ESAIM: COCV ESAIM: Con ol, Op imisa ion and Calculus o Va ia ions
July 2006, Vol. 12, 466–483 www.edpsciences.o g/coc
DOI: 10.1051/coc :2006011
EXACT CONTROLLABILITY TO THE TRAJECTORIES OF THE HEAT
EQUATION WITH FOURIER BOUNDARY CONDITIONS: THE SEMILINEAR
CASE ∗
En ique Fe n´
andez-Ca a1, Manuel Gonz´
alez-Bu gos1, Se gio Gue e o2
and Jean-Pie e Puel3
Abs ac . This pape is conce ned wi h he global exac con ollabili y o he semilinea hea equa ion
(wi h nonlinea e ms in ol ing he s a e and he g adien ) comple ed wi h bounda y condi ions o
he o m ∂y
∂n + (y) = 0. We conside dis ibu ed con ols, wi h suppo in a small se . The null
con ollabili y o simila linea sys ems has been analyzed in a p e ious i s pa o his wo k. In
his second pa we show ha , when he nonlinea e ms a e locally Lipschi z-con inuous and sligh ly
supe linea , one has exac con ollabili y o he ajec o ies.
Ma hema ics Subjec Classi ica ion. 35K20, 93B05.
Recei ed Feb ua y 17, 2005. Re ised May 30, 2005 and June 13, 2005.
1. In oduc ion
Le Ω ⊂RN(N≥1) be a bounded connec ed open se whose bounda y ∂Ω is egula enough ( o ins ance
∂Ω∈C2). Le ω⊂Ω be a (small) nonemp y open subse and le T>0. We will use he no a ion Q=Ω×(0,T)
and Σ = ∂Ω×(0,T) and we will deno e by n(x) he ou wa d uni no mal o Ω a he poin x∈∂Ω.
We will conside he semilinea hea equa ion wi h nonlinea Fou ie (o Robin) bounda y condi ions
⎧
⎪
⎪
⎨
⎪
⎪
⎩
y −∆y+F(y,∇y)= 1ωin Q,
∂y
∂n + (y)=0 onΣ,
y(x, 0) = y0(x)inΩ.
(1)
He e, we assume ha ∈L2(ω×(0,T)) (a leas ), 1ωis he cha ac e is ic unc ion o ω,y0∈L∞(Ω) and
F:R×RN→ Rand :R→ Ra e gi en unc ions. In (1), y=y(x, ) is he s a e and = (x, )is he
con ol; i is assumed ha we can ac on he sys em only h ough ω×(0,T).
Keywo ds and ph ases. Con ollabili y, hea equa ion, Fou ie bounda y condi ions, semilinea .
∗This wo k has been pa ially suppo ed by D.G.E.S. (Spain), G an s BFM2000–1317 and BFM2003–06446.
1Dp o. E.D.A.N., Uni e si y o Se illa, Ap do. 1160, 41080 Se illa, Spain; [email p o ec ed];[email p o ec ed]; [email p o ec ed]
2Labo a oi e Jacques-Louis Lions, Uni e si ´e Pie e e Ma ie Cu ie, boˆı e cou ie 187, 75035 Cedex 05, Pa is, F ance;
[email p o ec ed]
3Labo a oi e de Ma h´ema iques Appliqu´ees, Uni e si ´e de Ve sailles – S . Quen in, 45 a enue des ´
E a s-Unis, 78035 Ve sailles,
F ance; [email p o ec ed]e. c
EDP Sciences, SMAI 2006
A icle published by EDP Sciences and a ailable a h p://www.edpsciences.o g/coc o h p://dx.doi.o g/10.1051/coc :2006011
E. FERN´
ANDEZ-CARA ET AL. 467
Fo he exis ence, uniqueness, egula i y and gene al p ope ies o he solu ions o p oblems like (1), see o
ins ance [1, 2, 7]. An illus a i e in e p e a ion o he da a and a iables in (1) is he ollowing. The unc ion
y=y(x, ) can be iewed as he ela i e empe a u e o a medium (wi h espec o he ex e io su ounding ai )
subjec o anspo and chemical eac ions. The pa abolic equa ion in (1) means, among o he hings, ha a
hea sou ce 1ωis applied on a pa o he body. On he bounda y, −∂y
∂n can be iewed as he no mal hea flux,
inwa ds di ec ed, up o a posi i e coefficien . Thus, he equali y
−∂y
∂n = (y)
means ha his flux is a (nonlinea ) unc ion o he empe a u e. Acco dingly, i is easonable o assume ha
is nondec easing and (0) = 0.
A simplified linea model which was conside ed in a p e ious pape [10] is he ollowing:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
y −∆y+a(x, )y+B(x, )·∇y= 1ωin Q,
∂y
∂n +β(x, )y=0 onΣ,
y(x, 0) = y0(x)inΩ.
(2)
He e, i is assumed ha he coefficien s a,Band βsa is y
a∈L∞(Q),B∈L∞(Q)N,β∈L∞(Σ) (3)
and, o he easons abo e, i is also na u al o assume ha β≥0 (al hough his assump ion was no used
in [10]).
The main goal o his pape is o analyze he con ollabili y p ope ies o he nonlinea sys em (1). Mo e
p ecisely, we will y o each exac ly uncon olled solu ions o (1), i.e. unc ions y=y(x, ) sa is ying
⎧
⎪
⎪
⎨
⎪
⎪
⎩
y −∆y+F(y,∇y)=0 inQ,
∂y
∂n + (y)=0 onΣ,
y(x, 0) = y0(x)inΩ.
(4)
I will be said ha (1) is (globally) exac ly con ollable o he ajec o ies a ime Ti , o any solu ion o (4)
wi h “sui able” egula i y and any y0∈L∞(Ω), he e exis con ols ∈L2(ω×(0,T)) and associa ed solu ions
y∈C0([0,T]; L2(Ω)) such ha
y(x, T )=y(x, T )inΩ.(5)
He e, by sui able egula i y we mean he ollowing:
y∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)) ∩L∞(Q), y0∈L∞(Ω).(6)
The con ollabili y p ope ies o semilinea ime-dependen sys ems ha e been s udied in ensi ely hese las
yea s. See o ins ance [8,11–13,15,16], whe e nonlinea i ies o he o m (y) a e conside ed. See also he gene al
ea ise [14]. In pa icula , o pa abolic sys ems comple ed wi h Di ichle bounda y condi ions, nonlinea
e ms (y,∇y) depending on bo h he s a e and he g adien ha e been aken in o accoun in [6, 9]. Fo he
simila linea sys em (2), he null con ollabili y was analyzed mo e in de ail in [10]. In he case o (1), some
pa ial esul s ha e been gi en in [5].
468 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION
Ou main esul conce ns he global exac con ollabili y o he ajec o ies o (1). I is he ollowing:
Theo em 1. Le us assume ha Fand a e locally Lipschi z-con inuous and sa is y
lim
|s|→∞
|F(s, p)−F( , p)|
|s− |log3/2(1 + |s− |)=0,(7)
uni o mly in ( , p)∈[−K, K]×RN∀K>0,
⎧
⎪
⎨
⎪
⎩
∀L>0,∃M>0such ha
|F(s, p)−F( , p)|≤M|s− |,|F(s, p)−F(s, q)|≤M|p−q|
∀(s, , p, q)∈[−L, L]2×RN×RN
(8)
and
lim
|s|→∞
| (s)− ( )|
|s− |log1/2(1 + |s− |)=0 (9)
uni o mly in ∈[−K, K]∀K>0. Then, o each T>0, he nonlinea sys em (1) is exac ly con ollable o he
ajec o ies a ime Twi h L∞con ols.
Rema k 1. Condi ions (7)–(9) a e sa isfied i Fand a e globally Lipschi z con inuous. No ice ha (7)
means ha he unc ion Fcan only be sligh ly supe linea in s, uni o mly in p. In he simila case o Di ichle
bounda y condi ions, i is known ha condi ions like hese a e sha p. Indeed, o ins ance, when Fdoes no
depend on pand
|F(s)−F( )|∼|s− |logβ(1 + |s− |),β>2,
due o blow-up phenomena, he sys em ails o be con ollable whene e ω= Ω (see [11]). On he o he hand,
(9) is also a sligh ly supe linea g ow h assump ion o . I would be in e es ing o know whe he a mo e
supe linea leading o blow up in he absence o con ol can also be an obs uc ion o he null con ollabili y
o (1). Bu his ques ion does no seem ob ious and emains open.
Rema k 2. A esul p o ed in [5] says ha when F≡0, is smoo h nea ze o and
(s)s≥0∀s∈R,(10)
he nonlinea sys em (1) is null con ollable o la ge T. Tha is o say, unde hese assump ions, o each
y0∈L2(Ω) he e exis T(y0)>0andcon ols in L∞(ω×(0,T) such ha he associa ed s a es ysa is y
y(x, T (y0)) = 0 in Ω.
By inspec ion o he p oo o heo em 1, we see ha he same esul holds o (1) wi h F≡0 whene e is
locally Lipschi z-con inuous and sa isfies he good sign condi ion (10).
Fo he p oo o Theo em 1, we will fi s es ablish a null con ollabili y esul o (2) (see P op. 1 below).
This will be used, oge he wi h an app op ia e fixed poin a gumen , o deduce he desi ed esul .
This s a egy was in oduced in [15] in he amewo k o he exac con ollabili y o he semilinea wa e
equa ion. See also [8,12] o simila esul s conce ning he app oxima e and null con ollabili y o he semilinea
hea equa ion wi h Di ichle o Neumann bounda y condi ions.
Ou null con ollabili y esul o (2) is he ollowing:
P oposi ion 1. Fo e e y T>0,sys em(2) is null con ollable a ime T, wi h con ols in L∞(ω×(0,T)).
Mo e p ecisely, o each y0∈L2(Ω), he eexis s ∈L∞(ω×(0,T)) such ha he associa ed solu ion o (2)
sa isfies (5). Fu he mo e, he con ol can be ound sa is ying
 L∞(ω×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω) ,(11)
E. FERN´
ANDEZ-CARA ET AL. 469
whe e
K=1+1/T +a2/3
∞+B2
∞+β2
∞+T(1 + a∞+B2
∞+β2
∞).(12)
Fo he p oo o p oposi ion 1, we fi s in oduce a con ol L2(ω×(0,T)) which leads he solu ion o (2) o
ze o a ime T. In a second s ep, a guing as in Sec ion 2 in [4], a egula izing a gumen will lead o he
desi ed L∞con ol.
The es o his pape is o ganized as ollows. In Sec ion 2, we p o e P oposi ion 1. Sec ion 3 is de o ed
o he p oo o Theo em 1. Fo comple eness, we ha e also included an Appendix whe e he p oo o a a he
echnical local egula i y esul is gi en in de ail.
In he sequel, Cdeno es a gene ic posi i e cons an only depending on Ω and ω.
2. A null con ollabili y esul o he linea sys em
In his sec ion we p esen he p oo o P oposi ion 1.
Le y0∈L2(Ω) be gi en and le us in oduce wo open se s ωand ω,wi hω ⊂⊂ ω⊂⊂ ω. Then, we
can use he main esul in [10] (Th. 2) wi h con ol egion ω ×(0,T) o deduce he exis ence o a con ol
 ∈L2(ω ×(0,T)) such ha he associa ed solu ion o (2) e ifies (5) and also he es ima e
 L2(ω ×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω),(13)
whe e Kis o he o m (12).
Le us deno e by y he s a e associa ed o  . We now in oduce a cu -off unc ion η=η( ) sa is ying
η∈C∞([0,T]),η( )=1in(0,T/4),η( )=0in(3T/4,T)
and
0≤η( )≤1,|η( )|≤C
in (0,T)
and we deno e by χ he solu ion o he sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
χ −∆χ+a(x, )χ+B(x, )·∇χ=0 inQ,
∂χ
∂n +β(x, )χ=0 onΣ,
χ(x, 0) = y0(x)inΩ.
Then, he unc ion w=y−ηχ sa isfies
⎧
⎪
⎪
⎨
⎪
⎪
⎩w −∆w+a(x, )w+B(x, )·∇w=−η( )χ+ 1ω in Q,
∂w
∂n +β(x, )w=0 onΣ,
w(x, 0) = 0,w(x, T )=0 inΩ.
Ou aim is o cons uc a con ol ∈L∞(ω×(0,T)) which d i es he solu ion o (2) o ze o a ime =T.To
his end, we will need a local egula i y esul o he solu ions o linea hea equa ions wi h L∞coefficien s a
and B. This will be used below o he unc ions χand wand eads as ollows:
Lemma 1. Le us deno e by Y he space L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)).Le y∈Ybe a solu ion o he
equa ion
y −∆y+a(x, )y+B(x, )·∇y= , (14)
whe e a∈L∞(Q),B∈L∞(Q)Nand ∈L2(Q).Le O⊂Ωbe a nonemp y open se and assume ha is L∞
in he cylinde O×(0,T).Then
y∈L∞(δ, T ;W1,∞(O))
470 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION
o any δ∈(0,T)and any nonemp y open se O⊂⊂ O. Fu he mo e, he e exis s a posi i e cons an C(O)
such ha he ollowing es ima e holds:
yL∞(δ,T ;W1,∞(O)) ≤C(O)(T1/2+TN/2)1+δ−1+a∞+B∞N+1 yY+ L∞(O×(0,T )).(15)
The p e ious egula i y also holds wi h δ=0i , besides (14), we ha e y(x, 0) = 0 in Ω. In ha case, one has
an es ima e simila o (15) wi hou he e m in δ.
This lemma is implied by well known pa abolic egula i y heo y. Fo comple eness, i s p oo is gi en in an
Appendix, a he end o his pape .
Le us now conside an open se ω0wi h ω⊂⊂ ω0⊂⊂ ωand a cu -off unc ion ξ,wi h
ξ∈C2
0(ω0),ξ≡1inω
and le us se w=(1−ξ)w.Thenweha e:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
w −∆w+a(x, )w+B(x, )·∇w=−η( )χ+ 1ωin Q,
∂w
∂n +β(x, )w=0 onΣ,
w(x, 0) = 0,w(x, T )=0 inΩ,
wi h
=ηξχ+2∇ξ·∇w+∆ξw−B·∇ξw. (16)
Le us ema k ha supp ⊂ω×[0,T]. The e o e, i we p o e ha ∈L∞(ω×(0,T)), we will ha e ha he
unc ion y=w+ηχ sol es ( oge he wi h ) he null con ollabili y p oblem o (2).
Thus, le us check ha ∈L∞(ω×(0,T)) and le us es ima e i s no m in his space:
•The egula i y o he fi s e m in he igh hand side o (16) is implied by he in e io egula i y o χ
no only in space bu in ime as well. F om Lemma 1 wi h O=ω, we deduce ha χ∈L∞(ω0×(δ, T )) wi h
supp ξ⊂ω0⊂⊂ ω(we e en ha e χ∈L∞(δ, T ;W1,∞
loc (ω))) and
χL∞(ω0×(δ,T )) ≤C(T1/2+TN/2)1+δ−1+a∞+B∞N+1 χY;
ecall ha Y=L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)).
Consequen ly aking o ins ance δ=T/8, since η≡0in(0,T/4), we ge
ηξχL∞(ω×(0,T )) ≤CT−1(T1/2+TN/2)1+T−1+a∞+B∞N+1 χY.
•The egula i y o he o he h ee e ms in he igh hand side o (16) is ela ed o he in e io space
egula i y o w. Thus, le us in oduce ω1wi h ω0⊂⊂ ω1⊂⊂ ωand le us apply Lemma 1 wi h O=ω1 ω.
This gi es w∈L∞(0,T;W1,∞(ω0 ω)) and he es ima e
wL∞(0,T;W1,∞(ω0 ω)) ≤C(T1/2+TN/2)(1+a∞+B∞)N+1 (wY+ηχL∞(ω1×(0,T))),
whence 2∇ξ·∇w+∆ξw−B·∇ξwL∞(ω×(0,T )) ≤C(T1/2+TN/2)
×(1 + a∞+B∞)N+2 (wY+ηχL∞(ω1×(0,T))).

E. FERN´
ANDEZ-CARA ET AL. 471
Pu ing he p e ious es ima es oge he , we find ha ∈L∞(ω×(0,T)) and
 L∞(ω×(0,T)) ≤C(1 + TN−1)1+T−1+a∞+B∞2N+3 (wY+χY).(17)
A his poin , no ice ha o any ∈L2(Q)andanyy0∈L2(Ω) he solu ion y o he linea sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
y −∆y+a(x, )y+B(x, )·∇y= in Q,
∂y
∂n +β(x, )y=0 onΣ,
y(x, 0) = y0(x)inΩ
(18)
sa isfies
yY≤eCT(1+a∞+B2
∞+β2
∞)( L2(Q)+y0L2(Ω)).
Fo a de ailed p oo , see o example p oposi ion 1 in [10].
Thiscanbeused oes ima ewYand χYin e ms o  L2(ω×(0,T)) and y0L2(Ω). In iew o (17), we
see ha
 L∞(ω×(0,T)) ≤L( L2(ω ×(0,T )) +y0L2(Ω)),(19)
whe e
L=CT−1(1 + TN−1)1+T−1+a∞+B∞2N+3 exp{CT(1 + a∞+B2
∞+β2
∞)}.
Combining his es ima e and (13), we finally ob ain ha
 L∞(ω×(0,T)) ≤eCK(T,a∞,B∞,β∞)y0L2(Ω),(20)
whe e Kis gi en by (12).
This ends he p oo o P oposi ion 1.
3. Con ollabili y o he nonlinea sys em
In his sec ion we will p o e Theo em 1. The ollowing auxilia y esul will be needed:
P oposi ion 2. Le us assume ha , in (18), we ha e ∈L∞(Q)and y0∈L∞(Ω). Le us also assume ha
he coefficien s a,Band βsa is y (3).Theny∈L∞(Q)and
y∞≤eCT(1+a∞+B2
∞+β2
∞)y0∞+ ∞.(21)
o some C=C(Ω).
P oo . We will conside wo diffe en si ua ions:
Case 1. We will fi s assume ha a≥1andβ≥0 and we will es ablish (21) in his case. In ac , we will show
ha , unde hese assump ions,
y∞≤y0∞+ ∞.(22)
To his end, le us in oduce he sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
z −∆z+a(x, )z+B(x, )·∇z=hin Q,
∂z
∂n +β(x, )z=kon Σ,
z(x, 0) = z0(x)inΩ,
472 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION
whe e h∈L∞(Q), k∈L∞(Σ) and z0∈L∞(Ω) and le us show ha , i h,z0and ka e nonnega i e, hen his
is also he case o z.
Indeed, by mul iplying he equa ion sa isfied by zby z−(·, ) ( he nega i e pa o z(·, )) o each ∈(0,T)
and in eg a ing in Ω, a e se e al simplifica ions, we find:
1
2
d
d Ω
|z−(x, )|2dx+Ω
|∇z−(x, )|2dx
+∂Ω
β(x, )|z−(x, )|2dσ+∂Ω
k(x, )z−(x, )dσ+Ω
a(x, )|z−(x, )|2dx=
−Ω
h(x, )z−(x, )dx−Ω
B(x, )·∇z−(x, )z−(x, )dx.
F om his iden i y, in iew o he posi i eness o a,h,βand k, we easily deduce ha
d
d Ω
|z−(x, )|2dx≤B2
∞Ω
|z−(x, )|2dx,
whence z≥0inQ.
Now, le M>0 be a la ge cons an ( o be chosen below). The unc ion z=M−ysa isfies
⎧
⎪
⎪
⎨
⎪
⎪
⎩
z −∆z+a(x, )z+B(x, )·∇z=a(x, )M− in Q,
∂z
∂n +β(x, )z=β(x, )Mon Σ,
z(x, 0) = M−y0(x)inΩ.
The e o e, i we ake
M≥max{ L∞(Q),y0L∞(Ω)},
we can apply he p e ious a gumen and deduce ha y≤M. In a simila way, one can deduce ha y≥−M
and, consequen ly, |y|≤M. This p o es ha whene e a≥1andβ≥0, he es ima e (22) holds.
Case 2. We will now p o e (21) o gene al L∞coefficien s aand β.
Le γ∈C2(Ω) be a unc ion sa is ying
γ≥0inΩ,∂γ
∂n ≤−β∞on ∂Ω,γ∞≤1,
∇γ∞≤Cβ∞,D2γ∞≤Cβ2
∞.
(23)
We gi e he e a ske ch o he p oo o he exis ence o such a unc ion γ. To his end, le δ>0beapa ame e
(depending on Ω) such ha
x∈Ωδ→ dis (x, ∂Ω)
is C2,wi hΩ
δ={x∈Ω:dis (x, ∂Ω) <δ}. We dis inguish wo cases.
Le usfi s assume ha β∞≥1/δ.Thenwe akeγ(x)≡1inΩ Ωδ,γ(x)=β∞dis (x, ∂Ω) in Ωεwi h
ε=1/(2β∞) and a egula iza ion o γin Ωδ Ωε. This gi es he desi ed p ope ies o γ.
On he o he hand, i β∞<1/δ,we akeγ(x)=δβ∞in Ω Ωδ,γ(x)=β∞dis (x, ∂Ω) in Ωδ/2and a
egula iza ion in Ωδ Ωδ/2. This also p o ides a desi ed unc ion in his case.
E. FERN´
ANDEZ-CARA ET AL. 473
Le us now se y=e
γ(x)y.Thenysa isfies
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩y −∆y+a(x, )y+
B(x, )·∇y=e
γ(x) in Q,
∂y
∂n +
β(x, )y=0 onΣ,
y(x, 0) = eγ(x)y0(x)inΩ,
(24)
whe e a=a+∆γ−|∇γ|2−B·∇γ,

B=B+2∇γ, 
β=β−∂γ
∂n ≥0onΣ.
No ice ha , om he inequali ies (23) sa isfied by γ, we know ha
|a+∆γ−|∇γ|2−B·∇γ|≤C1(a∞+B2
∞+β2
∞)inQ
o some C1>0.
Now, le us se
y=e
−(C1(a∞+B2
∞+β2
∞)+1) y.
Then ysa isfies ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩y −∆y+a(x, )y+(B(x, )+2∇γ(x)) ·∇y=
in Q,
∂y
∂n +
β(x, )y=0 onΣ,
y(x, 0) = eγ(x)y0(x)inΩ,
(25)
whe e
a=a+∆γ−|∇γ|2−B·∇γ+C1(a∞+B2
∞+β2
∞)+1,

=e
−(C1(a∞+B2
∞+β2
∞)+1) +γ(x)
and 
β=
β.
Since a≥1and
β≥0, we can apply case 1 o y. This p o ides he es ima es
y∞≤y∞≤eCT(1+a∞+B2
∞+β2
∞)(y0∞+ ∞),
whence we deduce (21). 
Le us now s a wi h he p oo o Theo em 1. Le y0∈L∞(Ω) and ybe gi en and assume ha ysa isfies
(6) and (4) in he weak sense. Le us conside he nonlinea sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
w −∆w+F1(w,∇w;x, )w+F2(∇w;x, )·∇w= 1ωin Q,
∂w
∂n +F3(w;x, )w=0 onΣ,
w(x, 0) = y0(x)−y(x, 0) in Ω,
(26)
whe eweha eused heno a ion
F1(s, p;x, )=F(y(x, )+s, ∇y(x, )+p)−F(y(x, ),∇y(x, )+p)
s,(27)
F2=(F21,...,F
2N),F
2j(p;x, )=1
0
∂F
∂pj
(y(x, ),∇y(x, )+λp)dλ(28)
474 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION
and
F3(s;x, )= (y(x, )+s)− (y(x, ))
s(29)
o s∈Rand p∈RN.
We will p o e ha he e exis a con ol ∈L∞(ω×(0,T)) and an associa ed solu ion o (26) such ha
w(x, T )=0 in Ω.(30)
Wi h his con ol and he s a e y=w+y, we will ha e sol ed he exac con ollabili y p oblem o (1) and we
will ha e hus p o ed Theo em 1.
We will fi s assume ha he unc ions Fand a e con inuously diffe en iable. Then, by a densi y a gumen ,
we will be able o p o e he esul in he gene al case.
3.1. The case in which Fand a e C1
The idea o he p oo is well known: we in oduce an app op ia e (se - alued) fixed poin mapping and
we check ha i possesses a leas one fixed poin ; his will be a solu ion o he null con ollabili y p oblem
associa ed o (26).
Le R>0 be gi en and le us in oduce he ollowing unc ion:
MR(s)=⎧
⎪
⎨
⎪
⎩
−Ri s<−R,
si −R≤s≤R,
Ri s>R.
Le us deno e by Z he Hilbe space Z=L2(0,T;H1(Ω)) and le us se o each R>0andeachz∈Z
aR,z(x, )=F1(MR(z(x, )),∇z(x, ); x, ),
Bz(x, )=F2(∇z(x, ); x, )
and
βR,z(x, )=F3(MR(z(x, )); x, ).
Conside he linea null con ollabili y p oblem
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
w −∆w+aR,z(x, )w+Bz(x, )·∇w= 1ωin Q,
∂w
∂n +βR,z(x, )w=0 onΣ,
w(x, 0) = y0(x)−y(x, 0) in Ω,
(31)
oge he wi h (30).
Le us in oduce he unc ion w0,wi hw0(x)=y0(x)−y(x, 0) o all x∈Ω. F om (6), (8) and he ac ha
∈C1(R), we ha e
aR,z ∈L∞(Q),B
z∈L∞(Q)N,β
R,z ∈L∞(Σ).
Consequen ly, in iew o P oposi ion 1, (30)–(31) can be sol ed wi h con ols in L∞(ω×(0,T)).
We a e now going o selec a pa icula solu ion o (30)–(31) cons uc ed as in [11]. To do his, we fi s se
TR=min{T,a−1/3
R}>0, whe e
aR=sup
|s|≤R, p∈RN
ess sup
(x, )∈Q
|F1(s, p;x, )|.
E. FERN´
ANDEZ-CARA ET AL. 481
We ha e 0∈L2(Q). Consequen ly, ∆y0∈L2(Q), y0∈C0([0,T]; H1
0(Ω)) and app op ia e es ima es a e
sa isfied. Indeed, by mul iplying he equa ion sa isfied by y0by −∆y0and in eg a ing wi h espec o xin Ω,
we find 1
2
d
d ∇y0(·, )2
L2+Ω
|∆y0(x, )|2dx=−Ω
0(x, )∆y0(x, )dx. (45)
Since
 0L2≤C L2+(1+δ−1+a∞+B∞)yY,
we easily ob ain om (45) ha
y0C0([0,T];H1
0(Ω)) +∆y0L2(Q)≤C L2+1+δ−1+a∞+B∞yY.(46)
Clea ly, he same es ima e holds o
yL∞(δ/(N+1),T ;H1(O0)) +∆yL2(O0×(δ/(N+1),T )).
Le us now y o imp o e he local space egula i y p ope ies o y. To his end, we will use he ollowing
lemma:
Lemma 2. Le us se p0=2,le pibe defined by
1
pi
=1
pi−1
−1
2N
o 1≤i≤N−1and le us se pN=+∞. Le us deno e by Xi he space
Xi=L∞((i+1)δ/(N+1),T;W1,pi(Oi))
o 0≤i≤Nand suppose ha y∈Xj−1 o some j. Then we also ha e y∈Xjand
yXj≤C(O)(T1/2 ∞+D(T,δ,a∞,B∞)yXj−1),
whe e
D(T,δ,a∞,B∞)=(T1/4+T1/2)(1 + δ−1+a∞+B∞).(47)
P oo o Lemma 2. Le us in oduce ξj∈C2
c(Oj−1)andηj∈C1([0,T]), wi h
ξj(x)=1inOj,η
j( )=1in[(j+1)δ/(N+1),T],
ηj( )=0in[0,jδ/(N+1)],|ηj, ( )|≤C
δin (0,T)
and le us pu yj=ηjξjy.Thenyjsa isfies he ollowing:
⎧
⎪
⎨
⎪
⎩
yj, −∆yj= jin Q,
yj=0 onΣ,
yj(x, 0) = 0 in Ω
(48)
wi h
j= j,1+ j,2+ j,3,
whe e j,1=ηjξj , j,2=ηj, ξjy−ηj∆ξjy−aη
jξjy−ηj(B·∇ξj)y,
j,3=−2ηj∇ξj·∇y−ηjξjB·∇y.

482 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION
F om he ac ha he sys em (48) is linea , we see ha yjcan be w i en as he sum o h ee solu ions o
simila sys ems wi h igh hand sides j,1, j,2and j,3. Le us espec i ely deno e hem by yj,1,yj,2and yj,3.
We a e now going o deduce es ima es o yj,k in Xj o 1 ≤k≤3.
To his end, we will use he usual ep esen a ion o yj,k p o ided by he semig oup S( )associa ed o he
hea equa ion wi h homogeneous Di ichle condi ions, say
yj,k(·, )=
0
S( −s) j,k(·,s)ds
o all ∈(0,T).
Since ∈L∞(Q), we can w i e
yj,1(·, )W1,pj(Ω) ≤C
0
( −s)−1/2 j,1(·,s)Lpj(Ω) ds.
The e o e, om Young’s inequali y we find ha yj,1∈L∞(0,T;W1,pj(Ω)) and
yj,1L∞(0,T ;W1,pj(Ω)) ≤CT1/2 j,1L∞(0,T ;Lpj(Ω))
≤C(O)T1/2 L∞(O×(0,T)) .
Taking in o accoun ha j,2∈L∞(0,T;Lp∗
j−1(Ω)) wi h
p∗
j−1=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∞i j>N−1,
p(a bi a y in (1,+∞)) i j=N−1,
2N
N−j−1i j<N−1,
we see ha j,2is no wo se han j,1and, again,
yj,2(·, )W1,pj(Ω) ≤C
0
( −s)−1/2 j,2(·,s)Lpj(Ω) ds
o all . F om Young’s inequali y and he assump ion y∈Xj−1,wealsoge yj,2∈L∞(0,T;W1,pj(Ω)) and
yj,2L∞(0,T ;W1,pj(Ω)) ≤CT1/2 j,2L∞(0,T ;Lp∗
j−1(Ω))
≤C(O)T1/2(1 + δ−1+a∞)yXj−1.
In he defini ion o j,3, we find ∇y. Consequen ly, we can only ensu e ha j,3∈L∞(0,T;Lpj−1(Ω)). Since
−N
21
pj−1
−1
pj−1
2=−3
4,
we ha e
yj,3(·, )W1,pj(Ω) ≤C
0
( −s)−3/4 j,3(·,s)Lpj−1(Ω) ds
and now Young’s inequali y gi es yj,3∈L∞(0,T;W1,pj(Ω)) and
yj,3L∞(0,T ;W1,pj(Ω)) ≤CT1/4 j,3L∞(0,T ;Lpj−1(Ω))
≤C(O)T1/4(1 + B∞)yXj−1.
E. FERN´
ANDEZ-CARA ET AL. 483
Pu ing he es ima es o yj,kL∞(0,T;W1,pj(Ω)) oge he and aking in o accoun he defini ions o ηjand ξj,
we ob ain he desi ed inequali y o yXj.
This concludes he p oo o Lemma 2. 
Since we al eady had y∈X0, we deduce om Lemma 2 ha y∈XNand
yXN≤C(O)(T1/2 ∞+D(T,δ,a∞,B∞)yXN−1),
whe e, Dis gi en by (47).
We can apply Lemma 2 subsequen ly o j=N,N −1,...,1. The es ima es we find yield
yXN≤C(T1/2(1 + DN−1) L∞(O×(0,T )) +DNyX0).
This, oge he wi h (46), yields
yXN≤C(T1/2+TN/2)D(T,δ,a∞,B∞)N+1( L∞(O×(0,T )) +yY),
which is exac ly (15).
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