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
Trang 1Vietnam Journal of Mathematics 34:3 (2006) 357–368
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Robust Stability of Metzler Operator
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1Department of Mathematics, University of Pedagogy
280 An Duong Vuong Str Ho Chi Minh City, Vietnam
2 Institute 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
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 inLp([−1,0],X)
2000 Mathematics Subject Classification: 34K06, 93C73, 93D09
Keywords: Metzler operator, stability radius,C 0-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 stabilstabil-ity 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
Trang 2358 B T Anh, N K Son, and D D X Thanh (where A is 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 A is
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 A is subjected to affine multi-perturbations of the
form
A → A +
N
X
i=1
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 X be 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 A defined on ρ(A) The spectral radius r(A)
and the spectral bound s(A) of A are defined by
r(A) = sup
|λ| : λ ∈ σ(A)
<λ : λ ∈ σ(A)
.
Denote the open complex left half-plane by C− = {λ ∈ C : <λ < 0}.A closed operator A on X is 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 X and Y respectively; 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 = X then we use LR(X), L+(X) to denote the above spaces A closed operator A is said to
be a Metzler operator if there exists ω ∈ R such that (ω, ∞) ⊂ ρ(A) and R(t; A)
is positive for t ∈ (ω, ∞)).It is clear that if A ∈ L+(X) then A is a Metzler
operator
We recall some results of [5] and [6] which will be used in the sequel
i) r(T ) ∈ σ(T ).
ii) R(λ; T ) > 0 if and only if λ ∈ R and λ > r(T ).
Theorem 2.2 Let A be a Metzler operator on X Then
i) s(A) ∈ σ(A)
Trang 3ii) the function R(·; A) is positive and decreasing for t > s(A)
s(A) < t16 t2=⇒ 0 6 R(t2; A) 6 R(t2; A).
|ER(λ; A)x| 6 ER(<λ; A)|x|, <λ > s(A), x ∈ X.
(Remind that for x in a complex Banach lattice X, |x| denotes the modulus
of x : |x| = sup{x, −x}).
Now we assume that A is a Hurwitz stable closed operator on a complex Banach lattice X and that A is subjected to under multi-perturbations of the
form
A → A∆= A +
N
X
i=1
where Di ∈ L(Ui, X), Ei ∈ L(X, Yi), i ∈ N = {1, , N } are given linear
bounded operators determining the structure of perturbations and ∆i∈ L(Yi, Ui), i ∈
N are unknown disturbance operators.
The transfer function Gij : ρ(A) → L(Uj, Yi) associated with the triplet
(A, Ei, Dj) 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, Ui), i ∈ N If
N
X
i=1
||∆i|| < 1
max
i,j∈N
then A∆ is closed and λ ∈ ρ(A∆)
Proof Let us consider the Banach spaces U =QN
i=1Ui, Y =QN
i=1Yi provided with the norm
kuk =
N
X
i=1
kuik, u = (u1, , uN) ∈ U, ui∈ Ui, i ∈ N , (5)
kyk =
N
X
i=1
kyik, u = (y1, , yN) ∈ Y, yi∈ Yi, i ∈ N (6)
Let us define the linear operators E ∈ L(X, Y ), D ∈ L(U, X) by setting
Ex = (E1x, · · · , ENx), Du =
N
X
Diui, for x ∈ X, u = (u1, · · · , uN) ∈ U (7)
Trang 4360 B T Anh, N K Son, and D D X Thanh
For any ∆i ∈ L(Yi, Ui), i ∈ N we define the “block-diagonal” operator ∆ :
Y −→ U by setting
∆y = (∆1y1, · · · , ∆NyN), y = (y1, · · · , yN) ∈ Y, (8)
It is clear that ∆ ∈ L(Y, U ) Assume λ ∈ ρ(A), then, by definition, we have, for each u = (u1, · · · , uN) ∈ U ,
∆ER(λ; A)Du = (
N
X
j=1
∆1G1j(λ)uj, · · · ,
N
X
j=1
∆NGN j(λ)uj).
Therefore,
k∆ER(λ; A)Duk =
N
X
i=1
k∆i
N
X
j=1
Gij(λ)ujk 6 max
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 X and λI − A∆: D(A) → X is
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=1Di∆iEi), completing the proof.
Defition 2.4 Let A be Hurwitz stable The complex, the real and the positive
Hurwitz stability radii of A with respect to the multi-perturbations of the form
(2) is defined, respectively, by
rC= infnXN
i=1
||∆i|| : ∆i∈ L(Yi, Ui), i ∈ N , σ(A∆) 6⊂ C−
o
,
rR= infnXN
i=1
||∆i|| : ∆i∈ LR(Yi, Ui), i ∈ N , σ(A∆) 6⊂ C−
o
,
r+= infnXN
i=1
||∆i|| : ∆i∈ L+(Yi, Ui), 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 definiassump-tion,
Trang 5The following theorem gives a formula for calculation of the complex stability radius with respect to multi-perturbations
Theorem 2.5 Let A be Hurwitz stable Then
1 max
i,j∈N
sup
<s>0
||Gij(s)|| 6 rC6
1 max
i∈N
sup
<s>0
||Gii(s)|| . (10)
In particular, if Di= Dj or Ei= Ej for all i, j ∈ N , then
max
i∈N
sup
<s>0
Proof Assume to the contrary that the first inequality in (10) is not true, that is
max
i,j∈N
sup
<s>0
Then, by the definition of rC, there exist λ0, <λ0> 0 and ∆0= (∆0, , ∆0N), ∆0i ∈ L(Yi, Ui), i ∈ N such that λ0∈ σ(A∆ 0) and
N
X
i=1
k∆0ik < γ 6 1
max
i,j∈N
On the other hand, since A is 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
max
i,j∈N
sup
<s>0
We now prove that
max
i∈N
sup
<s>0
Let us fix λ ∈ C with <λ > 0, i ∈ N and ε > 0 Then, there exists ˆ ui ∈
Ui, kˆ uik = 1 satisfying kGii(λ)k > kGii(λ)ˆ uk > kGii(λ)k − ε By Hahn-Banach
theorem there exists ˆy∗
i ∈ Y∗
i such thatkˆ y∗
ik = 1, ˆ y∗
i(Gii(λ)ˆ ui) = kGii(λ)ˆ uik.
We define ∆i: Yi → Ui by setting
∆iyi= 1
k|Gii(λ)ˆ uk yˆ
∗
i(yi)ˆui, ∀yi ∈ Yi.
Then, it is clear that ∆i∈ L(Yi, Ui) and
kGii(λ)ˆ uk6
1
kGii(λ)k − ε
Now we define the disturbance ∆ = (∆ , , ∆ ) by setting, for j ∈ N ,
Trang 6362 B T Anh, N K Son, and D D X Thanh
∆j=
∆
i if j = i,
Then PN
j=1||∆j|| = ||∆i|| and, taking ˆ x = R(λ; A)D ˆ u ∈ D(A) we can easily
verify that ˆx 6= 0 and (A + PN
j=1Dj∆jEj)ˆx = A∆x = λˆˆ x. This implies
λ ∈ σ(A∆) and so σ(A∆) 6⊂ C− Consequently, by the definition of rC
rC6
N
X
j=1
Since the above inequality has been established for arbitrary λ ∈ C with <λ >
0, i ∈ N and ε > 0, the inequality (15) follows Furthermore, if Di= Dj, ∀i, j ∈
N (resp Ei= Ej, ∀i, j ∈ N ) then, by the definition Gii(s) = Gij(s), ∀i, j ∈ N (resp Gjj(s) = Gij(s), ∀i, j ∈ N ) Thus, in this case, (10) implies (11) The
Remark that for the Hurwitz stable operator A, the functions Gij(·) are
analytic in the complex right-half plane, therefore, by the maximum modulus principle, one has
sup
<s>0
||Gij(s)|| = sup
s∈R
||Gij(ıs)||
and hence the formulas (10) and (11) can be rewriten accordingly with the supremum is taken over the real line
Theorem 2.6 Let A be a Hurwitz Metzler stable operator and all operator
Di, Ei, i ∈ N are positive If Di = Dj (or) Ei= Ej for all i, j ∈ N then
max
i∈N
kGii(0)k .
Proof Since s(A) < 0 and Ei, Di, ∀i ∈ N , are positive operators, it follows
from Theorem 2.2 that all Gii(t) are decreasing for t > 0:
0 6 t16 t2 ⇒ 0 6 Gii(t2) 6 Gii(t1) and kGii(t2)k 6 kGii(t1)k, ∀i ∈ N
(18) Applying Lemma 2.3 and the lattice norm property, from (18), we get, for all
λ = t + ıω ∈ C with t = <λ > 0,
kGii(λ)k 6 kGii(t)k 6 kGii(0)k, ∀i ∈ N
Therefore, by formula (11),
max
i∈N
kGii(0)k .
To show that r+ 6 rC, let us fix i ∈ N and an arbitrary ε > 0. As in [6], using the Krein-Rutman Theorem, one can construct an one-rank positive
Trang 7destabilizing perturbation ∆ = (∆1, · · · , ∆N), ∆j ∈ L+(Yj, Uj), ∀j ∈ N such that k∆k = k∆ik < kGii(0)k−1+ ε This implies
r+6 k∆k < 1
kGii(0)k + ε = rC+ ε,
In this section, we apply the results of the above section to study robust stability for linear delay equations in Banach spaces
Assume that A0 is a generator of a uniformly continuous C0-semigroup
(T (t))t>0 on a complex Banach space X We also fix p ∈ [1, ∞) and non-negative real numbers 0 6 h1 < h2 < < hn =: h Given bounded linear operators A1, , Anon X, we will study the delay equation
˙
u(t) = A0u(t) +
n
P
i=1
Aiu(t − hi), t > 0
u(0) = x, u(t) = f (t), t ∈ [−h, 0).
(19)
Here, x ∈ X is the initial value and f ∈ Lp([−h, 0]; X) is the ‘history’ function.
A mild solution of (19) is the function u(·) ∈ Lploc([−h, ∞); X) satisfying u(t) =
f (t), t ∈ [−h, 0) and
u(t) = T (t)x +
Z t 0
T (t − s)
n
X
i=1
Aiu(s − hi)ds, t > 0. (20)
The delay equation (19) is called exponentially stable is there exist M > 0 and
ω > 0 such that the solution u(t) of (19) satisfies
ku(t)k 6 M e−ωt(kxkp+ kf kpLp ([−h,0];X)), t > 0.
In order to study the asymptotic behavior of solutions of (19) by semigroup methods, we introduce the product space
X := X × Lp([−h, 0]; X) (endowed with the norm k(x, f )kp= kxkp+ kf kpLp ([−h,0];X)) and the operator
A on X define by
A(x, f ) = (A0x +
n
X
i=1
Aif (−hi), f0),
with the domain
D(A) = {(x, f ) ∈ X : f ∈ W1,p([−h, 0]; X), f (0) = x ∈ D(A0)}
(here W1,p([−h, 0]; X) denotes the space of absolutely contiuous with X-valued functions f on [−h, 0] that are strongly differentiable a.e. with derivatives
Trang 8364 B T Anh, N K Son, and D D X Thanh
f0(t) ∈ Lp([−h, 0]; X)) Then, as it has been shown in [2], A generates a C0
-semigroup (T (t))t>0on X which is defined by
(T (t))(x, f ) = (u(t), ut), t > 0, where u(t) is a mild solution of (19) and ut(s) := u(t+s), s ∈ [−h, 0] Moreover, the delay equation (19) is exponentially stable if and only if ω0(T ) < 0 Note that s(A) = ω0(T ) because T (t) is uniformly continuous semigroup for t > h
(see [5], p.94)
In what follows we assume X a complex Banach lattice and we consider
Lp([−h, 0]; X) as the Banach lattice with respect to the pointwise order relation Then the product space X becomes a Banach lattice as well The following
result follows directy from the definition
for all i = 1, , n, then T is a positive C0-semigroup
Now we define an operator quasi-polynomial P (λ) = A0+
n
P
i=1
e−λhiAi The
spectral set, the resolvent set, and the spectral bound of P (·) are defined respec-tively by σ(P (·)) = {λ ∈ C : λ ∈ σ(P (λ))}, ρ(P (·)) = C\σ(P (·)), s(P (·)) = sup{<λ : λ ∈ σ(P (·))} Then, by definition, it is easy to show
Remark 3.1 We have σ(P (·)) = σ(A) and s(P (·)) = s(A).
The following result will address the properties about the monotonicity and
the positivity of the resolvent R(·, P (·)).
L+(X), for all i = 1, , n Then the resolvent R(·; P (·)) is positive and de-creasing for t > s(P (·)) = s(A) :
s(A) = s(P (·)) < t16 t2=⇒ 0 6 R(t2; P (t2)) 6 R(t1; P (t1)).
Proof By Proposition 3.1, A is a generator of a positive C0-semigroup This
implies that A is Metzler operator By Theorem 2.2, we have that the resolvent
R(·, A) is positive and decreasing for t > s(A) = s(P (·)) The assertion now
follows from the formula about relationship between R(·, A) and R(·; P (·)) (see
[5, Proposition 3.2])
L+(X) for all i = 1, , n For E ∈ L+(X, Y ), x ∈ X we have
|ER(λ; P (λ))x| 6 ER(<λ; P (<λ))|x|, <λ > s(A) = s(P (·)).
Proof We set the operator eE : X → Y defined by e E(x, f ) = Ex and we choose
the vector (x, 0) ∈ X Applying Lemma 2.5 we have
Trang 9| ER(λ; A)(x, 0)| 6 ee ER(<λ; A)(x, 0)|, <λ > s(A) = s(P (·)).
or equivalent
|ER(λ; P (λ))x| 6 ER(<λ; P (<λ))|x|, <λ > s(A) = s(P (·)).
Now we return to study the stability radii of the delay equation (19)
Sup-pose that the equation (19) is exponentially stable, or the operator A generates the exponentially stable C0-semigroup This is equivalent to ω0(T ) = s(A) =
s(P (·)) < 0 Suppose that the operators Ai, i = 0, 1, , n are subjected to
perturbations of the form
Ai → Ai+ Di∆iEi, i = 0, 1, , n, (21)
where Di ∈ L(Ui, X), Ei ∈ L(X, Yi), i = 0, 1, , n are given operators
deter-mining structure of perturbation and ∆i∈ L(Yi, Ui), i = 0, 1, , n are unknown
operators Then, the perturbed equation has the form
˙
u(t) = (A0u(t) + D0∆0E0) +
n
P
i=1
(Ai+ Di∆iEi)u(t − hi), t > 0
u(0) = x,
u(t) = f (t), t ∈ [−h, 0).
(22)
We also set
A∆(x, f ) = ((A0+ D0∆0E0)x +
n
X
i=1
(Ai+ Di∆iEi)f (−hi), f0),
and
P∆(λ) = (A0+ D0∆0E0) +
n
X
i=1
e−λhi(Ai+ Di∆iEi).
Definition 3.3 Let equation (19) be exponentially stable The complex, the
real and the positive stability radii of (19) under perturbations of the form (21) are defined respectively by
rC(DE)=inf
nXn
i=0
k∆ik : ∆i∈ L(Yi, Ui), i = 0, 1, , nand(22)is not exponentially stable
o
rR(DE)=inf
nXn
i=0
k∆ik : ∆i∈ LR(Yi, Ui), i = 0, 1, , nand (22)is not exponentially stable
o
r+(DE)=inf
nXn
i=0
k∆ik : ∆i∈ L+(Yi, Ui), i = 0, 1, , nand(22)is not exponentially stable
o
(we set inf ∅ = ∞).
Trang 10366 B T Anh, N K Son, and D D X Thanh Since A0 is the generator of a C0-semigroup which is uniformly continuous
for t > 0 and Di, ∆i, Ei are all bounded operators, it can be easily verified, by
using Lemma 2.4 in [5] that the operator A∆generates a uniformly continuous
C0-semigroup This follows that the equation (22) is not exponentially if and
only if s(A∆) = s(P∆(·)) > 0 Thus we can rewrite the definition of the stability
radii of (19) as follows
rC(DE)= infnXn
i=0
||∆i|| : ∆i∈ L(Yi, Ui), i = 0, n, s(P∆(·)) > 0o
rR(DE)= infnXn
i=0
||∆i|| : ∆i∈ LR(Yi, Ui), i = 0, n, s(P∆(·)) > 0o
r+(DE)= infnXn
i=0
||∆i|| : ∆i∈ L+(Yi, Ui), i = 0, n, s(P∆(·)) > 0o
Assume that the equation (19) is exponentially stable For λ ∈ ρ(P (·)) we introduce the transfer function associated with the triplet (P (λ), Dj, Ei)
Gij(λ) = EiR(λ; P (λ))Dj, i, j ∈ {0, 1, , n}.
Noticing that for λ ∈ C with <λ > 0, ke−λh i∆iGij(λ)k 6 maxi,jkGij(λ)k, ∀i, j ∈
{0, 1, , n} we can prove the following proposion similarly as it was done for
Proposition 2.1
0, 1, , n If
n
X
i=1
max
i,j∈{0,1, ,n}kGij(λ)k , (23) then λ ∈ ρ(P∆(λ)).
Using the above proposition we get the following theorem The proof is similar to that of Theorem 2.5 and is therefore omitted
Theorem 3.4 Let the equation (19) be exponentially stable Then
1 max
i,j∈{0,1, ,n}sup
s∈R
kGij(is)k 6 r
(DE)
max
i∈{0,1, ,n}sup
s∈R
kGii(is)k . (24)
In particular, if Di= Dj or Ei= Ej for all i, j ∈ {0, 1, , n}, then
max
i∈{0,1, ,n}sup
s∈R
In general, the three stability radii are distinct However, for the case of positive delay equations, they coinside and, moreover, can be computed easily,
as shown by the following