Some remarks on the exact controllability to trajectories for the nonlinear heat equation
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Some ema ks on he exac con ollabili y o
ajec o ies o he nonlinea hea equa ion
M. González-Bu gos
Wo kshop on Con ol and In e se P oblems
Besançon, June 2010
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
Con en s
1In oduc ion. S a emen o he p oblem
2Null Con ollabili y o he linea p oblem wi h egula con ols
Fi s app oach
Second app oach
Thi d app oach
3The “bes ” null con ol
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Le Ω⊂RNbe a bounded domain, N≥1, wi h bounda y ∂Ωo class
C2. Le ω⊆Ωbe an open subse an le us ix T>0.
We conside he linea and nonlinea p oblems o he hea equa ion:
(1)
∂ y−∆y+ay= 1ωin Q= Ω ×(0,T),
y=0 on Σ = ∂Ω×(0,T),
y(·,0) = y0in Ω,
(2) (∂ y−∆y+F(y) = 1ωin Q,
y=0 on Σ,y(·,0) = y0in Ω.
In (1) and (2), 1ω ep esen s he cha ac e is ic unc ion o he se ω,
y(x, )is he s a e, y0is he ini ial da um and is gi en in an app op ia e
space, and is he con ol unc ion (which is localized in ω
-dis ibu ed con ol-). In (1), a∈L∞(Q)is gi en. We will assume ha
F:R→Ris a gi en unc ion.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Le Ω⊂RNbe a bounded domain, N≥1, wi h bounda y ∂Ωo class
C2. Le ω⊆Ωbe an open subse an le us ix T>0.
We conside he linea and nonlinea p oblems o he hea equa ion:
(1)
∂ y−∆y+ay= 1ωin Q= Ω ×(0,T),
y=0 on Σ = ∂Ω×(0,T),
y(·,0) = y0in Ω,
(2) (∂ y−∆y+F(y) = 1ωin Q,
y=0 on Σ,y(·,0) = y0in Ω.
In (1) and (2), 1ω ep esen s he cha ac e is ic unc ion o he se ω,
y(x, )is he s a e, y0is he ini ial da um and is gi en in an app op ia e
space, and is he con ol unc ion (which is localized in ω
-dis ibu ed con ol-). In (1), a∈L∞(Q)is gi en. We will assume ha
F:R→Ris a gi en unc ion.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Le Ω⊂RNbe a bounded domain, N≥1, wi h bounda y ∂Ωo class
C2. Le ω⊆Ωbe an open subse an le us ix T>0.
We conside he linea and nonlinea p oblems o he hea equa ion:
(1)
∂ y−∆y+ay= 1ωin Q= Ω ×(0,T),
y=0 on Σ = ∂Ω×(0,T),
y(·,0) = y0in Ω,
(2) (∂ y−∆y+F(y) = 1ωin Q,
y=0 on Σ,y(·,0) = y0in Ω.
In (1) and (2), 1ω ep esen s he cha ac e is ic unc ion o he se ω,
y(x, )is he s a e, y0is he ini ial da um and is gi en in an app op ia e
space, and is he con ol unc ion (which is localized in ω
-dis ibu ed con ol-). In (1), a∈L∞(Q)is gi en. We will assume ha
F:R→Ris a gi en unc ion.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Rema k
In his alk we a e in e es ed in s udying he con ollabili y p ope ies o
sys ems (1) and (2) (con ollabili y o ajec o ies).
Linea P oblem: Fo e e y ωand Tsys em (1) is null con ollable
(equi alen ly exac ly con ollable o ajec o ies): Fo e e y y0∈L2(Ω)
he e is ∈L2(Q)s. . he solu ion y o (1) sa is ies y(T)≡0 in Ω.
1H.O. FATTORINI, D.L. RUSSELL,Exac con ollabili y heo ems o
linea pa abolic equa ions in one space dimension, A ch. Ra ional
Mech. Anal. 43 (1971), 272–292.
2G. LEBEAU, L. ROBBIANO,Con ôle exac de l’équa ion de la
chaleu , Comm. P.D.E. 20 (1995), no. 1-2, 335–356.
a≡0: ∈C∞
0(ω×(0,T)).
3O. YU. IMANUVILOV,Con ollabili y o pa abolic equa ions,
(Russian) Ma . Sb. 186 (1995), no. 6, 109–132; ansla ion in Sb.
Ma h. 186 (1995), no. 6, 879–900.
a∈L∞(Q): ∈L2(Q).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Rema k
In his alk we a e in e es ed in s udying he con ollabili y p ope ies o
sys ems (1) and (2) (con ollabili y o ajec o ies).
Linea P oblem: Fo e e y ωand Tsys em (1) is null con ollable
(equi alen ly exac ly con ollable o ajec o ies): Fo e e y y0∈L2(Ω)
he e is ∈L2(Q)s. . he solu ion y o (1) sa is ies y(T)≡0 in Ω.
1H.O. FATTORINI, D.L. RUSSELL,Exac con ollabili y heo ems o
linea pa abolic equa ions in one space dimension, A ch. Ra ional
Mech. Anal. 43 (1971), 272–292.
2G. LEBEAU, L. ROBBIANO,Con ôle exac de l’équa ion de la
chaleu , Comm. P.D.E. 20 (1995), no. 1-2, 335–356.
a≡0: ∈C∞
0(ω×(0,T)).
3O. YU. IMANUVILOV,Con ollabili y o pa abolic equa ions,
(Russian) Ma . Sb. 186 (1995), no. 6, 109–132; ansla ion in Sb.
Ma h. 186 (1995), no. 6, 879–900.
a∈L∞(Q): ∈L2(Q).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Rema k
In his alk we a e in e es ed in s udying he con ollabili y p ope ies o
sys ems (1) and (2) (con ollabili y o ajec o ies).
Linea P oblem: Fo e e y ωand Tsys em (1) is null con ollable
(equi alen ly exac ly con ollable o ajec o ies): Fo e e y y0∈L2(Ω)
he e is ∈L2(Q)s. . he solu ion y o (1) sa is ies y(T)≡0 in Ω.
1H.O. FATTORINI, D.L. RUSSELL,Exac con ollabili y heo ems o
linea pa abolic equa ions in one space dimension, A ch. Ra ional
Mech. Anal. 43 (1971), 272–292.
2G. LEBEAU, L. ROBBIANO,Con ôle exac de l’équa ion de la
chaleu , Comm. P.D.E. 20 (1995), no. 1-2, 335–356.
a≡0: ∈C∞
0(ω×(0,T)).
3O. YU. IMANUVILOV,Con ollabili y o pa abolic equa ions,
(Russian) Ma . Sb. 186 (1995), no. 6, 109–132; ansla ion in Sb.
Ma h. 186 (1995), no. 6, 879–900.
a∈L∞(Q): ∈L2(Q).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
1. In oduc ion. S a emen o he p oblem
Nonlinea P oblem: Unde app op ia e assump ions on he unc ion
F(which has a supe linea g ow h a in ini y) sys em (2) is exac ly
con ollable o ajec o ies a ime T:
1E. FERNÁNDEZ-CARA,Null con ollabili y o he semilinea hea
equa ion, ESAIM Con ol Op im. Calc. Va . 2 (1997), 87–103.
F(s)∼ |s|log(1+|s|).
2E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and app oxima e
con ollabili y o weakly blowing up semilinea hea equa ions,
Ann. Ins . H. Poinca é Anal. Non Linéai e 17 (2000), no. 5,
583–616.
F(s)∼ |s|logp(1+|s|),p∈[0,3/2).
3V. BARBU,Exac con ollabili y o he supe linea hea equa ion,
Appl. Ma h. Op im. 42 (2000), no. 1, 73–89.
F(s)∼ |s|logp(1+|s|)(p∈[0,3/2)), 1 ≤N<6 and a dissipa i i y
condi ion on he he nonlinea i y: sF(s)≥ −µo|s|2(µ0≥0).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
We conside he dis ibu ed con ollabili y p oblem o he linea
sys em:
(1) (∂ y−∆y+ay= 1ωin Q,
y=0 on Σ,y(·,0) = y0in Ω,
whe e ω⊂Ωis an open subse , ∈L2(Q)is he con ol and y0is
gi en in L2(Ω).
Le us ix ϕ0∈L2(Ω) and conside he adjoin p oblem
(3) (−∂ ϕ−∆ϕ+aϕ=0 in Q,
ϕ=0 on Σ, ϕ(T) = ϕ0in Ω.
I is well known:
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
Theo em
The ollowing condi ions a e equi alen :
1The e exis s Cs. . ∀y0∈L2(Ω), he e is ∈L2(Q), wi h
k k2
L2(Q)≤Cky0k2
L2(Ω),
s. . he solu ion y o (1) associa ed o y0and sa is ies
y (T) = 0in L2(Ω).
2The e exis s C>0s. . (obse abili y inequali y)
kϕ(0)k2
L2(Ω) ≤CZZω×(0,T)
|ϕ(x, )|2dx d ,
holds o e e y solu ion ϕ o he adjoin p oblem (3) associa ed o
ϕ0∈L2(Ω).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
The obse abili y inequali y o he adjoin p oblem wi h an explici
exp ession o Cwi h espec o he da a can be ob ained om a global
Ca leman inequali ies o he linea pa abolic p oblem:
(4) (−∂ ϕ−∆ϕ=F0in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω,
wi h F0∈L2(Q)and ϕ0∈L2(Ω) a e gi en.
In
V. A. FURSIKOV, O. YU. IMANUVILOV,Con ollabili y o Pa abolic
Equa ions, Lec u e No es Se ies 34, Seoul Na ional Uni e si y,
Resea ch Ins i u e o Ma hema ics, Seoul, 1996,
E. FERNÁNDEZ-CARA, E. ZUAZUA,The cos o app oxima e
con ollabili y o hea equa ions: he linea case. Ad . Di e en ial
Equa ions 5 (2000), no. 4-6, 465–514,
i is p o ed:
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
The obse abili y inequali y o he adjoin p oblem wi h an explici
exp ession o Cwi h espec o he da a can be ob ained om a global
Ca leman inequali ies o he linea pa abolic p oblem:
(4) (−∂ ϕ−∆ϕ=F0in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω,
wi h F0∈L2(Q)and ϕ0∈L2(Ω) a e gi en.
In
V. A. FURSIKOV, O. YU. IMANUVILOV,Con ollabili y o Pa abolic
Equa ions, Lec u e No es Se ies 34, Seoul Na ional Uni e si y,
Resea ch Ins i u e o Ma hema ics, Seoul, 1996,
E. FERNÁNDEZ-CARA, E. ZUAZUA,The cos o app oxima e
con ollabili y o hea equa ions: he linea case. Ad . Di e en ial
Equa ions 5 (2000), no. 4-6, 465–514,
i is p o ed:
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
Lemma
The e exis a egula and s ic ly posi i e unc ion, α0, and wo
cons an s C0yσ0(only depending on Ωand ω) s. .
I(ϕ)≡s−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
+sZZQ
e−2sα −1(T− )−1|∇ϕ|2+s3ZZQ
e−2sα −3(T− )−3|ϕ|2
≤C0 s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2+ZZQ
e−2sα|F0|2!,
∀s≥s0=σ0(Ω,ω)(T+T2), (ϕis he solu ion o (4)associa ed o
ϕ0∈L2(Ω)). The unc ion α=α(x, )is gi en by
α(x, ) = α0(x)/ (T− ).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
Coming back o he adjoin p oblem
(3) −∂ ϕ−∆ϕ+aϕ=0 in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω.
Lemma
The e exis C1>0and σ1>0(only depending on Ωand ω) s. .
(5)
I(ϕ) = s−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
+sZZQ
e−2sα −1(T− )−1|∇ϕ|2+s3ZZQ
e−2sα −3(T− )−3|ϕ|2
≤C1s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2. Linea null con ollabili y esul wi h egula con ols
Coming back o he adjoin p oblem
(3) −∂ ϕ−∆ϕ+aϕ=0 in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω.
Lemma
The e exis C1>0and σ1>0(only depending on Ωand ω) s. .
(5)
I(ϕ) = s−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
+sZZQ
e−2sα −1(T− )−1|∇ϕ|2+s3ZZQ
e−2sα −3(T− )−3|ϕ|2
≤C1s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
We ollow:
E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and app oxima e
con ollabili y o weakly blowing up semilinea hea equa ions,
Ann. Ins . H. Poinca é Anal. Non Linéai e 17, No. 5, (2000),
583–616.
F om he p e ious global Ca leman inequali y one has:
Theo em
Fo e e y a∈L∞(Q)and ϕ0∈L2(Ω) one has (obse abili y inequali y)
kϕ(0)k2
L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T)
|ϕ|2,
(ϕsolu ion o (3)) wi h C=C(Ω,ω)>0and M gi en by:
M(T,kak∞) = 1+1
T+Tkak∞+kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
We ollow:
E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and app oxima e
con ollabili y o weakly blowing up semilinea hea equa ions,
Ann. Ins . H. Poinca é Anal. Non Linéai e 17, No. 5, (2000),
583–616.
F om he p e ious global Ca leman inequali y one has:
Theo em
Fo e e y a∈L∞(Q)and ϕ0∈L2(Ω) one has (obse abili y inequali y)
kϕ(0)k2
L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T)
|ϕ|2,
(ϕsolu ion o (3)) wi h C=C(Ω,ω)>0and M gi en by:
M(T,kak∞) = 1+1
T+Tkak∞+kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
Rema k
This inequali y shows he null con ollabili y esul o he linea
sys em (1) wi h a con ol in L2(Q)(in ac , Supp ⊂ω×(0,T)) and
p o ides he ollowing es ima e o k kL2(Q):
k k2
L2(Q)≤exp [CM(T,kak∞)]ky0k2,
wi h M gi en as be o e.
Is i possible o sol e his p oblem wi h a con ol ∈L∞(Q)?YES.
The key poin is a be e obse abili y inequali y wi h a weake no m on
he igh hand-side:
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
Coupled pa abolic sys ems: Le us conside a “simple” coupled
pa abolic sys em
(∂ y−∆y=Ay+B 1ωin Q,
y=0 on Σ,y(0) = y0in Ω,(∂ ϕ+ ∆ϕ=−A∗ϕin Q,
ϕ=0 on Σ, ϕ(T) = ϕ0in Ω,
wi h A=0 0
1 0 ,B=1
0(one con ol o ce) and y0∈L2(Ω)2.
The pa icula s uc u e o Aand B(cascade sys em) gi es:
kϕ(0)k2
L2(Ω) ≤CZZω0×(T/4,3T/4)
|ϕ1|2,
o a cons an C>0. Then, he e is ∈L2(Q)s. . y (T) = 0 in Ωand
k k2
L2(Ω) ≤Cky0k2
L2(Ω)2.Con ol in L∞(Q)??
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
Coupled pa abolic sys ems: Le us conside a “simple” coupled
pa abolic sys em
(∂ y−∆y=Ay+B 1ωin Q,
y=0 on Σ,y(0) = y0in Ω,(∂ ϕ+ ∆ϕ=−A∗ϕin Q,
ϕ=0 on Σ, ϕ(T) = ϕ0in Ω,
wi h A=0 0
1 0 ,B=1
0(one con ol o ce) and y0∈L2(Ω)2.
The pa icula s uc u e o Aand B(cascade sys em) gi es:
kϕ(0)k2
L2(Ω) ≤CZZω0×(T/4,3T/4)
|ϕ1|2,
o a cons an C>0. Then, he e is ∈L2(Q)s. . y (T) = 0 in Ωand
k k2
L2(Ω) ≤Cky0k2
L2(Ω)2.Con ol in L∞(Q)??
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.1. Fi s app oach
Following his echnique, does he ollowing inequali y
ZZω0×(T/4,3T/4)
|ϕ1|2≤C ZZω×(0,T)
|ϕ1|!2
hold?? NO.
Rema k
This i s app oach canno be applied o he p e ious coupled sys em
since he local egula izing e ec o he linea adjoin p oblem
in ol es he unc ions ϕ1and ϕ2while he co esponding “ e ined”
obse abili y inequali y should only in ol e ϕ1( ecall ha he con ol
only appea s in i s equa ion o he di ec p oblem).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
We ollow
V. BARBU,Exac con ollabili y o he supe linea hea equa ion,
Appl. Ma h. Op im. 42 (2000), no. 1, 73–89.
We ecall he global Ca leman inequali y (∂Ω∈C2):
I(ϕ) = s−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
+sZZQ
e−2sα −1(T− )−1|∇ϕ|2+s3ZZQ
e−2sα −3(T− )−3|ϕ|2
≤C1s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3
∞, whe e C1=C1(Ω,ω)>0
and ϕ he solu ion o
(3) −∂ ϕ−∆ϕ+aϕ=0 in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0(·)in Ω.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
We ollow
V. BARBU,Exac con ollabili y o he supe linea hea equa ion,
Appl. Ma h. Op im. 42 (2000), no. 1, 73–89.
We ecall he global Ca leman inequali y (∂Ω∈C2):
I(ϕ) = s−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
+sZZQ
e−2sα −1(T− )−1|∇ϕ|2+s3ZZQ
e−2sα −3(T− )−3|ϕ|2
≤C1s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3
∞, whe e C1=C1(Ω,ω)>0
and ϕ he solu ion o
(3) −∂ ϕ−∆ϕ+aϕ=0 in Q,
ϕ=0 on Σ, ϕ(·,T) = ϕ0(·)in Ω.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
In his wo k, a con ol in Lp(Q), wi h p=p(N), is ob ained om he
p e ious global Ca leman inequali y (we ix
s=s1=σ1(Ω,ω)T+T2+T2kak2/3
∞).
Fi s S ep:
Lemma
Fo e e y a∈L∞(Q)and ϕ0∈L2(Ω) one has (obse abili y inequali y)
kϕ(0)k2
L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T)
e−2s1α −3(T− )−3|ϕ|2,
(ϕsolu ion o (3)) wi h C=C(Ω,ω)>0and M gi en by:
M(T,kak∞) = 1+1
T+Tkak∞+kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
In his wo k, a con ol in Lp(Q), wi h p=p(N), is ob ained om he
p e ious global Ca leman inequali y (we ix
s=s1=σ1(Ω,ω)T+T2+T2kak2/3
∞).
Fi s S ep:
Lemma
Fo e e y a∈L∞(Q)and ϕ0∈L2(Ω) one has (obse abili y inequali y)
kϕ(0)k2
L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T)
e−2s1α −3(T− )−3|ϕ|2,
(ϕsolu ion o (3)) wi h C=C(Ω,ω)>0and M gi en by:
M(T,kak∞) = 1+1
T+Tkak∞+kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
Second S ep: F om his obse abili y inequali y we deduce
P oposi ion
∀y0∈L2(Ω), he e is ∈Lp(N)(Q), wi h p(N)<∞i N =2and
p(N) = 2(N+2)
N−2i N ≥3, and
k k2
Lp(N)(Q)≤e[CM(T,kak∞)]ky0k2
L2(Ω),
s. . he solu ion y o (1) associa ed o y0and sa is ies
y (T) = 0in L2(Ω).
Ske ch o he p oo :1.- We conside he op imal con ol p oblem
min
∈L2(Q)1
2ZZQ
e2s1α 3(T− )3| (x, )|2dx d +1
2εky (T)k2
L2(Ω),
(y ∈L2(Q)2is he solu ion o (1) associa ed o y0and ).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
Second S ep: F om his obse abili y inequali y we deduce
P oposi ion
∀y0∈L2(Ω), he e is ∈Lp(N)(Q), wi h p(N)<∞i N =2and
p(N) = 2(N+2)
N−2i N ≥3, and
k k2
Lp(N)(Q)≤e[CM(T,kak∞)]ky0k2
L2(Ω),
s. . he solu ion y o (1) associa ed o y0and sa is ies
y (T) = 0in L2(Ω).
Ske ch o he p oo :1.- We conside he op imal con ol p oblem
min
∈L2(Q)1
2ZZQ
e2s1α 3(T− )3| (x, )|2dx d +1
2εky (T)k2
L2(Ω),
(y ∈L2(Q)2is he solu ion o (1) associa ed o y0and ).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Second app oach
This p oblem has a unique solu ion ε∈L2(Q)and, using he
op imali y sys em, i is cha ac e ized: ε=e−2s1α −3(T− )−3ϕε1ω
and (∂ yε−∆yε+ayε= ε1ωin Q,
yε=0 sob e Σ,yε(·,0) = y0in Ω,
−∂ ϕε−∆ϕε+aϕε=0 in Q,
ϕε=0 on Σ, ϕε(·,T) = −1
εyε(·,T)in Ω.
The p e ious obse abili y inequali y (Lemma 7) gi es:
ZZω×(0,T)
e−2s1α −3(T− )−3|ϕε|2+1
εkyε(T)k2
L2(Ω) ≤e[CM(T,kak∞)]ky0k2
L2(Ω).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
We ollow
O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Exis ence o
insensi izing con ols o a semilinea hea equa ion wi h a
supe linea nonlinea i y, Comm. P.D.E 29 (2004), no. 7-8,
1017–1050.
ASSUMPTION
Gi en y0∈L2(Ω), he e is e
∈L2(Q), wi h Supp e
⊂ω0and ω0⊂⊂ ω,
such ha he solu ion o (1) e
ysa is ies e
y(·,T)≡0 in Ω.
One has
e
y∈W(0,T) = {y∈L2(0,T;H1
0(Ω)) : ∂ y∈L2(0,T;H−1(Ω))}and a
explici es ima e ke
ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke
k2.
The unc ion e
yis egula excep nea =0 and nea ω0. The idea is
elimina e hese i egula pa s o e
y.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
We ollow
O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Exis ence o
insensi izing con ols o a semilinea hea equa ion wi h a
supe linea nonlinea i y, Comm. P.D.E 29 (2004), no. 7-8,
1017–1050.
ASSUMPTION
Gi en y0∈L2(Ω), he e is e
∈L2(Q), wi h Supp e
⊂ω0and ω0⊂⊂ ω,
such ha he solu ion o (1) e
ysa is ies e
y(·,T)≡0 in Ω.
One has
e
y∈W(0,T) = {y∈L2(0,T;H1
0(Ω)) : ∂ y∈L2(0,T;H−1(Ω))}and a
explici es ima e ke
ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke
k2.
The unc ion e
yis egula excep nea =0 and nea ω0. The idea is
elimina e hese i egula pa s o e
y.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
We ollow
O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Exis ence o
insensi izing con ols o a semilinea hea equa ion wi h a
supe linea nonlinea i y, Comm. P.D.E 29 (2004), no. 7-8,
1017–1050.
ASSUMPTION
Gi en y0∈L2(Ω), he e is e
∈L2(Q), wi h Supp e
⊂ω0and ω0⊂⊂ ω,
such ha he solu ion o (1) e
ysa is ies e
y(·,T)≡0 in Ω.
One has
e
y∈W(0,T) = {y∈L2(0,T;H1
0(Ω)) : ∂ y∈L2(0,T;H−1(Ω))}and a
explici es ima e ke
ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke
k2.
The unc ion e
yis egula excep nea =0 and nea ω0. The idea is
elimina e hese i egula pa s o e
y.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
Le us now in oduce wo cu -o unc ions η∈C∞([0,T]) and
θ∈C∞(Ω) such ha
(η≡1 in [0,T
4],η≡0 in [3T
4,T],0≤η≤1 in [0,T],|η0( )| ≤ C/T,∀ ;
θ≡1 in ω0,0≤θ≤1 in Ωand Supp θ⊂ω.
Le Ybe he solu ion o sys em (1) co esponding o ≡0:
(∂ Y−∆Y+aY=0 in Q,
Y=0 on Σ,Y(·,0) = y0(·)in Ω,
We now ake
(y= (1−θ)e
y+ηθYin Q,
= (∂ −∆ + a)y.
I is clea ha Supp (·, )⊆Supp θ⊂ω,yis he solu ion o (1)
co esponding o he con ol and, aking in o accoun ha e
y(T)≡0
in Ω, we ge y(·,T)≡0 in Ω.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
Le us now in oduce wo cu -o unc ions η∈C∞([0,T]) and
θ∈C∞(Ω) such ha
(η≡1 in [0,T
4],η≡0 in [3T
4,T],0≤η≤1 in [0,T],|η0( )| ≤ C/T,∀ ;
θ≡1 in ω0,0≤θ≤1 in Ωand Supp θ⊂ω.
Le Ybe he solu ion o sys em (1) co esponding o ≡0:
(∂ Y−∆Y+aY=0 in Q,
Y=0 on Σ,Y(·,0) = y0(·)in Ω,
We now ake
(y= (1−θ)e
y+ηθYin Q,
= (∂ −∆ + a)y.
I is clea ha Supp (·, )⊆Supp θ⊂ω,yis he solu ion o (1)
co esponding o he con ol and, aking in o accoun ha e
y(T)≡0
in Ω, we ge y(·,T)≡0 in Ω.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
In ac is a egula con ol and i s egula i y p ope ies a e
independen o y0and e
. Indeed, we can exp ess yand as
y≡(1−θ)q+η( )Y, ≡θη0Y+2∇θ· ∇q+ (∆θ)q,
whe e qis gi en by q=e
y−ηYand, he e o e, sa is ies
(∂ q−∆q+aq=e
1ω−η0Yin Q,
q=0 on Σ,q(·,0) = 0 in Ω.
Le us ix δ∈(0,T/4),p∈[2,∞)and O0,O1⊂⊂ Ωsuch ha
O1⊂⊂ Ω ω0(and, in pa icula , O1∩Supp e
=∅). I we deno e by
(Xp
0={y∈Lp(δ,T;W2,p(O0)) : ∂ y∈Lp(O0×(δ,T))},
Xp
1={y∈Lp(0,T;W2,p(O1)) : ∂ y∈Lp(Oi×(0,T))}
hen, Y∈Xp
0,q∈Xp
1and ∈Lp(0,T;W1,p
0(Ω)).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
In ac is a egula con ol and i s egula i y p ope ies a e
independen o y0and e
. Indeed, we can exp ess yand as
y≡(1−θ)q+η( )Y, ≡θη0Y+2∇θ· ∇q+ (∆θ)q,
whe e qis gi en by q=e
y−ηYand, he e o e, sa is ies
(∂ q−∆q+aq=e
1ω−η0Yin Q,
q=0 on Σ,q(·,0) = 0 in Ω.
Le us ix δ∈(0,T/4),p∈[2,∞)and O0,O1⊂⊂ Ωsuch ha
O1⊂⊂ Ω ω0(and, in pa icula , O1∩Supp e
=∅). I we deno e by
(Xp
0={y∈Lp(δ,T;W2,p(O0)) : ∂ y∈Lp(O0×(δ,T))},
Xp
1={y∈Lp(0,T;W2,p(O1)) : ∂ y∈Lp(Oi×(0,T))}
hen, Y∈Xp
0,q∈Xp
1and ∈Lp(0,T;W1,p
0(Ω)).
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.3. Thi d app oach
In ac , we can ob ain some hing be e : i p>N+2, one has
Xp
0,→C1+α,(1+α)/2(O0×[δ,T]) and Xp
1,→C1+α,(1+α)/2(O1×[0,T])
wi h α=1−(N+2)/p. Thus, ∈C0
0(Q)and
k kC0≤eC(1+T+Tkak∞)ke
ykW(0,T)
wi h C=C(Ω,T)>0.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Thi d app oach. Rema ks I
1The p e ious egula i y esul o is independen o he ini ial
da um y0, he con ol e
and he egula i y o he bounda y ∂Ω. We
ha e only used he local egula i y p ope ies o he ope a o
L≡∂ −∆ + a. In he case in which a≡0, we ob ain ∈C∞(Q)
(as in he pape o Lebeau-Robbiano).
2In ac we ha e p o ed: “Le us ix y0∈L2(Ω) and assume ha
he e exis s e
∈L2(Q)such ha he solu ion e
y o he linea
p oblem (1) sa is ies e
y(T)≡0in Ω. Then, he e exis s e
∈C0
0(Q)
s. . he solu ion y o (1) also sa is ies y (T)≡0in Ω”.
3This echnique can be applied i we conside a linea pa abolic
p oblem wi h a i s o de e m B· ∇yob aining he same
egula i y esul .
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
2.2. Thi d app oach. Rema ks II
4When Ωand ωa e unbounded open se s we can ob ain he same
esul :
L. DE TERESA, M. G.-B.,Some esul s on con ollabili y o linea
and nonlinea hea equa ions in unbounded domains,
Ad . Di . Eq. 12 (2007), no. 11, 1201–1240.
5This app oach also wo ks in he case o sys ems o wo coupled
pa abolic equa ions.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
We ake
ψ=s−5/2e−sα∗( ) 5/2(T− )5/2ϕ=ρ0( )ϕ.
Then, (∂ ψ+ ∆ψ=aρ0( )ϕ+∂ ρ0( )ϕin Q,
ψ=0 on Σ, ψ(·,T) = 0 in Ω.
I s≥s1=σ1T+T2+T2kak2/3
∞, we ha e ∂ ρ0( )ϕ∈H2,1(Q)and
k∂ ρ0( )ϕk2
H2,1≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2.
Bu , H2,1(Q),→Lp(N)(Q)wi h p(N) = 2(N+2)
N−2. Thus,
k∂ ρ0( )ϕkLp(N)(Q)≤Ck∂ ρ0( )ϕkH2,1
We can also p o e ha aρ0( )ϕ∈Lp(N)(Q)and
kaρ0( )ϕk2
Lp(N)(Q)≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
We ake
ψ=s−5/2e−sα∗( ) 5/2(T− )5/2ϕ=ρ0( )ϕ.
Then, (∂ ψ+ ∆ψ=aρ0( )ϕ+∂ ρ0( )ϕin Q,
ψ=0 on Σ, ψ(·,T) = 0 in Ω.
I s≥s1=σ1T+T2+T2kak2/3
∞, we ha e ∂ ρ0( )ϕ∈H2,1(Q)and
k∂ ρ0( )ϕk2
H2,1≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2.
Bu , H2,1(Q),→Lp(N)(Q)wi h p(N) = 2(N+2)
N−2. Thus,
k∂ ρ0( )ϕkLp(N)(Q)≤Ck∂ ρ0( )ϕkH2,1
We can also p o e ha aρ0( )ϕ∈Lp(N)(Q)and
kaρ0( )ϕk2
Lp(N)(Q)≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
The maximal pa abolic egula i y o he hea equa ion (∂Ω∈C2) gi es
ψ=s−5/2e−sα∗( ) 5/2(T− )5/2ϕ∈W2,1
p(N)(Q)and
kψk2
W2,1
p(N)(Q)≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
≤C2s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2.
Conclusion
We ha e ob ained a new Ca leman inequali y o he p oblem (3)
ks−5/2e−sα∗( ) 5/2(T− )5/2ϕk2
W2,1
p(N)(Q)+I(ϕ)
≤C2s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1T+T2+T2kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
The maximal pa abolic egula i y o he hea equa ion (∂Ω∈C2) gi es
ψ=s−5/2e−sα∗( ) 5/2(T− )5/2ϕ∈W2,1
p(N)(Q)and
kψk2
W2,1
p(N)(Q)≤Cs−1ZZQ
e−2sα (T− )|∂ ϕ|2+|∆ϕ|2
≤C2s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2.
Conclusion
We ha e ob ained a new Ca leman inequali y o he p oblem (3)
ks−5/2e−sα∗( ) 5/2(T− )5/2ϕk2
W2,1
p(N)(Q)+I(ϕ)
≤C2s3ZZω×(0,T)
e−2sα −3(T− )−3|ϕ|2,
∀s≥s1=σ1T+T2+T2kak2/3
∞.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
Co olla y
∀y0∈L2(Ω), he e is ∈W2,1
p(N)(Q), wi h p(N)<∞i N =2and
p(N) = 2(N+2)
N−2i N ≥3, and
k k2
W2,1
p(N)
≤e[CM(T,kak∞)]ky0k2
L2(Ω),
s. . he solu ion y o (1) associa ed o y0and sa is ies
y (T) = 0in L2(Ω).
Rema k
We can apply a boo -s ap a gumen and deduce ha he p e ious
esul is alid o e e y p ∈[2,∞). In his case he cons an Calso
depends on p.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
Co olla y
∀y0∈L2(Ω), he e is ∈W2,1
p(N)(Q), wi h p(N)<∞i N =2and
p(N) = 2(N+2)
N−2i N ≥3, and
k k2
W2,1
p(N)
≤e[CM(T,kak∞)]ky0k2
L2(Ω),
s. . he solu ion y o (1) associa ed o y0and sa is ies
y (T) = 0in L2(Ω).
Rema k
We can apply a boo -s ap a gumen and deduce ha he p e ious
esul is alid o e e y p ∈[2,∞). In his case he cons an Calso
depends on p.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
Re e ence
V. BARBU,Con ollabili y o pa abolic and Na ie -S okes
equa ions, Sci. Ma h. Jpn. 56 (2002), no. 1, 143–211.
F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A
Kalman ank condi ion o he localized dis ibu ed con ollabili y o
a class o linea pa abolic sys ems, J. E ol. Equ. 9 (2009), no. 2,
267–291.
M. G.-B., S. GUERRERO, J.-P. PUEL,Local exac con ollabili y o
he ajec o ies o he Boussinesq sys em ia a ic i ious con ol on
he di e gence equa ion, Commun. Pu e Appl. Anal. 8 (2009),
no. 1, 311–333.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion
3. The “bes ” null con ol
Re e ence
V. BARBU,Con ollabili y o pa abolic and Na ie -S okes
equa ions, Sci. Ma h. Jpn. 56 (2002), no. 1, 143–211.
F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A
Kalman ank condi ion o he localized dis ibu ed con ollabili y o
a class o linea pa abolic sys ems, J. E ol. Equ. 9 (2009), no. 2,
267–291.
M. G.-B., S. GUERRERO, J.-P. PUEL,Local exac con ollabili y o
he ajec o ies o he Boussinesq sys em ia a ic i ious con ol on
he di e gence equa ion, Commun. Pu e Appl. Anal. 8 (2009),
no. 1, 311–333.
M. González-Bu gos Rema ks on he con ollabili y o he nonlinea hea equa ion