and some C(Ω,O,T, ¯y) > 0. This can be p o ed using i s a global
Ca leman inequali y es ablished in [1]. A his poin |ϕ1|2appea s in he second e m o he igh hand side o (18).
Using hypo hesis (12) and he ac s ha ∇·ϕ=0inQand ϕ1=0onΣwe can ge id o his local e m in |ϕ1|2.
On he o he hand, we can also deduce a null con ollabili y esul o (16) whene e ¯y∈L∞(Q)2. To his end,
le us in oduce he s eamline- o ici y o mula ion o (16), namely



ω −ω +∇×(∇×ψ)·∇¯y=∇×( 1O), ψ =ωin Q,
ψ=0,ω=0onΣ,
ω(0)=∇×y0in Ω
(19)
and he associa ed adjoin sys em



−ρ −ρ −∇×(¯y·∇×)∇θ=0,θ=ρin Q,
θ=0,ρ=0onΣ,
ρ(T)=ρ0in Ω.(20)
Then he ask amoun s o p o e he obse abili y inequali y

∇θ(0)

2
L2⩽C
O×(0,T )
|∇θ|2dxd (21)
o he solu ions o (20). The p oo o (21) elies on some global Ca leman inequali ies es ablished in [3] and [4].

280 E. Fe nández-Ca a e al. / C. R. Acad. Sci. Pa is, Se . I 340 (2005) 275–280
3. The local exac con ollabili y o he Na ie –S okes sys ems (8) and (11)
Theo em 1.1 is p o ed by applying an in e se mapping heo em. In he amewo k o he Na ie –S okes
equa ions, his s a egy was in oduced by O.Yu. Imanu ilo in [2] and has been used ecen ly in [1], elaxing
conside ably he egula i y equi emen s on he ajec o ies (¯y, ¯p).
On he o he hand, in o de o p o e Theo em 1.3 we can use a ixed poin a gumen . This is possible because
he unique assump ion on ¯yneeded o he null con ollabili y o (16) is ¯y∈L∞(Q)2. A his le el, he ac ha
N=2 and he pa icula o m o he bounda y condi ions in (11) a e essen ial. Thus, we can in oduce a se - alued
mapping z→ Λ(z) whe e, o each z∈L2(Q)2,Λ(z) is he amily o unc ions ywhich sol e ( oge he wi h some
p) he linea sys em (16) wi h ¯y=TM(z) and sa is y y(T) =0 (and sui able es ima es). I can be seen ha an
app op ia e e sion o Kaku ani’s ixed poin heo em can be applied o Λ.
Rema k 1. Assume ha N=2. The a gumen s in [1] implici ly show ha , unde hypo heses (13), we can ind
con ols 1Owi h ∈Wsuch ha he associa ed solu ions o (8) sa is y y(T) =¯y(T). Obse e ha he assump ion
(12) on he con ol domain is no necessa y he e.
Rema k 2. Assume ha N=3. I is na u al o ask whe he a esul simila o Theo em 1.1 holds wi h con ols
ha ing wo ze o componen s. In gene al, he answe is no.
In ac , i seems di icul o iden i y he open se s Ωand Osuch ha one has null con ollabili y o all T>0,
e en o he linea p oblems, wi h a educed numbe o con ols. This is unknown e en o he classical S okes
equa ions o which, up o now, he unique known esul s conce n app oxima e con ollabili y; see [5].
Acknowledgemen s
The wo i s au ho s ha e been pa ially suppo ed by D.G.E.S. (Spain), G an BFM2003-06446. The hi d
au ho has been suppo ed by NSF G an DMS 0205148.
Re e ences
[1] E. Fe nández-Ca a, S. Gue e o, O.Yu. Imanu ilo , J.P. Puel, Local exac con ollabili y o he Na ie –S okes sys em, J. Ma h. Pu es Appl.,
in p ess.
[2] O.Yu. Imanu ilo , Rema ks on exac con ollabili y o he Na ie –S okes equa ions, ESAIM Con ol Op im. Calc. Va . 6 (2001) 39–72.
[3] O.Yu. Imanu ilo , J.P. Puel, Global Ca leman es ima es o weak ellip ic non homogeneous Di ichle p oblem, In . Ma h. Res. No ices 16
(2003) 883–913.
[4] O.Yu. Imanu ilo , M. Yamamo o, Ca leman Es ima e o a Pa abolic Equa ion in a Sobole Space o Nega i e O de and i s Applica ions,
Lec u e No es in Pu e and Appl. Ma h., ol. 218, Dekke , New Yo k, 2001.
[5] J.-L. Lions, E. Zuazua, A gene ic uniqueness esul o he S okes sys em and i s con ol heo e ical consequences, in: P. Ma cellini,
G. Talen i, E. Visen ini (Eds.), Pa ial Di e en ial Equa ions and Applica ions, in: Lec u e No es in Pu e and Appl. Ma h., ol. 177, Dekke ,
New Yo k, 1996, pp. 221–235.