Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Controllability problems for fluid equations
Jean-Pierre Puel
Laboratoire de Mathématiques de Versailles,
Université de Versailles Saint-Quentin,
45 avenue des Etats Unis, 78035 Versailles Cedex, France
[email protected]
CIMPA 2009, Pointe à Pitre, January 2009
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Outline
1
Introduction
Motivation
Fluid equations
Heat type equations
2
Exact controllability to trajectories for the linear heat equation
Optimal control problem : penalty method
Estimates
Passage to the limit
Observability Inequality
3
How to deal with the nonlinear problems. Results
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Motivation
Drive a fluid to rest, or to an ideal motion.
Oceanology, climatology, meteorology
Data assimilation problems (cf last lecture),
Fluid structure interaction : design of a plane, landing of a
plane, design of cars (aerodynamics), how to drive a (rigid)
body moving in a fluid to a prescribed position,
Stabilization of an unstable solution (stationnary),
etc....
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Navier-Stokes equations
x = (x1 , x2 , x3 ) ∈ Ω ⊂ IR3 : space variable
t ∈ (0, T ), T > 0 : time variable.
y = y(x, t) = (y1 , y2 , y3 ) : velocity of the fluid,
p = p(x, t) : pressure of the fluid.
Navier-Stokes equations :
3
X ∂yi
∂p
∂yi
− ν∆yi +
yj
+
= gi , in Ω × (0, T ), i = 1, · · · 3,
∂t
∂xj
∂xi
j=1
div y = 0 in Ω × (0, T ),
y = 0 on Γ × (0, T ), Γ = ∂Ω,
y(x, 0) = y0 (x) in Ω.
where g = (g1 , g2 , g3 ) is a source term and y0 is the initial
condition.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Navier-Stokes equations
x = (x1 , x2 , x3 ) ∈ Ω ⊂ IR3 : space variable
t ∈ (0, T ), T > 0 : time variable.
y = y(x, t) = (y1 , y2 , y3 ) : velocity of the fluid,
p = p(x, t) : pressure of the fluid.
Navier-Stokes equations :
3
X ∂yi
∂p
∂yi
− ν∆yi +
yj
+
= gi , in Ω × (0, T ), i = 1, · · · 3,
∂t
∂xj
∂xi
j=1
div y = 0 in Ω × (0, T ),
y = 0 on Γ × (0, T ), Γ = ∂Ω,
y(x, 0) = y0 (x) in Ω.
where g = (g1 , g2 , g3 ) is a source term and y0 is the initial
condition.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Navier-Stokes equations
x = (x1 , x2 , x3 ) ∈ Ω ⊂ IR3 : space variable
t ∈ (0, T ), T > 0 : time variable.
y = y(x, t) = (y1 , y2 , y3 ) : velocity of the fluid,
p = p(x, t) : pressure of the fluid.
Navier-Stokes equations :
3
X ∂yi
∂p
∂yi
− ν∆yi +
yj
+
= gi , in Ω × (0, T ), i = 1, · · · 3,
∂t
∂xj
∂xi
j=1
div y = 0 in Ω × (0, T ),
y = 0 on Γ × (0, T ), Γ = ∂Ω,
y(x, 0) = y0 (x) in Ω.
where g = (g1 , g2 , g3 ) is a source term and y0 is the initial
condition.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Here :
g = f + v1Iω
where f is a given source term and ω ⊂ Ω.
v is a control acting on ω × (0, T ).
For simplcity, the control is here distributed (inside the domain)
but we can also consider boundary control...
We then obtain (in a vector formulation)
∂y
− ν∆y + y.∇y + ∇p = f + v.1Iω in Ω × (0, T ),
∂t
div y = 0 in Ω × (0, T ),
y=0
on
Γ × (0, T )
y(0) = y0 in Ω,
Question :
Given y1 = y1 (x), can we chose a control v such that y(T ) = y1 ?
The answer is NO for general y1 (the system is dissipative,...) but
we will give a positive answer for particular y1 .
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Here :
g = f + v1Iω
where f is a given source term and ω ⊂ Ω.
v is a control acting on ω × (0, T ).
For simplcity, the control is here distributed (inside the domain)
but we can also consider boundary control...
We then obtain (in a vector formulation)
∂y
− ν∆y + y.∇y + ∇p = f + v.1Iω in Ω × (0, T ),
∂t
div y = 0 in Ω × (0, T ),
y=0
on
Γ × (0, T )
y(0) = y0 in Ω,
Question :
Given y1 = y1 (x), can we chose a control v such that y(T ) = y1 ?
The answer is NO for general y1 (the system is dissipative,...) but
we will give a positive answer for particular y1 .
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Exact controllability to trajectories
Consider an “ideal” solution of uncontrolled Navier-Stokes
equations (starting from a different initial data), for example a
stationnary solution :
∂ȳ
− ν∆ȳ + ȳ.∇ȳ + ∇p̄ = f in Ω × (0, T ),
∂t
div ȳ = 0 in Ω × (0, T ),
ȳ = 0 on Γ × (0, T )
ȳ(0) = ȳ0 in Ω.
Exact Controllability to Trajectories :
Can we find a control v such that
y(T ) = ȳ(T ) ?
i.e can we reach exactly in finite time the “ideal” trajectory ȳ?
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
By difference : z = y − ȳ, z0 = y0 − ȳ0 , q = p − p̄
∂z
− ν∆z + z.∇z + z.∇ȳ + ȳ.∇z + ∇q = v.1Iω in Ω × (0, T ),
∂t
div z = 0 in Ω × (0, T ),
z = 0 on Γ × (0, T )
z(0) = z0 in Ω,
We now want to find v such that z(T ) = 0.
Local version : same result provided ||y0 − ȳ0 || is small enough.
Remark
If there exists such a control v, then, after time T , just switch off
the control (v = 0) and the system can (will) stay on the “ideal”
trajectory.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
By difference : z = y − ȳ, z0 = y0 − ȳ0 , q = p − p̄
∂z
− ν∆z + z.∇z + z.∇ȳ + ȳ.∇z + ∇q = v.1Iω in Ω × (0, T ),
∂t
div z = 0 in Ω × (0, T ),
z = 0 on Γ × (0, T )
z(0) = z0 in Ω,
We now want to find v such that z(T ) = 0.
Local version : same result provided ||y0 − ȳ0 || is small enough.
Remark
If there exists such a control v, then, after time T , just switch off
the control (v = 0) and the system can (will) stay on the “ideal”
trajectory.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Linearized Navier-Stokes equations : LNS
Previous problem is very difficult. Begin with a simpler linearized
model (around ȳ):
∂y
− ν∆y + y.∇ȳ + ȳ.∇y + ∇p = f + v.1Iω in Ω × (0, T ),
∂t
div y = 0 in Ω × (0, T ),
y=0
on
y(0) = y0
Γ × (0, T )
in
Ω,
The question is now : can we find v such that y(T ) = 0 ?
We have to solve this problem first (already very, very difficult) and
then try to apply a kind of inverse mapping Theorem to obtain a
local controllability result.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Linearized Navier-Stokes equations : LNS
Previous problem is very difficult. Begin with a simpler linearized
model (around ȳ):
∂y
− ν∆y + y.∇ȳ + ȳ.∇y + ∇p = f + v.1Iω in Ω × (0, T ),
∂t
div y = 0 in Ω × (0, T ),
y=0
on
y(0) = y0
Γ × (0, T )
in
Ω,
The question is now : can we find v such that y(T ) = 0 ?
We have to solve this problem first (already very, very difficult) and
then try to apply a kind of inverse mapping Theorem to obtain a
local controllability result.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Nonlinear heat equation
Simpler case of a semilinear heat equation
∂y
− ∆y + g (y ) = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0
in
Ω.
Again we cannot expect to obtain at time T any final state y1
(dissipative effect, regularizing effect, ...). Consider an “ideal”
trajectory ȳ solution of the uncontrolled problem
∂ȳ
− ∆ȳ + g (ȳ ) = f in Ω × (0, T ),
∂t
ȳ = 0 on Γ × (0, T )
ȳ (0) = ȳ0
in
Ω.
Exact controllability to trajectories : Can we find a control v such
that
y (T ) = ȳ (T ).
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Motivation
equation How
Fluid to
equations
deal withHeat
the nonlinear
type equations
problems. Results
Linearized heat equations : LH
The problem is still very difficult and we need to start with a
linearized problem :
∂y
− ∆y + a(x, t).y = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0
in
Ω.
Question : can we find v such that at time T we have
y (T ) = 0.
We will explain the method to solve this linearized problem (with
suitable assumptons on a and f ).
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
We first consider an optimal control problem where the condition
y (T ) = 0 is replaced by a penalty term.
Take a ∈ L∞ (Ω × (0, T )), u0 ∈ L2 (Ω) and f ∈ L2 (Ω × (0, T )). For
every v ∈ L2 (ω × (0, T ) by standard results on the heat equation
we know that there exists a unique solution y = y (v ) of
∂y
− ∆y + a(x, t).y = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0
in
Ω.
For > 0 we take the “cost” functional
Z Z
Z Z
1
1 T
1 T
2
2
J (v ) = |y (v )(T )|L2 +
|v | dxdt +
|y (v )|2 dxdt.
2
2 0 ω
2 0 Ω
This is well defined, continuous, convex and coercive.
Consider the optimal control problem
Find v such that J (v ) = min J (v ).
v
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
We first consider an optimal control problem where the condition
y (T ) = 0 is replaced by a penalty term.
Take a ∈ L∞ (Ω × (0, T )), u0 ∈ L2 (Ω) and f ∈ L2 (Ω × (0, T )). For
every v ∈ L2 (ω × (0, T ) by standard results on the heat equation
we know that there exists a unique solution y = y (v ) of
∂y
− ∆y + a(x, t).y = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0
in
Ω.
For > 0 we take the “cost” functional
Z Z
Z Z
1
1 T
1 T
2
2
J (v ) = |y (v )(T )|L2 +
|v | dxdt +
|y (v )|2 dxdt.
2
2 0 ω
2 0 Ω
This is well defined, continuous, convex and coercive.
Consider the optimal control problem
Find v such that J (v ) = min J (v ).
v
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
We first consider an optimal control problem where the condition
y (T ) = 0 is replaced by a penalty term.
Take a ∈ L∞ (Ω × (0, T )), u0 ∈ L2 (Ω) and f ∈ L2 (Ω × (0, T )). For
every v ∈ L2 (ω × (0, T ) by standard results on the heat equation
we know that there exists a unique solution y = y (v ) of
∂y
− ∆y + a(x, t).y = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0
in
Ω.
For > 0 we take the “cost” functional
Z Z
Z Z
1
1 T
1 T
2
2
J (v ) = |y (v )(T )|L2 +
|v | dxdt +
|y (v )|2 dxdt.
2
2 0 ω
2 0 Ω
This is well defined, continuous, convex and coercive.
Consider the optimal control problem
Find v such that J (v ) = min J (v ).
v
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
By standard arguments of convex analysis, there exists a unique
solution v for this problem. This solution is characterized by an
optimality system (first order conditions) saying essentially that
DJ (v )[w ] = 0, ∀w .
The mapping v → y (v ) is affine so its derivative Dy (v )[w ] = z(w )
satisfies
∂z(w )
− ∆z(w ) + a(x, t).z(w ) = w 1Iω in Ω × (0, T ),
∂t
z(w ) = 0 on Γ × (0, T )
z(w )(0) = 0
in
Ω.
Writing y = y (v ) and computing DJ (v )(w ) we obtain
Z TZ
Z TZ
1
( y (T ), z(w )(T ))L2 +
v .wdxdt+
y .z(w )dxdt = 0, ∀w .
0
ω
0
Ω
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Adjoint state
In order to write this in a simpler way we introduce the adjoint
state ϕ solution of the following backward problem
∂ϕ
− ∆ϕ + a(x, t).ϕ = y in Ω × (0, T ),
∂t
ϕ = 0 on Γ × (0, T )
1
ϕ (T ) = y (T ) in Ω.
−
Multiplying by z(w ) and integrating by parts it is easy to show
that DJ (v )(w ) = 0 is equivalent to
Z TZ
ϕ + v ).wdxdt = 0, ∀w
0
ω
which is also equivalent to
ϕ + v = 0 in ω × (0, T ).
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Optimality system
We then obtain the optimality (coupled) system which
characterizes v :
∂y
− ∆y + a(x, t).y = f + v 1Iω in Ω × (0, T ),
∂t
y = 0 on Γ × (0, T )
y (0) = y0 in Ω,
∂ϕ
− ∆ϕ + a(x, t).ϕ = y in Ω × (0, T ),
−
∂t
ϕ = 0 on Γ × (0, T )
1
ϕ (T ) = y (T ) in Ω,
ϕ + v = 0 in ω × (0, T ).
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Now we want to obtain estimates (on the controls v ) independent
of .
Multiply equation for ϕ by y and integrate by parts, we obtain :
Z TZ
1
|y (T )|2L2 +
|y |2 dxdt = (ϕ (0), y0 )L2
0
Ω
Z TZ
Z TZ
+
f .ϕ dxdt +
ϕ .v dxdt
0
Ω
0
ω
Using
ϕ + v = 0 in ω × (0, T )
we obtain, choosing a weight ρ on Ω × (0, T )
Z TZ
2J (v ) = (ϕ (0), y0 )L2 +
ϕ dxdt
0
Ω
Z
Z TZ
1
2
≤ |ϕ (0)|L2 |y0 |L2 + (
ρ|f | dxdt) 2 (
0
Ω
Jean-Pierre Puel
0
T
Z
Ω
1
1
|ϕ |2 dxdt) 2 .
ρ
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Now suppose that we can choose a weight ρ such that we have the
following Observability Inequality on the adjoint equation
(OI )
Then as
Z
+
0
T
Z
T
Z
|ϕ (0)|2L2
Z
Ω
1
|ϕ |2 dxdt ≤ C
ρ
Z
2
T
Z
|ϕ | dxdt =
0
ω
0
we obtain, if we assume that
1
|y (T )|2L2 +
Z
0
T
Z
ω
Z
0
|ϕ |2 dxdt.
ω
|v |2 dxdt
ω
RT R
0
T
Z
2
Ω ρ|f | dxdt
|v |2 dxdt ≤ C (|y0 |2L2 +
< +∞,
Z
0
T
Z
ρ|f |2 dxdt).
Ω
This gives a bound (independent of ) on the controls v and on
the final states √1 y (T ).
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Now it is easy to pass to the limt : extract from v a subsequence
(still denoted in the same way) such that
v converges weakly to v ,
√
|y (T )|L2 ≤ C .
Then it can be shown that y → y (v ) and y (T ) → y (v )(T ) and
therefore we have
y (v )(T ) = 0,
so that v is a control satisfying the null controllability problem.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Comments
It can be shown easily that v minimizes the functional
J(v ) =
1
2
Z
0
T
Z
|v |2 dxdt +
ω
1
2
Z
0
T
Z
|y (v )|2 dxdt
Ω
among the controls v such that y (v )(T ) = 0.
Also the convergence of v to v is strong and for the whole
sequence.
In the functional J and also in the functional J one can
introduce weights in order to change the norms. This allows
to show that there exist controls v which are exponentially
decreasing when t → T and the same for the states y .
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
How to obtain the Observability Inequality ?
This is the hard mathematical part of the problem.
It has to be proved for each type of equation which are
considered.
It makes the whole difference between the case of heat type
equations and linearized Navier-Stokes equations.
The choice of the weight ρ is crucial. It is not true without
weight and requires a clever choice for the weight.
It gives an estimate on the solution of an evolution equation
from the knowledge of the solution on a subdomain, without
any knowledge of the initial data.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Carleman estimates
The main tool for obtaining the Observability Inequality is
Carleman estimates, in fact global Carleman estimates.
They give a weighted Sobolev estimate of the solution in terms of
values of the solution in a subdomain.
Very technical estimate which relies on a good choice of the
weight function.
The weight is complicated to write down. It is positive in
Ω × (0, T ) and it degenerates near t = T (for the backward
equation, essential) and also for t = 0 for technical reasons.
This estimate has to be obtained for each operator.
For the heat equation it is quite difficult. For the linearized
Navier-Stokes equation, the level of complexity is much higher.
This makes the difference and corresponds to the major
mathematical difficulty.
Details on Carleman estimates, on the weight etc can be found in
the annexed documents.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat
Optimal
equation
control
Howproblem
to deal :with
penalty
the nonlinear
method problems.
Estimates Results
Passage
Carleman estimates
The main tool for obtaining the Observability Inequality is
Carleman estimates, in fact global Carleman estimates.
They give a weighted Sobolev estimate of the solution in terms of
values of the solution in a subdomain.
Very technical estimate which relies on a good choice of the
weight function.
The weight is complicated to write down. It is positive in
Ω × (0, T ) and it degenerates near t = T (for the backward
equation, essential) and also for t = 0 for technical reasons.
This estimate has to be obtained for each operator.
For the heat equation it is quite difficult. For the linearized
Navier-Stokes equation, the level of complexity is much higher.
This makes the difference and corresponds to the major
mathematical difficulty.
Details on Carleman estimates, on the weight etc can be found in
the annexed documents.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Few words on the treatment of the nonlinear problems.
The first idea is to use a fixed point theorem, or a version of the
inversion mapping theorem, after having obtained the results for
the linearized problems..
In practice this requires to work on a class of controls and solutions
which decay exponentially in time when t → T (see comment
above).
For Navier-Stokes equations the class is even more complicated.
In general this gives local controllability results.
In some cases for the heat equation one can obtain global results
when the growth of nonlinearity is not too strong.
Global controllability for Navier-Stokes equation is still an open
problem in the general case.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Few words on the treatment of the nonlinear problems.
The first idea is to use a fixed point theorem, or a version of the
inversion mapping theorem, after having obtained the results for
the linearized problems..
In practice this requires to work on a class of controls and solutions
which decay exponentially in time when t → T (see comment
above).
For Navier-Stokes equations the class is even more complicated.
In general this gives local controllability results.
In some cases for the heat equation one can obtain global results
when the growth of nonlinearity is not too strong.
Global controllability for Navier-Stokes equation is still an open
problem in the general case.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Few words on the treatment of the nonlinear problems.
The first idea is to use a fixed point theorem, or a version of the
inversion mapping theorem, after having obtained the results for
the linearized problems..
In practice this requires to work on a class of controls and solutions
which decay exponentially in time when t → T (see comment
above).
For Navier-Stokes equations the class is even more complicated.
In general this gives local controllability results.
In some cases for the heat equation one can obtain global results
when the growth of nonlinearity is not too strong.
Global controllability for Navier-Stokes equation is still an open
problem in the general case.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Few words on the treatment of the nonlinear problems.
The first idea is to use a fixed point theorem, or a version of the
inversion mapping theorem, after having obtained the results for
the linearized problems..
In practice this requires to work on a class of controls and solutions
which decay exponentially in time when t → T (see comment
above).
For Navier-Stokes equations the class is even more complicated.
In general this gives local controllability results.
In some cases for the heat equation one can obtain global results
when the growth of nonlinearity is not too strong.
Global controllability for Navier-Stokes equation is still an open
problem in the general case.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Few words on the treatment of the nonlinear problems.
The first idea is to use a fixed point theorem, or a version of the
inversion mapping theorem, after having obtained the results for
the linearized problems..
In practice this requires to work on a class of controls and solutions
which decay exponentially in time when t → T (see comment
above).
For Navier-Stokes equations the class is even more complicated.
In general this gives local controllability results.
In some cases for the heat equation one can obtain global results
when the growth of nonlinearity is not too strong.
Global controllability for Navier-Stokes equation is still an open
problem in the general case.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Results
Nonlinear heat equation.
Theorem
In the case of nonlinear heat equation, if the function g is locally
Lipschitz, there is local exact controllability to trajectories. If g is
globally Lipschitz, then global exact controllability holds.
Theorem
If the function g is locally Lipschitz and satisfies
|g (s) − g (s̄)| ≤ |s − s̄|Log (1 + |s − s̄|)q , with q <
3
2
then we have global exact controlability to trajectories. There are
counterexamples for q > 2.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Results
Nonlinear heat equation.
Theorem
In the case of nonlinear heat equation, if the function g is locally
Lipschitz, there is local exact controllability to trajectories. If g is
globally Lipschitz, then global exact controllability holds.
Theorem
If the function g is locally Lipschitz and satisfies
|g (s) − g (s̄)| ≤ |s − s̄|Log (1 + |s − s̄|)q , with q <
3
2
then we have global exact controlability to trajectories. There are
counterexamples for q > 2.
Jean-Pierre Puel
Controllability problems for fluid equations
Introduction Exact controllability to trajectories for the linear heat equation How to deal with the nonlinear problems. Results
Navier-Stokes equations.
Theorem
2
If ȳ ∈ L∞ (Ω × (0, T )) and ∂ȳ
∂t ∈ L (Ω × (0, T )) then if y0 − ȳ0 is
small enough in the L4 (Ω) norm, there exists a control v such that
y(T ) − ȳ(T ) = 0.
Jean-Pierre Puel
Controllability problems for fluid equations