The T-D equation
The heat equation
On the uniform controllability of the
one-dimensional transport-diusion equation
Pierre Lissy
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie
New trends in Modeling, Control and Inverse Problems, CIMI,
Toulouse,
June 18, 2014
Conclusion
The T-D equation
1
The heat equation
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Introduction
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
Introduction
Notations
T > 0 time,
The heat equation
Conclusion
The T-D equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
The heat equation
Conclusion
The T-D equation
The heat equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
diusion coecient
ε>0
(destined to tend to 0),
Conclusion
The T-D equation
The heat equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
diusion coecient
length
L > 0,
ε>0
(destined to tend to 0),
Conclusion
The T-D equation
The heat equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
diusion coecient
ε>0
(destined to tend to 0),
L > 0,
y ∈ L (0, L) initial condition (or H − (0, L)),
length
0
2
1
Conclusion
The T-D equation
The heat equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
diusion coecient
ε>0
(destined to tend to 0),
L > 0,
y ∈ L (0, L) initial condition (or H − (0, L)),
control v ∈ L (0, T ),
length
0
2
1
2
Conclusion
The T-D equation
The heat equation
Introduction
Notations
T > 0 time,
M 6= 0 speed,
diusion coecient
ε>0
(destined to tend to 0),
L > 0,
y ∈ L (0, L) initial condition (or H − (0, L)),
control v ∈ L (0, T ),
Q := (0, T ) × (0, L).
length
0
2
1
2
Conclusion
The T-D equation
The heat equation
Conclusion
Introduction
The equation
The transport-diusion equation


yt − εyxx + Myx = 0 in Q ,




y (., 0) = v (t ) in (0, T ),

y (., L) = 0 in (0, T ),




y (0, .) = y 0 in (0, L).
(T-D)
The T-D equation
The heat equation
Conclusion
Introduction
The equation
The transport-diusion equation


yt − εyxx + Myx = 0 in Q ,




y (., 0) = v (t ) in (0, T ),

y (., L) = 0 in (0, T ),




y (0, .) = y 0 in (0, L).
(T-D)
Denition
Let T , L, M , ε be xed. Equation (T-D) is null-controllable at time
T if and only if for every y ∈ L (0, L), there exists v ∈ L (0, T )
such that y (T ) = 0.
0
2
2
Null-controllability in arbitrary small time: well-known for a long
time in the one-dimensional case (Fattorini-Russell`71).
The T-D equation
The heat equation
Conclusion
Introduction
The cost of the control
Denition
Cost of the control for
:
(T-D)
CTD (T , L, M , ε) := sup
y 0 (y ,v ) ver.
inf
(T-D)
||v ||L2 (0,T )
.
& y (T )=0 ||y 0 ||L2 (0,L)
The T-D equation
The heat equation
Conclusion
Introduction
The cost of the control
Denition
Cost of the control for
:
(T-D)
CTD (T , L, M , ε) := sup
y 0 (y ,v ) ver.
inf
(T-D)
||v ||L2 (0,T )
.
& y (T )=0 ||y 0 ||L2 (0,L)
CTD (T , L, M , ε) is the smallest constant C such that
for every y ∈ L (0, L), there exists v ∈ L (0, L) such that
Equivalently,
0
2
2
||v ||L2 (0,T ) 6 C ||y 0 ||L2 (0,L) .
CTD (T , L, M , ε) is the norm of the (linear) operator
y 7→ vHUM , where vHUM is the control given by the HUM method.
Equivalently,
0
The T-D equation
The heat equation
Conclusion
Introduction
Uniform controllability
Denition
Let T , L, M be xed. The family of equations (T-D) is uniformly
controllable in the vanishing viscosity limit if and only if there exists
ε > 0 and C > 0 such that for every ε ∈ (0, ε ),
0
0
CTD (T , L, M , ε) 6 C .
The T-D equation
The heat equation
Conclusion
Introduction
Uniform controllability
Denition
Let T , L, M be xed. The family of equations (T-D) is uniformly
controllable in the vanishing viscosity limit if and only if there exists
ε > 0 and C > 0 such that for every ε ∈ (0, ε ),
0
0
CTD (T , L, M , ε) 6 C .
Of course, if (T-D) is uniformly controllable at time
uniformly controllable at any time
T >T
T
0,
it is
0.
Question
What about the uniform controllability when
ε → 0,
precisely what about the minimal time that ensures
null-controllability?
and more
The T-D equation
The heat equation
The transport equation
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Conclusion
The transport equation
Convergence toward the transport equation (1)
Theorem (Coron-Guerrero'05)
Let y ∈ L (0, L). Let (εn )n∈N be a sequence of positive numbers
tending to 0. Let (vn )n∈N a sequence of function of L (0, T ) and
v ∈ L (0, T ) such that vn * v (in L ). At xed n, we denote yn
the unique solution of the problem
0
2
2
2
2


y − εn ynxx + Mynx = 0

 nt


yn (·, 0) = vn (t )

yn (·, L) = 0




y (0, ·) = y 0
in Q ,
on (0, T ),
on (0, T ),
in (0, L).
The T-D equation
The heat equation
Conclusion
The transport equation
Convergence toward the transport equation (2)
Theorem
Then yn ∈ L (Q ) and yn * y ∈ L (Q ) solution of
2


yt + Myx = 0



 y (·, 0) = v (t )

y (·, L) = 0



 y (0, ·) = y 0
2
in Q ,
on (0, T ) for M > 0,
on (0, T ) for M < 0,
in (0, L).
(Transp)
The T-D equation
The heat equation
Conclusion
The transport equation
Control of the transport equation (M > 0)
We consider (Transp).
We extend
y ,v
0
by 0.
t
Unique weak solution:
y (x − Mt ) + v (t − x /M ).
0
x > Mt : depends only on
y .
x < Mt : dépends only on
v.
0
t=
L
M
v
0
y0
L
x
The T-D equation
The heat equation
Conclusion
The transport equation
Control of the transport equation (M > 0)
We consider (Transp).
We extend
y ,v
0
by 0.
t
Unique weak solution:
y (x − Mt ) + v (t − x /M ).
0
T > L/M : x − MT < 0
for x ∈ (0, L) so
y (T , x ) ≡ 0
on
(0, L)
if
v = 0.
T
t=
>
L
M
L
M
v
0
y0
L
x
The T-D equation
The heat equation
Conclusion
The transport equation
Control of the transport equation (M > 0)
We consider (Transp).
We extend
y ,v
0
by 0.
t
Unique weak solution:
y (x − Mt ) + v (t − x /M ).
0
T < L/M : x − MT ∈
(0, L) and x ∈ (0, L)
i x
∈ (MT , L) so
y (T , x ) not necessary null
on (MT , L).
t=
L
M
T
v
0
y0
L
<
L
M
x
The T-D equation
The heat equation
The transport equation
An important consequence
Corollary
Equation (T-D) cannot be uniformly controllable in time
T < L/|M |.
Conclusion
The T-D equation
The heat equation
Conclusion
The transport equation
An important consequence
Corollary
Equation (T-D) cannot be uniformly controllable in time
T < L/|M |.
Proof: by contradiction. Assume
CTD (ε, T , L, M ) 6→ +∞.
There
(εn )n∈N positive tending to 0 such that
(CTD (εn , T , L, M ))n∈N is bounded. Let y 0 ∈ L2 (0, L) and let vn the
0
optimal control driving y to 0. Let T0 ∈ (T , L/M ). We extend yn
and vn by 0 on (T , T0 ). From inequality
exists
||vn ||L2 (0,T ) 6 CTD (εn )||y 0 ||L2 (0,T ) ,
(||vn ||)n∈N is bounded in L2 (0, T0 ), so we extract a subsequence
(vn )n∈N weakly converging to v ∈ L2 (0, T0 ). We deduce yn * y
solution of (Transp), and necessarily y ≡ 0 on (T , T0 ) × (0, L),
which gives a contradiction.
The T-D equation
The heat equation
Uniform controllability in the vanishing viscosity limit
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What should we expect and what is true?
Natural conjecture concerning (T-D):
What we expect
CTD (T , L, M , ε) → 0 if T > L/|M | and CTD (T , L, M , ε) → +∞ if
T < L/|M |.
Conjecture (partially) false:
Theorem (Coron-Guerrero'05)
1
2
C
If M > 0, then CTD (T , L, M , ε) > Ce ε when ε → 0 for
T < L/|M |.
C
If M < 0, then CTD (T , L, M , ε) > Ce ε when ε → 0 for
T < 2L/|M |.
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What should we expect and what is true? (2)
On the other hand, upper bounds:
Theorem (Coron-Guerrero'05)
1
2
−C
If M > 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 4.3L/|M |.
−C
If M < 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 57.2L/|M |.
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What should we expect and what is true? (2)
On the other hand, upper bounds:
Theorem (Coron-Guerrero'05)
1
2
−C
If M > 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 4.3L/|M |.
−C
If M < 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 57.2L/|M |.
Method: Carleman estimates and adapted dissipation estimate on
the adjoint system.
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What should we expect and what is true? (3)
Improvement:
Theorem (Glass'09)
1
2
−C
If M > 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 4.2L/|M |.
−C
If M < 0, CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 6.1L/|M |.
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What should we expect and what is true? (3)
Improvement:
Theorem (Glass'09)
1
2
−C
If M > 0, then CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 4.2L/|M |.
−C
If M < 0, CTD (T , L, M , ε) 6 Ce ε when ε → 0 for
T > 6.1L/|M |.
Method: similar to the moment method (relying essentially on the
application of the Paley-Wiener Theorem).
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What is conjectured
The numbers given in the previous theorems come from technical
restrictions.
The T-D equation
The heat equation
Conclusion
Uniform controllability in the vanishing viscosity limit
What is conjectured
The numbers given in the previous theorems come from technical
restrictions.
⇒
New natural conjecture concerning (T-D):
Conjecture (Coron-Guerrero`05)
1
2
If M > 0, CTD (T , L, M , ε) → 0 when ε → 0 for T > L/|M |,
If M < 0, CTD (T , L, M , ε) → 0 when ε → 0 for T > 2L/|M |.
The T-D equation
The heat equation
Introduction
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Conclusion
Introduction
The heat equation controlled on the left side


 zt − zxx = 0 in Q ,
z (., 0) = w (t ) on (0, T ),


z (., L) = 0 on (0, T ).
(Heat)
The T-D equation
The heat equation
Conclusion
Introduction
The heat equation controlled on the left side


 zt − zxx = 0 in Q ,
z (., 0) = w (t ) on (0, T ),


z (., L) = 0 on (0, T ).
(Heat)
Denition
Cost of the control for
CH (T , L) = sup
:
(Heat)
z 0 (z ,w ) ver.
inf
(Heat)
||w ||L2 (0,T )
.
& z (T ,.)=0 ||z 0 ||L2 (0,T )
The T-D equation
The heat equation
Conclusion
Introduction
The heat equation controlled on the left side


 zt − zxx = 0 in Q ,
z (., 0) = w (t ) on (0, T ),


z (., L) = 0 on (0, T ).
(Heat)
Denition
Cost of the control for
CH (T , L) = sup
:
(Heat)
z 0 (z ,w ) ver.
inf
(Heat)
||w ||L2 (0,T )
.
& z (T ,.)=0 ||z 0 ||L2 (0,T )
Question: what about the cost of fast controls (i.e. when
T → 0)?
The T-D equation
The heat equation
Introduction
The cost of fast controls for the heat equation
We introduce
α∗ := lim sup Tln(CH (T , L)).
T →0
α∗ := lim inf Tln(CH (T , L)).
T →0
Conclusion
The T-D equation
The heat equation
Conclusion
Introduction
The cost of fast controls for the heat equation
We introduce
α∗ := lim sup Tln(CH (T , L)).
T →0
α∗ := lim inf Tln(CH (T , L)).
T →0
Güichal`85:
Miller`04:
α∗ > 0
CH (T , L) & Ce C /T ).
L2 /( T ) ).
for T small CH (T , L) & Ce
(i.e. for
α∗ > L2 /4
(i.e.
T
small
4
The T-D equation
The heat equation
Conclusion
Introduction
The cost of fast controls for the heat equation
We introduce
α∗ := lim sup Tln(CH (T , L)).
T →0
α∗ := lim inf Tln(CH (T , L)).
T →0
CH (T , L) & Ce C /T ).
L2 /( T ) ).
Miller`04: α∗ > L /4 (i.e. for T small CH (T , L) & Ce
∗
C /T ).
Seidman`96: α < ∞ (i.e. for T small CH (T , L) . Ce
Güichal`85:
α∗ > 0
2
(i.e. for
T
small
4
The T-D equation
The heat equation
Conclusion
Introduction
The cost of fast controls for the heat equation
We introduce
α∗ := lim sup Tln(CH (T , L)).
T →0
α∗ := lim inf Tln(CH (T , L)).
T →0
CH (T , L) & Ce C /T ).
L2 /( T ) ).
Miller`04: α∗ > L /4 (i.e. for T small CH (T , L) & Ce
∗
C /T ).
Seidman`96: α < ∞ (i.e. for T small CH (T , L) . Ce
∗
Tenenbaum-Tucsnak`07: α 6 3L /4 (i.e. for T small
2 )+ /T
(
/
L
CH (T , L) . e
).
Güichal`85:
α∗ > 0
(i.e. for
T
small
2
4
2
3 4
The T-D equation
The heat equation
Conclusion
Introduction
The cost of fast controls for the heat equation
We introduce
α∗ := lim sup Tln(CH (T , L)).
T →0
α∗ := lim inf Tln(CH (T , L)).
T →0
CH (T , L) & Ce C /T ).
L2 /( T ) ).
Miller`04: α∗ > L /4 (i.e. for T small CH (T , L) & Ce
∗
C /T ).
Seidman`96: α < ∞ (i.e. for T small CH (T , L) . Ce
∗
Tenenbaum-Tucsnak`07: α 6 3L /4 (i.e. for T small
2 )+ /T
(
/
L
CH (T , L) . e
).
Güichal`85:
α∗ > 0
(i.e. for
T
small
2
4
2
3 4
Conjecture (Miller`04, Ervedoza-Zuazua`11, ...)
α∗ = L2 /4, i.e. CH (T , L) ≈ e (L
2 /4)+ /T
.
The T-D equation
The heat equation
Link between heat and transport-diusion equation
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Link between heat and transport-diusion equation
An relevant changing of unknowns
The fondamental remark is the following:
Lemma
y veries
(T-D)
with initial condition y and control v i
0
M 2 t Mx
t
z (t , x ) = e 4ε2 − 2ε y ( , x )
ε
veries (Heat) posed on (0, εT ) × (0, L) with initial condition
M2t
Mx
z (x ) := e − 2ε y (x ) and control w (t ) := e 4ε2 v ( εt ).
0
0
Conclusion
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
An relevant changing of unknowns
The fondamental remark is the following:
Lemma
y veries
(T-D)
with initial condition y and control v i
0
M 2 t Mx
t
z (t , x ) = e 4ε2 − 2ε y ( , x )
ε
veries (Heat) posed on (0, εT ) × (0, L) with initial condition
M2t
Mx
z (x ) := e − 2ε y (x ) and control w (t ) := e 4ε2 v ( εt ).
0
0
Proof: direct computations (dierentiate
and compare with
y .)
z
with respect to
t
and
x
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
An relevant changing of unknowns
The fondamental remark is the following:
Lemma
y veries
(T-D)
with initial condition y and control v i
0
M 2 t Mx
t
z (t , x ) = e 4ε2 − 2ε y ( , x )
ε
veries (Heat) posed on (0, εT ) × (0, L) with initial condition
M2t
Mx
z (x ) := e − 2ε y (x ) and control w (t ) := e 4ε2 v ( εt ).
0
0
Proof: direct computations (dierentiate
z
with respect to
t
and
x
y .)
CTD (T , L, M , ε) in the vanishing viscosity limit
by estimating CH (T , L) in small time!
and compare with
⇒
We can estimate
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Estimations on the costs of controls
Link between the two costs:
Lemma
For all T < T , one has
0






CTD (T , L, M , ε) 6
e−
M 2 T0
√
4ε
ε
CD (ε(T − T ), L) if M > 0,
0
M 2 T |M |L

− 4ε 0 + 2ε


e

 CTD (T , L, M , ε) 6
√
CD (ε(T − T0 ), L) if M < 0.
ε
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Estimations on the costs of controls
Link between the two costs:
Lemma
For all T < T , one has
0






CTD (T , L, M , ε) 6
e−
M 2 T0
√
4ε
ε
CD (ε(T − T ), L) if M > 0,
0
M 2 T |M |L

− 4ε 0 + 2ε


e

 CTD (T , L, M , ε) 6
√
CD (ε(T − T0 ), L) if M < 0.
ε
Proof: use the denitions of the costs, the previous changing of
unknowns and let System (T-D) evolve naturally on
the control is 0 on
(0, T0 )).
(0, T0 )
(i.e.
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Improving the constants (1)
We can deduce the following result for
M > 0:
Theorem (Lissy`12,C.R.A.S.)
√
Assume that M > 0 and T > M L > 3.47L/M. Then there exists
some constant K > 0, there exists some constant C > 0 such that,
for all ε > 0 and all y ∈ L (0, L), there exists a solution (yε , v ) of
the control problem (T-D) verifying yε (T , .) = 0 and
2
0
3
2
K
||vε ||L2 (0,T ) 6 Ce − ε ||y 0 ||L2 (0,L) .
Moreover, if we assume that the conjecture α∗ = L /4 is veried,
then one can state the same result as soon as
2
T>
2
L
|M |
.
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Improving the constants (2)
We have a similar result for
M < 0:
Theorem (Lissy`12, C.R.A.S.)
√
Assume that M < 0 and T > ( |M+| )L > 5.47L/M. Then there
exists some constant K > 0, there exists some constant C > 0 such
that, for all ε > 0 and all y ∈ L (0, L), there exists a solution
(yε , v ) of the control problem (T-D) verifying yε (T , .) = 0 and
2
0
3
2
2
K
||vε ||L2 (0,T ) 6 Ce − ε ||y 0 ||L2 (0,L) .
Moreover, if we assume that the conjecture α∗ = L /4 is veried,
then one can state the same result as soon as
2
T>
4
L
|M |
.
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Improving the constants (3)
Proof: use the previous Lemma and Tenenbaum-Tucsnak'07
= 3L2 /4) and optimize
T0 (for M > 0, T0 = T /2, for M < 0,
√
T0 = 1/2 + 1/(2 + 2 3)). See what happens if we assume
α∗ = L2 /4.
(α
∗
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Improving the constants (3)
Proof: use the previous Lemma and Tenenbaum-Tucsnak'07
= 3L2 /4) and optimize
T0 (for M > 0, T0 = T /2, for M < 0,
√
T0 = 1/2 + 1/(2 + 2 3)). See what happens if we assume
α∗ = L2 /4.
(α
∗
Remarks
1
Every better upper bound on
α∗
will automatically provide a
better lower bound for the time needed to ensure exponential
decay of
CTD (T , L, M , ε).
The T-D equation
The heat equation
Conclusion
Link between heat and transport-diusion equation
Improving the constants (3)
Proof: use the previous Lemma and Tenenbaum-Tucsnak'07
= 3L2 /4) and optimize
T0 (for M > 0, T0 = T /2, for M < 0,
√
T0 = 1/2 + 1/(2 + 2 3)). See what happens if we assume
α∗ = L2 /4.
(α
∗
Remarks
1
Every better upper bound on
α∗
will automatically provide a
better lower bound for the time needed to ensure exponential
decay of
2
CTD (T , L, M , ε).
Unfortunately, even with the best
α∗
possible we cannot
recover the conjecture of Coron and Guerrero!
The T-D equation
The heat equation
Another approach: integral observability inequalities
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (1)
From the controllability-observability duality, null-controllability is
equivalent to
Z L
0
where
ϕ
|ϕ(T , x )| dx 6 CH (T , L)
2
2
Z T
|∂x ϕ(t , 0)|2 dt ,
(1)
0
satises


ϕt − ϕxx = 0



 ϕ(·, 0) = 0





ϕ(·, L) = 0
ϕ(0, ·) = ϕ0
in
(0, T ) × (0, L),
on
(0, T ),
on
(0, T ),
in
(0, L).
(2)
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (1)
From the controllability-observability duality, null-controllability is
equivalent to
Z L
0
where
ϕ
|ϕ(T , x )| dx 6 CH (T , L)
2
2
Z T
|∂x ϕ(t , 0)|2 dt ,
(1)
0
satises


ϕt − ϕxx = 0



 ϕ(·, 0) = 0





ϕ(·, L) = 0
ϕ(0, ·) = ϕ0
in
(0, T ) × (0, L),
on
(0, T ),
on
(0, T ),
in
(2)
(0, L).
One possible method to prove this: Carleman inequalities
Z TZ L
0
0
e
C (L )
− 1t
|ϕ(t , x )| dxdt 6 C2 (T , L)
2
Z T
|∂x ϕ(t , 0)|2 dt .
0
(Obs-Fin)
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (1)
(Obs-Fin)
⇒
Observability. Moreover,
CH (T , L) . C (T , L)e
2
2
C 1 (L )+
T .
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (1)
(Obs-Fin)
⇒
Observability. Moreover,
CH (T , L) . C (T , L)e
2
2
C 1 (L )+
T .
Ervedoza-Zuazua'11: thanks to transmutation methods one can
prove
Z
0
∞Z
L
e
2
− L2t
0
|ϕ(t , x )| dxdt 6 C (L)
2
Z
∞
|∂x ϕ(t , 0)|2 dt ,
0
(Obs-Inf )
which is optimal.
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (1)
⇒
(Obs-Fin)
Observability. Moreover,
CH (T , L) . C (T , L)e
2
2
C 1 (L )+
T .
Ervedoza-Zuazua'11: thanks to transmutation methods one can
prove
Z
L
∞Z
0
e
2
− L2t
|ϕ(t , x )| dxdt 6 C (L)
2
0
Z
∞
|∂x ϕ(t , 0)|2 dt ,
0
(Obs-Inf )
which is optimal.
From (Obs-Inf) one can deduce
Z
0
∞Z
0
L
L2
e − 2t |ϕ(t , x )| dxdt 6 Cint (T , L)
2
Z T
0
|∂x ϕ(t , 0)|2 dt .
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (3)
The method used in Ervedoza-Zuazua`11 (argument by
contradiction) cannot give any estimation on
Cint (T , L).
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (3)
The method used in Ervedoza-Zuazua`11 (argument by
contradiction) cannot give any estimation on
Cint (T , L).
It is naturel to expect that this constant does not explode too
much.
Conjecture [Ervedoza-Zuazua'11]
For every
δ > 0,
one can choose
Cint (T , L) such that
Cint (T , L) = O (e T ).
δ
T →0
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Integral observability inequalities (3)
The method used in Ervedoza-Zuazua`11 (argument by
contradiction) cannot give any estimation on
Cint (T , L).
It is naturel to expect that this constant does not explode too
much.
Conjecture [Ervedoza-Zuazua'11]
For every
δ > 0,
one can choose
Cint (T , L) such that
Cint (T , L) = O (e T ).
δ
T →0
Link with the cost of the control?
Proposition
The previous conjecture implies α∗ = 1/4.
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Another result
Theorem (Lissy'14, System Control Lett.)
We assume that the Ervedoza-Zuazua conjecture is true. Then
there exists some C , K > 0 (independent on ε but maybe
dependent on T , M et L) such that for every ε ∈ (0, 1),
K
CTD (T , L, ε, M ) 6 Ce − ε
as soon as
1 T > L/M if M > 0,
√
2 T > (1 + 2)L/|M | if M < 0.
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Another result
Theorem (Lissy'14, System Control Lett.)
We assume that the Ervedoza-Zuazua conjecture is true. Then
there exists some C , K > 0 (independent on ε but maybe
dependent on T , M et L) such that for every ε ∈ (0, 1),
K
CTD (T , L, ε, M ) 6 Ce − ε
as soon as
1 T > L/M if M > 0,
√
2 T > (1 + 2)L/|M | if M < 0.
We recover the Coron-Guerrero conjecture for
M < 0.
M > 0, but not for
This result may question the validity of the Coron-Guerrero
conjecture for
M < 0.
The T-D equation
The heat equation
Conclusion
Another approach: integral observability inequalities
Another result
Theorem (Lissy'14, System Control Lett.)
We assume that the Ervedoza-Zuazua conjecture is true. Then
there exists some C , K > 0 (independent on ε but maybe
dependent on T , M et L) such that for every ε ∈ (0, 1),
K
CTD (T , L, ε, M ) 6 Ce − ε
as soon as
1 T > L/M if M > 0,
√
2 T > (1 + 2)L/|M | if M < 0.
We recover the Coron-Guerrero conjecture for
M < 0.
M > 0, but not for
This result may question the validity of the Coron-Guerrero
conjecture for
M < 0.
Proof: essentially relies on the same of transformation as before.
The T-D equation
The heat equation
Conclusion
1
The transport-diusion equation controlled on one side
Introduction
The transport equation
Uniform controllability in the vanishing viscosity limit
2
The heat equation controlled on one side
Introduction
Link between heat and transport-diusion equation
Another approach: integral observability inequalities
3
Conclusion
Conclusion
Conclusion
The T-D equation
The heat equation
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
(0, T0 ),
which is for sure not optimal.
Conclusion
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
Use Carleman estimates with some weights related to these
used here?
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
Use Carleman estimates with some weights related to these
used here?
Study of more complicated situations: speed depending on the
space/time, multidimensional case (Lebeau-Guerrero`07).
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
Use Carleman estimates with some weights related to these
used here?
Study of more complicated situations: speed depending on the
space/time, multidimensional case (Lebeau-Guerrero`07).
For the heat equation
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
Use Carleman estimates with some weights related to these
used here?
Study of more complicated situations: speed depending on the
space/time, multidimensional case (Lebeau-Guerrero`07).
For the heat equation
Improve choice of the multiplier in the moment method to
improve the estimate on
α∗ .
The T-D equation
The heat equation
Conclusion
Conclusion
Some perspectives
For the transport-diusion equation
We do nothing on
⇒
(0, T0 ),
which is for sure not optimal.
keep quite the same method and do something intelligent
on these intervals?
Use Carleman estimates with some weights related to these
used here?
Study of more complicated situations: speed depending on the
space/time, multidimensional case (Lebeau-Guerrero`07).
For the heat equation
Improve choice of the multiplier in the moment method to
improve the estimate on
α∗ .
For the Ervedoza-Zuazua conjecture: ?
The T-D equation
The heat equation
Conclusion
Conclusion
Conclusion
References
1
A link between the cost of fast controls for the 1-D heat
equation and the uniform controllability of a 1-D
transport-diusion equation, C. R. Math. Acad. Sci. Paris 350
(2012), no. 11-12, 591-595.
2
An application of a conjecture due to Ervedoza and Zuazua
concerning the observability of the heat equation in small
time to a conjecture due to Coron and Guerrero concerning
the uniform controllability of a convection-diusion equation
in the vanishing viscosity limit, System Control Lett., 2014.
The T-D equation
The heat equation
Conclusion
Conclusion
Conclusion
References
1
A link between the cost of fast controls for the 1-D heat
equation and the uniform controllability of a 1-D
transport-diusion equation, C. R. Math. Acad. Sci. Paris 350
(2012), no. 11-12, 591-595.
2
An application of a conjecture due to Ervedoza and Zuazua
concerning the observability of the heat equation in small
time to a conjecture due to Coron and Guerrero concerning
the uniform controllability of a convection-diusion equation
in the vanishing viscosity limit, System Control Lett., 2014.
Thank you for your attention!