Stability and robust stability of positive Volterra systems

Stability and robust stability of positive Volterra systemsAchim Ilchmann1and Pham Huu Anh Ngoc2,∗,† 1Institute of Mathematics, Ilmenau Technical University, Weimarer Straße 25, 98693 Il

Stability and robust stability of positive Volterra systems

Achim Ilchmann1and Pham Huu Anh Ngoc2,∗,†

1Institute of Mathematics, Ilmenau Technical University, Weimarer Straße 25, 98693 Ilmenau, Germany

2Department of Mathematics, International University, Thu Duc, Saigon, Vietnam

SUMMARY

We study positive linear Volterra integro-differential systems with infinitely many delays Positivity ischaracterized in terms of the system entries A generalized version of the Perron–Frobenius theorem isshown; this may be interesting in its own right but is exploited here for stability results: explicit spectral

criteria for L1-stability and exponential asymptotic stability Also, the concept of stability radii, determining

the maximal robustness with respect to additive perturbations to L1-stable system, is introduced and it

is shown that the complex, real and positive stability radii coincide and can be computed by an explicitformula Copyright 䉷 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;

where ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy

(A1) ∀i ∈ N0: A i∈ Rn ×n with

i 0A i<∞,(A2) 0= h0<h1<h2<· · ·<h k<h k+1<· · ·,

(A3) B( ·)∈ L1(R+, Rn ×n),

and, for (, x0)∈ L1((−∞,0); Rn)×Rn, the solution of (1) may satisfy the initial data

Roughly speaking, a system is called positive if, and only if, for any nonnegative initial condition,

the corresponding solution of the system is also nonnegative In particular, a dynamical systemwith state space Rn is positive if, and only if, any trajectory of the system starting at an initialstate in the positive orthant Rn+remains in Rn+ Positive dynamical systems play an important role

in the modelling of dynamical phenomena whose variables are restricted to be nonnegative

∗ Correspondence to: Pham Huu Anh Ngoc, Department of Mathematics, International University, Thu Duc, Saigon, Vietnam.

†E-mail: phanhngoc@yahoo.com

Published online 1 March 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/rnc.1712

The mathematical theory of positive systems is based on the theory of nonnegative matricesfounded by Perron and Frobenius, see for example [1–3] Recently, problems of positive systemshave attracted a lot of attention from researchers, see for example stability [4, 5], robustness [5, 6],Perron–Frobenius theorem [7].

Positive systems have been studied in many applications, such as Economics and PopulationDynamics [2, Section 13], [3, Section 6], Biology and Chemistry [8, 9], Biology and Physiology[10, 11], Nuclear Reactors [12, p 298]

By setting B(·)≡ 0, (1) encompasses the subclass

time-By setting A i = 0 for all i ∈ N, (1) encompasses the subclass

under-The purpose of this paper is to develop a complete theory of positive systems (1), which includesthe definition of positivity and characterizations thereof, a Perron–Frobenius theorem, explicitcriteria for stability and robust stability Several results are also new for the subclasses (3) and (4).The paper is organized as follows In Section 2 we collect some well-known results on thesolution theory of (1) In Section 3 we characterize positivity in terms of the system data Ageneralized version of the classical Perron–Frobenius theorem is shown in Section 4 This resultmay be worth knowing in its own right as a result in Linear Algebra However, we utilize it forproving stability results in the following sections In Section 5 we investigate various stability

concepts and give, beside other characterizations, explicit spectral criteria for L1-stability andexponential asymptotic stability of positive linear Volterra integro-differential systems with delays

(1) Finally, Section 6 is on robustness of the L1-stability of (1) For this we introduce the concept

of complex, real and positive stability radius, show that, for positive systems, all the three areequal and present a simple formula to determine the stability radius

2 SOLUTION THEORY

In this section we recall the well-known facts on the solution theory of equations of the form (1)

Definition 2.1

Let (, x0)∈ L1((−∞,0); Rn)×Rn Then, a function x : R→ Rn is said to be a solution of the

initial value problem (1), (2) if, and only if

• x is locally absolutely continuous on [0,∞),

• x satisfies the initial condition (2) on (−∞,0],

• x satisfies (1) for almost all t ∈ [0,∞).

This solution is denoted by x ( ·; 0,, x0)

A fundamental solution for (1) is given as follows

The following proposition gives the Variation of Constants formula for (1).

For any initial data (, x0)∈ L1((−∞,0); Rn)×Rn , there exists a solution x ( ·; 0,, x0,g) : R→ Rn

of the initial value problem (6), (2); this solution is unique and, invoking the fundamental solution

X of (1), satisfies, for all t 0

has a unique solution x ( ·; ,, x0) : R→ Rn and this solution satisfies, invoking the fundamental

on a set S⊂ C where it exists, see e.g [18, p 742]

(ii) For ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfying (A1)–(A3), the function

Suppose ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy (A1)–(A3) Then, an application of the Gronwall

inequality to (7) for g ≡ 0 yields for the fundamental solution X of (1)

∃M>0∃∈ R ∀t0 : X(t)Me t.Thus applying the Laplace transform to the first equation in (5) gives

Definition 3.1

System (1) is said to be positive if, and only if, for every nonnegative initial data (, , x0)∈

R+× L1((−∞,); Rn+)×Rn+, the solution of the initial value problem (8) is nonnegative

We are now in the position to state the main result of this section

Theorem 3.2

Let ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy (A1)–(A3) Then, the system (1) is positive if, and only if

(i) A0is a Metzler matrix,

(ii) A i0 for all i∈ N,

(iii) B(·)0

Proposition 3.3

Suppose ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy (A1)–(A3) and (1) is positive Then for any nonnegative

initial data (, x0)∈ L1((−∞,0); Rn+)×Rn+ and any nonnegative inhomogenity g: R→ Rn+, the

solution of the initial value problem (6), (2) is nonnegative: x ( ·; 0,, x0,g): R→ Rn+

Remark 3.4

It may be worth noting that positivity of (1) implies monotonicity in the sense that if

(k,x k,g k)∈ L1((−∞,0); Rn+)×Rn+× L1(R+, Rn), k= 1,2satisfy

12, x1x2, g1g2

then

x (t; 0,1,x1,g1)x (t; 0,2,x2,g2) ∀t0.

This follows immediately from (7) and since X (t )0 for all t 0 by Proposition 3.3.

In the remainder of this section we prove Theorem 3.2 and Proposition 3.3 For this, some

technical lemmata are needed Throughout, we assume that ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy (A1)–(A3)

Lemma 3.5

Let 0, and consider, for nonnegative B( ·)∈ L1([0, ], Rn+×n ), g ∈ L1([0, ], Rn+), x0∈ Rn+and the

Metzler matrix A0∈ Rn ×n the initial value problem

This implies the existence of some k∗∈ N so that T k is a contraction By the contraction

mapping principle, the sequence (Tℓ ∗)ℓ ∈N converges in the space C([0, ], R n), for arbitrary

∈ C([0,],R n ) to the unique solution x ( ·; 0, x0,g ) of x = T x Choose  ≡ x0∈ C([0,],R n

+)

Since A0is a Metzler matrix, I n + A0 is nonnegative for some 0 This implies that eteA0t=

e(I n +A0)t0 for all t∈ [0,] Hence, eA0t0 for all t∈ [0,] and nonnegativity of g and B yields

Since G(t ) = 0 for all t ∈ [0,h1), Lemma 3.5 gives X (t )0 for all t ∈ [0,h1) Hence, G(t )0 for all

t ∈ [0,2h1), and a repeated application of Lemma 3.5 gives X (t )0 for all t ∈ [0,2h1) Proceeding

in this way, we arrive at X (t )0 for all t ∈ [0,kh1), for all k∈ N This completes the proof 

Proof of Theorem 3.2

‘⇐’: This direction follows immediately from Lemma 3.6 and (9)

‘⇒’:

Step 1 : We show that A0 is a Metzler matrix

By Remark 2.7 and Lemma 3.6 we have, for some ∈ R

it follows that, for all i, j ∈ n with i = j,

Thus, A0is a Metzler matrix

Step 2 : We show that Aℓ0 for all ℓ∈ N

Let ℓ∈ N be fixed and consider Aℓ:= (c ij)∈ Rn ×n Fix i, j ∈ n Define an L1-function

Since (1) is positive, the initial value problem (1), (x|(−∞,0),x(0))= (,0) has, by Proposition

2.4, a unique solution x ( ·)= x(·; 0,,0) with x(t)0 for all t0 Note that for k ∈ N, x(·)= (x1(·), , x n(·))T is locally absolutely continuous on (0, 1/ k), x ( ·)0, x(0+)= 0 and x(·) satisfies (1) almost everywhere on (0, 1/ k) Thus, invoking Newton–Leibniz’s formula, we may choose t k∈

(0, 1/ k) such that ˙x i (t k )0 and x ( ·) satisfies (1) at t k Since limk→∞˙x(t k)= Aℓ(−hℓ)= Aℓe j0,

we obtain, in particular, limk→∞˙x i (t k)= eTi Aℓe j = c ij0 Since i, j∈ n are arbitrary, it follows that

Aℓ∈ Rn+×n

Step 3: We show that

∀i, j ∈ n for a.a t ∈ R+: eTi B (t )e j0

Fix i , j ∈ n Choose

∈ L1((−∞,h1), R+) with |(−∞,0]= 0and set

Thus, invoking Newton–Leibniz’s formula, for every k ∈ N there exists t k ∈ (h1,h1+1/k) such that

eTi ˙x(t k )0 and the differential equation in (12) is satisfied at t = t k This implies that

eTi B (h1−s)e j (s) ds. (13)Assume on the contrary that

e iTB (hℓ−s)e j (s) ds0. (14)

It follows from (13) and (14) that



However, since mess(N)>0, this contradicts −eT

i B (t )e j>0t∈ N Hence, B(t)0 for a.a t ∈ [0, h1]

By a similar argument, we can show that B(t )0 for a.a t ∈ [h1,2h1] Proceeding in this way,

we obtain B(t )0 for a.a t ∈ [kh1,(k +1)h1] and for arbitrary k∈ N This completes the proof

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 5 and 6 Note that theassumptions (A1)–(A3) are relaxed in this section

i 1e−h iA i+∞

0 e−t B (t ) dt )x x ],

(iii) > 0⇐⇒ H()−10

Theorem 4.1 is a generalization of the Perron–Frobenius theorem for positive

• linear time-delay differential systems, proved in [16];

• linear Volterra integro-differential system of convolution type (4), proved in [5];

• linear systems ˙x(t)= A0x (t ), proved in [19].

The latter case is quoted next because it will be used in several proofs

Proposition 4.2

For a Metzler matrix A0∈ Rn ×n , and A i = 0 for all i ∈ N and B ≡ 0 in (1), the spectral abscissa

(see Nomenclature) satisfies

(i) (A0)= [A0,0, 0],

and thus, invoking Lemma 4.3

∀k ∈ N: kℜz k (A0+R(z k )) (A0+R(ℜz k )) (A0+R(k)), which gives, by k→ ∞

∞ lim

k→∞ (A0+R(k))= (A0)<∞,which is a contradiction

Step 2: We show that

0= (A0 0)) Since A0 0) is a Metzler matrix, we may apply Proposition 4.2(i)

0) 0> 0

Step 4: We are now ready to show (i)–(iii)

By Step 3, we have 0= (A0+R( 0)), and an application of Proposition 4.2(i) yields (i)

1 2, (A0 2)) (A0 1)), and so

0)= 0, (ii) and (iii) follow from Proposition 4.2(iii)

limt→∞x (t ; ,, x0)= 0uniformly asymptotically stable :⇔ uniformly stable and ∃>0∀ε>0∃T (ε)>0∀(,, x0)∈

R+× L1((−∞,); Rn)×Rn with L1+x0<

∀tT (ε)+ : x(t; ,, x0)<εexponentially asymptotically stable :⇔ ∃M,>0∀(,, x0)∈ R+× L1((−∞,); Rn)×

Rn ∀t : x(t; ,, x0)Me −(t−)[L1+x0]

L p -stable, p∈ [1,∞] :⇔ X ∈ L p(R+; Rn ×n ), where X denotes the fundamental

solution of (1)

The following proposition shows how these stability concepts are related and how they can

be characterized in terms of the fundamental solution X (·) and characteristic matrix H(·), seeDefinition 2.6

Proposition 5.2

Consider a system (1) satisfying (A1)–(A3) Then, the statements

(i) (1) is asymptotically stable,

(ii) (1) is L1-stable,

(iii) (1) is L p -stable for all p∈ [1,∞],

(iv) ∀z ∈ C0: det H(z)= 0, where H is the characteristic matrix of (1),

(v) (1) is uniformly asymptotically stable,

(vi) ∃M,>0∀t0 : X(t)Me −t , where X is the fundamental solution of (1),

(vii) (1) is exponentially asymptotically stable

are related as follows:

(v)⇐ (vii)⇒ (vi)⇒ (iv)⇐⇒ (iii)⇐⇒ (ii)⇒ (i)

Proof

By definition, it is easy to see that (v)⇐ (vii)⇒ (vi) and (vi)⇒ (iii) The proof of implications(iv)⇐⇒ (iii)⇐⇒ (ii) can be found in [12, p 303] It remains to show that (ii)⇒ (i) Since

the convolution of an L1-function with an L p -function belongs to L p (see e.g [26, p 172]),

it follows from (1) that ˙X ∈ L1(R+; Rn+×n ) Therefore, X , ˙ X ∈ L1(R+; Rn+×n) and [27, Lemma2.1.7] yields limt→∞X (t )= 0 Finally, an application of the Variation of Constants formula (9)

Remark 5.3

In particular, for linear Volterra integro-differential systems (4) without delay and B(·)∈

L1(R+, Rn ×n), Miller [28] has shown

(iv)⇐⇒ (v)

For (1), by a standard argument, we can show that (iv) implies that (1) is uniformly stable and

asymptotically stable Conversely, (v) implies that det H(z) = 0 for z ∈ C0 and in particular (v)implies (iv) provided supi∈Nh i<∞ However, in general, it is an open problem whether theimplications

(iv)⇒ (v) or (v)⇒ (iv)hold true

Finally, Murakami [25] showed that even for (4)

(iv)⇒ (vi) and (v) ⇒ (vi)

These implications are claimed in [17, Theorem 2.2.2]; the proof in [17] however is based on thesemigroup property of the fundamental solution; in Remark 2.3 we showed that this property doesnot hold

Next, we present a sufficient condition under which most of the statements in Proposition 5.2are equivalent

then the statements (ii), (iii), (iv), (vi) and (vii) in Proposition 5.2 are equivalent

(ii) If ((A i)i∈N 0,(h i)i∈N 0,B(·)) satisfy (A1)–(A3), A i∈ Rn+×n for all i ∈ N, B(·)0 and (vi) in

Proposition 5.2 holds, then (17) is valid

Theorem 5.4 extends [25, Theorems 1 and 2] and [29, Theorems 3.1 and 3.2], where linearintegro-differential systems without delay (4) are considered, to the linear Volterra integro-differential systems with delays (1) The proof of Theorem 5.4(i) is different from [25], the proof

of Theorem 5.4(ii) is based on ideas of the proof of [29, Theorem 3.2]

Proof of Theorem 5.4

(i) In view of Proposition 5.2, it suffices to show ‘(i v) ⇒ (vii)’ Let X(·) and H(·) be the

fundamental solution and the characteristic matrix of (1), respectively

∀z ∈ C with −ℜz0 and |ℑz|T0+1 : detH(z)= 0.

Since det H(·) is analytic on C−, it has at most a finite number of zeros in

˙y(t)= (A0+εI n )y(t )+

i 1

eεh i A i y (t −h i)+

 t

0

eε(t−s) B (t −s)y(s)ds for a.a t0. (19)

Since det H(z) = 0 for all z ∈ C−ε, it follows that det Hε(z) = 0 for all z ∈ C0 Applying Proposition

5.2 to (19), we may conclude that Y (·)= eε ·X(·)∈ L∞(R+; Rn ×n) This gives (18)

is analytic on B 0(0) Note that R(z)=

i 1e−zh i A i + ˆB(z), z ∈ C0 Since A1, A3hold, by standardproperties of the Laplace transform and of sequences of analytic functions [30, p 230], we have

Invoking the properties A i = (A(i) p,q)∈ Rn+×n for all i ∈ N and B(·)0 yields, for h>0 sufficiently

If the second inequality in (24) is valid, then, by invoking the second inequality in (25) and (26),

we arrive at the contradiction

Therefore, (23) holds for m= 1

If (23) holds for m, then it can be shown analogously as in the previous paragraph for m= 1

that (23) holds for m +1 by replacing B(t), A i, R(·) by t m B (t ), h m i A i, R(m)(·), resp This provesStep 1

of Theorem 5.4(ii) is based on ideas of the proof of [29, Theorem...

Proof of Theorem 5.4

(i) In view of Proposition 5.2, it suffices to show ‘(i v) ⇒ (vii)’ Let X(·) and H(·) be the

fundamental solution and the characteristic matrix of. .. B(·)0 and (vi) in

Proposition 5.2 holds, then (17) is valid

Theorem 5.4 extends [25, Theorems and 2] and [29, Theorems 3.1 and 3.2], where linearintegro-differential systems