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TRANSCRIPT
Elgersburg 2013: ∞ -dimensional systems
Infinite Dimensional Systems
Hans Zwart, University of Twente
– 1 –
Elgersburg 2013: ∞ -dimensional systems CONTENTS
Contents
1 Motivation 6
1.1 Potato storage . . . . . . . . . . . . . . . . . . . . . 7
1.2 UV-disinfection . . . . . . . . . . . . . . . . . . . . 10
1.3 Platoon systems . . . . . . . . . . . . . . . . . . . . 14
2 Semigroups and Generators 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Generators . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Which A generates a C0-semigroup? . . . . . . . . . 40
– 2 –
Elgersburg 2013: ∞ -dimensional systems CONTENTS
2.5 Platoon systems . . . . . . . . . . . . . . . . . . . . 42
2.6 Contraction semigroup . . . . . . . . . . . . . . . . 46
2.7 Semigroups and solutions of p.d.e.’s . . . . . . . . . . 61
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . 64
3 Inputs and Outputs 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Inputs . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Outputs . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Boundary control . . . . . . . . . . . . . . . . . . . 85
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . 97
– 3 –
Elgersburg 2013: ∞ -dimensional systems CONTENTS
4 Transfer Functions 103
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Exponential solutions . . . . . . . . . . . . . . . . . 106
4.3 Impulse response . . . . . . . . . . . . . . . . . . . 123
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . 135
5 Stability and Stabilizability 138
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 139
5.2 Stability notions . . . . . . . . . . . . . . . . . . . . 140
5.3 Stability of platoon models . . . . . . . . . . . . . . . 142
5.4 Exponential stability and Lyapunov equations . . . . . 144
5.5 Strong stability . . . . . . . . . . . . . . . . . . . . 153
– 4 –
Elgersburg 2013: ∞ -dimensional systems CONTENTS
5.6 Stabilizability . . . . . . . . . . . . . . . . . . . . . 162
5.7 Stabilization of the platoon model . . . . . . . . . . . 170
5.8 Transfer function and stability . . . . . . . . . . . . . 172
5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . 174
6 Linear Port Hamiltonian Systems 175
6.1 A typical example . . . . . . . . . . . . . . . . . . . 176
7 Homogeneous solutions of P.H.S. 181
8 Well-posedness for PHS 192
8.1 Well-posedness for simple PHS . . . . . . . . . . . . 195
8.2 Characterization of well-posedness for PHS . . . . . . 198
– 5 –
Elgersburg 2013: ∞ -dimensional systems Motivation
1 Motivation
– 6 –
Elgersburg 2013: ∞ -dimensional systems Motivation
1.1 Potato storage
– 7 –
Elgersburg 2013: ∞ -dimensional systems Motivation
Schematic view point:
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replacements
0
L
shaftbulk
Tp(x, t)Ta(x, t) v(t) x
Tc(t)
T0(t)
Φ(t)
– 8 –
Elgersburg 2013: ∞ -dimensional systems Motivation
The model:
dT0
dt(t) = M1α(Φ(t)) [Ta(L, t) − Tc(t)] +
M2Φ(t) [Ta(L, t) − T0(t)] +
M3 [Tout(t) − T0(t)]
∂Ta
∂t(x, t) = −v(t)
∂Ta
∂x(x, t) + M4 [Tp(x, t) − Ta(x, t)]
∂Tp
∂t(x, t) = M5Tp(x, t) + M6Ta(x, t)
Ta(0, t) = T0(t).
– 9 –
Elgersburg 2013: ∞ -dimensional systems Motivation
1.2 UV-disinfection
– 10 –
Elgersburg 2013: ∞ -dimensional systems Motivation
Schematic view point:
C1
r
r
r1 r2
L
x
C0(t)
– 11 –
Elgersburg 2013: ∞ -dimensional systems Motivation
The model for the velocity
∂vx
∂t(r, t) = β(t) +
1
Re
[
1
r
∂
∂r
(
r∂vx
∂r(r, t)
)]
vx(r1, t) = vx(r2, t) = 0
and for the concentration
– 12 –
Elgersburg 2013: ∞ -dimensional systems Motivation
∂C
∂t(x, r, t) = −vx(r, t)
∂C
∂x(x, r, t) +
1
Pe
[
1
r
∂
∂r
(
r∂C
∂r(x, r, t)
)
+∂2C
∂x2(x, r, t)
]
−Da K(r, t)C(x, r, t)
∂C
∂x(0, r, t) = 0
C(x, r1, t) = C(x, r2, t) = 0
C(0, r, t) = C0(t).
– 13 –
Elgersburg 2013: ∞ -dimensional systems Motivation
1.3 Platoon systems
Consider an infinite string of vehicles (a platoon)
didi+1di+2 di-1
i+1 i i-1
s i+1
s i
s i-1
LiLi+1 Li-1
The aim is to control the distance di between the cars. Every car
controls its jerk, i.e, the change of its force.
– 14 –
Elgersburg 2013: ∞ -dimensional systems Motivation
Standard modeling gives the following model for the i’th car
d
dt
di(t)
vi(t)
ai(t)
=
vi−1(t) − vi(t)
ai(t)
−τ−1ai(t) + τ−1ui(t)
, i ∈ Z.
where τ is a constant.
– 15 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2 Semigroups and Generators
– 16 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.1 Introduction
The aim of this part is to learn the concepts semigroups and
infinitesimal generator.
The reason for this is that we want to know if the partial diffe rential
equations (p.d.e.’s) or the coupled ordinary differential equations
(o.d.e.’s) have a solution.
Note there is a difference between knowing the existence of a solution
and having the form/expression of the solution.
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example (Transport equation)
On the spatial domain [0, 1] consider the p.d.e.
∂z
∂t(x, t) =
∂z
∂x(x, t), x ∈ [0, 1], t ≥ 0,
z(1, t) = 0
z(x, 0) = z0(x) (given).
�
– 18 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example (Diffusion equation)
On the spatial domain [0, 1] consider the p.d.e.
∂z
∂t(x, t) =
∂2z
∂x2(x, t), x ∈ [0, 1], t ≥ 0,
∂z
∂x(0, t) = 0
∂z
∂x(1, t) = 0
z(x, 0) = z0(x) (given).
�
– 19 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.2 Semigroups
Throughout this course, we assume that
• Z is a (complex) Hilbert space with inner product 〈·, ·〉.
• L(Z) denotes the set of linear and bounded operators from Z to
Z .
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Definition
A strongly continuous semigroup is an operator-valued func tion from
[0, ∞) to L(Z) which satisfies
• T (0) = I
• T (t)T (s) = T (t + s), t, s ∈ [0, ∞)
• For all z0 ∈ Z there holds
limt↓0
T (t)z0 = z0.
�
Notation : (T (t))t≥0; Short : C0-semigroup.
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
To motivate this definition, think of
z0 7→ T (t)z0
as a solution of a time-invariant, linear differential equa tion.
• T (0) = I Trivial.
• For fixed t, T (t) is linear Is linearity of diff. eq.
• T (t)T (s) = T (t + s) Is time invariance.
• T (t)z0 → z0 if t ↓ 0 Is strong continuity.
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example
Let A be an n × n-matrix, then eAt is a C0-semigroup on Cn or R
n.
For instance, if
A =
1 2
0 3
,
then
eAt =
et e3t − et
0 e3t
.
�
– 23 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example
Let Z = L2(0, 1) and define for t ≥ 0 and x ∈ [0, 1]
(T (t)f) (x) =
f(t + x) x + t ≤ 1
0 x + t > 1
0 1x →
f
0 1x →
T (t)f
�
– 24 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
An important property of a C0-semigroup is that it is exponentially
bounded.
Lemma
There exist M ≥ 1 and ω ∈ R such that for all t ≥ 0
‖T (t)‖ ≤ Meωt.
�
– 25 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
The proof is as follows.
• supt∈[0,1] ‖T (t)‖ ≤ M .
This follows from the strong continuity and the uniform
boundedness theorem.
Note M ≥ 1.
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• Choose t > 1 and write t = n + t0 with t0 ∈ [0, 1).
‖T (t)‖ = ‖T (n + t0)‖= ‖T (n)T (t0)‖≤ ‖T (n)‖‖T (t0)‖≤ M‖T (n)‖= M‖T (n − 1)T (1)‖≤ M‖T (1)‖‖T (n − 1)‖≤ MM‖T (n − 1)‖≤ Mn+1 = Melog Mn
≤ Melog Mt.
– 27 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.3 Generators
Consider the finite-dimensional semigroup
eAt =
et e3t − et
0 e3t
.
Question
What is A?
– 28 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Answer
Evaluate the derivative of the semigroup at t = 0.
Since ddt
eAt = AeAt, we have
d
dteAt |t=0= A.
�
So given the (general) C0-semigroup (T (t))t≥0, we could try to find
A by differentiating it at t = 0. However, in general (T (t))t≥0 is
only (strongly) continuous.
– 29 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Definition
If the following limit exists,
limt↓0
T (t)z0 − z0
t,
then z0 is in the domain of A, D(A).
Furthermore, for z0 ∈ D(A), we define
Az0 = limt↓0
T (t)z0 − z0
t.
A is named the infinitesimal generator of the C0-semigroup
(T (t))t≥0. �
– 30 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example
Consider the shift semigroup.
(T (t)f) (x) =
f(t + x) x + t ≤ 1
0 x + t > 1
00 11x →x →
fT (t)f
– 31 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
00 11x →x →
fT (t)f
Choose x < 1, then for t ∈ (0, x), we have that
(
T (t)f − f
t
)
(x) =f(t + x) − f(x)
t.
Hence if the limit exists, then f must be differentiable, and the limit is
the derivative.
– 32 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Now take x = 1. For any t > 0, we have that(
T (t)f − f
t
)
(1) =0 − f(1)
t.
This limit can only exist when f(1) = 0.
– 33 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Concluding, we have that
D(A) ={
f ∈ L2(0, 1) | f is absolutely continuous,
withdf
dx∈ L2(0, 1) and f(1) = 0
}
.
Furthermore,
Af =df
dx.
�
– 34 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Lemma
If z0 ∈ D(A), then T (t)z0 ∈ D(A), and
d
dt[T (t)z0] = AT (t)z0.
�
Hence for z0 ∈ D(A), we have that z(t) := T (t)z0 is a (classical)
solution of the abstract differential equation
z(t) = Az(t), z(0) = z0.
How does this abstract differential equation relate to a p.d .e.?
– 35 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
We make the following observations:
• The function z is at every time instant an element of a Hilbert
space, i.e., z(t) ∈ Z .
• Assume that the Hilbert space consists of functions, i.e., f ∈ Z
means a.o. f := x 7→ f(x).
For instance, Z = L2(0, 1).
– 36 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• This implies that z(t) is for every t a function of x. We write
(z(t)) (x) = z(x, t).
• Using this last form we see that
(z(t)) (x) =∂z
∂t(x, t).
– 37 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• For A = ddx
, the abstract differential equation
z(t) = Az(t)
becomes∂z
∂t(x, t) =
∂z
∂x(x, t).
• In the domain of A are also the boundary conditions:
z(1, t) = 0.
– 38 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• For a given p.d.e. we (roughly) do the following
– The state z is identified, and the p.d.e. is written as
∂z
∂t= . . .
– The right-hand side defines together with the boundary
conditions A.
– 39 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.4 Which A generates a C0-semigroup?
So we have that every semigroup has an infinitesimal generato r, but
you would like to know which operator A generates a C0-semigroup.
The answer is given by the Hille-Yosida Theorem. This theore m we will
not treat. We focus on two special cases
• A bounded, and
• T (t) a contraction
– 40 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
If A is a bounded (linear) operator, i.e., there exists an m ≥ 0 such
that ‖Az‖ ≤ m‖z‖ for all z ∈ Z , then A generates a
C0-semigroup, given by
T (t) = eAt = I + At +A2t2
2!+
A3t3
3!+ · · ·
– 41 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.5 Platoon systems
For zero input our platoon system is given by
d
dt
dr(t)
vr(t)
ar(t)
=
vr−1(t) − vr(t)
ar(t)
−τ−1ar(t)
, r ∈ Z.
where τ is a constant.
As state space we choose ℓ2(Z; C3) and as state
z(t) =
· · · ,
d−1(t)
v−1(t)
a−1(t)
,
d0(t)
v0(t)
a0(t)
,
d1(t)
v1(t)
a1(t)
, · · · ,
– 42 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
For the sequence (fr)r∈Z ∈ ℓ2(Z; Cn) define the
discrete Fourier transform as
f(φ) =
∞∑
r=−∞frφ
−r, φ ∈ ∂D (unit circle)
This Fourier transform maps ℓ2(Z; Cn) onto L2(∂D; C
n).
Furthermore,
‖(fr)‖2 =
∞∑
r=−∞‖fr‖2 =
1
2π
∫ 2π
0
‖f(eθ)‖2dθ = ‖f‖2
– 43 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
For differential equations of the following type
zr(t) =N∑
l=−N
Alzr−l(t), r ∈ Z
the Fourier transform gives;
∂z
∂t(φ, t) =
[
N∑
l=−N
Alφ−l
]
z(φ, t), φ ∈ ∂D
= A(φ)z(φ, t).
Hence a parametrized finite-dimensional system.
– 44 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
It is easy to see that A(φ) =∑N
l=−N Alφ−l defines a bounded
linear operator on L2(∂D; Cn).
So A generates the C0-semigroup on L2(∂D; Cn)
(
eAt)
(θ) = eA(θ)t, θ ∈ ∂D.
Thus in the Fourier domain it is easy to find the solutions of
zr(t) =N∑
l=−N
Alzr−l(t), r ∈ Z
– 45 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.6 Contraction semigroup
Definition
The C0-semigroup (T (t))t≥0 is contraction semigroup if
‖T (t)z0‖ ≤ ‖z0‖ for all t ≥ 0.
�
What can we say about these semigroups?
– 46 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
0 t →
‖z0‖
‖T (t)z0‖↑
– 47 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
We known that
‖T (t)z0‖2 = 〈T (t)z0, T (t)z0〉.
For z0 ∈ D(A), we have that the derivative of T (t)z0 equals
AT (t)z0.
So if we differentiate ‖T (t)z0‖2, we find
d
dt‖T (t)z0‖2 = 〈AT (t)z0, T (t)z0〉 + 〈T (t)z0, AT (t)z0〉.
– 48 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
At time equal to zero, we find
d
dt
(
‖T (t)z0‖2)
|t=0= 〈Az0, z0〉 + 〈z0, Az0〉.
We know that at t = 0, ‖T (t)z0‖ = ‖z0‖, and so if T (t) is a
contraction semigroup, then
〈Az0, z0〉 + 〈z0, Az0〉 =d
dt‖T (t)z0‖2 |t=0≤ 0.
This holds for all z0 ∈ D(A).
– 49 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Theorem (Lumer-Phillips)
Let A be a densely defined operator, then A generates a contraction
semigroup on Z if and only if
1. 〈Az0, z0〉 + 〈z0, Az0〉 ≤ 0 for all z0 ∈ D(A).
2. The range of A − I is the whole of Z .
�
– 50 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example
Consider A which is given as
Az =dz
dx, x ∈ [0, 1]
with the domain
D(A) ={
z ∈ L2(0, 1) | z is absolutely continuous,
dz
dx∈ L2(0, 1) and z(1) = 0
}
.
Let us check the properties
• A is densely defined in L2(0, 1).
– 51 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
•
〈Az, z〉+〈z, Az〉
=
∫ 1
0
dz
dx(x)z(x)dx +
∫ 1
0
z(x)dz
dx(x)dx
=
∫ 1
0
d
dx
[
z(x)z(x)]
dx
= |z(x)|2∣
∣
1
0
= 0 − |z(0)|2 ≤ 0.
– 52 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• To see if the range of (A − I) is everything, we have for every
f ∈ L2(0, 1) to solve (A − I)z = f .
This means solving
dz
dx(x) − z(x) = f(x), x ∈ (0, 1)
with boundary condition z(1) = 0.
The solution of this differential equation with the given bo undary
value is
z(x) = −∫ 1
x
ex−ξf(ξ)dξ.
�
– 53 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Example
Consider A given by
Az =d2z
dx2x ∈ [0, 1]
with the domain
D(A) =
{
z ∈ L2(0, 1) | z,dz
dxare absolutely continuous,
anddz
dx(0) = 0 =
dz
dx(1)
}
.
Clearly A is densely defined, and so it remains to check the other
conditions.
– 54 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
We have that
〈Az, z〉 + 〈z, Az〉
=
∫ 1
0
d2z
dx2(x)z(x) + z(x)
d2z
dx2(x)dx
=
[
dz
dx(x)z(x) + z(x)
dz
dx(x)
]1
0
− 2
∫ 1
0
∣
∣
∣
∣
dz
dx(x)
∣
∣
∣
∣
2
dx
≤ 0
for z ∈ D(A).
– 55 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
To see that
(A − I)z = f
is solvable for all f ∈ L2(0, 1) we have to solve an o.d.e.
The solution is given by
z(x) = cosh(x)z(0) +
∫ x
0
sinh(x − ξ)f(ξ)dξ,
with
z(0) =−1
sinh(1)
∫ 1
0
cosh(1 − ξ)f(ξ)dξ.
Thus A generates a contraction semigroup.
– 56 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Now we know that A generates a C0-semigroup, but can we find this
semigroup?
For this we could return to the corresponding p.d.e.
∂z
∂t(x, t) =
∂2z
∂x2(x, t), x ∈ [0, 1], t ≥ 0,
∂z
∂x(0, t) = 0
∂z
∂x(1, t) = 0
z(x, 0) = z0(x) (given).
This we can solve by the separation of variables principle. W e choose
another approach.
– 57 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• A is a self-adjoint operator and the inverse of (A − I) is
compact.
• Hence A has an orthonormal basis of eigenfunctions.
– 58 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• Solving Aφn = λnφn gives
φn(x) =
1 λ0 = 0√
2 cos(nπx) λn = −n2π2, n ∈ N
• Hence the solution of
z(t) = Az(t), z(0) = φn
is given by
z(t) = eλntφn
• This must be equal to T (t)φn.
– 59 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• Since {φn, n ∈ N ∪ {0}} is an orthonormal basis, we know that
z0 =∞∑
n=0
〈z0, φn〉φn
Hence
T (t)z0 = T (t)
( ∞∑
n=0
〈z0, φn〉φn
)
=
∞∑
n=0
〈z0, φn〉T (t)φn
=∞∑
n=0
〈z0, φn〉eλntφn.
�
– 60 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.7 Semigroups and solutions of p.d.e.’s
We have that for any z0 ∈ D(A), the function z(t) := T (t)z0 is
the solution of
z(t) = Az(t), z(0) = z0.
How for general z0?
Note that in general Az(t) has no meaning and so has z(t).
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
To solve this question we have first to introduce the adjoint o f A.
Definition
Let A be a densely defined operator with domain D(A). The domain
of A∗, D(A∗), is defined as consisting of those w ∈ Z for which
there exists a v ∈ Z such that
〈w, Az〉 = 〈v, z〉 for all z ∈ D(A).
If w ∈ D(A∗), then A∗ is defined as
A∗w = v.
A∗ is named the adjoint of A. �
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
Lemma
Let z0 ∈ Z , and define z(t) = T (t)z0. Then for every
w ∈ D(A∗), there holds that
d
dt〈w, z(t)〉 = 〈A∗w, z(t)〉.
�
This implies that z(t) := T (t)z0 is the weak solution of
z(t) = Az(t), z(0) = z0.
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Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
2.8 Summary
We have seen the following
• A p.d.e. like
∂z
∂t(x, t) =
∂z
∂x(x, t)
z(1, t) = 0
will be written as
z(t) = Az(t)
where z is the state, see figure
– 64 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
– 65 –
Elgersburg 2013: ∞ -dimensional systems Semigroups and Generators
• The state is a function of the spatial variable.
• The mapping initial state to state at time t is a strongly continuous
semigroup.
• There are conditions which tell you when the operator A
generates a strongly continuous semigroup.
• The function z(t) = T (t)z0 is always a (weak) solution of the
abstract differential equation
z(t) = Az(t), z(0) = z0.
– 66 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3 Inputs and Outputs
– 67 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.1 Introduction
From the previous part we know what we mean by the solution of t he
abstract differential equation
z(t) = Az(t), z(0) = z0
However, what do we mean by the solution of
z(t) = Az(t) + Bu(t), z(0) = z0,
y(t) = Cz(t) + Du(t).
To find the answer to this question is the aim of this part.
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.2 Inputs
In this section, we want to solve the abstract differential e quation
z(t) = Az(t) + Bu(t), z(0) = z0.
We assume that
• A generates the C0-semigroup (T (t))t≥0 on the Hilbert space
Z .
• B ∈ L(U, Z).
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
So we want to find a (candidate) solution for
z(t) = Az(t) + Bu(t), z(0) = z0.
Choose t1 > 0 and let t ∈ [0, t1].
We multiply the differential equation by T (t1 − t), and bring z to the
left-hand side
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
T (t1 − t)z(t) − T (t1 − t)Az(t) = T (t1 − t)Bu(t).
The left-hand side equals
d
dt[T (t1 − t)z(t)] = T (t1 − t)Bu(t).
Hence∫ t1
0
T (t1 − t)Bu(t)dt = [T (t1 − t)z(t)]t10
= z(t1) − T (t1)z(0).
– 71 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Theorem
Consider the abstract differential equation
z(t) = Az(t) + Bu(t), z(0) = z0,
where A generates the C0-semigroup (T (t))t≥0 on the Hilbert
space Z , B ∈ L(U, Z), and u ∈ L1loc((0, ∞); U). Then the
(weak) solution is given by
z(t) = T (t)z0 +
∫ t
0
T (t − τ )Bu(τ )dτ.
If u is continuously differentiable and z0 ∈ D(A), then it is the
classical solution. �
– 72 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Example
Consider the (controlled) p.d.e.
∂z
∂t(x, t) =
∂z
∂x(x, t) + u(t)
z(1, t) = 0.
We can write this as
z(t) = Az(t) + Bu(t)
with
– 73 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Az =dz
dx,
D(A) =
{
z ∈ L2(0, 1) | dz
dx∈ L2(0, 1) and z(1) = 0
}
and
Bu = 1 · u.
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Using the semigroup generated by A, we find that the solution of the
p.d.e. is given by
z(x, t) = z0(x+t)1[0,1](x+t)+
∫ t
0
1[0,1](x+t−τ )u(τ )dτ.
with 1[0,1](ξ) =
1 ξ ∈ [0, 1]
0 ξ 6∈ [0, 1]
�
– 75 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Example
We consider the heated bar which is heated uniformly in the in terval
[1/2, 1].
∂z
∂t(x, t) =
∂2z
∂x2(x, t) + 1[1/2,1](x)u(t)
∂z
∂x(0, t) =
∂z
∂x(0, t) = 0.
This we can write as
z(t) = Az(t) + Bu(t)
with
(Bu) (x) = 1[1/2,1](x) · u
– 76 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
and A as before
Az =d2z
dx2,
with domain
D(A) =
{
z ∈ L2(0, 1) |dz
dxand
d2z
dx2∈ L2(0, 1),
dz
dx(0) = 0 =
dz
dx(1)
}
.
�
– 77 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.3 Outputs
Now we have solved the input problem, the understanding of th e
output equation
y(t) = Cz(t) + Du(t)
is easy.
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
If C ∈ L(Z, Y ), and D ∈ L(U, Y ), with Y a Hilbert space, then
using the solution for z(t) we find
y(t) = C
[
T (t)z0 +
∫ t
0
T (t − τ )Bu(τ )dτ
]
+ Du(t)
= CT (t)z0 +
∫ t
0
CT (t − τ )Bu(τ )dτ + Du(t).
– 79 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Example
We take our heated bar, and we measure the (average) temperat ure in
the other half of the bar.
∂z
∂t(x, t) =
∂2z
∂x2(x, t) + 1[1
2,1](x)u(t)
∂z
∂x(0, t) =
∂z
∂x(0, t) = 0.
y(t) =
∫ 1
2
0
z(x, t)dx.
This we can written as
– 80 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
z(t) = Az(t) + Bu(t)
y(t) = Cz(t)
with A and B as before, and the operator C ∈ L(Z, C) given by
Cz =
∫ 1
2
0
z(x)dx.
�
– 81 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.4 Feedback
We have defined the open loop system, schematically given by
u y
z0
z(t) = Az(t) + Bu(t)y(t) = Cz(t)
However, for control we would like to close the loop.
– 82 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
u y
z0
z(t) = Az(t) + Bu(t)y(t) = Cz(t)
K
This gives the differential equation
z(t) = Az(t) + BKCz(t) = (A + BKC) z(t).
– 83 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Theorem
If A generates the C0-semigroup (T (t))t≥0 on the Hilbert space Z ,
and B ∈ L(U, Z), F ∈ L(Z, U), then A + BF also generates a
C0-semigroup.
Furthermore, if we denotes this (closed-loop) semigroup by
(S(t))t≥0, then
S(t)z0 = T (t)z0 +
∫ t
0
T (t − τ )BFS(τ )z0dτ.
�
– 84 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.5 Boundary control
Consider the following system
∂z
∂t(x, t) =
∂z
∂x(x, t), x ∈ [0, 1]
z(1, t) = u(t).
We cannot write this in the form z(t) = Az(t) + Bu(t), with
B ∈ L(Z, C).
– 85 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
∂z
∂t(x, t) =
∂z
∂x(x, t), x ∈ [0, 1]
z(1, t) = u(t).
We perform the following trick.
Define v(x, t) = z(x, t) − u(t).
Then v satisfies the following partial differential equation
∂v
∂t(x, t) =
∂v
∂x(x, t) − u(t), x ∈ [0, 1]
v(1, t) = 0.
This we can write as v(t) = Av(t) + Bu(t), for u = u.
– 86 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Definition
The system
z(t) = Az(t), z(0) = z0
Bz(t) = u(t)
is a boundary control system if
• A is a linear operator from D(A) ⊂ Z to Z , B is a linear
operator from D(B) ⊂ Z to U , and D(A) ⊂ D(B).
• The operator A defined as Az = Az with domain
D(A) = D(A) ∩ ker B generates a C0-semigroup on Z .
• The range of B equals U .
�
– 87 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
• Since B is surjective, we can find a B such that for all u ∈ U
– Bu ∈ D(B) and
– BBu = u.
• Define v(t) = z(t) − Bu(t). Then
– If z ∈ D(A), then v ∈ D(A), and
– since Bv = 0, we have that v ∈ D(A).
• v satisfies the following abstract differential equation
v(t) = Az(t) − Bu(t)
= Av(t) + ABu(t) − Bu(t)
= Av(t) + ABu(t) − Bu(t).
– 88 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
So if z(t) is the classical solution of
z(t) = Az(t), z(0) = z0
Bz(t) = u(t),
then v(t) = z(t) − Bu(t) is a classical solution of
v(t) = Av(t) + ABu(t) − Bu(t)
v(0) = z0 − Bu(0).
The reserve direction also holds.
– 89 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
For this last differential equation:
v(t) = Av(t) + ABu(t) − Bu(t)
v(0) = z0 − Bu(0)
we know the solution
v(t) = T (t)v(0) +
∫ t
0
T (t − τ ) [ABu(τ ) − Bu(τ )] dτ.
– 90 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Example
We apply this to the transport equation with boundary contro l
∂z
∂t(x, t) =
∂z
∂x(x, t), x ∈ [0, 1]
z(1, t) = u(t).
Here we have that
Az =dz
dx, D(A) = {z ∈ L2(0, 1) | dz
dx∈ L2(0, 1)}.
Bz = z(1), D(B) = D(A).
Let us check the assumptions.
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Az =dz
dx, D(A) = {z ∈ L2(0, 1) | dz
dx∈ L2(0, 1)}.
Bz = z(1), D(B) = D(A).
• Both operators are linear.
• A on the domain D(A) ∩ ker B generates a C0-semigroup.
• The range of B is C = U .
• Choose B = 1, then BBu = u.
– 92 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Since AB = 0, we find
z(t) − 1 · u(t) = v(t) = T (t)v0 −∫ t
0
T (t − τ )1u(τ )dτ.
Using the expression of the (shift) semigroup, we find that th e solution
of the boundary controlled p.d.e. is
z(x, t) − u(t) = v0(x + t)1[0,1](x + t) −∫ t
0
1[0,1](x + t − τ )u(τ )dτ.
This we can simplify.
– 93 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
For t > 1 this expression becomes
z(x, t) − u(t) = v0(x + t)1[0,1](x + t) −∫ t
0
1[0,1](x + t − τ )u(τ )dτ
= 0 − [u(τ )]tx+t−1
= −u(t) + u(x + t − 1).
– 94 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
For t ∈ [0, 1] we have to distinguish two cases:
• x ∈ [0, 1 − t]:
z(x, t) − u(t) = v0(x + t)1[0,1](x + t) −∫ t
0
1[0,1](x + t − τ )u(τ )dτ
= z0(x + t) − u(0) − [u(τ )]t0
= z0(x + t) − u(t).
• x ∈ [1 − t, 1]:
z(x, t) − u(t) = 0 − [u(τ )]tx+t−1
= −u(t) + u(x + t − 1).
– 95 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
This we can summarize as follows
z(x, t) =
z0(x + t) x + t < 1
u(x + t − 1) x + t > 1
The solution for t < 1 is depicted for all x ∈ [0, 1].
1100 xx
z0-part u-part
Hence for the control transport equation we have found a (wea k)
solution for all L2-input functions. �
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
3.6 Summary
We have seen the following
• A controlled p.d.e. like
∂z
∂t(x, t) =
∂2z
∂x2(x, t) + 1[1/2,1](x)u(t)
∂z
∂x(0, t) =
∂z
∂x(0, t) = 0
y(t) =
∫ 1
2
0
z(x, t)dx
can be written as
– 97 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
z(t) = Az(t) + Bu(t)
y(t) = Cz(t) + Du(t)
with B, C , and D bounded operators.
• We know what we mean by the solution of this (controlled)
abstract differential equation, and we have a formula for it .
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
• If the control appears at the boundary, like
∂z
∂t(x, t) =
∂z
∂x(x, t)
z(1, t) = u(t)
we can write this in the abstract form
z(t) = Az(t)
Bz(t) = u(t)
• We know that we have a solution (for smooth inputs) of such a
boundary control system.
– 99 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
With this knowledge let us revisit the UV-reactor model disc ussed in
the first part.
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Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
Schematic view point of UV-reactor:
C1
r
r
r1 r2
L
x
C0(t)
– 101 –
Elgersburg 2013: ∞ -dimensional systems Inputs and Outputs
∂C
∂t(x, r, t) = −vx(r, t)
∂C
∂x(x, r, t) +
1
Pe
[
1
r
∂
∂r
(
r∂C
∂r(x, r, t)
)
+∂2C
∂x2(x, r, t)
]
−Da K(r, t) C(x, r, t)
∂C
∂x(0, r, t) = 0
C(x, r1, t) = C(x, r2, t) = 0
C(0, r, t) = C0(t).
– 102 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
4 Transfer Functions
– 103 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
4.1 Introduction
The aim of this part is to define transfer function for systems
described by partial differential equations.
We derive these transfer functions via a very simple calcula tion.
Furthermore, the relation with the impulse response is give n.
– 104 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Consider the simple ordinary differential equation
y(t) + 5y(t) = 3u(t)
The transfer function of this system is given by
G(s) =3
s + 5
How do you come to this?
• Laplace transform, or
• Exponential solutions.
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Elgersburg 2013: ∞ -dimensional systems Transfer Functions
4.2 Exponential solutions
One way for obtaining the transfer function of
y(t) + 5y(t) = 3u(t)
is to take u(t) = est, s ∈ C, and to try to find a solution of the same
format, i.e., y(t) = αest.
Substituting this in the differential equation, gives
– 106 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
sαest + 5αest = 3est.
Since est is non-zero, we may divide by it, and we find
sα + 5α = 3.
If s 6= −5, this is solvable;
α =3
s + 5.
– 107 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Definition
Given an (abstract) differential equation in the variables
(u(t), z(t), y(t)), where u(t), z(t), and y(t) take their values in
the (Hilbert) spaces U , Z , and Y , respectively.
Let s ∈ C. If for every u0 ∈ U , there exists a unique solution of the
form (u0est, z0e
st, y0est), and the mapping u0 7→ y0 is linear and
bounded, then this mapping is called the transfer function at s, and
will be denoted by G(s). �
– 108 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Consider the abstract differential equation
z(t) = Az(t) + Bu(t)
y(t) = Cz(t) + Du(t)
with B, C , and D bounded operators.
Let s ∈ C, and u0 ∈ U . We try to find a solution of the form
(u(t), z(t), y(t)) = (u0est, z0e
st, y0est).
Substituting, this in the abstract differential equation g ives
sz0est = Az0e
st + Bu0est
y0est = Cz0e
st + Du0est.
– 109 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
(sI − A)z0 = Bu0
y0 = Cz0 + Du0.
If sI − A is (boundedly) invertible, then we find
y0 = C(sI − A)−1Bu0 + Du0.
This clearly defines a bounded linear mapping from u0 to y0, and so
the transfer function at s is given by
G(s) = C(sI − A)−1B + D.
This holds for all s ∈ ρ(A).
– 110 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Example
We take our heated bar. We heat it uniformly at one half, and we
measure (half) the average temperature in the other half.
∂z
∂t(x, t) =
∂2z
∂x2(x, t) + 1[1
2,1](x)u(t)
∂z
∂x(0, t) =
∂z
∂x(1, t) = 0
y(t) =
∫ 1
2
0
z(x, t)dx.
We obtain the transfer function, by two approaches.
– 111 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Method 1. :
The p.d.e. can be written as
z(t) = Az(t) + Bu(t)
y(t) = Cz(t)
with Cz =∫ 1
2
0z(x)dx, Bu = 1[1
2,1](x)u, and Az = d2z
dx2with
domain
D(A) =
{
z ∈ L2(0, 1) | dz
dx,d2z
dx2∈ L2(0, 1),
dz
dx(0) =
dz
dx(1) = 0
}
.
– 112 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Now we know that
G(s) = C(sI − A)−1B.
Using the fact that A has an orthonormal basis of eigenfunctions, i.e.,
Aφn = λnφn, we find that
(sI − A)−1z =
∞∑
n=0
1
s − λn
〈z, φn〉φn.
Hence
(sI − A)−1Bu =∞∑
n=0
1
s − λn
〈Bu, φn〉φn
and
C(sI − A)−1Bu =∞∑
n=0
1
s − λn
〈Bu, φn〉Cφn.
– 113 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Thus
G(s) =∞∑
n=0
1
s − λn
〈B, φn〉Cφn.
We know that λn = −n2π2, n=0,1,2,. . . , and
φn(x) =
1 n = 0√
2 cos(nπx) n ≥ 1.
Using this we find that
– 114 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
G(s) =∞∑
n=0
1
s − λn
〈B, φn〉Cφn
=1
s
∫ 1
1/2
1 · 1dx
∫ 1/2
0
1 · 1dx +
∞∑
n=1
1
s + n2π2
∫ 1
1/2
1 ·√
2 cos(nπx)dx ·∫ 1/2
0
1 ·√
2 cos(nπx)dx.
Evaluating the integrals gives
G(s) =1
4s− 2
∞∑
n=1
sin(nπ 12)2
n2π2(s + n2π2).
– 115 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Method 2 :
We try to find an exponential solution of the p.d.e. This gives the
following equation
sz0(x)est =d2z0
dx2(x)est + 1[1
2,1](x)u0e
st
dz0
dx(0)est =
dz0
dx(1)est = 0
y0est =
∫ 1
2
0
z0(x)estdx.
– 116 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Hence
sz0(x) =d2z0
dx2(x) + 1[1
2,1](x)u0
dz0
dx(0) =
dz0
dx(1) = 0
y0 =
∫ 1
2
0
z0(x)dx.
The first two lines represent an o.d.e. with boundary conditi ons.
– 117 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
The solution of
sz0(x) =d2z0
dx2(x) + 1[1
2,1](x)u0
dz0
dx(0) =
dz0
dx(1) = 0
is given as
z0(x) = cosh(√
sx)z0(0) −1
√s
∫ x
0
sinh(√
s(x − ξ))1[1/2,1](ξ)u0dξ
with
z0(0) =sinh(
√s/2)u0
s sinh(√
s)=
u0
2s cosh(√
s/2).
– 118 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Using this we find that
y0 =
∫ 1
2
0
z0(x)dx
=sinh(
√s/2)u0
2s√
s cosh(√
s/2).
Hence the transfer function is given by
G(s) =tanh(
√s/2)
2s√
s.
– 119 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
The transfer function is unique, and so we find that
tanh(√
s/2)
2s√
s= G(s) =
1
4s− 2
∞∑
n=1
sin(nπ 12)2
n2π2(s + n2π2).
�
– 120 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Example
Consider the system with boundary control and observation
∂z
∂t(x, t) =
∂z
∂x(x, t)
z(1, t) = u(t)
y(t) = z(0, t).
Substituting exponential functions for all signals, gives
sz0(x)est =dz0
dx(x)est
z0(1)est = u0e
st
y0est = z0(0)e
st.
Thus
– 121 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
sz0(x) =dz0
dx(x)
z0(1) = u0
y0 = z0(0).
This is an ordinary differential equation with given (end) c ondition, u0
and unknown (initial) condition, y0.
The solution equals z0(x) = es(x−1)u0. Thus y0 = e−su0.
The transfer function equals
G(s) = e−s s ∈ C.
�
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Elgersburg 2013: ∞ -dimensional systems Transfer Functions
4.3 Impulse response
Definition
Let the (abstract) differential equation have the transfer function
G(s). If this transfer function exists on some right-half plane a nd
possesses an inverse (one-sided) Laplace transform, then w e call the
inverse Laplace transform, the impulse response. �
– 123 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Theorem
Consider the abstract differential equation
z(t) = Az(t) + Bu(t)
y(t) = Cz(t) + Du(t)
with B, C , and D bounded operators.
The impulse response of this differential equation equals
h(t) = CT (t)B + Dδ(t), t ≥ 0−.
�
– 124 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Question
If we take the Laplace transform of the impulse response do we get
the transfer function? �
– 125 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Answer
No.
Example
Let Z = ℓ2(Z), and consider the following differential equation
dz
dt(n, t) = z(n + 1, t), n ∈ Z \ {0}
dz
dt(0, t) = z(1, t) + u(t)
y(t) = z(1, t).
– 126 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Substituting the exponential functions, leads to
sz0(n) = z0(n + 1), n ∈ Z \ {0}sz0(0) = z0(1) + u0
y0 = z0(1).
From the first equation, we obtain
z0(n) =
sn−1z0(1) n ≥ 1
snz0(0) n ≤ 0.
– 127 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Since we must have that z0 ∈ ℓ2(Z), we find that
z0(n) =
sn−1z0(1) = 0 n ≥ 1 and |s| ≥ 1
snz0(0) = 0 n ≤ 0 and |s| ≤ 1.
Using the equation
sz0(0) = z0(1) + u0,
we find for |s| > 1
z0(n) =
0 n ≥ 1
sn−1u0 n ≤ 0
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Elgersburg 2013: ∞ -dimensional systems Transfer Functions
For |s| < 1
z0(n) =
−sn−1u0 n ≥ 1
0 n ≤ 0
So we have that
y0 = z0(1) =
0 |s| > 1
−u0 |s| < 1.
The transfer function is given by
– 129 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
1
iC
Yellow means G(s) = −1. White means G(s) = 0.
– 130 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
The inverse Laplace transform of G(s) gives h(t) = 0.
If we take of this the Laplace transform, then it is obviously zero
everywhere, and thus not equals to G(s) for all s. �
Remark
The system can be written in the standard format
z(t) = Az(t) + Bu(t), y(t) = Cz(t) + Du(t) with all
operators bounded. �
– 131 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Theorem
Consider the system
z(t) = Az(t) + Bu(t)
y(t) = Cz(t) + Du(t),
where B, C , and D are bounded operators, and A generates the
C0-semigroup (T (t))t≥0. If this semigroup satisfies
‖T (t)‖ ≤ Meωt, for all t > 0,
then the transfer function and the Laplace transform of the i mpulse
response exist in {s ∈ C | Re(s) > ω}. Furthermore, they are
equal on this set. �
– 132 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Example (revisited)
In our previous example we have that the semigroup is bounded by et,
and so the theorem tells us that on the right-half plane
{s ∈ C | Re(s) > 1} the transfer function and the Laplace
transform of the impulse response are the same.
Since h(t) = 0 and since G(s) = 0 on this right-half plane, we are
in agreement with the theorem.
We also see that (in general) a larger half-plane is not possi ble. �
– 133 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Theorem
If the transfer function G(s) is a meromorphic function on C, then
the meromorphic continuation of the Laplace transform of th e impulse
response is equal to G(s) on C. �
– 134 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
4.4 Summary
In this part we have seen the following
• Transfer functions are easy to find by using exponential solu tions.
• You don’t have to write a p.d.e. as an abstract differential e quation
in order to obtain the transfer function.
• It is not hard to find the transfer function of a system with
boundary control and observation .
• The Laplace transform of the impulse response only equals th e
transfer function on some right-half plane.
• For scalar transfer function the design rules of Bode and Nyq uist
still apply.
– 135 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
Here we show the Nyquist plot of e−st.
1−1
iC
Re →
Im ↑
– 136 –
Elgersburg 2013: ∞ -dimensional systems Transfer Functions
The (magnitude) Bode plot of
G(s) = −2∞∑
n=1
sin(nπ 12)2
n2π2(s + n2π2)
−1
−1−4 0 1 2 3log(ω) →
log
|G(i
ω)|
– 137 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5 Stability and Stabilizability
– 138 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.1 Introduction
In this part we want to define stability and show how these noti ons
relate to similar notions for ordinary differential equati ons.
Furthermore, we show how we can stabilize systems.
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Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.2 Stability notions
There are several stability notions for the abstract differ ential equation
z(t) = Az(t), z(0) = z0.
We assume that A is the infinitesimal generator of a strongly
continuous semigroup (T (t))t≥0 on the Hilbert space Z .
– 140 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Definition
The semigroup (T (t))t≥0 is exponentially stable if there exists
M, ω > 0 such that
‖T (t)‖ ≤ Me−ωt, t ≥ 0.
�
Definition
The semigroup (T (t))t≥0 is strongly stable if for all z0 ∈ Z
limt→∞
T (t)z0 = 0.
�
– 141 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.3 Stability of platoon models
Consider the the platoon model
z(t) = Az(t), z(0) = z0 ∈ Z
where Z = L2(∂D; C), and(
Az(t))
(φ) = A(φ)z(φ, t),
φ ∈ ∂D.
Theorem
Assume that A is a continuous function of φ. The above system is
exponentially stable if and only if there exists a δ > 0 such that for all
φ ∈ ∂D the eigenvalues of A(φ) have real part less or equal to −δ.
– 142 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Example
Consider our platoon model
d
dt
dr(t)
vr(t)
ar(t)
=
vr−1(t) − vr(t)
ar(t)
−τ−1ar(t)
, r ∈ Z.
For this model A is given by
A(φ) =
0 φ−1 − 1 0
0 0 1
0 0 −τ−1
, φ ∈ ∂D.
It is clear that this is not exponentially stable (also not st rongly
stable). �
– 143 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.4 Exponential stability and Lyapunov equations
The following theorem relates exponential stability with t he solution of
a Lyapunov equation.
Theorem
The following are equivalent.
• (T (t))t≥0 is exponentially stable.
• For all z0 ∈ Z we have that T (t)z0 ∈ L2((0, ∞); Z).
• There exists a positive L ∈ L(Z) satisfying the following
Lyapunov equation
〈Az, Lz〉 + 〈Lz, Az〉 = −〈z, z〉 z ∈ D(A).
�
– 144 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Some remarks concerning the Lyapunov equation.
Lemma
For a positive operator L ∈ L(Z) the following are equivalent:
• L satisfies
〈Az, Lz〉 + 〈Lz, Az〉 = −〈z, z〉 ∀z ∈ D(A).
• L satisfies
〈Az1, Lz2〉+〈Lz1, Az2〉 = −〈z1, z2〉 ∀z1, z2 ∈ D(A).
• L maps the domain of A into the domain of A∗, and L satisfies
A∗L + LA = −I on D(A).
– 145 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Proof of the equivalence between the last two items.
Assume that z0 ∈ L2((0, ∞); Z) for all z0 ∈ Z . For z1, z2 ∈ Z ,
define
〈z1, Lz2〉 =
∫ ∞
0
〈T (t)z1, T (t)z2〉dt.
This operator is well-defined, self-adjoint, and positive. Furthermore,
for z1, z2 ∈ D(A)
〈Az1, Lz2〉 + 〈z1, LAz2〉
=
∫ ∞
0
〈T (t)Az1, T (t)z2〉dt +
∫ ∞
0
〈T (t)z1, T (t)Az2〉dt.
– 146 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Hence
〈Az1, Lz2〉 + 〈z1, LAz2〉
=
∫ ∞
0
〈T (t)Az1, T (t)z2〉dt +
∫ ∞
0
〈T (t)z1, T (t)Az2〉dt
=
∫ ∞
0
d
dt[〈T (t)z1, T (t)z2〉] dt
= 0 − 〈z1, z2〉.
– 147 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
If the Lyapunov equation has a positive solution, then
∫ t1
0
‖T (t)z0‖2dt
=
∫ t1
0
〈T (t)z0, T (t)z0〉dt =
∫ t1
0
〈T (t)z0, IT (t)z0〉dt
= −∫ t1
0
〈AT (t)z0, LT (t)z0〉dt
−∫ t1
0
〈T (t)z0, LAT (t)z0〉dt
= −∫ t1
0
d
dt[〈T (t)z0, LT (t)z0〉] dt
= −〈T (t1)z0, LT (t1)z0〉 + 〈z0, Lz0〉.
– 148 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Hence for all t1 > 0∫ t1
0
‖T (t)z0‖2dt ≤ ‖L‖‖z0‖2
and so T (t)z0 ∈ L2((0, ∞); Z).
Thus we have proved the equivalence between the last two item s.
– 149 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Proof of the equivalence between the first two items.
If T (t)z0 ∈ L2((0, ∞), Z) for all z0, then
• ‖T (t)‖ ≤ M for all t > 0
• The following estimate holds
t‖T (t)z0‖2 =
∫ t
0
‖T (t)z0‖2dt
=
∫ t
0
‖T (t − τ )T (τ )z0‖2dτ
≤ M2
∫ t
0
‖T (τ )z0‖2dτ ≤ K‖z0‖2.
– 150 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Thus
‖T (t)z0‖2 ≤K
t‖z0‖2.
Now using the semigroup property, we find that (T (t))t≥0 is
exponentially stable. �
– 151 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
There are examples of unstable semigroups for which the infin itesimal
generator A has no spectrum in the set {s ∈ C | Re(s) ≥ 0}.
Hence, in general, we cannot conclude stability by only look ing at the
spectrum of A. However,
Theorem
The semigroup (T (t))t≥0 is exponentially stable if and only if
sup{s∈C|Re(s)>0}
‖(sI − A)−1‖ < ∞.
�
– 152 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.5 Strong stability
We may also conclude exponential stability when there exist s a
positive L such that
A∗L + LA ≤ −εI, ε > 0.
Question
Is the semigroup strongly stable when there exists an L ∈ L(Z),
L > 0 such that
A∗L + LA < 0 on D(A)?
Answer
No.
– 153 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Example
Take the Hilbert space
Z =
{
f : [0, ∞) 7→ C |∫ ∞
0
|f(x)|2[
e−x + 1]
dx < ∞}
.
It inner product is given by:
〈f, g〉 =
∫ ∞
0
f(x)g(x)[
e−x + 1]
dx.
Note that Z “is” L2(0, ∞).
– 154 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
As semigroup, we choose the shift:
(T (t)f) (x) =
f(x − t) x > t
0 x ∈ [0, t)
= f(x − t)1[0,∞](x − t).
– 155 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
There holds
‖T (t)f‖2 =
∫ ∞
0
|f(x − t)1[0,∞](x − t)|2[
e−x + 1]
dx
=
∫ ∞
0
|f(ξ)|2[
e−(ξ+t) + 1]
dξ
≥∫ ∞
0
|f(ξ)|2dξ
≥ 1
2
∫ ∞
0
|f(ξ)|2[
e−ξ + 1]
dξ
=1
2‖f‖2.
Hence (T (t))t≥0 is not strongly stable.
– 156 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
The infinitesimal generator is given by
Af = −df
dx
with domain
D(A) = {f ∈ Z | f is absolutely continuous
df
dx∈ Z, and f(0) = 0
}
.
Next we evaluate
〈Az, z〉 + 〈z, Az〉for z ∈ D(A) and z 6= 0.
– 157 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
For A = − ddx
with boundary condition z(0) = 0, we find
〈Az, z〉 + 〈z, Az〉
=
∫ ∞
0
(−1)dz
dx(x)z(x)
[
e−x + 1]
dx +
∫ ∞
0
z(x)(−1)dz
dx(x)
[
e−x + 1]
dx
= −∫ ∞
0
d
dx
(
|z(x)|2) [
e−x + 1]
dx.
– 158 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Hence
〈Az, z〉 + 〈z, Az〉= −
[
|z(x)|2[
e−x + 1]]∞
0+
∫ ∞
0
|z(x)|2[
−e−x]
dx
= 0 −∫ ∞
0
|z(x)|2e−xdx
< 0.
– 159 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Concluding, we have constructed an infinitesimal generator A for
which the semigroup is not strongly stable, but the Lyapunov
inequality
〈Az, z〉 + 〈z, Az〉 < 0, z ∈ D(A), z 6= 0
holds. Note that L = I . �
Remark
If the trajectories, t 7→ T (t)z0, are pre-compact, then this Lyapunov
inequality implies strong stability. �
– 160 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Theorem
Let (T (t))t≥0 is a strongly continuous semigroup on the Hilbert
space Z . If
• The semigroup is uniformly bounded, i.e., ‖T (t)‖ ≤ M
• A has no eigenvalues on the imaginary axis
• The spectrum on the imaginary axis is countable,
then (T (t))t≥0 is strongly stable.
– 161 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.6 Stabilizability
In this section we study the question whether there exist a fe edback,
u = Fz which stabilizes the system
z(t) = Az(t) + Bu(t).
That is, the operator A + BF generates an exponentially stable
semigroup.
– 162 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Theorem
Consider z(t) = Az(t) + Bu(t), with B ∈ L(Cm, Z). The
following are equivalent.
• There exists an F ∈ L(Z, Cm) such that A + BF generates
an exponentially stable semigroup on Z .
• Z can be decomposed as Z = Zu ⊕ Zs with
– dim(Zu) < ∞.
– For all t > 0 we have that T (t)Zu ⊂ Zu and T (t)Zs ⊂ Zs.
– T (t)|Zs is exponentially stable.
– The system restricted to Zu is controllable.
– 163 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
1
iC
Red denotes the spectrum.
– 164 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
1
iC
Zu
Zs
– 165 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Example
We consider the heated bar. We heat it uniformly at one half, a nd we
measure (half) the average temperature in the other half.
∂z
∂t(x, t) =
∂2z
∂x2(x, t) + 1[1
2,1](x)u(t)
∂z
∂x(0, t) =
∂z
∂x(1, t) = 0
y(t) =
∫ 1
2
0
z(x, t)dx.
We know that the corresponding A has the eigenvalues
– 166 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
λn = −n2π2, n = 0, 1, 2, . . ., and the eigenfunctions
φn(x) =
1 n = 0√
2 cos(nπx) n = 1, 2, . . .
This A has an unstable eigenvalue, λ0 = 0. Now we define
Zu = span{φ0}
Zs = spann=1,2,...{φn} = Z⊥u .
Let us check the conditions.
– 167 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
• dim(Zu) = 1 < ∞.
• T (t)Zu ⊂ Zu and T (t)Zs ⊂ Zs.
•
T (t)|Zs =
∞∑
n=1
e−n2π2t〈·, φn〉φn.
This is exponentially stable.
• A restricted to Zu is Au = [0], and the projection of B onto Zu
is Bu = 〈B, φ0〉 = [1/2].
The pair (Au, Bu) is controllable.
Hence the system is stabilizable.
– 168 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
You can stabilize it by stabilizing the finite-dimensional p art, and not
effecting the stable part.
A feedback that works is
Fz = −k〈z, φ0〉φ0,
with k > 0. �
– 169 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.7 Stabilization of the platoon model
After Fourier transform our platoon model becomes
∂z
∂t(φ, t) = A(φ)z(φ, t) + B(φ)u(φ, t),
with φ ∈ ∂D, and
A(φ) =
0 φ−1 − 1 0
0 0 1
0 0 −τ−1
, B(φ) =
0
0
τ−1
.
Note that U = L2(∂D). So infinite-dimensional .
– 170 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Suppose that we apply a state feedback of the type
u(φ, t) = F (φ)z(φ, t) with
F (φ) =(
f1(φ) f2(φ) f3(φ))
Then A(φ) + B(φ)F (φ) equals
0 φ−1 − 1 0
0 0 1
τ−1f1(φ) τ−1f2(φ) τ−1f3(φ) − τ−1
.
Exponentially stable?
– 171 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.8 Transfer function and stability
One can check stability by looking at the transfer function. We can do
that for the state-space system.
z(t) = Az(t) + Bu(t)
y(t) = Cz(t) + Du(t).
– 172 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
Theorem
Consider the (standard) state space system with B, C , and D
bounded operators. Let the input and output space be
finite-dimensional.
Assume further that there exists a F ∈ L(Z, U) and a
K ∈ L(Y, Z) such that A + BF and A + KC generate
exponentially stable semigroups.
The semigroup generated by A is exponentially stable if and only if
the transfer function G(s) = C(sI − A)−1B + D is analytic and
bounded in {s ∈ C | Re(s) > 0}. �
– 173 –
Elgersburg 2013: ∞ -dimensional systems Stability and Stabilizability
5.9 Summary
• We have introduce different notions of stability for a
C0-semigroup.
• For exponential stability there are nice equivalent condit ions.
• For strong stability there are nice sufficient conditions, b ut they
are not sufficient.
• If the input is finite-dimensional, then there is a complete
characterizing of stabilizability.
– 174 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
6 Linear Port Hamiltonian Systems
Warning: x will denote the state from the state space X , and ζ
denotes the spatial variable.
– 175 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
6.1 A typical example
V (a)
I(a)
V (b)
I(b)a b
Consider the transmission line on the spatial interval [a, b]
∂Q
∂t(ζ, t) = − ∂
∂ζ
φ(ζ, t)
L(ζ)
∂φ
∂t(ζ, t) = −
∂
∂ζ
Q(ζ, t)
C(ζ).
We write x1 = Q (charge) and x2 = φ (flux), and we find that
– 176 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
∂
∂t
x1
x2
(ζ, t) =
0 −1
−1 0
∂
∂ζ
1C(ζ)
x1(ζ, t)
1L(ζ)
x2(ζ, t)
=
0 −1
−1 0
∂
∂ζ
1C(ζ)
0
0 1L(ζ)
x1(ζ, t)
x2(ζ, t)
= P1
∂
∂ζ(Hx) (ζ, t)
So we have, J = P1∂∂ζ
(skew-symmetric) and the Hamiltonian
(energy) equals H = 12
∫ b
axTHxdζ (positive).
To use this structure we differentiate the Hamiltonian (ene rgy) along
trajectories.
– 177 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
dH
dt(t) =
1
2
∫ b
a
∂x
∂t(ζ, t)TH(ζ)x(ζ, t)dz+
1
2
∫ b
a
x(ζ, t)TH(ζ)∂x
∂t(ζ, t)dz
=1
2
∫ b
a
(
P1
∂
∂ζ(Hx) (ζ, t)
)T
H(ζ)x(ζ, t)dζ+
1
2
∫ b
a
x(ζ, t)TH(ζ, t)(P1
∂
∂ζ(Hx) (ζ, t)dζ
=1
2
∫ b
a
∂
∂ζ
[
(Hx)T (ζ, t)P1 (Hx) (ζ, t)]
dζ
=1
2
[
(Hx)T (ζ, t)P1 (Hx) (ζ, t)]b
a,
where we have used the symmetry of P1 and H.
– 178 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
So we have that the time-change of Hamiltonian satisfies
dH
dt(t) =
1
2
[
(Hx)T (ζ, t)P1 (Hx) (ζ, t)]b
a. (1)
That is the change of internal energy goes via the boundary (p orts).
Note the we only used that P1 and H are symmetric. We did not need
the specific form of P1 or H.
The balance equation (1) also holds for the system
∂x
∂t(ζ, t) = P1
∂Hx
∂ζ(ζ, t) + P0 [Hx] (ζ, t) (2)
with P0 anti-symmetric, i.e., P T0 = −P0.
Many (hyperbolic) systems can be written in this format.
– 179 –
Elgersburg 2013: ∞ -dimensional systems Linear Port Hamiltonian Systems
Example
Consider transmission lines in a network
VV
IITrans. line I
Trans. line II
Trans. line III
Trans. line IVKK
In the coupling parts K , we have that Kirchhoff laws holds. Hence
charge flowing out of the transmission line I, enters II and II I, etc.
The P1 of the big system is the diagonal matrix, build from the
uncoupled P1’s (which are all the same). The H of the coupled
system is the diagonal matrix of the uncoupled H’s.
The coupling is written down as boundary conditions of the pde.
– 180 –
Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
7 Homogeneous solutions of P.H.S.
– 181 –
Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Consider the p.d.e. of our Port-Hamiltonian system
∂x
∂t(ζ, t) = P1
∂
∂ζ[H(ζ)x(ζ, t)] + P0 [H(ζ)x(ζ, t)] .
with energy (Hamiltonian)
E(t) =1
2
∫ b
a
x(ζ, t)TH(ζ)x(ζ, t)dζ.
We assume that
• P1 is real and symmetric, i.e., P T1 = P1,
• P0 is real and anti-symmetric, i.e., P T0 = −P0,
• H(ζ) is a positive (real) symmetric matrix, uniformly satisfyin g
0 < mI ≤ H(ζ) ≤ MI , and continuously differentiable.
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
The energy
E(t) =1
2
∫ b
a
x(ζ, t)TH(ζ)x(ζ, t)dζ.
can be seen as a weighted L2-norm.
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Associated to this p.d.e. we define the state space
X = L2((a, b); Cn)
with inner product
〈f, g〉 =1
2
∫ b
a
f(ζ)TH(ζ)g(ζ)dζ.
and the differential operator
Ax := P1
d
dζ[Hx] + P0 [Hx]
with domain
D(A) = {x ∈ X | x is absolutely continuous, anddx
dζ∈ X}.
Thus the squared norm of x equals the energy of this state.
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
The following balance equation holds
Lemma
On the space X , the operator Ax = P1ddζ
[Hx] + P0 [Hx]
satisfies
〈Ax, x〉 + 〈x, Ax〉
=1
2
[
(Hx)T (b)P1(Hx) (b) − (Hx)T (a)P1(Hx) (a)]
for x ∈ D(A). �
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Proof (for P0 = 0) Using the definition of the inner product, we find
〈Ax, x〉 + 〈x, Ax〉
=1
2
∫ b
a
[
P1
∂
∂ζ(Hx) (ζ)
]T
H(ζ)x(ζ)dζ +
1
2
∫ b
a
x(ζ)TH(ζ)
[
P1
∂
∂ζ(Hx) (ζ)
]
dζ.
Using the fact that P1, H(ζ) are real symmetric, we write the last
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
expression as
1
2
∫ b
a
[
d
dζ(Hx) (ζ)
]T
P1H(ζ)x(ζ)+
[H(ζ)x(ζ)]T[
P1
d
dζ(Hx) (ζ)
]
dζ
=1
2
∫ b
a
d
dζ
[
(Hx)T (ζ)P1(Hx) (ζ)]
dζ
=1
2
[
(Hx)T (b)P1(Hx) (b) − (Hx)T (a)P1(Hx) (a)]
.
�
– 187 –
Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Theorem
Consider the operator A which is defined as
Ax = Ax = P1
∂
∂ζ(Hx) + P0 (Hx)
on the domain D(A) = D(A) ∩ ker B, where
Bx = M
(Hx)(b)
(Hx)(a)
.
If M is n × 2n-matrix of full rank, then A generates a contraction
semigroup on X if and only if
〈Ax, x〉 + 〈x, Ax〉 ≤ 0 for all x ∈ D(A).
�
– 188 –
Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Hence by posing n boundary conditions, we only have to check the
inequality.
Since we have the power balance this can be done quickly.
If H is non-constant, it can be very hard (impossible) to obtain t he
expression for the C0-semigroup.
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
Example
Consider the transmission line.
Ax =
0 −1
−1 0
d
dζ
1C(ζ)
0
0 1L(ζ)
x1(ζ)
x2(ζ)
and
〈Ax, x〉 + 〈x, A〉
=1
2
[
(Hx)T (b)P1(Hx) (b) − (Hx)T (a)P1(Hx) (a)]
= V (a)I(a) − V (b)I(b),
since x1/C = Q/C = V and x2/L = φ/L = I .
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Elgersburg 2013: ∞ -dimensional systems Homogeneous solutions of P.H.S.
If we choose V (a) = 0 and V (b) = RI(b), R ≥ 0, then
M =
0 0 1 0
1 −R 0 0
.
This matrix has rank n = 2.
Furthermore, using these boundary conditions, we have that
V (a)I(a) − V (b)I(b) ≤ 0.
Hence the operator A generates a contraction semigroup on Z . �
– 191 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
8 Well-posedness for PHS
For t ≥ 0 and ζ ∈ [a, b] we consider the system:
∂x
∂t(ζ, t) = P1
∂
∂z(Hx) (ζ, t), x(0, z) = x0(ζ) (3)
0 = M11 (Hx) (b, t) + M12 (Hx) (a, t) (4)
u(t) = M21 (Hx) (b, t) + M22 (Hx) (a, t) (5)
y(t) = C1 (Hx) (b, t) + C2 (Hx) (a, t). (6)
We assume that x takes values in Cn, P T
1 = P1, det(P1) 6= 0,
H > 0, and that rank[
M11 M12
M21 M22
C1 C2
]
= n + rank [ C1 C2 ].
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
Definition
Consider the system (3)–(6). This system is well-posed if there exists
a tf > 0 and mf such that the following holds:
1. The homogeneous p.d.e., i.e., u ≡ 0 has for any initial condition,
x0, a unique (weak) solution.
2. The following inequality holds for all smooth initial con ditions, and
all smooth inputs
‖x(tf)‖2+
∫ tf
0
‖y(t)‖2dt ≤ mf
[
‖x0‖2 +
∫ tf
0
‖u(t)‖2dt
]
,
where ‖ · ‖2 is the energy/Hamiltonian: 12
∫ b
axTHxdζ.
�
– 193 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
Remark
• The semigroup need not to be a contraction.
• So well-posedness gives you that you have a unique solution f or
every square integrable input function and every initial co ndition.
Furthermore, the output signal is always square integrable .
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
8.1 Well-posedness for simple PHS
Consider the (very) simple port-Hamiltonian system
∂x
∂t(ζ, t) = v
∂x
∂ζ(ζ, t)
vx(b, t) = u(t)
y(t) = vx(a, t),
with v a positive constant.
The norm (energy/Hamiltonian) is given by 12
∫ b
av‖f(ζ)‖2dζ.
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
We know that for smooth inputs and smooth initial conditions , there
exists a solution. For this (classical) solution the balanc e equation
reads as
d
dt‖x(t)‖2 =
1
2
[
|vx(ζ, t)|2]b
a
=1
2
[
|u(t)|2 − |y(t)|2]
Hence, we have that for all tf > 0
‖x(tf)‖2 +
∫ tf
0
‖y(t)‖2dt =
[
‖x0‖2 +
∫ tf
0
‖u(t)‖2dt
]
.
Since the smooth inputs and smooth initial conditions are de nse in X
and L2(0, tf), respectively, we find that this system is well-posed.
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
A similar result holds for the other (very) simple port-Hami ltonian
system
∂x
∂t(ζ, t) = −v
∂x
∂ζ(ζ, t)
vx(a, t) = u(t)
y(t) = vx(b, t),
with v a positive constant.
The energy is the same as for the previous p.d.e.
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
8.2 Characterization of well-posedness for PHS
Theorem
Under the conditions we have imposed, the following holds:
• If condition 1 holds, then automatically condition 2 holds.
• That is: If the homogeneous p.d.e., i.e., u ≡ 0, has a weak
solution, then the system (3)–(6) is well-posed.
• There is a matrix condition for checking condition 1.
• Same theorem holds for∂x∂t
(ζ, t) = P1∂∂ζ
(Hx) (ζ, t) + P0 (Hx) (ζ, t).
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
We want to give an idea of the proof. Therefor we do it for our
transmission line system with the following choice of bound ary input
and output
u(t) =
V (a, t)
V (b, t)
, y(t) =
I(a, t)
I(b, t)
. (7)
Now we preform the following steps:
1. Since V = Q/C = x1/C and I = φ/L = x2/L, we have
that
[M21, M22]=
0 0 1C
0
1C
0 0 0
, [C1, C2]=
0 0 0 1L
0 1L
0 0
.
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
2. Write the system in the ”characteristic”. That is diagona lize
P1H =(
0 −1
C−1
L0
)
. Hence
P1H=
1 1
−√
LC
√
LC
√
1LC
0
0 −√
1LC
1 −√
CL
1√
CL
1
2.
In the new variables ( characteristics )
ξ1
ξ2
:=
1 −√
CL
1√
CL
x1
x2
the transmission line becomes
∂ξ1
∂t=
√
1
LC
∂ξ1
∂ζ,
∂ξ2
∂t= −
√
1
LC
∂ξ2
∂ζ.
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Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
The “perfect” input and output for this system is
us(t) =
ξ1(b, t)
ξ2(a, t)
, ys(t) =
ξ1(a, t)
ξ2(b, t)
. (8)
3. Write our input-output pair in this input-output pair.
u(t) =
0 12C
12C
0
us(t) +
12C
0
0 12C
ys(t)
= Kus(t) + Qys(t), (9)
– 201 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
y(t) =
0 1
2√
LC
−1
2√
LC0
us(t) +
−1
2√
LC0
0 1
2√
LC
ys(t)
= O1us(t) + O2ys(t). (10)
– 202 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
We regard the system with input/output u, y as a feedback of the
system with input/output us, ys, i.e.,
−u(t)
us(t)
y(t)
ys(t)Gs(s)
Q
K−1 O2
O1
Now Gs has feed-through zero, and so any feedback is allowed.
Only condition: K−1 exists.
– 203 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
References
1. R.F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional
Linear Systems Theory , Springer-Verlag, New York, 1995.
2. Y. Le Gorrec, H. Zwart, and B. Maschke, “Dirac structures a nd
boundary control systems associated with skew-symmetric
differential operators,” SIAM J. Control and Optim. , vol. 44, no. 2,
pp. 1864–1892, 2005.
3. J.A. Villegas, A Port-Hamiltonian Approach to Distributed
Parameter Systems , Ph.D. thesis, Department of Applied
Mathematics, University of Twente, Enschede, The Netherla nds,
May 2007, available at
http://www.eng.ox.ac.uk/control/peopleAJV.shtml.
– 204 –
Elgersburg 2013: ∞ -dimensional systems Well-posedness for PHS
4. B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on
Infinite-Dimensional Spaces , Operator Theory: Advances and
Applications, vol. 223, Birkh auser, 2012.
– 205 –