Mikael Kurula - Åbo Akademi

2.1 Abstract input/state/output systems 11
We give examples of system nodes in Examples 2.6 and 2.14.
A&B
Definition 2.5. The system node on (U,X,Y) is (L
C&D
2-)well-posed if there
exists a T >0 and a
constant KT, such that all classical input/state/output trajectories
A&B
(u,x,y) of satisfy
C&D
∀t ∈[0,T]: x(t) 2 X +
t
0
y(s) 2 Yds ≤KT
x(0) 2 X +
t
0
u(s) 2
U ds . (2.15)
Example 2.6. The operator A in (2.10) is unbounded on X =L2 (R + ;R2 ) but it can be
shown that it is maximally dissipative, i.e., that Re(x,Ax)X ≤0 for all x ∈Dom(A)
and Ran(1−A)=X. By the Lumer-Phillips Theorem [Paz83, Thm 1.4.3], this implies
that its semigroup A in (2.9), with Dom(At )=L 2 (R + ;R2 ), is a contraction semigroup.
In Examples 5.3 and 5.5 of [KS09] we showed that if A is maximally dissipative but
unbounded operator on X, and A|X is its unique extension to a continuous operator
from X to X−1, then the linear operator
A|X A|X Dom(S)
S :=
with domain
Dom(S)=
−A|X −A|X
x
u
∈
X x+u
∈Dom(A)
X
is a system node on (X,X,X) which is not L 2 -well posed.
The ill-posedness was proved by noting that the transfer function of S is
D(λ)=−A(λ−A) −1 , λ ∈C + , (2.16)
which tends to −A as λ → ∞ in R + . This transfer function is thus not bounded on any
complex right-half plane, and therefore S is ill-posed, as is well-known; see e.g. [Sta05,
Lem. 4.6.2].
See [KS09, Sect. 5] for more information on system nodes and their trajectories.
Also note that the system node in Example 2.6 is symmetric with respect to X and
U = Y. This is in general not the case, because B can in general not be recovered from
{0}
the restriction of A&B to Dom(S)∩ as is the case for A, cf. (2.13). Indeed, for
U
0
system nodes S of boundary control type we have ∈Dom(S) only if u=0; see the
u
text before (2.24) below.