Vol. 68 (2010) Perturbations of Regular Systems and Delay systems 361
for any u, v ∈ L
p
loc
(R
+
,U)andτ ≥ 0. By the representation theorem in [27],
there exists a unique operator C ∈ L(X
1
,Y), called admissible observation
operator for A, such that for any t ≥ 0andx ∈ D(A), CT(t)x =(Ψ
∞
x)(t). In
this case, we say that (A, C) generates an abstract linear observation system
on (X, Y ) and denote Ψ = Ψ
A,C
for brief.
We say that the well-posed linear system Σ is regular if the limit
lim
t→0
1
t
t
0
(F
∞
u
0
)(s)ds = Dz (2.5)
exists in Y for the constant input u
0
(t)=z, z ∈ U, t ≥ 0. In this case, we also
say that the quadruple Σ = (T,Φ, Ψ,F) is a regular linear system on (X, U, Y )
generated by (A, B, C, D), and we denote Σ
A,B,C,D
=ΣandF
A,B,C
= F .
Furthermore, we denote (A, B, C)=(A, B, C, 0) and Σ
A,B,C
=Σ
A,B,C,0
for
brief.
In order to introduce the representation theorem of regular linear sys-
tem, Weiss [30] introduced an extension of C, called Λ-extension with respect
to A, which is defined by
C
Λ
x = lim
λ→∞
CλR(λ, A)x (2.6)
with the domain D(C
Λ
)={x ∈ X : this above limit exists in Y }. It follows
from [27, Theorem 4.5 and Proposition 4.7] that for any x ∈ X, y(t)=
C
Λ
T (t)x a.e. in t ≥ 0 whenever C is admissible for A.
With the above Λ-extension, for well-posed linear system Σ
A,B,C
,the
regularity condition (2.5) is equivalent to each of the following two conditions:
• Range(R(λ, A
−1
)B) ⊂ D(C
A
Λ
) holds for some (and hence for all) λ ∈
ρ(A).
• For any u ∈ U, G(λ)u has a limits when λ →∞, where G is the transfer
function associated to F (∞).
In this case, the transfer function G is given explicitly by
G(λ)=C
A
Λ
R(λ, A
−1
)B + D, Re(λ) >w
0
(T ), (2.7)
where w
0
(T ) is the growth bound of the semigroup T , and we denote
G
A,B,C
= G.
In order to state the following theorem, we define
D
p
(M)
={f (·)∈L
p
loc
(R
+
,X):f ∈D(M) for a.e. t ≥0,and Mf(·)∈ L
p
loc
(R
+
,X)}.
Theorem 2.2. [28] Let Σ be a regular linear system with generating operator
A, B, C and D on (X, U, Y ). Then, for given (x
0
,u) ∈ X × L
p
(R
+
,U), the
state trajectory x(·) of Σ, given by x(t):=T (t)x
0
+Φ
A,B
u, t ≥ 0,isa.e.
differential in X
−1
and
˙x(t)=A
−1
x(t)+Bu(t),x(0) = x
0
for a.e. t ≥ 0. (2.8)
Furthermore, x(t) ∈ D(C
A
Λ
) for a.e. t ≥ 0 and the output function y =
Ψ
A,C
(∞)x + F
A,B,C,D
(∞)u of Σ is given by
y(t)=C
A
Λ
x(t)+Du(t) f or a.e. t ≥ 0. (2.9)
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