Con ollabili y o non-scala pa abolic sys ems: Some
ecen esul s and phenomena
Manuel González-Bu gos,
UNIVERSIDAD DE SEVILLA
Ma akesh Wo kshop on Con ol, In e se P oblems and S abiliza ion o In ini e Dimensional Sys ems
Ma akesh, Decembe 2016
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
Gene al objec i e:
S udy some null con ollabili y p oblems o non-scala pa abolic sys ems.
Non-scala pa abolic sys ems: a ise in chemical eac ions, when we model
p oblems om he Biology and in a wide a ie y o physical si ua ions.
In his cou se we will deal wi h non-scala sys ems which in ac a e coupled
pa abolic scala equa ions. We do no p esen esul s ela ing o he
con ollabili y p oblems o sys ems which come om luid mechanics as
S okes, Na ie -S okes, ...
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
GOAL:
1Show he impo an di e ences be ween scala and non-scala p oblems.
2Gi e necessa y and su icien condi ions (Kalman condi ions) which
cha ac e ize he con ollabili y p ope ies o hese sys ems.
3Show some hype bolic phenomena ela ed o he con ollabili y
p ope ies o hese sys ems.
We will only deal wi h
1Linea sys ems
2In gene al, “simple” Pa abolic Sys ems.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
Con en s
1In oduc ion
2The pa abolic scala case
The one-dimensional case: The momen me hod
Gene al case: Ca leman Inequali ies
Final commen s in he scala case
3Fini e-dimensional sys ems
4Dis ibu ed con ollabili y o 2 ×2 linea sys ems
5Bounda y con ollabili y o a 2 ×2 linea sys em
6A gene aliza ion: Cascade sys ems
7The Kalman condi ion o a class o pa abolic sys ems. Dis ibu ed
con ols
8The Kalman condi ion o a class o pa abolic sys ems. Bounda y con ols
9New phenomena: Minimal ime o con ollabili y
10 New phenomena: Dependence on he posi ion o he con ol se
11 Fu he esul s
12 Commen s and open p oblems
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
1. In oduc ion
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

1. In oduc ion
Le us ix T>0 and le Hand Ube wo sepa able Hilbe spaces. Le us
conside he au onomous sys em:
(1) (y0=Ay+Buon (0,T),
y(0) = y0∈H.
Aand Ba e “app op ia e” ope a o s, y0∈His he ini ial da um a =0 and
u∈L2(0,T;U)is he con ol (exe ed by means o he ope a o B).
Assume he p oblem is well-posed: ∀(y0,u) he e exis s a unique weak
solu ion y∈C0([0,T]; H) o (1) which depends con inuously on he da a.
Le us deno e by y( ;y0,u)∈H he solu ion o he sys em a ime ∈[0,T].
Example
H=Rn(n≥1), U=Rm(m≥1), A∈ L(Rn)and B∈ L(Rm;Rn): o dina y
di e en ial sys em wi h n a iables and mcon ols.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
1. In oduc ion
Exac Con ollabili y: Sys em (1) is exac ly con ollable a ime Ti
∀(y0,y1)∈H×H, he e exis s u∈L2(0,T;U)s. . he solu ion yo (1)
sa is ies y(T;y0,u) = y1.
Con ollabili y o ajec o ies: Sys em (1) is con ollable o
ajec o ies a ime Ti ∀(y0,by0)∈H×Hand bu∈L2(0,T;U), he e
exis s u∈L2(0,T;U)s. . he co esponding weak solu ion o (1)
sa is ies y(T;y0,u) = y(T;by0,bu).
Null Con ollabili y: Sys em (1) is null con ollable a ime Ti
∀y0∈H he e exis s u∈L2(0,T;U)s. . y(T;y0,u) = 0.
Linea case: Con ollabili y o ajec o ies and null con ollabili y a e
equi alen .
App oxima e Con ollabili y: Sys em (1) is app oxima ely
con ollable a ime Ti ∀(y0,y1)∈H×H, and e e y ε>0, he e
exis s u∈L2(0,T;U)s. .
ky(T;y0,u)−y1kH≤ε.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
Rema k
P oblem (1) is linea . Then, Sys em (1) is null con ollable a ime Ti and
only i he sys em is exac ly con ollable o he ajec o ies a ime T.
Rema k
We will deal wi h pa abolic p oblems. So, due o he egula izing e ec o
hese p oblems, i is well-known ha he exac con ollabili y esul ails.
The e o e, in his cou se we will s udy null o app oxima e con ollabili y
esul s o he sys em unde conside a ion.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
Theo em
Le us ix T >0. The ollowing condi ions a e equi alen
1Fo any Ω⊂RN, bounded open se wi h Ωha ing a C2bounda y, any
ω⊂Ω, nonemp y open subse , and any coe icien s αij (1≤i,j≤N), D
and c, sa is ying (3) and (4), Sys em (5) is null con ollable in L2(Ω) a
ime T >0wi h dis ibu ed con ols ∈L2(QT).
2Fo any Ω⊂RN, bounded open se wi h Ωha ing a C2bounda y, any
Γ0⊂∂Ω, nonemp y ela i e open subse , and any coe icien s αij
(1≤i,j≤N), Dand c, sa is ying (3) and (4), Sys em (6) is null
con ollable in L2(Ω) a ime T >0wi h bounda y con ols
h∈L2(0,T;H1/2(∂Ω)).
P oo : We will use in a undamen al way ha he p oblem unde
conside a ion is scala (in ac , same numbe o equa ions and con ols). We
ollow some ideas om [BODART,G.-B.,PÉREZ-GARCÍA] Comm. PDE
(2004) and [G.-B.,PÉREZ-GARCÍA] Asymp. Anal. (2006). ···
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

2. The pa abolic scala case
Theo em
Le us ix T >0. The ollowing condi ions a e equi alen
1Fo any Ω⊂RN, bounded open se wi h Ωha ing a C2bounda y, any
ω⊂Ω, nonemp y open subse , and any coe icien s αij (1≤i,j≤N), D
and c, sa is ying (3) and (4), Sys em (5) is null con ollable in L2(Ω) a
ime T >0wi h dis ibu ed con ols ∈L2(QT).
2Fo any Ω⊂RN, bounded open se wi h Ωha ing a C2bounda y, any
Γ0⊂∂Ω, nonemp y ela i e open subse , and any coe icien s αij
(1≤i,j≤N), Dand c, sa is ying (3) and (4), Sys em (6) is null
con ollable in L2(Ω) a ime T >0wi h bounda y con ols
h∈L2(0,T;H1/2(∂Ω)).
P oo : We will use in a undamen al way ha he p oblem unde
conside a ion is scala (in ac , same numbe o equa ions and con ols). We
ollow some ideas om [BODART,G.-B.,PÉREZ-GARCÍA] Comm. PDE
(2004) and [G.-B.,PÉREZ-GARCÍA] Asymp. Anal. (2006). ···
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
Rema k (Regula izing e ec )
The p e ious p oo shows ha i he dis ibu ed and bounda y null
con ollabili y esul s o Sys ems (5) and (6) a e alid wi h con ols in
L2(QT)and L2(0,T;H1/2(∂Ω)), hen he p e ious sys ems a e null
con ollable wi h con ols in L∞(QT)and L∞(ΣT)(and e en be e o
egula coe icien s).
Rema k
In he p oo o Theo em 1 we ha e s ongly used ha he ope a o ∂ +L( )is
scala . We will see ha he p e ious equi alence is no alid o non-scala
pa abolic ope a o s.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
F om now on, we will concen a e on he dis ibu ed con ol p oblem (5).
Le us in oduce he adjoin p oblem
(7) (−∂ ϕ+L∗( )ϕ=0 in QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕTin Ω,
whe e ϕT∈L2(Ω) is gi en and L∗( )is he ope a o gi en by
L∗( )ϕ=−
N
X
i,j=1
∂
∂xiαij(x, )∂ϕ
∂xj−∇·(Dϕ) + c(x, )ϕa.e. in QT.
This p oblem is also well-posed and he solu ion depends con inuously on
ϕT: he e exis s a cons an e
C>0 such ha ∀ϕT∈L2(Ω) Sys em (7) has only
one solu ion ϕ∈L2(0,T;H1
0(Ω)) ∩C0([0,T]; L2(Ω)) and i sa is ies
kϕkL2(0,T;H1
0(Ω)) +kϕkC0([0,T];L2(Ω)) ≤e
CkϕTkL2(Ω).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
Theo em (Obse abili y Inequali y)
Unde he p e ious assump ions, Sys em (5) is null con ollable a ime T >0
i and only i he e exis s a cons an CT>0s. .
(8) kϕ(·,0)k2
L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxd ,∀ϕT∈L2(Ω),
whe e ϕis he solu ion o (7) associa ed o ϕT.
Rema k
The Obse abili y Inequali y (8) in pa icula implies a be e esul : I (8)
holds hen, ∀y0∈L2(Ω) he e is a dis ibu ed con ol ∈L2(QT)s. .
k k2
L2(QT)≤CTky0k2
L2(Ω) and y(·,T) = 0,
being y he solu ion o (5) co esponding o y0and CT>0 he cons an in (8).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
Theo em (Obse abili y Inequali y)
Unde he p e ious assump ions, Sys em (5) is null con ollable a ime T >0
i and only i he e exis s a cons an CT>0s. .
(8) kϕ(·,0)k2
L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxd ,∀ϕT∈L2(Ω),
whe e ϕis he solu ion o (7) associa ed o ϕT.
Rema k
The Obse abili y Inequali y (8) in pa icula implies a be e esul : I (8)
holds hen, ∀y0∈L2(Ω) he e is a dis ibu ed con ol ∈L2(QT)s. .
k k2
L2(QT)≤CTky0k2
L2(Ω) and y(·,T) = 0,
being y he solu ion o (5) co esponding o y0and CT>0 he cons an in (8).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

2. The pa abolic scala case
Rema k (Con ol cos )
The p e ious ema k and inequali y (8) p o ide an es ima e o he cos o he
con ol o sys em (5): I (8) holds a ime T>0, hen
ZT(y0) := { ∈L2(QT) : y(T;y0, ) = 0} 6=∅,∀y0∈L2(Ω).
We can hen de ine he con ol cos o sys em (5) a ime Tas
K(T) = sup
ky0kL2(Ω)=1in
∈ZT(y0)k kL2(QT),∀T>0.
Thus, K(T)≤√CT. On he o he hand, i ZT(y0)6=∅, o any y0∈L2(Ω),
hen, he obse abili y inequali y (8) o he adjoin sys em (7) holds wi h
CT=K(T)2. I is hen clea ha
K(T) = in npCT:CT>0 is such ha (8) holdso.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
1. The one-dimensional case: The momen me hod
We ollow [FATTORINI,RUSSELL] A ch. Ra . Mech. Anal. (1971).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
1. The one-dimensional case: The momen me hod
Conside he bounda y null con ollabili y p oblem o he classical
one-dimensional hea equa ion in (0, π)( o simplici y):
(9) 




y −yxx =0 in QT= (0, π)×(0,T),
y(0,·) = ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
wi h y0∈H−1(0, π)and ∈L2(0,T). The p oblem is well-posed and he
solu ion (de ined by ansposi ion) depends con inuously on he da a y0and .
The ope a o −∂xx on (0, π)wi h homogenous Di ichle bounda y condi ions
admi s a sequence o eigen alues and no malized eigen unc ions gi en by
λk=k2,φk(x) = 2
πsin kx,k≥1,x∈(0, π)
which is a Hilbe basis o L2(0, π). In he sequel, we will use he no a ion
yk= (y,φk)L2(0,π),∀y∈L2(0, π).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
1. The one-dimensional case: The momen me hod
The idea o he momen me hod is simple: Gi en y0∈H−1(0, π),
ϕT∈H1
0(0, π)and ∈L2(0,T), hen
hy(·,T), ϕTi−hy0, ϕ(·,0)i=ZT
0
( )ϕx(0, )d .
whe e yis he solu ion o (9) and ϕis he solu ion o he adjoin p oblem
(−ϕ −ϕxx =0 in QT,
ϕ=0 on {0,1}×(0,T), ϕ(·,T) = ϕTin (0, π).
P ope y
∈L2(0, π)is a null con ol o sys em (9) (i.e., ∈L2(0,T)is a con ol
s. . he solu ion y o (9) sa is ies y(·,T) = 0 in (0, π)) i and only i
−hy0, ϕ(·,0)i=ZT
0
( )ϕx(0, )d ,∀ϕT∈H1
0(0, π).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
1. The one-dimensional case: The momen me hod
Consequence:
The p e ious esul is alid o any nonemp y bounded in e al (a,b)and o
any second o de ope a o sel -adjoin ellip ic ope a o
Ly=−(α(x)yx)x+c(x)y,
wi h α∈C1([a,b]) and α>0 in (a,b), and c∈C0([a,b]).Then, i we apply
Theo em 1, we also ge a dis ibu ed con ollabili y esul o he p oblem





y +Ly= 1ωin QT= (a,b)×(0,T),
y(a,·) = 0,y(b,·) = 0 on (0,T),
y(·,0) = y0in (a,b),
wi h y0∈L2(0, π)and ω⊆(a,b), a nonemp y open subse .
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
We ollow [FURSIKOV,IMANUVILOV] 1996 and
[IMANUVILOV,YAMAMOTO] 2003.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
We will conside he ollowing pa abolic equa ion:
(10) 






−∂ z+L0( )z=F0+
N
X
i=1
∂Fi
∂xi
in QT,
z=0 on ΣT,z(·,T) = zTin Ω,
wi h zT∈L2(Ω),Fi∈L2(QT),i=0,1,...,N, and L0( ) he sel -adjoin
pa abolic ope a o gi en by
L0( )y=−
N
X
i,j=1
∂
∂xiαij(x, )∂y
∂xj
wi h coe icien s αij sa is ying (3) ( egula i y) and (4) (uni o m ellip ic
condi ion).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Lemma
Le B⊂Ωbe a nonemp y open subse and d ∈R. Then, ∃β0∈C2(Ω)
(posi i e and only depending on Ωand B) and e
C0,eσ0>0(only depending on
Ω,Band d) s. . o e e y zT∈L2(Ω), he solu ion z o (10)sa is ies
(11)











I(d,z)≤e
C0 sdZZB×(0,T)
e−2sβγ( )d|z|2
+sd−3ZZQT
e−2sβγ( )d−3|F0|2+sd−1
N
X
i=1ZZQT
e−2sβγ( )d−1|Fi|2!,
∀s≥es0=eσ0(T+T2);γ( ) = −1(T− )−1,β(x, ) = β0(x)/ (T− )
and I(d,z)≡sd−2ZZQT
e−2sβγ( )d−2|∇z|2+sdZZQT
e−2sβγ( )d|z|2.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Lemma
When Fi≡0 o 1≤i≤N, ∃e
C1and eσ1(which only depend on Ω,Band d)
s. ., ∀zT∈L2(Ω), he solu ion z o (10)sa is ies
(12)
I1(d,z)≤e
C1 sdZZB×(0,T)
e−2sβγ( )d|z|2+sd−3ZZQT
e−2sβγ( )d−3|F0|2!,
o all s ≥es1=eσ1(T+T2)whe e
I1(d,z)≡sd−4ZZQT
e−2sβγ( )d−4
|∂ z|2+
N
X
i,j=1
∂2z
∂xi∂xj
2
+I(d,z).
P oo : See [FURSIKOV,IMANUVILOV] 1996; [IMANUVILOV,YAMAMOTO]
(2003) and [FERNÁNDEZ-CARA,GUERRERO] SICON (2006).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Recall ha ou objec i e is o p o e a null con ollabili y esul a ime T o
(5) (∂ y+L( )y= 1ωin QT,
y=0 on ΣT,y(·,0) = y0in Ω,
wi h L( )gi en by:







L( )y=−
N
X
i,j=1
∂
∂xiαij(x, )∂y
∂xj+D(x, )·∇y+c(x, )y
=L0( )y+D(x, )·∇y+c(x, )y,
wi h coe icien s αij sa is ying (3) and (4). We also know ha his is
equi alen o he obse abili y inequali y (8)
kϕ(·,0)k2
L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxd ,∀ϕT∈L2(Ω),
o he solu ions o he adjoin p oblem (7).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Co olla y
The e exis s a cons an C0=C0(Ω,ω)>0such ha ∀ϕT∈L2(Ω) and ϕ he
co esponding solu ion o (7), he obse abili y inequali y (8) holds wi h
CT=exp C01+1
T+kck2/3
∞+Tkck∞+ (1+T)kDk2
∞.
P oo : We ollow [FERNÁNDEZ-CARA,ZUAZUA] Ann. IHP (2000) and
[DOUBOVA,FERNÁNDEZ-CARA,MG-B,ZUAZUA] SICON (2002).
The Ca leman inequali y (11) applied o p oblem (7) implies (B≡ω,d=3
and −∂ ϕ+L0( )ϕ=∇·(Dϕ)−c(x, )ϕ) ha ∀s≥es0=eσ0(T+T2):
sZZQT
e−2sβγ( )|∇ϕ|2+s3ZZQT
e−2sβγ( )3|ϕ|2
≤e
C0 s3ZZω×(0,T)
e−2sβγ( )3|ϕ|2
+kck2
∞ZZQT
e−2sβ|ϕ|2+s2kDk2
∞ZZQT
e−2sβγ( )2|ϕ|2.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Co olla y
The e exis s a cons an C0=C0(Ω,ω)>0such ha ∀ϕT∈L2(Ω) and ϕ he
co esponding solu ion o (7), he obse abili y inequali y (8) holds wi h
CT=exp C01+1
T+kck2/3
∞+Tkck∞+ (1+T)kDk2
∞.
P oo : We ollow [FERNÁNDEZ-CARA,ZUAZUA] Ann. IHP (2000) and
[DOUBOVA,FERNÁNDEZ-CARA,MG-B,ZUAZUA] SICON (2002).
The Ca leman inequali y (11) applied o p oblem (7) implies (B≡ω,d=3
and −∂ ϕ+L0( )ϕ=∇·(Dϕ)−c(x, )ϕ) ha ∀s≥es0=eσ0(T+T2):
sZZQT
e−2sβγ( )|∇ϕ|2+s3ZZQT
e−2sβγ( )3|ϕ|2
≤e
C0 s3ZZω×(0,T)
e−2sβγ( )3|ϕ|2
+kck2
∞ZZQT
e−2sβ|ϕ|2+s2kDk2
∞ZZQT
e−2sβγ( )2|ϕ|2.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
As a consequence we can p o e ha o
s≥C1(T+T2+T2(kck2/3
∞+kDk2
∞)) (C1=C1(Ω,ω)) one has
[sγ( )]3−e
C0kck2
∞−e
C0[sγ( )]2kDk2
∞≥1
2[sγ( )]3.
Consequen ly, o s=C1(T+T2+T2(kck2/3
∞+kDk2
∞)) ha
ZZQT
e−2sβ −3(T− )−3|ϕ|2≤e
C1ZZω×(0,T)
e−2sβ −3(T− )−3|ϕ|2
and he e o e
ZZΩ×(T/4,3T/4)|ϕ|2≤eC(1+1/T+kck2/3
∞+kDk2
∞)ZZω×(0,T)|ϕ|2.
This las inequali y combined wi h ene gy es ima es (C=C(a0)>0)
d
d eC(kck∞+kDk2
∞) ZΩ|ϕ|2(·, )≥0∀ ∈[0,T]
implies (8) and he p oo is comple e.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
2. The pa abolic scala case
2. Gene al case: Ca leman Inequali ies
Co olla y
Le us ix T >0,Ω⊂RN,ω⊆Ωand Γ0⊆∂Ω(a bi a y) as be o e. Then,
he e exis posi i e cons an s C0=C0(Ω,ω)and b
C0=b
C0(Ω,Γ0)s. .
1∀y0∈L2(Ω) he e is a con ol ∈L2(Ω) which sa is ies
k k2
L2(QT)≤eC01+1/T+kck2/3
∞+Tkck∞+(1+T)kDk2
∞ky0k2
L2(Ω),
and y(·,T) = 0in Ω, (y is he solu ion o (5) associa ed o y0and ).
2∀y0∈L2(Ω) he e is a con ol h∈L2(0,T;H1/2(Ω)) which sa is ies
khk2
L2(0,T;H1/2(Ω)) ≤eb
C01+1/T+kck2/3
∞+Tkck∞+(1+T)kDk2
∞ky0k2
L2(Ω),
and y(·,T) = 0in Ω, (y is he solu ion o (6) associa ed o y0and and,
in ac , y ∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω))).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
3. Fini e-dimensional sys ems
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

3. Fini e-dimensional sys ems
Le us conside he au onomous linea sys em
(13) y0=Ay+Buon [0,T],y(0) = y0,
whe e A∈ L(Cn)and B∈ L(Cm,Cn)a e cons an ma ices, y0∈Cnand
u∈L2(0,T;Cm)is he con ol.
P oblem:
Gi en y0,yd∈Cn, is he e a con ol u∈L2(0,T;Cm)such ha he solu ion y
o he p oblem sa is ies
y(T) = yd????
Le us de ine (con ollabili y ma ix)
[A|B]=(B,AB ,A2B,··· ,An−1B)∈ L(Cnm;Cn).
On he o he hand, le {θl}1≤l≤ˆ
p⊂Cbe he se o dis inc eigen alues o A∗.
Fo l:1≤l≤ˆ
p, we deno e by ml he geome ic mul iplici y o θl. The
sequence {wl,j}1≤j≤mlwill deno e a basis o he eigenspace associa ed o θl.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
3. Fini e-dimensional sys ems
Le us conside he au onomous linea sys em
(13) y0=Ay+Buon [0,T],y(0) = y0,
whe e A∈ L(Cn)and B∈ L(Cm,Cn)a e cons an ma ices, y0∈Cnand
u∈L2(0,T;Cm)is he con ol.
P oblem:
Gi en y0,yd∈Cn, is he e a con ol u∈L2(0,T;Cm)such ha he solu ion y
o he p oblem sa is ies
y(T) = yd????
Le us de ine (con ollabili y ma ix)
[A|B]=(B,AB ,A2B,··· ,An−1B)∈ L(Cnm;Cn).
On he o he hand, le {θl}1≤l≤ˆ
p⊂Cbe he se o dis inc eigen alues o A∗.
Fo l:1≤l≤ˆ
p, we deno e by ml he geome ic mul iplici y o θl. The
sequence {wl,j}1≤j≤mlwill deno e a basis o he eigenspace associa ed o θl.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
3. Fini e-dimensional sys ems
The ollowing classical esul can be ound in
R. KALMAN, Y.-CH. HO, K. NARENDRA,Con ollabili y o linea
dynamical sys ems,1963.
and gi es a comple e answe o he p oblem o con ollabili y o ini e
dimensional au onomous linea sys ems:
Theo em
Unde he p e ious assump ions, he ollowing condi ions a e equi alen
1Sys em (13) is exac ly con ollable a ime T, o e e y T >0.
2The e exis s T >0such ha sys em (13) is exac ly con ollable a ime T.
3 ank [A|B] = n o ke [A|B]∗={0}(Kalman ank condi ion).
4Hau us es : ank A∗−θlIn
B∗=n,∀l:1≤l≤ˆ
p.
5 ank [B∗wl,1,B∗wl,2,··· ,B∗wl,ml] = ml, o e e y l :1≤l≤ˆ
p.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
3. Fini e-dimensional sys ems
Rema k
1The ou con ollabili y concep s (exac ,exac o ajec o ies,null and
app oxima e con ollabili y) o Sys em (13) a e equi alen
( ini e-dimensional space).
2Obse e ha {B∗wl,1,B∗wl,2,...,B∗wl,ml} ⊂ Cm. Condi ion 5 in
Theo em 4 says his se is linea ly independen o any l:1≤l≤ˆ
p. In
pa icula , ml≤m∀l:1≤l≤ˆ
p.
3Gi en he o.d.s. (adjoin p oblem)
−ϕ0=A∗ϕin [0,T], ϕ(T) = ϕT∈Cn,
i is no di icul o p o e he ollowing esul : “Sys em (13) is exac ly
con ollable a ime T i and only i he ollowing p ope y o he
adjoin p oblem holds (unique con inua ion p ope y)
I B∗ϕ(·) = 0 on [0,T], hen ϕT≡0."
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
3. Fini e-dimensional sys ems
Goal
We ha e a comple e cha ac e iza ion o he con ollabili y esul s o
ini e-dimensional linea o dina y di e en ial sys ems (a Kalman condi ion).
Is i possible o ob ain simila esul s o Pa ial Di e en ials Sys ems? We
will ocus on coupled linea pa abolic sys ems.
Wha a e he possible gene aliza ions o Sys ems o
Pa abolic Equa ions?
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

4. Dis ibu ed con ollabili y o 2×2
linea sys ems
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Le us conside he 2 ×2 linea eac ion-di usion sys em (QT= Ω ×(0,T))
(14) 




∂ y1+L1
0( )y1+a11y1+a12y2= 1ωin QT,
∂ y2+L2
0( )y2+a21y1+a22y2=0 in QT,
yi=0 on ΣT=∂Ω×(0,T),yi(·,0) = yi
0in Ω,1≤i≤2,
whe e Ω,ωand Ta e as be o e, aij =aij(x, )∈L∞(QT)(1 ≤i,j≤2),
yi
0∈L2(Ω) (1 ≤i≤2) and Lk
0( )is, o e e y 1 ≤k≤2, he second o de
ope a o Lk
0( )y=−
N
X
i,j=1
∂
∂xiαk
ij(x, )∂y
∂xjwhe e αk
ij sa is y (3) and (4).
Rema k
Sys em (14) is con olled by means o a scala dis ibu ed con ol exe ed on
he igh -hand side o he i s equa ion. The second equa ion is indi ec ly
con olled by he coupling e m a21y1.Necessa y condi ion a21 6≡ 0
(a21 ∈L∞(QT)).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Equi alen ly, he p e ious sys em can be w i en as
(15) (∂ y+b
L( )y+Ay=B 1ωin QT,
y=0 on ΣT,y(·,0) = y0in Ω,
whe e b
L( )is he ma ix ope a o gi en by b
L( ) = diag (L1
0( ),L2
0( )),
y= (yi)1≤i≤2is he s a e and whe e
(y0= (yi
0)1≤i≤2∈L2(Ω; Rn),A(·,·)=(aij(·,·))1≤i,j≤2∈L∞(QT;L(Rn)),
and B≡e1= (1,0)∗∈R2
a e gi en. Le us obse e ha , o each y0∈L2(Ω; R2)and ∈L2(QT),
Sys em (15) admi s a unique weak solu ion
y∈L2(0,T;H1
0(Ω; R2)) ∩C0([0,T]; L2(Ω; R2)).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Assump ion
We assume ha he coupling coe icien a21 ∈L∞(QT)sa is ies
(16) a21 ≥c0>0 o −a21 ≥c0>0 in ω0×(0,T),
wi h ω0⊆ωa new open subse .
As in he scala case, he con ollabili y esul o sys em (15) is equi alen o
he obse abili y inequali y:∃CT>0 such ha
kϕ1(·,0)k2
L2+kϕ2(·,0)k2
L2≤CTZZω×(0,T)|ϕ1(x, )|2dx d ,
whe e ϕis he solu ion associa ed o ϕ0∈L2(Ω; R2)o he adjoin p oblem:
(17) −ϕ +b
L( )ϕ+A∗ϕ=0 in QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Summa izing
We ha e p o ed ha he solu ions o he adjoin sys em
(17) −ϕ +b
L( )ϕ+A∗ϕ=0 in QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω.
sa is y he Ca leman inequali y C0=C0(Ω,ω0,c0,ka21k∞,d)
I1(d+3, ϕ1) + I1(d, ϕ2)≤C0sd+4ZZω×(0,T)
e−2sαγ( )d+4|ϕ1|2,
∀s≥s0=σ0hT+T2+T2ka11k2/3
∞+ka12k1/3
∞+ka22k2/3
∞i.
(C0=C0(Ω,ω0,c0,ka21k∞,d)and σ0=σ0(Ω,ω0,c0,ka21k∞,d)a e
posi i e cons an s).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
As in he scala case, combining he p e ious esul and ene gy inequali ies
sa is ied by he solu ions o he adjoin sys em i is possible o p o e an
obse abili y inequali y o he adjoin sys em and deduce:
Co olla y
Le us assume (16). Then, he e exis s a posi i e cons an C(only depending
on Ω,ω,c0and ka21k∞) such ha o e e y y0∈L2(Ω; R2) he e is a con ol
∈L2(Ω) which sa is ies
k |k2
L2(QT)≤eCHky0k2
L2(Ω;R2),
and y(·,T) = 0in Ω, wi h y he solu ion o (15) associa ed o y0and . In he
p e ious inequali y, His gi en by
H ≡ 1+T+1
T+ka11k2/3
∞+ka12k1/3
∞+ka22k2/3
∞+Tmax
1≤i,j≤2kaijk∞.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Rema k
Sys em (14) is always con ollable i we exe a con ol in each equa ion
( wo con ols).
The con ollabili y esul o sys em (14) is independen o he ope a o s
L1
0( )and L2
0( ). We will see ha he si ua ion is mo e in ica e i in he
sys em a gene al con ol ec o B∈R2is conside ed.
The same esul can be ob ained o he dis ibu ed app oxima e
con ollabili y a ime T. The e o e, app oxima e and null
con ollabili y a e equi alen concep s (dis ibu ed case).
Using a di e en echnique ( ic i ious con ols), i is possible o p o e a
null con ollabili y esul as in he p e ious co olla y when he coupling
ma ix A∈L∞(QT;L(R2)) sa is ies: The e exis an open subse
ω0⊂⊂ ωand a posi i e cons an a0s. .
|a21(x, )| ≥ a0>0 in ω0×(0,T).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
4. Dis ibu ed con ollabili y o 2 ×2 linea sys ems
Re e ences
1L. DE TERESA,Insensi izing con ols o a semilinea hea equa ion,
Comm. Pa ial Di e en ial Equa ions 25 (2000), no. 1–2, 39–72.
2F. AMMAR KHODJA, A. BENABDALLAH, C. DUPAIX ET I. KOSTIN,
Con ollabili y o he ajec o ies o phase- ield models by one con ol
o ce, SIAM J. Con ol Op im. 42 (2003), no. 5, 1661-1689.
3M. G.-B., R. PÉREZ-GARCÍA,Con ollabili y esul s o some
nonlinea coupled pa abolic sys ems by one con ol o ce, Asymp o .
Anal. 46 (2006), no. 2, 123–162.
4M. G.-B., L. DE TERESA,Con ollabili y esul s o cascade sys ems o
m coupled pa abolic PDEs by one con ol o ce, Po . Ma h. 67 (2010),
no. 1, 91–113.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
5. Bounda y con ollabili y o a 2×2
linea sys em
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
5. Bounda y con ollabili y o a 2 ×2 linea sys em
Le us now conside he bounda y con ollabili y p oblem o he
one-dimensional linea eac ion-di usion sys em:
(18) 






y −Dyxx =Ayin QT= (0, π)×(0,T),
y|x=0=1
0 ,y|x=π=0 on (0,T),
y(·,0) = y0in (0, π),
wi h y0∈H−1(0, π;R2), ∈L2(0,T)is he con ol and
D=d10
0d2,d1,d2>0,(d16=d2),and A=0 0
1 0 .
Exis ence and uniqueness
Fo any y0∈H−1(0, π;R2)and ∈L2(0,T), sys em (18) has a unique
solu ion y∈L2(QT)∩C0([0,T]; H−1(0, π;R2)) de ined by ansposi ion.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

5. Bounda y con ollabili y o a 2 ×2 linea sys em
Le us now conside he bounda y con ollabili y p oblem o he
one-dimensional linea eac ion-di usion sys em:
(18) 






y −Dyxx =Ayin QT= (0, π)×(0,T),
y|x=0=1
0 ,y|x=π=0 on (0,T),
y(·,0) = y0in (0, π),
wi h y0∈H−1(0, π;R2), ∈L2(0,T)is he con ol and
D=d10
0d2,d1,d2>0,(d16=d2),and A=0 0
1 0 .
Ques ion
A e he con ollabili y p ope ies o sys em (18) independen o d1and d2???
NO.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
5. Bounda y con ollabili y o a 2 ×2 linea sys em
As be o e, sys em (18) is null con ollable a ime Ti and only i he
obse abili y inequali y
kϕ1(·,0)k2
H1
0(0,π)+kϕ2(·,0)k2
H1
0(0,π)≤CTZT
0|ϕ1,x(0, )|2d ,
holds. Again ϕis he solu ion associa ed o ϕ0∈H1
0(0, π;R2)o he adjoin
p oblem:
(19) 


−ϕ −Dϕxx =A∗ϕin QT,
ϕ|x=0=ϕ|x=π=0 on (0,T),
ϕ(·,T) = ϕ0in (0, π).
Le us see ha , in gene al, his inequali y ails (e en i a21 =16=0!!!!!).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
5. Bounda y con ollabili y o a 2 ×2 linea sys em
A necessa y condi ion:
P oposi ion
Assume ha sys em (18) is null con ollable a ime T (d16=d2). Then
(λk=k2),
d1λk6=d2λj,∀k,j≥1(⇐⇒ pd1/d26∈ Q).
P oo : By con adic ion, assume ha d1λk=d2λj o some k,jand ake
K=max{k,j}. The idea is ans o ming sys em (19) in o an o.d.s.
Recall ha λkand φka e he eigen alues and no malized eigen unc ions o
−∂xx on (0, π)wi h homogenous Di ichle bounda y condi ions:
λk=k2, φk(x) = 2
πsin kx,k≥1,x∈(0, π).
Idea: Take ϕ0∈XK={ϕ0=PK
`=1a`φ`:a`∈R2} ⊂ H1
0(0, π;R2).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
5. Bounda y con ollabili y o a 2 ×2 linea sys em
Conside also
BK=


B
.
.
.
B


∈R2K,(B=1
0)and
L∗
K=diag (−λ1D+A∗,−λ2D+A∗,··· ,−λKD+A∗)∈ L(R2K).
Taking in (19) a bi a y ini ial da a ϕ0,K=PK
`=1a`φ`∈H1
0(0, π;R2)whe e
a`∈R2,i is no di icul o see ha sys em (19) is equi alen o he
o.d. sys em
(20) −Z0=L∗
KZon [0,T],Z(0) = Z0∈R2K.
F om he obse abili y inequali y o sys em (19) we deduce he unique
con inua ion p ope y o he solu ions o (20):
B∗
KZ(·) = 0 in (0,T)=⇒Z≡0.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
We conside he linea pa abolic sys em































∂ y1+L1
0( )y1+
n
X
j=1
C1j·∇yj+
n
X
j=1
a1jyj= 1ωin QT= Ω ×(0,T),
∂ y2+L2
0( )y2+
n
X
j=1
C2j·∇yj+
n
X
j=1
a2jyj=0 in QT,
···
∂ yn+Ln
0( )yn+
n
X
j=1
Cnj ·∇yj+
n
X
j=1
anjyj=0 in QT,
yi=0 on ΣT=∂Ω×(0,T),yi(·,0) = yi
0in Ω,1≤i≤n,
whe e aij =aij(x, )∈L∞(QT),Cij =Cij(x, )∈L∞(QT;RN)(1 ≤i,j≤n),
yi
0∈L2(Ω) (1 ≤i≤n) and Lk
0( )is, o e e y 1 ≤k≤n, he second o de
ope a o Lk
0( )y=−
N
X
i,j=1
∂
∂xiαk
ij(x, )∂y
∂xjwhe e αk
ij sa is y (3) and (4) o
e e y k.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

6. A gene aliza ion: Cascade sys ems
Objec i e
Con ollabili y p ope ies o he sys em: nequa ions con olled wi h a unique
dis ibu ed con ol.
Equi alen ly, he p e ious sys em can be w i en as
(21) (∂ y+b
L( )y+C·∇y+Ay=B 1ωin QT,
y=0 on ΣT,y(·,0) = y0in Ω,
whe e b
L( )is he ma ix ope a o gi en by b
L( ) = diag (L1
0( ),··· ,Ln
0( )),
y= (yi)1≤i≤nis he s a e and ∇y= (∇yi)1≤i≤n, and whe e
(y0= (yi
0)1≤i≤n∈L2(Ω; Rn),A(·,·)=(aij(·,·))1≤i,j≤n∈L∞(QT;L(Rn)),
C(·,·)=(Cij(·,·))1≤i,j≤n∈L∞(QT;L(Rn;RNn)) and B≡e1= (1,0, ..., 0)∗
a e gi en. Le us obse e ha , o each y0∈L2(Ω; Rn)and ∈L2(QT),
Sys em (21) admi s a unique weak solu ion
y∈L2(0,T;H1
0(Ω; Rn)) ∩C0([0,T]; L2(Ω; Rn)).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
By cascade sys em we mean ha ma ices Aand Cha e he ollowing
s uc u e:
A=






a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
0a32 a33 ... a3n
.
.
..
.
........
.
.
0 0 ... an,n−1ann







,C=




C11 C12 ... C1n
0C22 ... C2n
.
.
..
.
.....
.
.
0 0 ... Cnn





wi h aij ∈L∞(QT)and Cij ∈L∞(QT;RN)and he coe icien s ai,i−1sa is y
ai,i−1≥c0>0 o −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n,
wi h ω0⊆ωa new open subse .
Rema k
I is na u al o assume ha ai,i−16≡ 0 o any i:2≤i≤n. The p e ious
assump ion is s onge bu will p o ide he con ollabili y esul .
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
In his case, he co esponding adjoin p oblem has he o m





















−∂ ϕi+Li
0( )ϕi−
i
X
j=1
[∇·(Cjiϕj)−ajiϕj] = −ai+1,iϕi+1in QT,
··· (1≤i≤n−1),
−∂ ϕn+Ln
0( )ϕn−
n
X
j=1
[∇·(Cjnϕj)−ajnϕj] = 0 in QT,
ϕi=0 on ΣT, ϕi(·,T) = ϕi,Tin Ω,1≤i≤n,
whe e ϕi,T∈L2(Ω) (1 ≤i≤n). Again, he null con ollabili y o
Sys em (21) (wi h L2-con ols) a ime Tis equi alen o he exis ence o a
cons an CT>0 such ha he so-called obse abili y inequali y
kϕ(·,0)k2
L2(Ω;Rn)≤CTZZω×(0,T)|ϕ1(x, )|2
holds o e e y solu ion ϕ= (ϕ1, . . . , ϕn)∗ o he adjoin p oblem.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
Theo em
Unde he p e ious assump ions, le M0=max2≤i≤nkai,i−1k∞. Then, he e
exis a posi i e unc ion α0∈C2(Ω) (only depending on Ωand ω0), wo
posi i e cons an s C0and σ0(only depending on Ω,ω0,c0, M0and d) and
l≥0(only depending on n) such ha , o e e y ϕT∈L2(QT;Rn), he
solu ion ϕ o he adjoin p oblem sa is ies
n
X
i=1I(d+3(n−i), ϕi)≤C0sd+lZZω0×(0,T)
e−2sαγ( )d+l|ϕ1|2,
∀s≥s0=σ0T+T2+T2max
i≤jkaijk
2
3(j−i)+3
∞+kCijk
2
3(j−i)+1
∞. In he
p e ious inequali y, γ( ) = −1(T− )−1,α(x, ) = α0(x)/ (T− )and
I(d,z)is gi en in Lemma 2.3 (wi h αins ead o β).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
Combining he p e ious esul and ene gy inequali ies sa is ied by he
solu ions o he adjoin sys em i is possible o p o e an obse abili y
inequali y o he adjoin sys em (as in he scala case). Summa izing, we
ge
Co olla y
Unde assump ions o he p e ious esul , he e exis s a posi i e cons an C
(only depending on Ω,ω,n,c0and M0) such ha o e e y y0∈L2(Ω; Rn)
he e is a con ol ∈L2(Ω) which sa is ies
k |k2
L2(QT)≤eCHky0k2
L2(Ω;Rn),
and y(·,T) = 0in Ω, wi h y he solu ion o (21) associa ed o y0and . In he
p e ious inequali y, His gi en by
H ≡ 1+T+1
T+max
i≤jkaijk
2
3(j−i)+3
∞+kCijk
2
3(j−i)+1
∞+Tkaijk∞+kCijk2
∞.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

6. A gene aliza ion: Cascade sys ems
Ske ch o he p oo o Theo em 6.1: Gi en ω0⊂ω, we choose ω1⊂⊂ ω0.
Le α0∈C2(Ω) be he unc ion p o ided by Lemma 2.3 and associa ed o Ω
and B≡ω1. We will do he p oo in wo s eps:
S ep 1. Le ϕbe he solu ion o adjoin sys em associa ed o ϕT. Each
componen sa is ies
−∂ ϕi+Li
0( )ϕi=
i
X
j=1
[∇·(Cjiϕj)−ajiϕj]−ai+1,iϕi+1.
We begin applying inequali y (11) wi h B=ω1 o each unc ion ϕiwi h
L0≡Li
0,d=d+3(n−i)and he co esponding igh -hand side. Now i we
ake
s≥s0=σ0T+T2+T2max
i≤jkaijk
2
3(j−i)+3
∞+kCijk
2
3(j−i)+1
∞,
wi h σ0=σ0(Ω,ω0,c0,M0)>0, we ob ain he exis ence o a posi i e
cons an s C1=C1(Ω,ω0,c0,M0)such ha i s≥s0, hen
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
n
X
i=1I(d+3(n−i), ϕi)≤C1
n
X
i=1
ss+3(n−i)ZZω1×(0,T)
e−2sαγ( )s+3(n−i)|ϕi|2.
S ep 2. Thanks o he assump ion
ai,i−1≥c0>0 o −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n,
wi h ω0⊆ωan open subse , and he cascade s uc u e
ai,i−1ϕi=∂ ϕi−1−Li−1
0( )ϕi−1+
i−1
X
j=1
[∇·(Cj,i−1ϕj)−aj,i−1ϕi−1]in QT,
can elimina e he local e ms o 2 ≤i≤n. In o de o ca y his p ocess ou ,
we will need he ollowing esul :
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
Lemma
Unde assump ions o Theo em 6.1 and gi en l ∈N,ε > 0, k ∈ {2, ..., n}and
wo open se s O0and O1such ha ω1⊂O1⊂⊂ O0⊂ω0, he e exis a
cons an Ck(only depending on Ω,O0,O1,c0and M0) and lkj ∈N,
1≤j≤k−1(only depending on l, n, k and j), such ha , i s ≥s0, one has
slZZO1×(0,T)
e−2sαγ( )l|ϕk|2≤ε[I(d+3(n−k), ϕk) + I(d+3(n−k−1), ϕk+1)]
+Ck1+1
εk−1
X
j=1
slkj ZZO0×(0,T)
e−2sαγ( )lkj |ϕj|2.
(In his inequali y we ha e aken ϕk+1≡0when k =n).
The p oo o Theo em 6.1 is a consequence o his Lemma 6.3. Fo he
de ails, see [DE TERESA], Comm. PDE (2000), [G.-B., PÉREZ-GARCÍA],
Asymp. Anal. (2006) and [G.-B., DE TERESA], Po . Ma h. (2010).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
6. A gene aliza ion: Cascade sys ems
Rema k
1Cascade sys ems appea in he con ex o exis ence o insensi izing
con ols o a scala pa abolic equa ion: Equi alen o a null
con ollabili y esul o a 2 ×2 pa abolic sys em (n=2) wi h one
equa ion o wa d in ime and he o he one backwa d. The coupling
coe icien a21 is 1Owi h O⊆Ωan open se and O∩ω6=∅.
2The p e ious p oo uses he assump ion
ai,i−1≥c0>0 o −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n,
in a c ucial way. When ai,i−1a e cons an , his assump ion is necessa y.
Is his condi ion necessa y in he gene al case??? No.
3Is i possible o p o ide a necessa y and su icien (Kalman condi ion)
condi ion o he null con ollabili y o non-scala sys ems? YES in
some cons an coe icien sys ems.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Le us conside {λk}k≥1 he sequence o eigen alues o L0wi h
homogeneous Di ichle bounda y condi ions and {φk}k≥0 he co esponding
no malized eigen unc ions.
Theo em (A Necessa y Condi ion)
I sys em (22) is null con ollable a ime T hen
(24) ank [−λkD+A|B] = n,∀k≥1.
whe e
[−λkD+A|B]=[B,(−λkD+A)B,(−λkD+A)2B,··· ,(−λkD+A)n−1B].
P oo : Reasoning by con adic ion: ∃k≥1 such ha
ank [−λkD+A|B]<n. Then he o.d.s. −Z0= (−λkD+A∗)Zin (0,T),is
no B∗-obse able a ime T.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

7. The Kalman condi ion o a class o pa abolic sys ems
The e exis s Z0∈Rn,Z06=0, such ha he solu ion Z o he p e ious sys em
sa is ies B∗Z(·) = 0 on (0,T). Bu ϕ(x, ) = Z( )φk(x)is he solu ion o
adjoin p oblem
−∂ ϕ+DL0ϕ=A∗ϕin QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω,
associa ed o ϕ0(x) = Z0φk6≡ 0 and B∗ϕ(·,·)≡0 in QT. Then, he
obse abili y inequali y
kϕ(·,0)k2
L2(Ω) ≤CTZZω×(0,T)|B∗ϕ(x, )|2,
ails and he sys em is no null con ollable a ime T.
Rema k
I condi ion (24) is no sa is ied, hen sys em (22) is nei he app oxima ely
con ollable no null con ollable a ime T( o any T>0) e en i ω≡Ω.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Ques ion:
Is condi ion (24) ank [−λkD+A|B] = n,∀k≥1, a su icien condi ion o
he null con ollabili y o sys em (22)???
Le us now in oduce he unbounded ma ix ope a o
K= [DL0+A|B] = [B,(−DL0+A)B,··· ,(−DL0+A)n−1B],
(K:D(K)⊂L2(Ω; Rnm)→L2(Ω; Rn),wi h
D(K) := {y∈L2(Ω; Rnm) : Ky∈L2(Ω; Rn)}.
Then,
P oposi ion
ke K∗={0}i and only i condi ion (24), ank [−λkD+A|B] = n, ∀k≥1,
holds.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Ques ion:
Is condi ion (24) ank [−λkD+A|B] = n,∀k≥1, a su icien condi ion o
he null con ollabili y o sys em (22)???
Le us now in oduce he unbounded ma ix ope a o
K= [DL0+A|B] = [B,(−DL0+A)B,··· ,(−DL0+A)n−1B],
(K:D(K)⊂L2(Ω; Rnm)→L2(Ω; Rn),wi h
D(K) := {y∈L2(Ω; Rnm) : Ky∈L2(Ω; Rn)}.
Then,
P oposi ion
ke K∗={0}i and only i condi ion (24), ank [−λkD+A|B] = n, ∀k≥1,
holds.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
(22) (∂ y+DL0y=Ay+B 1ωin QT,
y=0 on ΣT,y(·,0) = y0(·)in Ω,
Theo em (Kalman condi ion)
Sys em (22) is exac ly con ollable o ajec o ies a ime T i and only i
Sys em (22) is app oxima ely con ollable a ime T i and only i
ke K∗={0}(⇐⇒ ank [−λkD+A|B] = n, ∀k≥1).
Rema k
One can p o e, ei he he e exis s k0≥1 such ha
ank [−λkD+A|B] = n,∀k≥k0
o
ank [−λkD+A|B]<n,∀k≥1.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Con ollabili y (ou side a ini e dimensional space) i and only i he
algeb aic Kalman condi ion ank [−λkD+A|B] = nis sa is ied o one
equency k≥1.
Rema k
Sys em (22) can be exac ly con olled o he ajec o ies wi h one con ol
o ce (m=1 and B∈Rn) e en i A≡0 . Indeed, le us assume ha
B= (bi)1≤i≤n∈Rn. Then,
[(−λkD+A)|B] = 




b1(−λkd1)b1··· (−λkd1)n−1b1
b2(−λkd2)b2··· (−λkd2)n−1b2
.
.
..
.
.....
.
.
bn(−λkdn)bn··· (−λkdn)n−1bn




∈ L(Rn),
and (24) holds i and only i bi6=0 o e e y iand dia e dis inc .
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

7. The Kalman condi ion o a class o pa abolic sys ems
Idea o he p oo : We ha e p o ed he necessa y condi ion. The e o e, le
us p o e ha ank [−λkD+A|B] = n, o any k, is a su icien condi ion
o he null con ollabili y a ime To he sys em.
Then, he objec i e is o p o e he obse abili y inequali y:
kϕ(·,0)k2
L2(Ω) ≤CZZω×(0,T)|B∗ϕ(x, )|2,
o he solu ions o he adjoin p oblem.
To his end we use wo a gumen s:
P o e a global Ca leman es ima e o a scala pa abolic equa ion o o de
nin ime.
P o e a coe ci i y p ope y o he Kalman ope a o K.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Le us ix ϕ0∈D(Li
0),∀i≥0 and conside ϕ he co esponding solu ion o
he adjoin sys em (23)
−∂ ϕ+DL0ϕ=A∗ϕin QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω.
Le us ake Φ =
n
X
i=1
aiϕi,wi h ai∈R(1 ≤i≤n). Then, Φis a egula
solu ion (Li
0∂j
Φ∈L2(QT),∀i,j) o he linea pa abolic scala equa ion o
o de nin ime
de (Id∂ −DL0+A∗) Φ = 0 in QT,
Li
0Φ = 0 on ΣT,∀i≥0.
The key poin is o p o e a Ca leman inequali y o he solu ions o he
p e ious p oblem. Fix ω0⊂⊂ ωa nonemp y open subse . Recall Lemmas 2.3
and 2.4:
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Lemma
The e exis a α0∈C2(Ω) (posi i e), and wo cons an s C0,σ0>0(only
depending on Ω,ω0and d) s. .



















I1(d, φ)≡ZZQT
e−2sα[sγ( )]d−4|φ |2+|L0φ|2
+ZZQT
e−2sα[sγ( )]d−2|∇φ|2+ZZQT
e−2sα[sγ( )]d|φ|2
≤C0 ZZω0×(0,T)
e−2sα[sγ( )]d|φ|2+ZZQT
e−2sα[sγ( )]d−3|φ ±L0φ|2!,
∀s≥s0=σ0(Ω,ω)(T+T2),∀φ∈L2(0,T;H1
0(Ω)) s. . φ ±L0φ∈L2(QT).
γ( ) = −1(T− )−1,α(x, ) = α0(x)/ (T− ).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Theo em
Le n,k1,k2∈Nand d ∈R. The e exis wo cons an s Cand σ(only
depending on Ω,ω,n,D,A, k1, k2and d), and 0= 0(n)∈Nsuch ha
k1
X
i=0
k2
X
j=0J(d−4(i+j),Li
0∂j
Φ) ≤CZZω×(0,T)
e−2sα[sγ( )]3+ 0|Φ|2, ,
∀s≥s=σ(Ω,ω)(T+T2),Φsolu ion o he p e ious p oblem and
J(τ, z) := I1(τ+3(n−1),z) +
n
X
i=1I1(τ+3(n−2),Piz)
+
n−1
X
p=2X
1≤i1<···<ip≤nI1(τ+3(n−p−1),Pip···Pi1z).
(Pi≡∂ −diL0)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Conclusion
I ϕis a egula solu ion o he adjoin p oblem
−∂ ϕ+DL0ϕ=A∗ϕin QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω,
hen, any linea combina ion Φ = Pn
i=1aiϕisa is ies Theo em 10. In
pa icula any componen o B∗ϕ.
Recall K= [DL0+A|B]=[B,(−DL0+A)B,··· ,(−DL0+A)n−1B], hen
K∗ϕ(·, ) = [B∗ϕ , B∗(−DL0+A∗)ϕ , ··· ,B∗(−DL0+A∗)n−1ϕ] (·, )
= [B∗ϕ , −∂ (B∗ϕ),··· ,(−1)n−1∂n−1
(B∗ϕ)] (·, )∈Rnm.
We apply Theo em 10 wi h k1=n−1 and k2=k≥0. Then, a e some
compu a ions, we deduce (d=3)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

7. The Kalman condi ion o a class o pa abolic sys ems
Conclusion
I ϕis a egula solu ion o he adjoin p oblem
−∂ ϕ+DL0ϕ=A∗ϕin QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω,
hen, any linea combina ion Φ = Pn
i=1aiϕisa is ies Theo em 10. In
pa icula any componen o B∗ϕ.
Recall K= [DL0+A|B]=[B,(−DL0+A)B,··· ,(−DL0+A)n−1B], hen
K∗ϕ(·, ) = [B∗ϕ , B∗(−DL0+A∗)ϕ , ··· ,B∗(−DL0+A∗)n−1ϕ] (·, )
= [B∗ϕ , −∂ (B∗ϕ),··· ,(−1)n−1∂n−1
(B∗ϕ)] (·, )∈Rnm.
We apply Theo em 10 wi h k1=n−1 and k2=k≥0. Then, a e some
compu a ions, we deduce (d=3)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Then, a e some compu a ions, we deduce (d=3)
ZT
0
e
−2sM0
(T− )[sγ( )]3kLk
0K∗ϕk2
L2(Ω)nm ≤CZZω×(0,T)
e−2sα[sγ( )]3+ 0|B∗ϕ|2
o e e y s≥σT+T2. In his inequali y, M0=maxΩα0and 0≥0 is an
in ege only depending on n.
Rema k
The p e ious inequali y is a pa ial obse abili y es ima e. I is alid e en i
he Kalman condi ion does no hold, i.e., e en i ke K∗6={0}.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
The coe ci i y p ope y o K∗:
Theo em
Assume ha ke K∗={0}and conside k = (n−1)(2n−1). Then he e
exis s C>0such ha i z ∈L2(Ω)nsa is ies K∗z∈D(Lk
0)nm, one has
kzk2
L2(Ω)n≤CkLk
0K∗zk2
L2(Ω)nm .
So, om he p e ious inequali y we ge
ZT
0
e
−2sM0
(T− )[sγ( )]3kϕk2
L2(Ω)nm ≤CZZω×(0,T)
e−2sα[sγ( )]3+ 0|B∗ϕ|2
and he obse abili y inequali y:
kϕ(·,0)k2
L2(Ω) ≤CZZω×(0,T)|B∗ϕ(x, )|2.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Summa izing
1We ha e es ablished a Kalman condi ion
ke K∗={0}
which cha ac e izes he con ollabili y p ope ies o sys em (22).
2The Kalman condi ion o sys em (22) ke K∗={0}gene alizes he
algeb aic Kalman condi ion ke [A|B]∗={0} o o.d.s.
3This Kalman condi ion is also equi alen o he app oxima e
con ollabili y o sys em (22) a ime T. Again, app oxima e and null
con ollabili y a e equi alen concep s o sys em (22).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
7. The Kalman condi ion o a class o pa abolic sys ems
Re e ences
1F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A
gene aliza ion o he Kalman ank condi ion o ime-dependen coupled
linea pa abolic sys ems, Di e . Equ. Appl. 1(2009), no. 3, 139–151.
D=Id,A=A( )and B=B( ).
2F. 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.
Ddiagonal ma ix,Aand Bcons an ma ices.
3E. FERNÁNDEZ-CARA, M. G.-B, L. DE TERESA,Con ollabili y o
linea and semilinea non-diagonalizable pa abolic sys ems, ESAIM
Con ol Op im. Calc. Va . 21 (2015), no. 4, 1178–1204.
Dnon-diagonalizable ma ix wi h Jo dan blocks o dimension ≤4,
Aand Bcons an ma ices.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

7. The Kalman condi ion o a class o pa abolic sys ems
Open p oblems
Null con ollabili y p ope ies o
(22) (∂ y+DL0y=A( )y+B( ) 1ωin QT,
y=0 on ΣT,y(·,0) = y0(·)in Ω,
when A( )and B( )depend on ( o ins ance, A∈C∞([0,T]; L(Rn))
and B∈C∞([0,T]; L(Rm,Rn))) and D=diag (d1,d2,··· ,dn)∈ L(Rn)
wi h di>0.
Null con ollabili y p ope ies o
(22) (∂ y+DL0y=Ay+B 1ωin QT,
y=0 on ΣT,y(·,0) = y0(·)in Ω,
when Aand Ba e cons an ma ices and Dis a gene al
non-diagonalizable ma ix (de ini e posi i e).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o
pa abolic sys ems. Bounda y con ols
[AMMAR-KHODJA,BENABDALLAH,G.-B.,DE TERESA], J. Ma h. Pu es
Appl. (2011).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
Le us conside he bounda y con ollabili y p oblem:
(25) 




y =yxx +Ayin QT= (0, π)×(0,T),
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
whe e A∈ L(Cn)and B∈ L(Cm;Cn)a e wo gi en ma ices and
y0∈H−1(0, π;Cn)is he ini ial da um. In sys em (25), ∈L2(0,T;Cm)is
he con ol unc ion ( o be de e mined).
Simple p oblem: One-dimensional case and D=Id.
This p oblem has been s udied in he case n=2:
E. FERNÁNDEZ-CARA, M. G.-B., L. DE TERESA,Bounda y
con ollabili y o pa abolic coupled equa ions, J. Func . Anal. 259
(2010), no. 7, 1720–1758.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
We conside again {λk}k≥1 he sequence o eigen alues o −∂xx in (0, π)
wi h homogenuous Di ichle bounda y condi ions and {φk}k≥0 he
co esponding no malized eigen unc ions:
λk=k2, φk(x) = 2
πsin kx,k≥1,x∈(0, π).
Theo em (n=2, m=1)
Le A∈ L(C2)and B∈C2be gi en and le us deno e by µ1and µ2 he
eigen alues o A∗. Then (25) is exac ly con ollable o he ajec o ies a any
ime T >0i and only i ank [A|B] = 2and
λk−λj6=µ1−µ2∀k,j∈Nwi h k 6=j.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
Rema k (One con ol, m=1)
When m=1, he Kalman condi ion (27) is equi alen o ank [A|B] = n
and λk−λl6=µi−µj o any k,l∈Nand 1 ≤i,j≤pwi h (k,i)6= (l,j),
whe e {µi}1≤i≤p⊂Cis he se o dis inc eigen alues o A∗. We gene alize
he esul s o [FERNÁNDEZ-CARA,G.-B.,DE TERESA], J. Func . Anal.
(2010).
One con ol, m=1
We ha e imposed wo condi ions:
1 ank [A|B] = n: Sys em (25) is no decoupled.
2λk−λl6=µi−µj: The adjoin sys em can be w i en (R0=Id∂xx +A∗)
(26) −ϕ =R0ϕin QT,
ϕ=0 on ΣT, ϕ(·,T) = ϕ0in (0, π),
and he eigen alues o R0a e simple.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
Be o e p o ing he esul , le us analyze he Kalman condi ion (27)
ank Kk=nk,∀k≥1:
P oposi ion
Le us deno e by {µi}1≤i≤p⊂C he se o dis inc eigen alues o A∗. Then,
1The e exis s an in ege k0=k0(A)∈N, only depending on A, such ha ,
λk−λl6=µi−µj,∀k>k0,l≥1,k6=l,and 1≤i,j≤p.
2The ollowing condi ions a e equi alen :
(a) ankKk=nk o e e y k ≥1.
(b) ankKk=nk o e e y k :1≤k≤k0.
(c) ankKk0=nk0.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
Necessa y implica ion. We eason as be o e: i ank Kk<nk, o some
k≥1, hen he o.d.s.
−Z0=L∗
kZon (0,T),Z(T) = Z0∈Cnk
is no B∗
k-obse able on (0,T), i.e., he e exis s Z06=0 s. . B∗
kZ( ) = 0 o
e e y ∈(0,T). F om Z0i is possible o cons uc ϕ0∈H1
0(0, π;Cn)wi h
ϕ06≡ 0 such ha he co esponding solu ion o he adjoin p oblem (27)
sa is ies
B∗ϕx(0, ) = 0∀ ∈(0,T).
As a consequence: The unique con inua ion p ope y and he p e ious
obse abili y inequali y o he adjoin p oblem ail:
Nei he app oxima e no null con ollabili y a any T o sys em (25).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
Su icien implica ion. Fo he p oo we ollow he ideas om
H.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.
Two “big” s eps:
(I) We e o mula e he null con ollabili y p oblem o sys em (25) as a
ec o momen p oblem.
(II) Exis ence and bounds o a amily bio hogonal o app op ia e complex
ma ix exponen ials.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(I) The ec o momen p oblem: As in he scala case, ∈L2(0,T;Cm)is a
null con ol o sys em
(25) 




y =yxx +Ayin QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
(i.e., he solu ion y o (25) sa is ies y(·,T) = 0 in (0, π))⇐⇒ sa is ies
−hy0, ϕ(·,0)i=ZT
0
( ( ),B∗ϕx(0, ))Cmd ,∀ϕ0∈H1
0(0, π;Cn),
whe e ϕis he solu ion o he adjoin p oblem
(26) 




−ϕ =ϕxx +A∗ϕin QT,
ϕ(0,·) = ϕ(π, ·) = 0 on (0,T),
ϕ(·,T) = ϕ0in (0, π).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(I) The ec o momen p oblem:
Thus, he idea is o ake i s ly ϕ0∈Xk0,
(Xk0={ϕ0:ϕ0=Pk0
i=1aiφiwi h ai∈Cn}) and hen ϕ0=aφk, wi h k>k0
and a∈Cn. The e o e, we wan ∈L2(0,T;Cm)s. .







ZT
0
( (T− ),B∗
k0eL∗
k0 Φ0)Cmd =F(Y0,Φ0),∀Φ0∈Cnk0,
ZT
0
( (T− ),B∗e(−λkId+A∗) a)Cmd = k(y0,a),∀a∈Cn,∀k>k0,
In some sense, has o sol e an in ini e numbe o null con ollabili y
p oblems o app op ia e o.d. sys ems:



Y0=Lk0Y+Bk0 on (0,T),Y(0) = Y0;
Z0= (−λkId+A)Z+B on (0,T),Z(0) = y0k:= (y0, φk),∀k>k0.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(II) Bio hogonal amilies o app op ia e complex ma ix exponen ials.
F om he p e ious s ep, we ha e ob ained he complex ma ix exponen ials
eL∗
k0 and {e(−λkId+A∗) }k>k0.
Le us deno e {γ`}1≤`≤e
p⊂C he se o dis inc eigen alues o L∗
k0and ecall
ha {µi}1≤i≤p⊂Cis he se o dis inc eigen alues o A∗. Then, he se
Λ={γ`}1≤`≤e
p∪{−λk+µi}k>k0,1≤i≤pis he se o eigen alues o he
ope a o ∂xxId +A∗. Thus, ou nex pu pose is:
Objec i e
As in he scala case, cons uc ion o a bio hogonal amily in L2(0,T;C) o
n jeγ` , je(−λk+µi) :1≤`≤ep,1≤i≤p,0≤j≤η−1,k>k0o,
which sa is ies app op ia e bounds (see (22)). In he p e ious exp ession, ηis
he maximal dimension o he Jo dan blocks associa ed o γ`and µi.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(II) Bio hogonal amilies o app op ia e complex ma ix exponen ials.
Le us ix η≥1, an in ege , T∈(0,∞]and {Λk}k≥1⊂C+a sequence s. .
Λk6=Λj,∀k,j≥wi h k6=j.
Le us ecall ha he amily {qk,j}k≥1,0≤j≤η−1⊂L2(0,T;C)is bio hogonal
o { je−Λk }k≥1,0≤j≤η−1i one has
ZT
0
je−Λk q∗
l,i( )d =δklδij,∀(k,j),(l,i) : k,l≥1,0≤i,j≤η−1.
In addi ion, we wan he amily {qk,j}k≥1,0≤j≤η−1⊂L2(0,T;C) o sa is y he
p ope y:
Fo any ε>0, he e is C(ε,T)>0 s. . kqk,jkL2(0,T;C)≤C(ε,T)eε<Λk,
∀k≥1 and 0 ≤j≤η−1.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(II) Bio hogonal amilies o app op ia e complex ma ix exponen ials.
Theo em
Le us ix T ∈(0,∞]and assume ha o wo posi i e cons an s δand ρone
has 




<Λk≥δ|Λk|,|Λk−Λl| ≥ ρ|k−l|,∀k,l≥1,
X
k≥1
1
|Λk|<∞.
Then, ∃{qk,j}k≥1,0≤j≤η−1bio hogonal o  je−Λk k≥1,0≤j≤η−1such ha , o
e e y ε>0, he e exis s C(ε,T)>0sa is ying
kqk,jkL2(0,T;C)≤C(ε,T)eε<Λk,∀(k,j) : k≥1,0≤j≤η−1.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
8. The Kalman condi ion o a class o pa abolic sys ems.
Bounda y con ols
(II) Bio hogonal amilies o app op ia e complex ma ix exponen ials.
P oo :
The p oo o his esul is e y echnical. I can be ound in
[AMMAR-KHODJA,BENABDALLAH,G.-B.,DE TERESA], The Kalman
condi ion o he bounda y con ollabili y o coupled pa abolic sys ems.
Bounds on bio hogonal amilies o complex ma ix exponen ials, J. Ma h.
Pu es Appl. (2011).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
We a e going o e isi ed p oblem (18). Wi h a sligh ly change o no a ions,
his p oblem is:
(18) 




y −Dyxx +A0y=0 in QT= (0, π)×(0,T),
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
whe e D=diag (1,d),A0=0 1
0 0 ,B=0
1.When d=1 (i.e.,
D=Id), we saw
Theo em (d=1)
Le A0∈ L(C2)and B∈C2be gi en and le us deno e by µ1and µ2 he
eigen alues o A∗
0. Then (18) is app oxima e and null con ollable a any
ime T >0i and only i ank [A|B] = 2and (λk=k2)
λk−λj6=µ1−µ2∀k,j∈Nwi h k 6=j.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT= (0, π)×(0,T),
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
whe e D=diag (1,d),A0=0 1
0 0 ,B=0
1.
Theo em (d6=1)
Unde he p e ious assump ions, sys em (18) is app oxima e con ollable a
ime T >0i and only i √d6∈ Q.
The e o e:
1I d=1, (18) is app oxima e and null con ollable a any T>0.
2I d6=1, we only know ha sys em (18) is app oxima e con ollable a
ime T>0i and only i √d6∈ Q.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
whe e D=diag (1,d),A0=0 1
0 0 ,B=0
1
Assump ion
In he sequel, D=diag (1,d)wi h d6=1 and √d6∈ Q.
Goal
Analyze he null con ollabili y p ope ies a ime T>0 o sys em (18).
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
Le ϕbe a solu ion o he adjoin p oblem:





−ϕ −Dϕxx +A∗
0ϕ=0 in QT,
ϕ(0,·) = ϕ(π, ·) = 0 on (0,T),
ϕ(·,T) = ϕ0∈H1
0(0, π)2in (0, π).
I yis a solu ion o he di ec p oblem, hen
hy(T), ϕ0i−hy0, ϕ(0)i=ZT
0
( )B∗Dϕx(0, )d
Thus y(T) = 0⇐⇒ ∃ ∈L2(0,T)such ha
ZT
0
( )B∗Dϕx(0, )d =−hy0, ϕ(0)i,∀ϕ0∈H1
0(0, π;R2)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
Le ϕbe a solu ion o he adjoin p oblem:





−ϕ −Dϕxx +A∗
0ϕ=0 in QT,
ϕ(0,·) = ϕ(π, ·) = 0 on (0,T),
ϕ(·,T) = ϕ0∈H1
0(0, π)2in (0, π).
I yis a solu ion o he di ec p oblem, hen
hy(T), ϕ0i−hy0, ϕ(0)i=ZT
0
( )B∗Dϕx(0, )d
Thus y(T) = 0⇐⇒ ∃ ∈L2(0,T)such ha
ZT
0
( )B∗Dϕx(0, )d =−hy0, ϕ(0)i,∀ϕ0∈H1
0(0, π;R2)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
Fa o ini-Russell Me hod
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems

9. New phenomena: Minimal ime o con ollabili y
Fa o ini-Russell Me hod
σ(−D∂2
xx +A∗
0) = Sk≥1k2,dk2:= Sk≥1{λk,1,λk,2}.
{Φk,i}a (Riesz) basis o H1
0(0, π)2, whe e Φk,i=Vk,isin kx,i=1,2 a e
eigen unc ions o he ope a o −D∂2
xx +A∗
0.
Vk,1and Vk,2: eigen ec o s o he ma ix k2D+A∗
0associa ed o he
eigen alues k2,dk2.
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
Objec i e: Exis ence o ∈L2(0,T)s. .
ZT
0
( )B∗Dϕx(0, )d =−hy0, ϕ(0)i,∀ϕ0∈H1
0(0, π;R2)
Choosing ϕ0=Φk,i,we ha e ϕ(·, ) = e−λk,i(T− )Φk,iand
ϕ(x,0) = e−λk,iTΦk,i(x), ϕx(0, ) = ke−λk,i(T− )Vk,i
The iden i y connec ing yand ϕw i es (momen p oblem)
kB∗DVk,iZT
0
(T− )e−λk,i d =−e−λk,iThy0,Φk,ii,∀(k,i)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
Objec i e: Exis ence o ∈L2(0,T)s. .
ZT
0
( )B∗Dϕx(0, )d =−hy0, ϕ(0)i,∀ϕ0∈H1
0(0, π;R2)
Choosing ϕ0=Φk,i,we ha e ϕ(·, ) = e−λk,i(T− )Φk,iand
ϕ(x,0) = e−λk,iTΦk,i(x), ϕx(0, ) = ke−λk,i(T− )Vk,i
The iden i y connec ing yand ϕw i es (momen p oblem)
kB∗DVk,iZT
0
(T− )e−λk,i d =−e−λk,iThy0,Φk,ii,∀(k,i)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems
9. New phenomena: Minimal ime o con ollabili y
(18) 




y −Dyxx +A0y=0 in QT,
y(0,·) = B ,y(π, ·) = 0 on (0,T),
y(·,0) = y0in (0, π),
App oxima e con ollabili y: a necessa y condi ion (I)
kB∗DVk,iZT
0
(T− )e−λk,i d =−e−λk,iThy0,Φk,ii,∀(k,i)
A necessa y condi ion: B∗DVk,i6=0 o all k≥1,i=1,2
Recall d6=1 ,
B∗= (0,1),Vk,1= 1
1
(d−1)k2!,Vk,2=0
1,∀k≥1.
So, he e B∗DVk,i6=0,∀k≥1,i=1,2 (algeb aic Kalman
condi ion)
M. González-Bu gos Con ollabili y o non-scala pa abolic sys ems