Vietnam Journal of Mathematics 34:3 (2006) 357–368
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Robust Stability of Metzler Operator
and Delay Equation in Lp(-h, 0;X)
B. T. Anh1, N. K. Son2, and D. D. X. Thanh3
1Department of Mathematics, University of Pedagogy
280 An Duong Vuong Str. Ho Chi Minh City, Vietnam
2Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
3Department of Mathematics, University of Ton Duc Thang
98 Ngo Tat To Str. Ho Chi Minh City, Vietnam
Received February 16, 2006
Abstract. In this paper we study how the spectral bound of Mezler operator changes
under multi-perturbations. Characterizations of the stability radius of Metzler opera-
tors with respect to this type of disturbances are established. We will then apply the
obtained results to study the stability radius of delays equation in Lp([−1,0],X).
2000 Mathematics Subject Classification: 34K06, 93C73, 93D09.
Keywords: Metzler operator, stability radius, C0-semigroup, delay equations.
1. Introduction
In the last two decades, a considerable attention has been paid to problems of
robust stability of dynamic systems in infinite-dimensional spaces. The inter-
ested readers are referred to [3, 5, 6, 9, 15] and the biography therein for further
references. One of the most important problems in the study of robust stabil-
ity is the calculation of the stability radius of a dynanmic system subjected to
various classes of parameter perturbations. In [5, 15] explicit formulas for the
complex stability radius of a given (uniformly) exponentially stable linear system
˙x(t)=Ax(t) under structured perturbations of the form
A→A+D∆E(1)
358 B. T. Anh, N. K. Son, and D. D. X. Thanh
(where Ais a closed unbounded operator in a Banach space X,D∈L(U, X),E ∈
L(X, Y ) are given linear bounded operators and ∆ ∈L(Y, U) is unknown pertur-
bation) have been established, extending the classical results in finite-dimensional
case obtained by Hinrichsen and Pritchard in [8]. The case of time-varying sys-
tems has been considered in [9] and [3] where various formulas and estimates
of complex stability radius have been obtained for evolution operators. In [6] it
was shown that, for the case of structured perturbation (1), if the operator Ais
a Metzler operator (i.e. the resolvent R(λ;A)=(λI −A)−1is positive operator),
then the real stability radius coincide with the complex stability radius and can
be calculated by a simple formula.
The main purpose of paper is to extend the main result of [6] to the case
where the system operator Ais subjected to affine multi-perturbations of the
form
A→A+
N
X
i=1
Di∆iEi.(2)
The result is then applied to study the stability radii of delay equations in the
Banach space Lp([−h, 0]; X). To simplify the presentation, we shall make use of
the notation used in [6].
2. Main Result
Let Xbe a complex Banach space. For a closed linear operator A, let σ(A)
denote the spectrum of A,ρ(A)=C\σ(A) the resolvent set of A, and R(λ;A)=
(λI −A)−1∈L(X) the resolvent of Adefined on ρ(A). The spectral radius r(A)
and the spectral bound s(A)ofAare defined by
r(A) = sup |λ|:λ∈σ(A),s(A) = sup ℜλ:λ∈σ(A).
Denote the open complex left half-plane by C−={λ∈C:ℜλ<0}.A closed
operator Aon Xis said to be Hurwitz stable if σ(A)⊂C−and strictly Hurwitz
stable if s(A)<0. Clearly, every strictly Hurwitz stable operator is Hurwitz
stable. Let X, Y be complex Banach lattices and X+,Y+denote positive cones
of Xand Yrespectively; and LR(X, Y )(L+(X, Y ) ) are the set of all the
real (the positive ) linear operators from X to Y, respectively. If Y=Xthen we
use LR(X),L+(X) to denote the above spaces. A closed operator Ais said to
be a Metzler operator if there exists ω∈Rsuch that (ω, ∞)⊂ρ(A) and R(t;A)
is positive for t∈(ω, ∞)).It is clear that if A∈L
+(X) then Ais a Metzler
operator.
We recall some results of [5] and [6] which will be used in the sequel.
Theorem 2.1. Suppose T∈L
+(X). Then
i) r(T)∈σ(T).
ii) R(λ;T)>0if and only if λ∈Rand λ>r(T).
Theorem 2.2. Let Abe a Metzler operator on X. Then
i) s(A)∈σ(A)
Robust Stability of Metzler Operator and Delay Equation in Lp([−h, 0]; X)359
ii) the function R(·;A)is positive and decreasing for t>s(A)
s(A)<t
16t2=⇒06R(t2;A)6R(t2;A).
Lemma 2.3. Let Abe a Metzler operator on Xand E∈L
+(X, Y ). Then
|ER(λ;A)x|6ER(ℜλ;A)|x|,ℜλ>s(A),x∈X.
(Remind that for xin a complex Banach lattice X, |x|denotes the modulus
of x:|x|= sup{x, −x}).
Now we assume that Ais a Hurwitz stable closed operator on a complex
Banach lattice Xand that Ais subjected to under multi-perturbations of the
form
A→A∆=A+
N
X
i=1
Di∆iEi(3)
where Di∈L(Ui,X),E
i∈L(X, Yi),i ∈N={1, ..., N}are given linear
bounded operators determining the structure of perturbations and ∆i∈L(Yi,U
i),i∈
Nare unknown disturbance operators.
The transfer function Gij :ρ(A)→L(Uj,Y
i) associated with the triplet
(A, Ei,D
j) is defined by
Gij(λ)=EiR(λ;A)Dj,λ∈ρ(A),i,j∈N.
It is clear that each Gij(·) is analytic on ρ(A). We have the following result.
Proposition 2.1. Let λ∈ρ(A)and ∆i∈L(Yi,U
i),i∈N.If
N
X
i=1
||∆i|| <1
max
i,j∈N
||Gij(λ)||,(4)
then A∆is closed and λ∈ρ(A∆).
Proof. Let us consider the Banach spaces U=QN
i=1Ui,Y=QN
i=1Yiprovided
with the norm
kuk=
N
X
i=1
kuik,u=(u1,... ,u
N)∈U, ui∈Ui,i∈N, (5)
kyk=
N
X
i=1
kyik,u=(y1,... ,y
N)∈Y, yi∈Yi,i∈N. (6)
Let us define the linear operators E∈L(X, Y ),D∈L(U, X) by setting
Ex =(E1x, ··· ,E
Nx),Du=
N
X
i=1
Diui,for x∈X, u =(u1,··· ,u
N)∈U. (7)
360 B. T. Anh, N. K. Son, and D. D. X. Thanh
For any ∆i∈L(Yi,U
i),i ∈Nwe define the “block-diagonal” operator ∆ :
Y−→ Uby setting
∆y=(∆
1y1,···,∆NyN),y=(y1,···,y
N)∈Y, (8)
It is clear that ∆ ∈L(Y, U). Assume λ∈ρ(A), then, by definition, we have,
for each u=(u1,···,u
N)∈U,
∆ER(λ;A)Du =(
N
X
j=1
∆1G1j(λ)uj,···,
N
X
j=1
∆NGNj(λ)uj).
Therefore,
k∆ER(λ;A)Duk=
N
X
i=1
k∆i
N
X
j=1
Gij(λ)ujk6max
i,j∈N
||Gij(λ)||
N
X
i=1
||∆i||kuk,
and hence, by (4), k∆ER(λ;A)Dk<1. It follows that the operator [I−
∆ER(λ;A)D] is invertible and [I−∆ER(λ;A)D]−1∈L(U). Therefore, [I−
D∆ER(λ;A)] is invertible and [I−D∆ER(λ;A)]−1∈L(X). Since, obviously,
[I−D∆ER(λ;A)](λI −A)=λI −A−D∆E=λI −A∆,(9)
it follows that λI −A∆is a closed operator on Xand λI −A∆:D(A)→Xis
invertible. Moreover, by (9),
(λI −A∆)−1=R(λ;A)[I−D∆ER(λ;A)]−1∈L(X),
which implies that λ∈ρ(A∆)=ρ(A+PN
i=1 Di∆iEi), completing the proof.
Defition 2.4. Let Abe Hurwitz stable. The complex, the real and the positive
Hurwitz stability radii of Awith respect to the multi-perturbations of the form
(2) is defined, respectively, by
rC= inf nN
X
i=1
||∆i|| :∆
i∈L(Yi,U
i),i∈N, σ(A∆)6⊂ C−o,
rR= inf nN
X
i=1
||∆i|| :∆
i∈L
R(Yi,U
i),i∈N, σ(A∆)6⊂ C−o,
r+= inf nN
X
i=1
||∆i|| :∆
i∈L
+(Yi,U
i),i ∈N, σ(A∆)6⊂ C−o,
where we set inf ∅=∞.
Note that the first two stability radii are well defined without the assump-
tion that the underlying spaces are Banach lattices. Moreover, by definition,
rC6rR6r+.
Robust Stability of Metzler Operator and Delay Equation in Lp([−h, 0]; X)361
The following theorem gives a formula for calculation of the complex stability
radius with respect to multi-perturbations.
Theorem 2.5. Let Abe Hurwitz stable. Then
1
max
i,j∈N
sup
ℜs>0
||Gij(s)|| 6rC61
max
i∈N
sup
ℜs>0
||Gii(s)|| .(10)
In particular, if Di=Djor Ei=Ejfor all i, j ∈N, then
rC=1
max
i∈N
sup
ℜs>0
||Gii(s)|| .(11)
Proof. Assume to the contrary that the first inequality in (10) is not true, that
is
rC<1
max
i,j∈N
sup
ℜs>0
||Gij(s)|| =: γ. (13)
Then, by the definition of rC, there exist λ0,ℜλ0>0 and ∆0=(∆
0
1, ..., ∆0
N),∆0
i∈
L(Yi,U
i),i ∈Nsuch that λ0∈σ(A∆0) and
N
X
i=1
k∆0
ik<γ61
max
i,j∈N
kGij(λ0)k.(13)
On the other hand, since Ais Hurwitz stable, λ0∈ρ(A) and hence, by Propo-
sition 2.1, it follows from (13) that λ0∈ρ(A∆0), a contradiction. Thus we
have
rC>1
max
i,j∈N
sup
ℜs>0
kGij(s)k(14)
We now prove that
rC61
max
i∈N
sup
ℜs>0
kGii(s)k.(15)
Let us fix λ∈Cwith ℜλ>0,i ∈Nand ε>0. Then, there exists ˆui∈
Ui,kˆuik= 1 satisfying kGii(λ)k>kGii(λ)ˆuk>kGii(λ)k−ε. By Hahn-Banach
theorem there exists ˆy∗
i∈Y∗
isuch thatkˆy∗
ik=1,ˆy∗
i(Gii(λ)ˆui)=kGii(λ)ˆuik.
We define ∆i:Yi→Uiby setting
∆iyi=1
k|Gii(λ)ˆukˆy∗
i(yi)ˆui,∀yi∈Yi.
Then, it is clear that ∆i∈L(Yi,U
i) and
k∆ik61
kGii(λ)ˆuk61
kGii(λ)k−ε
Now we define the disturbance ∆ = (∆1, ..., ∆N) by setting, for j∈N,
