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

  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, ; X)   B T Anh1, N K Son2, and D D X Thanh3  Department of Mathematics, University of Pedagogy  280 An Duong Vuong Str Ho Chi Minh City, Vietnam Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam 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 operators 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 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 interested 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 stability 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 perturbation) have been established, extending the classical results in finite-dimensional case obtained by Hinrichsen and Pritchard in [8] The case of time-varying systems 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] 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) < 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 ) 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 =⇒ 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| 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 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 ||∆i|| < i=1 , max ||Gij (λ)|| (4) i,j∈N then A∆ is closed and λ ∈ ρ(A∆ ) Proof Let us consider the Banach spaces U = with the norm N i=1 Ui , Y = N i=1 Yi provided N u = ui , u = (u1, , uN ) ∈ U, ui ∈ Ui , i ∈ N , (5) yi , u = (y1 , , yN ) ∈ Y, yi ∈ Yi , i ∈ N (6) i=1 N y = 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 , · · · , j=1 ∆N GN j (λ)uj ) j=1 Therefore, N N ∆ER(λ; A)Du = ∆i i=1 N Gij (λ)uj max ||Gij (λ)|| i,j∈N j=1 ||∆i|| u , i=1 and hence, by (4), ∆ER(λ; A)D < 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 + N i=1 Di ∆i Ei), 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 N rC = inf ||∆i|| : ∆i ∈ L(Yi , Ui ), i ∈ N , σ(A∆ ) ⊂ C− , i=1 N ||∆i|| : ∆i ∈ LR (Yi , Ui ), i ∈ N, σ(A∆ ) ⊂ C− , ||∆i|| : ∆i ∈ L+ (Yi , Ui), i ∈ N , rR = inf σ(A∆ ) ⊂ C− , i=1 N r+ = inf i=1 where we set inf ∅ = ∞ Note that the first two stability radii are well defined without the assumption 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 rC max sup ||Gij (s)|| max sup ||Gii(s)|| i,j∈N s i∈N s In particular, if Di = Dj or Ei = Ej for all i, j ∈ N , then rC = max sup ||Gii(s)|| i∈N (10) (11) s Proof Assume to the contrary that the first inequality in (10) is not true, that is rC < =: γ (13) max sup ||Gij (s)|| i,j∈N s Then, by the definition of rC , there exist λ0 , λ0 L(Yi , Ui), i ∈ N such that λ0 ∈ σ(A∆0 ) and N ∆0 < γ i i=1 and ∆0 = (∆0 , , ∆0 ), ∆0 ∈ N i max Gij (λ0 ) (13) i,j∈N On the other hand, since A is Hurwitz stable, λ0 ∈ ρ(A) and hence, by Proposition 2.1, it follows from (13) that λ0 ∈ ρ(A∆0 ), a contradiction Thus we have rC (14) max sup Gij (s) i,j∈N s We now prove that rC max sup Gii(s) i∈N (15) s Let us fix λ ∈ C with λ 0, i ∈ N and ε > Then, there exists ui ∈ ˆ Ui , ui = satisfying Gii(λ) ˆ Gii(λ)ˆ u 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 ∆i yi = u y∗ (yi )ˆi , ∀yi ∈ Yi ˆ |Gii(λ)ˆ i u Then, it is clear that ∆i ∈ L(Yi , Ui ) and ∆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 ∆i ∆j = N j=1 ||∆j || Then if j = i, if j = i (16) = ||∆i|| and, taking x = R(λ; A)Dˆ ∈ D(A) we can easily ˆ u N verify that x = and (A + j=1 Dj ∆j Ej )ˆ = A∆ x = λˆ This implies ˆ x ˆ x λ ∈ σ(A∆ ) and so σ(A∆ ) ⊂ C− Consequently, by the definition of rC N rC Gii(λ) − ε ∆j = ∆i j=1 (17) 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 s 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 rC = rR = r+ = max Gii(0) i∈N Proof Since s(A) < and Ei , Di, ∀i ∈ N , are positive operators, it follows from Theorem 2.2 that all Gii(t) are decreasing for t 0: t1 t2 ⇒ 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(t1 ) and Gii(t) Gii(t2 ) Gii(0) , ∀i ∈ N Therefore, by formula (11), rC = max Gii(0) i∈N To show that r+ rC , let us fix i ∈ N and an arbitrary ε > 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 r+ ∆ < + ε = rC + ε, Gii(0) concluding the proof 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 on a complex Banach space X We also fix p ∈ [1, ∞) and nonnegative real numbers 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  ˙  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) i=1 The delay equation (19) is called exponentially stable is there exist M > and ω > such that the solution u(t) of (19) satisfies u(t) M e−ωt ( x p + f p Lp ([−h,0];X) ), t 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 (x, f) A on X define by p = x p + f p Lp ([−h,0];X) ) and the operator 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 C0semigroup (T (t))t 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 ) < 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 Now we define an operator quasi-polynomial P (λ) = A0 + e−λhi Ai The i=1 spectral set, the resolvent set, and the spectral bound of P (·) are defined respectively 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 decreasing for t > s(P (·)) = s(A) : s(A) = s(P (·)) < t1 t2 =⇒ 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| ER( 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) |ER(λ; A)(x, 0)| ER( λ; A)(x, 0)|, λ > s(A) = s(P (·)) ER( λ; P ( λ))|x|, 365 λ > s(A) = s(P (·)) or equivalent |ER(λ; P (λ))x| The proof is complete Now we return to study the stability radii of the delay equation (19) Suppose 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 (·)) < 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 determining 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 (Ai + Di ∆i Ei)u(t − hi ), t 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) rR ∆i : ∆i ∈ LR(Yi , Ui ), i = 0, 1, , n and (22) is not exponentially stable = inf i=0 n (DE) r+ ∆i : ∆i ∈ L+ (Yi , Ui), i = 0, 1, , n and (22) is not exponentially stable = 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 > 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∆ (·)) 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∆ (·)) i=0 n (DE) rR ||∆i|| : ∆i ∈ LR (Yi , Ui ), i = 0, n, s(P∆ (·)) ||∆i|| : ∆i ∈ L+ (Yi , Ui), i = 0, n, s(P∆ (·)) = inf i=0 n (DE) r+ = inf 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, 1, , n If n ∆i < i=1 and ∆i ∈ L(Yi, Ui ), for all i = max i,j∈{0,1, ,n} Gij (λ) , (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 max sup Gij (is) (DE) rC i,j∈{0,1, ,n} s∈R max sup Gii(is) (24) i∈{0,1, ,n} s∈R In particular, if Di = Dj or Ei = Ej for all i, j ∈ {0, 1, , n}, then (DE) rC = max sup Gii(is) (25) 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 (DE) (DE) (DE) rC = rR = r+ = max Gii(0) i∈{0,1, ,n} (26) = max Ei(−A0 − A1 − − An)−1 Di 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 W Arendt, Resolvent of positive operators, Proc London Math Soc 54 (2001) 321–349 A B´tkai and S Piazzera, Semigroups and linear partial differential equations, a J Math Anal Appl 264 (2001) 1–20 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 R Datko, Representation of solutions and stability of linear differential-difference equations in Banach space, J Different Equations 29 (1978) 105–166 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 A Fischer, D Hinrichsen, and N K Son, Stability radii of Metzler operators, Vietnam J Math 26 (1998) 147–163 J K Hale, Functional Differential Equations, Springer–Verlag, Berlin, 1971 D Hinrichsen and A J Pritchard, Stability radii of linear systems, Systems & Control Letters (1986) 1–10 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 368 B T Anh, N K Son, and D D X Thanh 11 S Nagakiri, Structural properties of functional differential equations in Banach spaces, Osaka J Math 25 (1988) 353–398 12 J van Neerven The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhăuser, Basel, 1996 a 13 P H A Ngoc and N K Son, Stability radii of positive linear functional differential equations under multi-perturbations, SIAM J Control and Optim 43 (2005) 2278–2295 14 A Pazy Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983 15 A J Pritchard and S Townley, Robustness of linear systems, J Differen Equations 77 (1989) 254–286 16 N K Son and D Hinrichsen Stability radii of positive discrete- time systems under affine perturbations, Inter J Robust and Nonlinear Control (1998) 1169– 1188 17 N K Son and D Hinrichsen, Robust stability of positive continuous-time systems, Numer Funct Anal Optim 17 (1996) 649–659 18 N K Son and P H A Ngoc, Robust stability of positive linear time-delay systems under affine parameter perturbations, Acta Math Vietnam 24 (1999) 353–371 19 N K Son and P H A Ngoc, Robust stability of functional differential equations, Adv Stud Contemp Math (2001) 43–59  ... vector (x, 0) ∈ X Applying Lemma 2.5 we have Robust Stability of Metzler Operator and Delay Equation in Lp ([ −h, 0] ; X) |ER(? ?; A)(x, 0) | ER( ? ?; A)(x, 0) |, λ > s(A) = s (P (? ?)) ER( ? ?; P ( λ))|x|,... a positive C0-semigroup and Ai ∈ L+ (X), for all i = 1, , n Then the resolvent R(? ?; P (? ?)) is positive and decreasing for t > s (P (? ?)) = s(A) : s(A) = s (P (? ?)) < t1 t2 =⇒ R(t 2; P (t2)) R(t 1; P. .. 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