Báo cáo toán học: " Robust Stability of Metzler Operato p and Delay Equation in L ( -h, 0 ; X)"

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Trong bài báo này chúng tôi nghiên cứu quang phổ ràng buộc các nhà điều hành thay đổi Mezler theo đa-nhiễu loạn. Characterizations của bán kính ổn định của các nhà khai thác Metzler đối với loại rối loạn được thành lập. Sau đó chúng tôi sẽ áp dụng các kết quả thu được nghiên cứu bán kính ổn định của phương trình chậm trễ trong Lp (-1,0], X).

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Vietnam Journal of Mathematics 34:3 (2006) 357–368 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 9$ 67 Robust Stability of Metzler Operator p and Delay Equation in L ( -h, 0 ; X) B. T. Anh1, N. K. Son2, and D. D. X. Thanh3 1 Department 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 3 Department 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 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)−1 is 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 N A →A+ Di ∆iEi . (2) 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) , s(A) = sup 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. Theorem 2.1. Suppose T ∈ L+ (X ). Then 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)
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) < t1 t2 =⇒ 0 R(t2 ; A) R(t2 ; A). Lemma 2.3. Let A be a Metzler operator on X and E ∈ L+ (X, Y ). Then |ER(λ; A)x| E R( 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 N A → A∆ = A + Di ∆i Ei (3) 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 1 ||∆i|| < , (4) max ||Gij (λ)|| i=1 i,j ∈N then A∆ is closed and λ ∈ ρ(A∆ ). N N Proof. Let us consider the Banach spaces U = i=1 Ui , Y= i=1 Yi provided with the norm N u= ui , u = (u1, . . . , uN ) ∈ U, ui ∈ Ui , i ∈ N , (5) i=1 N y= yi , u = (y1 , . . . , yN ) ∈ Y, yi ∈ Yi , i ∈ N. (6) i=1 Let us deﬁne the linear operators E ∈ L(X, Y ), D ∈ L(U, X ) by setting N Ex = (E1x, · · · , EN x), Du = Di ui , for x ∈ X, u = (u1, · · · , uN ) ∈ U. (7) i=1
360 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 , · · · , ∆N yN ), 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 , N N ∆ER(λ; A)Du = ( ∆1G1j (λ)uj , · · · , ∆N GN j (λ)uj ). j =1 j =1 Therefore, N N N ∆ER(λ; A)Du = ∆i Gij (λ)uj max ||Gij (λ)|| ||∆i|| u , i,j ∈N i=1 j =1 i=1 and hence, by (4), ∆ER(λ; A)D < 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 ), N which implies that λ ∈ ρ(A∆ ) = ρ(A + Di ∆i Ei), completing the proof. i=1 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 N rC = inf ||∆i|| : ∆i ∈ L(Yi , Ui ), i ∈ N , σ(A∆ ) ⊂ C− , i=1 N ||∆i|| : ∆i ∈ LR (Yi , Ui ), i ∈ N, rR = inf σ(A∆ ) ⊂ C− , i=1 N ||∆i|| : ∆i ∈ L+ (Yi , Ui), i ∈ N , r+ = inf σ(A∆ ) ⊂ C− , i=1 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, rC rR r+ .
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 A be Hurwitz stable. Then 1 1 rC . (10) max sup ||Gij (s)|| max sup ||Gii(s)|| i,j ∈N s0 i∈N s0 In particular, if Di = Dj or Ei = Ej for all i, j ∈ N , then 1 rC = . (11) max sup ||Gii(s)|| i∈N s0 Proof. Assume to the contrary that the ﬁrst inequality in (10) is not true, that is 1 rC < =: γ. (13) max sup ||Gij (s)|| i,j ∈N s0 0 and ∆0 = (∆0 , ..., ∆0 ), ∆0 ∈ Then, by the deﬁnition of rC , there exist λ0 , λ0 1 N i L(Yi , Ui), i ∈ N such that λ0 ∈ σ(A∆0 ) and N 1 ∆0 < γ . (13) i max Gij (λ0 ) i=1 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 1 rC (14) max sup Gij (s) i,j ∈N s0 We now prove that 1 rC . (15) max sup Gii(s) i∈N s0 Let us ﬁx λ ∈ C with λ 0, i ∈ N and ε > 0. Then, there exists ui ∈ˆ Ui , ui = 1 satisfying Gii(λ) ˆ Gii(λ)ˆu Gii(λ) − ε. By Hahn-Banach theorem there exists yi ∈ Yi∗ such that yi = 1, yi (Gii(λ)ˆi ) = Gii(λ)ˆi . ˆ∗ ˆ∗ ˆ∗ u u We deﬁne ∆i : Yi → Ui by setting 1 y∗ (yi )ˆi , ∀yi ∈ Yi . ∆i yi = ˆ u |Gii(λ)ˆ i u Then, it is clear that ∆i ∈ L(Yi , Ui ) and 1 1 ∆i Gii(λ)ˆ u Gii(λ) − ε Now we deﬁne the disturbance ∆ = (∆1, ..., ∆N ) by setting, for j ∈ N ,
362 B. T. Anh, N. K. Son, and D. D. X. Thanh if j = i, ∆i ∆j = (16) 0 if j = i. N Then j =1 ||∆j || = ||∆i|| and, taking x = R(λ; A)Du ∈ D(A) we can easily ˆ ˆ N verify that x = 0 and (A + j =1 Dj ∆j Ej )ˆ = A∆ x = λx. This implies ˆ x ˆ ˆ λ ∈ σ(A∆ ) and so σ(A∆ ) ⊂ C−. Consequently, by the deﬁnition of rC N 1 rC ∆j = ∆i . (17) Gii(λ) − ε 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 proof is complete. 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 ||Gij (s)|| = sup ||Gij (ıs)|| s∈R s0 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 1 rC = rR = r+ = . max Gii(0) i∈N 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 t1 t2 ⇒ 0 Gii(t2 ) Gii(t1 ) and Gii(t2 ) Gii(t1 ) , ∀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, Gii(λ) Gii(t) Gii(0) , ∀i ∈ N . Therefore, by formula (11), 1 rC = . max Gii(0) i∈N To show that r+ 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
Robust Stability of Metzler Operator and Delay Equation in Lp ([−h, 0]; X ) 363 destabilizing perturbation ∆ = (∆1, · · · , ∆N ), ∆j ∈ L+ (Yj , Uj ), ∀j ∈ N such that ∆ = ∆i < Gii(0) −1 + ε. This implies 1 r+ ∆< + ε = rC + ε, Gii(0) concluding the proof. 3. Stability Radii of Delay Equation in Lp ([−h, 0]; X ) 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 h1 < h2 < ... < hn =: h. Given bounded linear operators A1, ..., An on X , we will study the delay equation  n  u(t) = A0 u(t) + ˙ Ai u(t − hi ), t 0  i=1 (19)  u(0) = x,   u(t) = f (t), t ∈ [−h, 0). 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(·) ∈ Lp ([−h, ∞); X ) satisfying u(t) = loc f (t), t ∈ [−h, 0) and n t u( t ) = T ( t ) x + T (t − s) Ai u(s − hi )ds, t 0. (20) 0 i=1 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 p M e−ωt ( x p u( t ) +f Lp ([−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 ) p p p (endowed with the norm (x, f ) =x +f Lp ([−h,0];X ) ) and the operator A on X deﬁne by n A(x, f ) = (A0x + Ai f (−hi ), f ), i=1 with the domain D(A) = {(x, f ) ∈ X : f ∈ W 1,p ([−h, 0]; X ), f (0) = x ∈ D(A0 )} (here W 1,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
364 B. T. Anh, N. K. Son, and D. D. X. Thanh f (t) ∈ Lp ([−h, 0]; X )). Then, as it has been shown in [2], A generates a C0- semigroup (T (t))t 0 on 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. Proposition 3.1. If A0 generates a positive C0-semigroup and Ai ∈ L+ (X ), for all i = 1, ..., n, then T is a positive C0 -semigroup. n e−λhi Ai . The Now we deﬁne an operator quasi-polynomial P (λ) = A0 + i=1 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 (·)). Lemma 3.2. Suppose that A0 generates a positive C0-semigroup and Ai ∈ 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 (·)) < t1 t2 =⇒ 0 R(t2; P (t2)) 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]). Proposition 3.2. Suppose that A generates a positive C0-semigroup and Ai ∈ L+ (X ) for all i = 1, ..., n. For E ∈ L+ (X, Y ), x ∈ X we have |ER(λ; P (λ))x| E R( λ; P ( λ))|x|, λ > s(A) = s(P (·)). Proof. We set the operator E : X → Y deﬁned by E (x, f ) = Ex and we choose the vector (x, 0) ∈ X . Applying Lemma 2.5 we have
Robust Stability of Metzler Operator and Delay Equation in Lp ([−h, 0]; X ) 365 |ER(λ; A)(x, 0)| E R( λ; A)(x, 0)|, λ > s(A) = s(P (·)). or equivalent |ER(λ; P (λ))x| E R( λ; P ( λ))|x|, λ > s(A) = s(P (·)). The proof is complete. 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  n  u(t) = (A0u(t) + D0 ∆0E0 ) + ˙ (Ai + Di ∆i Ei)u(t − hi ), t 0  i=1 (22)  u(0) = x,   u(t) = f (t), t ∈ [−h, 0). We also set n A∆ (x, f ) = ((A0 + D0 ∆0E0 )x + (Ai + Di ∆i Ei)f (−hi ), f ), i=1 and n e−λhi (Ai + Di ∆iEi ). P∆ (λ) = (A0 + D0 ∆0E0 ) + i=1 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 n (DE ) rC = inf ∆i : ∆i ∈ L(Yi, Ui ), i = 0, 1, ..., n and (22) is not exponentially stable i=0 n (DE ) ∆i : ∆i ∈ LR(Yi , Ui ), i = 0, 1, ..., n and (22) is not exponentially stable rR = inf i=0 n (DE ) ∆i : ∆i ∈ L+ (Yi , Ui), i = 0, 1, ..., n and (22) is not exponentially stable r+ = inf i=0 (we set inf ∅ = ∞).
366 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 n (DE ) rC = inf ||∆i|| : ∆i ∈ L(Yi , Ui ), i = 0, n, s(P∆ (·)) 0 i=0 n (DE ) ||∆i|| : ∆i ∈ LR (Yi , Ui ), i = 0, n, s(P∆ (·)) rR = inf 0 i=0 n (DE ) ||∆i|| : ∆i ∈ L+ (Yi , Ui), i = 0, n, s(P∆ (·)) r+ = inf 0 i=0 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, e−λhi ∆iGij (λ) maxi,j Gij (λ) , ∀i, j ∈ {0, 1, ..., n} we can prove the following proposion similarly as it was done for Proposition 2.1. Propostion 3.3. Let λ ∈ ρ(P (·)), λ 0 and ∆i ∈ L(Yi, Ui ), for all i = 0, 1, ..., n. If n 1 ∆i < , (23) max Gij (λ) i=1 i,j ∈{0,1,...,n} 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 1 (DE ) rC . (24) max sup Gij (is) max sup Gii(is) i,j ∈{0,1,...,n} s∈R i∈{0,1,...,n} s∈R In particular, if Di = Dj or Ei = Ej for all i, j ∈ {0, 1, ..., n}, then 1 (DE ) rC = . (25) max sup Gii(is) i∈{0,1,...,n} 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
Robust Stability of Metzler Operator and Delay Equation in Lp ([−h, 0]; X ) 367 Theorem 3.5. Suppose that equation (19) is exponentially stable, A0 generates a positive C0-semigroup and the equation is subjected to aﬃne perturbations of the form (21) with Ai ∈ L+ (X ), Di ∈ L+ (Ui , X ), Ei ∈ L+ (X, Yi ), ∀i = 1, ..., n. If Di = Dj or Ei = Ej for all i, j ∈ {0, 1, ..., n}, then 1 (DE ) (DE ) (DE ) rC = rR = r+ = max Gii(0) i∈{0,1,...,n} (26) 1 = . Ei(−A0 − A1 − ... − An)−1 Di max i∈{0,1,...,n} The proof similar to that of Theorem 2.6 and is omitted. We note that the results obtained in Theorems 2.5 and 2.6 generalize results in [6] to multi-perturbations, while Theorems 3.4 and 3.5 extend results due to [18] to the case of delay systems in Banach spaces. Similarly, we can also consider the case of delay systems in Banach spaces where operators Ai , i ∈ {0, 1, ..., n} are subjected to multi-perturbations of the form k Ai −→ Ai + Dij ∆ij Eij , i ∈ {0, 1, ..., n}. j =0 It is a natural open problem to extend the results regarding stability radii of functional diﬀerential equations obtained recently in [19, 13] to Banach spaces. References 1. W. Arendt, Resolvent of positive operators, Proc. London Math. Soc. 54 (2001) 321–349. 2. A. B´tkai and S. Piazzera, Semigroups and linear partial diﬀerential equations, a J. Math. Anal. Appl. 264 (2001) 1–20. 3. S. Clark, Y. Latushkin, S. Montgomery-Smith, and T. Randolph, Stability radius and versus external stability in Banach spaces: An evolution semigroup approach, SIAM J. Control and Optim. 36 (2000) 1757–1793. 4. R. Datko, Representation of solutions and stability of linear diﬀerential-diﬀerence equations in Banach space, J. Diﬀerent. Equations 29 (1978) 105–166. 5. A. Fischer, van Neerven, J. M. A. M, Robust stability of C0-semigroups and an application to stability of delay equations, J. Math. Anal. Appl. 226(1998) 1169–1188. 6. A. Fischer, D. Hinrichsen, and N. K. Son, Stability radii of Metzler operators, Vietnam J. Math. 26 (1998) 147–163. 7. J. K. Hale, Functional Diﬀerential Equations, Springer–Verlag, Berlin, 1971. 8. D. Hinrichsen and A. J. Pritchard, Stability radii of linear systems, Systems & Control Letters 7 (1986) 1–10. 9. D. Hinrichsen and A. J. Pritchard, Robust stability of linear evolution operators on Banach spaces, SIAM J. Control and Optim. 32 (1994) 1503–1541. 10. R. Nagel (Ed.), One-Parameter Semigroups of Positive Operators, Springer–Verlag, Berlin, 1986.
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