An introduction to linear control theory
Guillaume OLIVE
Seminar Applied Mathematics
Jagiellonian University Krakow, November 15 2016
Outline
1
Introduction
2
Controllability in nite dimension
Duality
Kalman rank condition, Hautus test
Time-dependent systems
Control cost
3
Controllability in innite dimension : the heat equation
Introduction
Towards systems of heat equations
G. Olive (IMB)
Introduction to control theory
2 / 31
Outline
1
Introduction
2
Controllability in nite dimension
Duality
Kalman rank condition, Hautus test
Time-dependent systems
Control cost
3
Controllability in innite dimension : the heat equation
Introduction
Towards systems of heat equations
G. Olive (IMB)
Introduction to control theory
3 / 31
Regulation of the heat in a room
y 0, y 1
u
Heater
y (T )
Heat of the room
Figure Open-loop control
• y0
• y 1 : Desired temperature (e.g. 22◦ C)
• y (T ) : Temperature at time T , after the action of the control.
: Initial temperature (e.g.
• u : Heater (control),
G. Olive (IMB)
15◦ C),
Introduction to control theory
4 / 31
Regulation of the heat in a room
Disturbances
y 0, y 1
u
Heater
y (T )
Heat of the room
Figure Open-loop control
• y0
• y 1 : Desired temperature (e.g. 22◦ C)
• y (T ) : Temperature at time T , after the action of the control.
: Initial temperature (e.g.
• u : Heater (control),
Drawback :
15◦ C),
Open-loop controls do not check if the output has achieved the desired goal.
G. Olive (IMB)
Introduction to control theory
4 / 31
Regulation of the heat in a room
Disturbances
y 0, y 1
u
Heater
y (T )
Heat of the room
F (y )
Thermostat
Figure Closed-loop control
• y0
• y 1 : Desired temperature (e.g. 22◦ C)
• y (T ) : Temperature at time T , after the action of the control.
: Initial temperature (e.g.
• u : Heater (control),
Drawback :
15◦ C),
Open-loop controls do not check if the output has achieved the desired goal.
• F (y ) : Thermostat (feedback).
G. Olive (IMB)
Introduction to control theory
4 / 31
Regulation of the heat in a room
Disturbances
y 0, y 1
u
Heater
y (T )
Heat of the room
F (y )
Thermostat
Figure Closed-loop control
• y0
• y 1 : Desired temperature (e.g. 22◦ C)
• y (T ) : Temperature at time T , after the action of the control.
: Initial temperature (e.g.
• u : Heater (control),
Drawback :
15◦ C),
Open-loop controls do not check if the output has achieved the desired goal.
• F (y ) : Thermostat (feedback).
In this talk, only open-loop controls !
However theoretically closed-loop controls can be obtained by open loop controls
("controllability =⇒ stabilization").
G. Olive (IMB)
Introduction to control theory
4 / 31
Questions in control theory
In control theory, we address the following issues :
Existence of a control
Existence of controls with various physical constraints
Estimate the cost of the control
Design of the control
G. Olive (IMB)
Introduction to control theory
5 / 31
Conuence of the mathematics
In control theory, we use several elds of Mathematics :
Dierential Equations
Spectral theory
Nonlinear analysis
Functional analysis
Numerical analysis
Stochastic processes
G. Olive (IMB)
Introduction to control theory
6 / 31
Some reference books
Figure Control and Nonlinearity, J.-M.
Coron (2007)
Figure Observation and Control for Operator
Semigroups, M. Tucsnak and G. Weiss
(2009)
G. Olive (IMB)
Figure Contr
olabilite exacte, perturbations et
stabilisation de systemes distribues, J.-L. Lions
(1988)
Figure Mathematical Control Theory, J.
Zabczyk (1992)
Introduction to control theory
7 / 31
Outline
1
Introduction
2
Controllability in nite dimension
Duality
Kalman rank condition, Hautus test
Time-dependent systems
Control cost
3
Controllability in innite dimension : the heat equation
Introduction
Towards systems of heat equations
G. Olive (IMB)
Introduction to control theory
8 / 31
System description
Consider
d
y
dt

y (0)


=
Ay + Bu,
=
y 0,
t ∈ (0, T ),
(ODE)
where
T > 0 is the time of control.
y = (y1 , . . . , yn ) is the state.
y 0 ∈ Rn is the initial data.
A ∈ Rn×n is a coupling matrix.
u ∈ L2 (0, T )m are the m controls (possibly m < n).
B ∈ Rn×m localizes algebraically the controls.
G. Olive (IMB)
Introduction to control theory
9 / 31
System description
Consider
d
y
dt

y (0)


=
Ay + Bu,
=
y 0,
t ∈ (0, T ),
(ODE)
where
T > 0 is the time of control.
y = (y1 , . . . , yn ) is the state.
y 0 ∈ Rn is the initial data.
A ∈ Rn×n is a coupling matrix.
u ∈ L2 (0, T )m are the m controls (possibly m < n).
B ∈ Rn×m localizes algebraically the controls.
Well-posedness :
For every y 0 ∈ Rn and u ∈ L2 (0, T )m , there exists a unique solution
y (t) = e tA y 0 +
t
Z
0
e (t−s)A Bu(s) ds.
Note that y ∈ C 0 ([0, T ])n .
G. Olive (IMB)
Introduction to control theory
9 / 31
Notions of controllability
y (T ; 0)
y1
y0
Figure Uncontrolled trajectory
y 0 : initial state,
y 1 : target,
y (T ; u) : value of the solution to (ODE) at time T with control u .
G. Olive (IMB)
Introduction to control theory
10 / 31
Notions of controllability
y (T ; 0)
y 1 = y (T ; u)
y0
Figure Trajectory controlled exactly
y 0 : initial state,
y 1 : target,
y (T ; u) : value of the solution to (ODE) at time T with control u .
Denition
(ODE) is exactly controllable at time T if
∀y 0 , y 1 ∈ Rn , ∃u ∈ L2 (0, T )m ,
G. Olive (IMB)
Introduction to control theory
y (T ) = y 1 .
10 / 31
Notions of controllability
y (T ; 0)
0 = y (T ; u)
y0
Figure Trajectory controlled to 0
y 0 : initial state,
y 1 : target,
y (T ; u) : value of the solution to (ODE) at time T with control u .
Denition
(ODE) is null-controllable at time T if
∀y 0 ∈ Rn , ∃u ∈ L2 (0, T )m ,
G. Olive (IMB)
y (T ) = 0.
Introduction to control theory
10 / 31
Notions of controllability
y (T ; 0)
y1
ε
y (T ; u)
y0
Figure Trajectory controlled approximately
y 0 : initial state,
y 1 : target,
y (T ; u) : value of the solution to (ODE) at time T with control u .
Denition
(ODE) is approximately controllable at time T if
∀y 0 , y 1 ∈ Rn , ∀ε > 0, ∃u ∈ L2 (0, T )m ,
G. Olive (IMB)
Introduction to control theory
ky (T ) − y 1 kRn ≤ ε.
10 / 31
Reformulation
Let
Rn
−→
Rn
y0
7−→
y (T ),
L2 (0, T )m
−→
Rn
u
7−→
yb(T ),
S(T )
:
and
GT
:
d
y
dt

y (0)


d
yb
dt

yb(0)


t ∈ (0, T ),
=
Ay ,
=
y 0,
=
Ab
y + Bu,
=
0,
t ∈ (0, T ),
so that y (T ) = S(T )y 0 + GT u .
G. Olive (IMB)
Introduction to control theory
11 / 31
Reformulation
Let
Rn
−→
Rn
y0
7−→
y (T ),
L2 (0, T )m
−→
Rn
u
7−→
yb(T ),
S(T )
:
d
y
dt

y (0)


and
GT
:
d
yb
dt

yb(0)


t ∈ (0, T ),
=
Ay ,
=
y 0,
=
Ab
y + Bu,
=
0,
t ∈ (0, T ),
so that y (T ) = S(T )y 0 + GT u . Therefore,
(ODE) is exactly controllable at time T if, and only if,
Im GT
= Rn .
(1)
(ODE) is null-controllable at time T if, and only if,
Im S(T )
⊂ Im GT .
(2)
(ODE) is approximately controllable at time T if, and only if,
Im GT
G. Olive (IMB)
= Rn .
Introduction to control theory
(3)
11 / 31
Reformulation
Let
Rn
−→
Rn
y0
7−→
y (T ),
L2 (0, T )m
−→
Rn
u
7−→
yb(T ),
S(T )
:
d
y
dt

y (0)


and
GT
:
d
yb
dt

yb(0)


t ∈ (0, T ),
=
Ay ,
=
y 0,
=
Ab
y + Bu,
=
0,
t ∈ (0, T ),
so that y (T ) = S(T )y 0 + GT u . Therefore,
(ODE) is exactly controllable at time T if, and only if,
Im GT
= Rn .
(1)
(ODE) is null-controllable at time T if, and only if,
Im S(T )
⊂ Im GT .
(2)
(ODE) is approximately controllable at time T if, and only if,
Im GT
= Rn .
(3)
All these notions are equivalent.
(1) ⇐⇒ (2) since Im S(T ) = Rn .
(1) ⇐⇒ (3) since dim Im GT < +∞.
Remark :
G. Olive (IMB)
Introduction to control theory
11 / 31
Duality
Note that GT ∈ L(L2 (0, T )m , Rn ). Thus,
Im GT
= Rn
⇐⇒
ker GT∗ = {0} .
(4)
Let us compute GT∗ .
G. Olive (IMB)
Introduction to control theory
12 / 31
Duality
Note that GT ∈ L(L2 (0, T )m , Rn ). Thus,
Im GT
= Rn
ker GT∗ = {0} .
⇐⇒
(4)
Let us compute GT∗ . Multiplying (ODE) by z , solution to

 −d z
dt

z(T )
=
A∗ z,
=
z 1,
we obtain
y (T ) · z 1 − y 0 · z(0) =
G. Olive (IMB)
t ∈ (0, T ),
T
Z
0
u(t) · B ∗ z(t) dt.
Introduction to control theory
12 / 31
Duality
Note that GT ∈ L(L2 (0, T )m , Rn ). Thus,
Im GT
= Rn
ker GT∗ = {0} .
⇐⇒
(4)
Let us compute GT∗ . Multiplying (ODE) by z , solution to

 −d z
dt

z(T )
=
A∗ z,
=
z 1,
we obtain
y (T ) · z 1 − y 0 · z(0) =
T
Z
0
This shows that
GT∗
G. Olive (IMB)
:
t ∈ (0, T ),
u(t) · B ∗ z(t) dt.
Rn
−→
L2 (0, T )m
z1
7−→
B ∗ z.
Introduction to control theory
12 / 31
Duality
Note that GT ∈ L(L2 (0, T )m , Rn ). Thus,
Im GT
= Rn
ker GT∗ = {0} .
⇐⇒
(4)
Let us compute GT∗ . Multiplying (ODE) by z , solution to

 −d z
dt

z(T )
=
A∗ z,
=
z 1,
we obtain
y (T ) · z 1 − y 0 · z(0) =
T
Z
0
This shows that
GT∗
:
t ∈ (0, T ),
u(t) · B ∗ z(t) dt.
Rn
−→
L2 (0, T )m
z1
7−→
B ∗ z.
Using (4), (ODE) is controllable at time T if, and only if,
∀z 1 ∈ Rn ,
G. Olive (IMB)
B ∗ z(t) = 0,
t ∈ (0, T ) =⇒ z 1 = 0.
Introduction to control theory
12 / 31
Duality
Note that GT ∈ L(L2 (0, T )m , Rn ). Thus,
Im GT
= Rn
ker GT∗ = {0} .
⇐⇒
(4)
Let us compute GT∗ . Multiplying (ODE) by z , solution to

 −d z
dt

z(T )
=
A∗ z,
=
z 1,
we obtain
y (T ) · z 1 − y 0 · z(0) =
T
Z
0
This shows that
GT∗
:
t ∈ (0, T ),
u(t) · B ∗ z(t) dt.
Rn
−→
L2 (0, T )m
z1
7−→
B ∗ z.
Using (4), (ODE) is controllable at time T if, and only if,
∀z 1 ∈ Rn ,
Remark :
analytic.
B ∗ z(t) = 0,
t ∈ (0, T ) =⇒ z 1 = 0.
The controllability of (ODE) does not depend on T since z(t) = e (T −t)A z 1 is
G. Olive (IMB)
∗
Introduction to control theory
12 / 31
Controllability Gramian - HUM operator
Proposition (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if, the n × n matrix
ΛT = GT GT∗
G. Olive (IMB)
is invertible.
Introduction to control theory
13 / 31
Controllability Gramian - HUM operator
Proposition (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if, the n × n matrix
ΛT = GT GT∗
Proof :
is invertible.
(ODE) is controllable if, and only if,
ker GT∗ = {0} .
This is equivalent to
G. Olive (IMB)
ker GT GT∗ = {0} .
Introduction to control theory
13 / 31
Controllability Gramian - HUM operator
Proposition (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if, the n × n matrix
ΛT = GT GT∗
Proof :
is invertible.
(ODE) is controllable if, and only if,
ker GT∗ = {0} .
This is equivalent to
Remark :
ker GT GT∗ = {0} .
We can explicitly compute ΛT .
G. Olive (IMB)
Introduction to control theory
13 / 31
Controllability Gramian - HUM operator
Proposition (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if, the n × n matrix
ΛT = GT GT∗
Proof :
is invertible.
(ODE) is controllable if, and only if,
ker GT∗ = {0} .
This is equivalent to
Remark :
ker GT GT∗ = {0} .
We can explicitly compute ΛT . Indeed,
Z
GT u =
0
T
e (T −t)A Bu(t) dt,
so that
Z
ΛT =
G. Olive (IMB)
0
T
GT∗ z 1 (t) = B ∗ e (T −t)A z 1 ,
∗
∗
e (T −t)A BB ∗ e (T −t)A dt.
Introduction to control theory
13 / 31
Kalman rank condition
d
y
dt

y (0)


=
Ay + Bu,
=
y 0.
t ∈ (0, T ),
(ODE)
Theorem (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if,
n−1
rank (B|AB| · · · |A
B)
G. Olive (IMB)
= n.
Introduction to control theory
14 / 31
Kalman rank condition
d
y
dt

y (0)


=
Ay + Bu,
=
y 0.
t ∈ (0, T ),
(ODE)
Theorem (Kalman, Ho and Narendra, 1963)
(ODE) is controllable if, and only if,
n−1
rank (B|AB| · · · |A
B)
Remark :
= n.
This condition does not depend on T (as expected).
The 2 × 2 system
Example :





d
y
dt 1
d
y
dt 2
= αy1 + βy2 + u,
= γy1 + δy2 ,



 y1 (0) = y 0 , y2 (0) = y 0 ,
1
2
is controllable if, and only if,
G. Olive (IMB)
t ∈ (0, T ),
γ 6= 0.
Introduction to control theory
14 / 31
Proof of the Kalman condition
We have to establish that
∀z 1 ∈ Rn ,
where z is the solution to
G. Olive (IMB)
B ∗ z(t) = 0,

 −d z
dt

z(T )
t ∈ (0, T ) =⇒ z 1 = 0,
=
A∗ z,
=
z 1.
t ∈ (0, T ),
Introduction to control theory
15 / 31
Proof of the Kalman condition
We have to establish that
∀z 1 ∈ Rn ,
where z is the solution to
Since z is analytic,
B ∗z

 −d z
dt

z(T )
=
A∗ z,
=
z 1.
B ∗ (A∗ )k z 1 = 0,
G. Olive (IMB)
t ∈ (0, T ),
= 0 if, and only if,
dk
(B ∗ z)(T ) = 0,
dt k
that is,
t ∈ (0, T ) =⇒ z 1 = 0,
B ∗ z(t) = 0,
∀k ∈ {0, 1, . . .} ,
∀k ∈ {0, 1, . . .} .
Introduction to control theory
15 / 31
Proof of the Kalman condition
We have to establish that
∀z 1 ∈ Rn ,
where z is the solution to
Since z is analytic,
B ∗z

 −d z
dt

z(T )
=
A∗ z,
=
z 1.
t ∈ (0, T ),
= 0 if, and only if,
dk
(B ∗ z)(T ) = 0,
dt k
that is,
t ∈ (0, T ) =⇒ z 1 = 0,
B ∗ z(t) = 0,
B ∗ (A∗ )k z 1 = 0,
∀k ∈ {0, 1, . . .} ,
∀k ∈ {0, 1, . . .} .
By the Cayley-Hamilton theorem, this is equivalent to
B ∗ (A∗ )k z 1 = 0,
G. Olive (IMB)
∀k ∈ {0, 1, . . . , n − 1} .
Introduction to control theory
15 / 31
Proof of the Kalman condition
We have to establish that
∀z 1 ∈ Rn ,
where z is the solution to
Since z is analytic,
B ∗z
t ∈ (0, T ) =⇒ z 1 = 0,
B ∗ z(t) = 0,

 −d z
dt

z(T )
=
A∗ z,
=
z 1.
t ∈ (0, T ),
= 0 if, and only if,
dk
(B ∗ z)(T ) = 0,
dt k
that is,
B ∗ (A∗ )k z 1 = 0,
∀k ∈ {0, 1, . . .} ,
∀k ∈ {0, 1, . . .} .
By the Cayley-Hamilton theorem, this is equivalent to
B ∗ (A∗ )k z 1 = 0,
Thus,




z 1 ∈ ker 



and
Im
B∗
B ∗ A∗
..
.
∀k ∈ {0, 1, . . . , n − 1} .




 = ker B|AB| · · · |An−1 B ∗ =



Im
B|AB| · · · |An−1 B
⊥
,
B ∗ (A∗ )n−1
⊥
B|AB| · · · |An−1 B
= (Rn )⊥ = {0}.
G. Olive (IMB)
Introduction to control theory
15 / 31
Higher order ODEs
Consider the second order ODE :

2

 d y = Ay + Bu, t ∈ (0, T ),
dt 2
d

 y (0) = y 0 ,
y (0) = ẏ 0 .
dt
(5)
Corollary
(5) is controllable if, and only if,
n−1
B)
rank (B|AB| · · · |A
G. Olive (IMB)
= n.
Introduction to control theory
16 / 31
Higher order ODEs
Consider the second order ODE :

2

 d y = Ay + Bu, t ∈ (0, T ),
dt 2
d

 y (0) = y 0 ,
y (0) = ẏ 0 .
dt
(5)
Corollary
(5) is controllable if, and only if,
n−1
B)
rank (B|AB| · · · |A
Proof :
= n.
1) Introduce

ỹ = 
y
d
y
dt

 ∈ R2n ,

à = 
2) We have
rank (B̃|ÃB̃| · · · |Ã
G. Olive (IMB)

0
Id
A
0
2n−1 B̃) =
 ∈ R2n×2n ,
 
0
B̃ =   ∈ R2n×m .
B
2 rank (B|AB| · · · |An−1 B) = 2n.
Introduction to control theory
16 / 31
Hautus test
There is a
dual condition
to the Kalman rank condition :
Theorem (Fattorini, 1966 ; Hautus, 1969)
(ODE) is controllable if, and only if,
ker(λ − A∗ ) ∩ ker B ∗ = {0} ,
G. Olive (IMB)
Introduction to control theory
∀λ ∈ C.
(6)
17 / 31
Hautus test
There is a
dual condition
to the Kalman rank condition :
Theorem (Fattorini, 1966 ; Hautus, 1969)
(ODE) is controllable if, and only if,
ker(λ − A∗ ) ∩ ker B ∗ = {0} ,
Proof :
Let
We have to establish that
G. Olive (IMB)
N = z 1 ∈ Rn ,
B ∗ z(t) = 0,
∀λ ∈ C.
(6)
t ∈ (0, T ) .
N = {0} .
Introduction to control theory
17 / 31
Hautus test
There is a
to the Kalman rank condition :
dual condition
Theorem (Fattorini, 1966 ; Hautus, 1969)
(ODE) is controllable if, and only if,
ker(λ − A∗ ) ∩ ker B ∗ = {0} ,
Proof :
Let
N = z 1 ∈ Rn ,
We have to establish that
Taking t = T , we see that
B ∗ z(t) = 0,
∀λ ∈ C.
(6)
t ∈ (0, T ) .
N = {0} .
N ⊂ ker B ∗ .
Taking the derivative at time t = T , we have
A∗ N ⊂ N.
G. Olive (IMB)
Introduction to control theory
17 / 31
Hautus test
There is a
to the Kalman rank condition :
dual condition
Theorem (Fattorini, 1966 ; Hautus, 1969)
(ODE) is controllable if, and only if,
ker(λ − A∗ ) ∩ ker B ∗ = {0} ,
Proof :
Let
N = z 1 ∈ Rn ,
We have to establish that
B ∗ z(t) = 0,
∀λ ∈ C.
(6)
t ∈ (0, T ) .
N = {0} .
Taking t = T , we see that
N ⊂ ker B ∗ .
Taking the derivative at time t = T , we have
A∗ N ⊂ N.
Thus, if N 6= {0}, there exist λ ∈ C and ξ ∈ Cn such that
ξ 6= 0,
ξ ∈ ker(λ − A∗ ) ∩ ker B ∗ ,
a contradiction with (6).
G. Olive (IMB)
Introduction to control theory
17 / 31
Silverman-Meadows rank condition
Let us consider
with
d
y
dt

y (0)


=
A(t)y + B(t)u,
=
y 0,
A ∈ C ∞ ([0, T ])n×n ,
G. Olive (IMB)
t ∈ (0, T ),
(7)
B ∈ C ∞ ([0, T ])n×m .
Introduction to control theory
18 / 31
Silverman-Meadows rank condition
Let us consider
d
y
dt

y (0)


with
=
A(t)y + B(t)u,
=
y 0,
A ∈ C ∞ ([0, T ])n×n ,
t ∈ (0, T ),
(7)
B ∈ C ∞ ([0, T ])n×m .
Theorem (Silverman and Meadows, 1967)
(7) is controllable at time T if
∃τ ∈ [0, T ],
where
rank (B0 (τ )|B1 (τ )| · · · |Bn−1 (τ ))

 B0 (t) = B(t),
 Bi (t) = − d Bi−1 (t) + A(t)Bi−1 (t),
dt
G. Olive (IMB)
= n,
(8)
∀i ∈ {1, . . . , n − 1} .
Introduction to control theory
18 / 31
Silverman-Meadows rank condition
Let us consider
d
y
dt

y (0)


with
=
A(t)y + B(t)u,
=
y 0,
A ∈ C ∞ ([0, T ])n×n ,
t ∈ (0, T ),
(7)
B ∈ C ∞ ([0, T ])n×m .
Theorem (Silverman and Meadows, 1967)
(7) is controllable at time T if
∃τ ∈ [0, T ],
where
rank (B0 (τ )|B1 (τ )| · · · |Bn−1 (τ ))

 B0 (t) = B(t),
 Bi (t) = − d Bi−1 (t) + A(t)Bi−1 (t),
dt
= n,
(8)
∀i ∈ {1, . . . , n − 1} .
Remarks :
(B0 (t)|B1 (t)| · · · |Bn−1 (t))= (B|AB| · · · |An−1 B) if A(t) = A and B(t) = B .
(8) is necessary if A and B are analytic.
G. Olive (IMB)
Introduction to control theory
18 / 31
Optimal control
Assume that (ODE) is controllable.
The control is not unique !
Consider the minimization problem
min
u∈UT
where
1
kuk2L2 (0,T )m ,
2
UT = u ∈ L2 (0, T )m ,
y (T ) = y 1 .
There exists a unique solution uopt = projUT (0), characterized by
huopt , uopt − uiL2 ≤ 0,
G. Olive (IMB)
∀u ∈ UT .
Introduction to control theory
(9)
19 / 31
Optimal control
Assume that (ODE) is controllable.
The control is not unique !
Consider the minimization problem
min
u∈UT
where
1
kuk2L2 (0,T )m ,
2
UT = u ∈ L2 (0, T )m ,
y (T ) = y 1 .
There exists a unique solution uopt = projUT (0), characterized by
huopt , uopt − uiL2 ≤ 0,
(9)
∀u ∈ UT .
Proposition (Kalman, Ho and Narendra, 1963)
∗
1
uopt (t) = B ∗ e (T −t)A Λ−
y 1 − e TA y 0 .
T
Remarks :
uopt ∈ C ∞ ([0, T ])m (uopt is even analytic). Obviously, y 1 = e TA y 0 =⇒ uopt = 0.
G. Olive (IMB)
Introduction to control theory
19 / 31
Optimal control
Assume that (ODE) is controllable.
The control is not unique !
Consider the minimization problem
min
u∈UT
where
1
kuk2L2 (0,T )m ,
2
UT = u ∈ L2 (0, T )m ,
y (T ) = y 1 .
There exists a unique solution uopt = projUT (0), characterized by
huopt , uopt − uiL2 ≤ 0,
(9)
∀u ∈ UT .
Proposition (Kalman, Ho and Narendra, 1963)
∗
1
uopt (t) = B ∗ e (T −t)A Λ−
y 1 − e TA y 0 .
T
uopt ∈ C ∞ ([0, T ])m (uopt is even analytic). Obviously, y 1 = e TA y 0 =⇒ uopt = 0.
Since UT = uopt + ker GT , (9) gives
Remarks :
Proof :
huopt , uiL2 = 0,
∀u ∈ ker GT .
Thus, uopt ∈ (ker GT )⊥ = Im GT∗ : GT∗ zn1 → uopt . But (zn1 )n∈N converges (uopt ∈ UT ) :
zn1 −−−−−→ (GT GT∗ )−1 y 1 − S(T )y 0 .
n→+∞
G. Olive (IMB)
Introduction to control theory
19 / 31
Cost of controllability
Let y 0 = 0. We introduce the control cost
CT =
sup kuopt kL2 (0,T )m .
ky 1 k=1
We have
CT is decreasing in T .
CT → +∞ as T → 0+ .
G. Olive (IMB)
Introduction to control theory
20 / 31
Cost of controllability
Let y 0 = 0. We introduce the control cost
CT =
sup kuopt kL2 (0,T )m .
ky 1 k=1
We have
CT is decreasing in T .
CT → +∞ as T → 0+ .
More precisely,
Theorem (Seidman, 1988)
1
CT ∼ γ
T
K+ 1
2
as T → 0+ ,
where γ > 0 and K is the smallest exponent such that
G. Olive (IMB)
rank (B|AB| · · · |AK B)
Introduction to control theory
= n.
20 / 31
Miscellaneous properties of controls
Assume that (ODE) is controllable.
Proposition
One can nd controls u (6= uopt ) such that, in addition, given u 0 , u 1 ∈ Rm ,
u(0) = u 0 ,
G. Olive (IMB)
u(T ) = u 1 .
Introduction to control theory
21 / 31
Miscellaneous properties of controls
Assume that (ODE) is controllable.
Proposition
One can nd controls u (6= uopt ) such that, in addition, given u 0 , u 1 ∈ Rm ,
u(0) = u 0 ,
Proof :
u(T ) = u 1 .
Consider the (n + m) × (n + m) system





d
y
dt
d
u
dt
= Ay + Bu,
= v,



 y (0) = y 0 ,
G. Olive (IMB)
t ∈ (0, T ),
u(0) = u 0 .
Introduction to control theory
21 / 31
More nite dimensional control theory
Important subjects we did not discuss here :
Nonlinear control theory
Stabilization
Controllability with constraints
Numerical analysis of controlled systems
G. Olive (IMB)
Introduction to control theory
22 / 31
Outline
1
Introduction
2
Controllability in nite dimension
Duality
Kalman rank condition, Hautus test
Time-dependent systems
Control cost
3
Controllability in innite dimension : the heat equation
Introduction
Towards systems of heat equations
G. Olive (IMB)
Introduction to control theory
23 / 31
The heat equation
Let T > 0 and Ω ⊂ RN a bounded open subset and set
QT = (0, T ) × Ω,
The heat equation is
ΣT = (0, T ) × ∂Ω.

∂ y = ∆y + 1ω u

 t
y =0


y (0) = y 0
in QT ,
on ΣT ,
(heat)
in Ω,
where
y is the state, y 0 the initial data,
u ∈ L2 (QT ) is the control,
ω ⊂ Ω localizes in space the control.
1ω (x) = 1 if x ∈ ω and 1ω (x) = 0 otherwise.
Well-posedness :
For every y 0 ∈ L2 (Ω) and u ∈ L2 (QT ), there exists a unique (weak) solution
y ∈ C 0 ([0, T ]; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)).
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Introduction to control theory
24 / 31
Remarks
As before, we say that :
(heat) is exactly controllable at time T if
∀y 0 , y 1 ∈ L2 (Ω), ∃u ∈ L2 (QT ),
y (T ) = y 1 .
(heat) is null-controllable at time T if
∀y 0 ∈ L2 (Ω), ∃u ∈ L2 (QT ),
y (T ) = 0.
(heat) is approximately controllable at time T if
∀y 0 , y 1 ∈ L2 (Ω), ∀ε > 0, ∃u ∈ L2 (QT ),
G. Olive (IMB)
Introduction to control theory
ky (T ) − y 1 kL2 (Ω) ≤ ε.
25 / 31
Remarks
As before, we say that :
(heat) is exactly controllable at time T if
∀y 0 , y 1 ∈ L2 (Ω), ∃u ∈ L2 (QT ),
y (T ) = y 1 .
(heat) is null-controllable at time T if
∀y 0 ∈ L2 (Ω), ∃u ∈ L2 (QT ),
y (T ) = 0.
(heat) is approximately controllable at time T if
∀y 0 , y 1 ∈ L2 (Ω), ∀ε > 0, ∃u ∈ L2 (QT ),
ky (T ) − y 1 kL2 (Ω) ≤ ε.
Remarks :
Exact controllability to a state y 1 6∈ C ∞ is impossible (regularizing eect in Ω\ω ),
Approximate controllability does not depend on T (analyticity in time of the adjoint system).
G. Olive (IMB)
Introduction to control theory
25 / 31
Duality
Let us introduce the adjoint system to (heat) :

−∂t z = ∆z


z=0


z(T ) = z 1
in QT ,
on ΣT ,

∂ y = ∆y + 1ω u

 t
y =0


y (0) = y 0
in Ω,
in QT ,
on ΣT ,
(heat)
in Ω.
Multiplying (heat) by z we have
G. Olive (IMB)
y (T ), z 1
L2
− y 0 , z(0) L2 =
Z
0
T
hu(t), 1ω z(t)iL2 dt.
Introduction to control theory
26 / 31
Duality
Let us introduce the adjoint system to (heat) :

−∂t z = ∆z


z=0


z(T ) = z 1
in QT ,
on ΣT ,

∂ y = ∆y + 1ω u

 t
y =0


y (0) = y 0
in Ω,
in QT ,
on ΣT ,
(heat)
in Ω.
Multiplying (heat) by z we have
y (T ), z 1
L2
− y 0 , z(0) L2 =
Z
T
0
hu(t), 1ω z(t)iL2 dt.
Theorem (Dolecki and Russell, 1977)
(heat) is approximately controllable at time T if, and only if,
∀z 1 ∈ L2 (Ω),
1ω z(t) = 0,
t ∈ (0, T ) =⇒ z 1 = 0.
(heat) is null-controllable at time T if, and only if,
∃CT > 0,
kz(0)k2L2 ≤ CT2
T
Z
0
k1ω z(t)k2L2 dt,
∀z 1 ∈ L2 (Ω).
CT is again the control cost !
G. Olive (IMB)
Introduction to control theory
26 / 31
Controllability of the heat equation
Theorem
(heat) is null-controllable at time T for any T > 0 and any nonempty open subset ω ⊂ Ω.
Remark :
The proof is easy if ω = Ω, but very complex if ω ( Ω.
G. Olive (IMB)
Introduction to control theory
27 / 31
Controllability of the heat equation
Theorem
(heat) is null-controllable at time T for any T > 0 and any nonempty open subset ω ⊂ Ω.
The proof is easy if ω = Ω, but very complex if ω ( Ω.
Proved by :
Fattorini and Russell (1971), using the method of moments (proof in dimension N = 1),
Lebeau and Robbiano (1995), using elliptic Carleman estimates,
Fursikov and Imanuvilov (1996), using parabolic Carleman estimates,
Everdoza and Zuazua (2011), using the transmutation method (from the wave equation
to the heat equation), introduced by Miller (2006).
Remark :
G. Olive (IMB)
Introduction to control theory
27 / 31
Boundary controllability
Boundary controllability :

∂ y = ∆y

 t
y = 1γ u


y (0) = y 0
in QT ,
on ΣT ,
(10)
in Ω,
where γ ⊂ ∂Ω is a nonempty relative open subset.
G. Olive (IMB)
Introduction to control theory
28 / 31
Boundary controllability
Boundary controllability :

∂ y = ∆y

 t
y = 1γ u


y (0) = y 0
in QT ,
on ΣT ,
(10)
in Ω,
where γ ⊂ ∂Ω is a nonempty relative open subset.
The controllability of (10) is a consequence of the one of (heat) (and vice versa).
1) We extend the domain.
Remark :
Ω
γ
ω
2) We take the trace of the controlled extended solution as control.
G. Olive (IMB)
Introduction to control theory
28 / 31
Systems of coupled heat equations
Consider the system of 2 equations with only 1 distributed control :


∂t y1 = ∆y1 + 1ω u




 ∂t y = ∆y + a(x)y
2
2
1
 y1 = y2 = 0





y1 (0) = y10 , y2 (0) = y20 ,
in QT ,
in QT ,
on ΣT ,
(syst)
in Ω,
where
(y1 , y2 ) is the state and (y10 , y20 ) ∈ L2 (Ω)2 the initial data,
u ∈ L2 (QT ) is the control,
ω ⊂ Ω localizes in space the control,
a ∈ C 0 (Ω) is the coupling.
G. Olive (IMB)
Introduction to control theory
29 / 31
Systems of coupled heat equations
Consider the system of 2 equations with only 1 distributed control :


∂t y1 = ∆y1 + 1ω u




 ∂t y = ∆y + a(x)y
2
2
1
 y1 = y2 = 0





y1 (0) = y10 , y2 (0) = y20 ,
in QT ,
in QT ,
on ΣT ,
(syst)
in Ω,
where
(y1 , y2 ) is the state and (y10 , y20 ) ∈ L2 (Ω)2 the initial data,
u ∈ L2 (QT ) is the control,
ω ⊂ Ω localizes in space the control,
a ∈ C 0 (Ω) is the coupling.
We can show that :
If a(x) = a is constant, (syst) is null-controllable at time T if, and only if,
a 6= 0.
More generally, (syst) is null-controllable at time T if
a 6= 0 in ω.
If a = 0 in ω , everything may happen.
G. Olive (IMB)
Introduction to control theory
29 / 31
Unexpected behaviors for systems

 ∂t y = ∆y + 1ω u
1
1
 ∂t y2 = ∆y2 + a(x)y1
in QT ,
in QT .
(syst)
• Distributed 6= boundary controls : There exists a(x) such that (syst) is distributed
approximately controllable but NOT boundary approximately controllable.
G. Olive (IMB)
Introduction to control theory
30 / 31
Unexpected behaviors for systems

 ∂t y = ∆y + 1ω u
1
1
 ∂t y2 = ∆y2 + a(x)y1
in QT ,
(syst)
in QT .
• Distributed 6= boundary controls : There exists a(x) such that (syst) is distributed
approximately controllable but NOT boundary approximately controllable.
•
Outbreak of the geometry :
ω
ω
ω
supp a
supp a
Figure Case n◦ 1 : ω is connected
Figure Case n◦ 2 : ω is NOT connected
There exists a(x) such that :
(syst) is NOT approximately controllable in Case n◦ 1.
(syst) is approximately controllable in Case n◦ 2.
G. Olive (IMB)
Introduction to control theory
30 / 31
Unexpected behaviors for systems

 ∂t y = ∆y + 1ω u
1
1
 ∂t y2 = ∆y2 + a(x)y1
in QT ,
(syst)
in QT .
• Distributed 6= boundary controls : There exists a(x) such that (syst) is distributed
approximately controllable but NOT boundary approximately controllable.
•
Outbreak of the geometry :
ω
ω
ω
supp a
supp a
Figure Case n◦ 1 : ω is connected
Figure Case n◦ 2 : ω is NOT connected
There exists a(x) such that :
(syst) is NOT approximately controllable in Case n◦ 1.
(syst) is approximately controllable in Case n◦ 2.
•
There exists a(x) such that, for some T ∗ > 0,
For every T >
(syst) is null-controllable at time T .
For every T < T ∗ , (syst) is NOT null-controllable at time T .
Minimal time of control :
T ∗,
G. Olive (IMB)
Introduction to control theory
30 / 31
Unexpected behaviors for systems

 ∂t y = ∆y + 1ω u
1
1
 ∂t y2 = ∆y2 + a(x)y1
in QT ,
(syst)
in QT .
• Distributed 6= boundary controls : There exists a(x) such that (syst) is distributed
approximately controllable but NOT boundary approximately controllable.
•
Outbreak of the geometry :
ω
ω
ω
supp a
supp a
Figure Case n◦ 1 : ω is connected
Figure Case n◦ 2 : ω is NOT connected
There exists a(x) such that :
(syst) is NOT approximately controllable in Case n◦ 1.
(syst) is approximately controllable in Case n◦ 2.
•
There exists a(x) such that, for some T ∗ > 0,
For every T >
(syst) is null-controllable at time T .
For every T < T ∗ , (syst) is NOT null-controllable at time T .
Minimal time of control :
T ∗,
Null 6= approximate controllability : There exists a(x) such that (syst) is approximately
controllable but NOT null-controllable at any time T .
•
G. Olive (IMB)
Introduction to control theory
30 / 31
Some references
Insensitizing controls for a semilinear heat equation, L.
Dierential Equations, 25 (2000), 39-72.
de Teresa,
Boundary controllability of parabolic coupled equations, E.
Gonzalez-Burgos and L. de Teresa,
Comm. Partial
Fernandez-Cara, M.
J. Funct.Anal., 259 (2010), 1720-1758.
Boundary approximate controllability of some linear parabolic systems, G.
Equ. Control Theory 3 (2014), no. 1, 167-189.
Olive,
Evol.
Approximate controllability conditions for some linear 1D parabolic systems with
space-dependent coecients, F. Boyer and G. Olive, Math. Control Relat. Fields 4
(2014), no. 3, 263-287.
Minimal time for the null controllability of parabolic systems : The eect of the
condensation index of complex sequences, F. Ammar-Khodja, A. Benabdallah,
Gonzalez-Burgos, and L. de Teresa,
J. Funct. Anal., 267 (2014), pp. 20772151.
M.
Dziekuje za uwage !
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Introduction to control theory
31 / 31