Stability and robust stability of positive Volterra systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int J Robust Nonlinear Control 2012; 22:604–629 Published online March 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/rnc.1712 Stability and robust stability of positive Volterra systems Achim Ilchmann1 and Pham Huu Anh Ngoc2, ∗, † Institute of Mathematics, Ilmenau Technical University, Weimarer Straße 25, 98693 Ilmenau, Germany Department of Mathematics, International University, Thu Duc, Saigon, Vietnam SUMMARY We study positive linear Volterra integro-differential systems with infinitely many delays Positivity is characterized in terms of the system entries A generalized version of the Perron–Frobenius theorem is shown; this may be interesting in its own right but is exploited here for stability results: explicit spectral criteria for L -stability and exponential asymptotic stability Also, the concept of stability radii, determining the maximal robustness with respect to additive perturbations to L -stable system, is introduced and it is shown that the complex, real and positive stability radii coincide and can be computed by an explicit formula Copyright ᭧ 2011 John Wiley & Sons, Ltd Received 21 May 2009; Revised 30 December 2010; Accepted 13 January 2011 KEY WORDS: linear Volterra system with delay; positive system; Perron–Frobenius theorem; stability; stability radius INTRODUCTION We study positive linear Volterra integro-differential systems with infinitely many delays of the form t x(t) ˙ = A0 x(t)+ i Ai x(t −h i )+ B(t −s)x(s) ds for a.a t 0, (1) where ((Ai )i∈N0 , (h i )i∈N0 , B(·)) satisfy (A1) ∀i ∈ N0 : Ai ∈ Rn×n with i Ai p Since k lim s→∞ k s −(k−1) Copyright ᭧ 2011 John Wiley & Sons, Ltd Ai e i −shi ˆ + B(s) = A0 Int J Robust Nonlinear Control 2012; 22:604–629 DOI: 10.1002/rnc 610 A ILCHMANN AND P H A NGOC it follows that, for all i, j ∈ n with i = j , ⎡ k lim eT ⎣s In + s −(k−1) s→∞ i k i ˆ Ai e−shi + B(s) ⎤ ⎦ e j = eiT A0 e j Thus, A0 is a Metzler matrix Step 2: We show that Aℓ for all ℓ ∈ N Let ℓ ∈ N be fixed and consider Aℓ := (cij ) ∈ Rn×n Fix i, j ∈ n Define an L -function ⎧ 0, t0 to satisfy |[0,h ] = N N, eiT B(s)e j ds = Copyright ᭧ 2011 John Wiley & Sons, Ltd where h1 N ∀t ∈ N : eiT B(t)e j 0, this contradicts −eiT B(t)e j >0t ∈ N Hence, B(t) for a.a t ∈ [0, h ] By a similar argument, we can show that B(t) for a.a t ∈ [h , 2h ] Proceeding in this way, we obtain B(t) for a.a t ∈ [kh , (k +1)h ] and for arbitrary k ∈ N This completes the proof of the theorem Proof of Proposition 3.3 The proof of Proposition 3.3 is an immediate consequence of Lemma 3.6 combined with (7) and Theorem 3.2 PERRON–FROBENIUS THEOREM It is well known that Perron–Frobenius-type theorems are principle tools for analyzing stability and robust stability of positive systems There are many extensions of the classical Perron–Frobenius theorem, see e.g [5, 7, 16, 21, 22] and the references therein In this section, we present a Perron–Frobenius theorem for positive systems (1) This may also be interesting in its own right as a result in Linear Algebra However, we will apply the Perron– Frobenius theorem to prove stability and robustness results in Sections and Note that the assumptions (A1)–(A3) are relaxed in this section Theorem 4.1 If ((Ai )i∈N0 , (h i )i∈N0 , B(·)) satisfy n×n ˜ ∀i ∈ N : Ai ∈ R+ ( A1) , and A0 ∈ Rn×n is a Metzler matrix, ˜ ∀i ∈ N : h i 0, ( A2) n×n ˜ ( A3) B(·) : R+ → Rn×n is Lebesgue measurable and, for a.a t ∈ R+ , B(t) ∈ R+ , ∞ − t −h ˜ ( A4) := inf{ ∈ R| i e i Ai + e B(t) dt0 ∀t : X (t) (18) Note that by (17) det H(z) = for some z ∈ C− ⇒ |z| T0 := A0 + e hi ∞ Ai + i e t B(t) dt and hence ∀z ∈ C with − ℜz and |ℑz| T0 +1 : detH(z) = Since det H(·) is analytic on C− , it has at most a finite number of zeros in D := {z ∈ C|− /2 ℜz 0, |ℑz| T0 +1} and thus det H(z) = for all z ∈ C0 yields c0 := sup{ℜz|z ∈ C, det H(z) = 0}0, ∀t : X (t − −h i −u)Ai (u) du i −h i t− + X (t − −u) 0 B(u + −s) (s) ds du K e−ε(t− ) L1 (20) Then, the exponential asymptotic stability of (1) follows from (18) and the Variation of Constants formula (9) By (18), we have, for all t 0 i −h i X (t − −h i −u)Ai (u) du K e−ε(t− −h i −u) Ai (u) du i −h i eεh i Ai K i e−ε(t− ) L1 and t− X (t − −u) Copyright ᭧ 2011 John Wiley & Sons, Ltd B(u + −s) (s) ds du Int J Robust Nonlinear Control 2012; 22:604–629 DOI: 10.1002/rnc 616 A ILCHMANN AND P H A NGOC t− K e−ε(t− −u) B(u + −s) K e−ε(t− t− ) K e−ε(t− t−s ) (s) K eε(s− ) eεu B(u) du ds −s ∞ eεu B(u + −s) du ds (s) (s) ds du eεu B(u) du e−ε(t− ) L1 Now combining the above two chains of inequalities gives (20) This completes the proof of Assertion (i) n×n (ii): Assume that ((Ai )i∈N0 , (h i )i∈N0 , B(·)) satisfy (A1)–(A3), Ai ∈ R+ for all i ∈ N, B(·) and ∃M, Me− t >0 ∀t : X (t) (21) Choose ∈ (0, ) Then, (21) implies that Xˆ (·) is analytic on C− Clearly, H(z) Xˆ (z) = In , z ∈ C0 ˆ Thus, det Xˆ (0) = Since the function z → det X(z) is continuous at z = 0, there exists ∈ (0, ) ˆ −1 exists on B (0) Since the entries of Xˆ (·) ˆ = for all z ∈ B (0) Thus X(·) such that det X(z) ˆ −1 Therefore are analytic on B (0), so must be the entries of X(·) R: B (0) → Cn×n , z → R(z) := z In − A0 − Xˆ (z)−1 ˆ is analytic on B (0) Note that R(z) = i e−zh i Ai + B(z), z ∈ C0 Since A1 , A3 hold, by standard properties of the Laplace transform and of sequences of analytic functions [30, p 230], we have ∞ ◦ ∀m ∈ N∀s ∈ B (0)∩ C0 : R(m) (s) = (−1)m t m e−st B(t) dt + i −shi Ai hm i e (22) We may consider in the following, without restriction of generality, the norm n U := i, j =1 Step 1: |u ij | for U := (u ij ) ∈ Cn×n We show, by induction, that ∀m ∈ N: (t → t m B(t)) ∈ L (R+ , Rn×n ) and i hm i Ai 1 : Choose >0 B(t) (t −1) dt>M, ∃N1 ∈ N : N1 i=1 h i Ai >M + Ai (24) i sufficiently small such that ∀h ∈ (0, )∀t ∈ [0, T ] : 1−e−ht t −1, h Copyright ᭧ 2011 John Wiley & Sons, Ltd ∀i ∈ N1 : 1−e−hh i h h i −1 (25) Int J Robust Nonlinear Control 2012; 22:604–629 DOI: 10.1002/rnc STABILITY AND ROBUST STABILITY OF POSITIVE VOLTERRA SYSTEMS 617 n×n Invoking the properties Ai = (A(i) p,q ) ∈ R+ for all i ∈ N and B(·) yields, for h>0 sufficiently small ˆ ˆ R(h)−R(0) B(h)− B(0) = + h h i ∞ n = ∞ n = Bpq (t) p,q=1 e−hh i −1 Ai h Bpq (t) p,q=1 e−hhi −1 (A(i) p,q ) h e−ht −1 dt + h i 1−e−ht dt + h i 1−e−hhi (A(i) p,q ) h (26) If the first inequality in (24) is valid, then, by invoking the first inequality in (25), (26) and continuity of the norm, we arrive at the contradiction M = R′ (0) = lim h→0+ T R(h)−R(0) h B(t) (t −1) dt>M If the second inequality in (24) is valid, then, by invoking the second inequality in (25) and (26), we arrive at the contradiction n M lim h→0+ p,q=1 i N1 1−e−hhi (A(i) p,q ) h h→0+ p,q=1 i=1 1−e−hhi (A(i) p,q ) h N1 n lim (h i −1) Ai >M i=1 Therefore, (23) holds for m = If (23) holds for m, then it can be shown analogously as in the previous paragraph for m = (m) that (23) holds for m +1 by replacing B(t), Ai , R(·) by t m B(t), h m (·), resp This proves i Ai , R Step Step 2: We show (17) By (22) and (23), we have for any m ∈ N ∞ R(m) (0) = lim R(m) (s) = (−1)m s→0+ t m B(t) dt + i hm i Ai Since R(·) is analytic on B (0), Maclaurin’s series R(k) (0) k s k! k=0 ∞ is, for some absolutely convergent in B (0) Therefore >0, ∞ k k=0 k! ∞ t k Bpq (t) dt + i h ki (A(i) p,q ) = ∞ (k) |Rpq (0)| k! k=0 k