Mikael Kurula - Åbo Akademi

2.1 Abstract input/state/output systems 7
We ask that the solutions U and I of (2.3) have a distribution partial derivative in
X :=L 2 (R + ;R) with respect to z and that they have continuous partial derivatives with
respect to t. In this case the equations (2.3) hold for all t>0 if and only if they hold
for all t ∈R +
U0
:=[0,∞). The initial state should lie in H1 (R + ;R2 ); see Definition
I0
B.3. We finally note that we have the implicit boundary condition U(∞)=I(∞)=0
at infinity, because U,I ∈H 1 (R + ;R).
One often sees the telegrapher’s equations (2.2) written as
∂
∂t
∂
U(z,t) 0 −
= ∂z
I(z,t) − ∂ 0 ∂z
U(z,t)
.
I(z,t)
The difference between the former and the latter convention is that the direction of
the current is reversed. In the latter case positive current flows into the transmission
line. It is also more common to consider a transfer line of finite length, but we use the
infinite transfer line because we need it later, in Chapter 5.
2.1 Abstract input/state/output systems
In this section we give some definitions from the abstract input/state/output system
theory. Although it is not completely obvious, Example 2.1 is a special case of this
theory, as we will see in Section 2.3. Comprehensive expositions of the theory of infinitedimensional
linear input/state/output systems can be found in [CZ95] and [Sta05].
Definition 2.2. Let X be a Hilbert space. A family t →A t , t ≥0, of bounded linear
operators on X is a semigroup on X if A 0 =1 and A s+t =A s A t for all s,t ≥0.
The semigroup is strongly continuous, or shorter C0, if lim t→0 +A t x0 =x0 for all
x0 ∈ X.
The semigroup is a contraction semigroup if A t L(X) ≤1 for all t ≥0, where ·L(X)
denotes the operator norm.
The generator A:X ⊃Dom(A) → X of A is the (in general unbounded) linear op-
erator defined by
1
Ax0 := lim
t→0 + t (Atx0 −x0), (2.4)
with Dom(A) consisting of those x0 ∈ X for which the limit (2.4) exists in X. The
domain of A is usually equipped with the inner product
(x 1 ,x 2 )Dom(A) =(x 1 ,x 2 )X +(Ax 1 ,Ax 2 )X. (2.5)
The generator A of a C0 semigroup on X is closed and Dom(A) is dense in X;
see [Paz83, Thm 1.2.7]. In particular, Dom(A) is then a Hilbert space with the inner
product (2.5). It follows immediately from (2.5) that A is a bounded operator from
Dom(A) to X. Moreover, Dom(A) is invariant under A: A t x0 ∈Dom(A) for all x0 ∈
Dom(A) and t ≥0; see [Sta05, Thm 3.2.1(iii)].