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Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices
Advances in Difference Equations volume 2008, Article number: 598964 (2008)
Abstract
For positive linear Volterra difference equations in Banach lattices, the uniform asymptotic stability of the zero solution is studied in connection with the summability of the fundamental solution and the invertibility of the characteristic operator associated with the equations. Moreover, the robust stability is discussed and some stability radii are given explicitly.
1. Introduction
A dynamical system is called positive if any solution of the system starting from nonnegative states maintains nonnegative states forever. In many applications where variables represent nonnegative quantities we often encounter positive dynamical systems as mathematical models (see [1, 2]), and many researches for positive systems have been done actively; for recent developments see, for example, [3] and the references therein.
In this paper we treat the Volterra difference equations

together with

in a (complex) Banach lattice X, where is a sequence of compact linear operators on
satisfying the summability condition
, and we study stability properties of (1.1) and (1.2) under the restriction that the operators
, are positive. In fact, the restriction on
yields the positivity for the above equations (whose notion is introduced in Section 2). Also, without the restriction, in [4] the authors characterized the uniform asymptotic stability of the zero solution of (1.1), together with (1.2), in connection with the invertibility of the characteristic operator

of (1.1) for any complex numbers such that
. In Section 3, we will prove that under the restriction that the operators
, are positive, the invertibility of the characteristic operator reduces to that of the operator
, and consequently the uniform asymptotic stability of the zero solution for positive equations is equivalent to the condition which is much easier than the one for the characteristic operator in checking (Theorem 3.6). Moreover, we will discuss in Section 4 the robust stability of (1.1) and give explicit formulae of some stability radii.
2. Preliminaries
Let ,
,
,
,
,
and
be the sets of natural numbers, nonnegative integers, nonpositive integers, integers, nonnegative real numbers, real numbers and complex numbers, respectively.
To make the presentation self-contained, we give some basic facts on Banach lattices which will be used in the sequel (see, e.g., [5]). Let be a real vector space endowed with an order relation
Then
is called an ordered vector space. Denote the positive elements of
by
. If furthermore the lattice property holds, that is, if
for
then
is called a vector lattice. It is important to note that
is generating, that is,

Then, the modulus of is defined by
If
is a norm on the vector lattice
satisfying the lattice norm property, that is, if

then is called a normed vector lattice. If, in addition,
is a Banach space, then
is called a (real) Banach lattice.
We now extend the notion of Banach lattices to the complex case. For this extension all underlying vector lattices are assumed to be relatively uniformly complete, that is, if for every sequence
in
satisfying
and for every
and every sequence
in
, it holds that

Now, let be a relatively uniformly complete vector lattice. The complexification of
is defined by
The modulus of
is defined by

A complex vector lattice is defined as the complexification of a relatively uniformly complete vector lattice equipped with the modulus (2.4). If is normed, then

defines a norm on satisfying the lattice norm property; in fact, the norm restricted to
is equivalent to the original norm in
, and we use the same symbol
to denote the (new) norm. If
is a Banach lattice, then
equipped with the modulus (2.4) and the norm (2.5) is called a complex Banach lattice, and
is called the real part of
.
For Banach spaces and
, we denote by
the Banach space of all bounded linear operators from
to
equipped with the operator norm, and use the notation
in place of
. Let
and
be Banach lattices with real parts
and
, respectively. An operator
is called real if
. A linear operator
from
to
is called positive, denoted by
, if
holds. Such an operator is necessarily bounded (see [5]) and hence real. Denote by
and
the sets of real operators and positive operators between
and
, respectively:

Then we observe that

Indeed, it is clear that the inequality holds true for any , the real part of
. Let
with
and
. Since
by (2.4), we get

with and
. Then, it follows from the modulus (2.4) that

as required. We also emphasize the simple fact that

(see, e.g., [5, page 230]). Moreover, by the symbol we mean
for
Throughout this paper,
is assumed to be a complex Banach lattice with the real part
and the positive convex cone
.
For any interval we use the same notation
meaning the discrete one
, for example,
for
. Also, for an
-valued function
on a discrete interval
, its norm is denoted by
. Let
and a function
be given. We denote by
the solution
of (1.1) satisfying
on
. Similarly, for
and a function
, we denote by
the solution
of (1.2) satisfying
on
. We then recall the representation formulae of solutions for initial value problems of (1.1) and (1.2) (see [6–8]). Let
be the fundamental solution of (1.1) (or (1.2)), that is, the sequence in
satisfying

for Then, the solution
is given by

for arbitrary initial function , and also for arbitrary initial function
, the solution
of (1.2) is given by

for , where we promise
for
.
Here, we give the definition of the positivity of Volterra difference equations.
Definition 2.1.
Equation (1.1) is said to be positive if for any and
, the solution
for
. Similarly,(1.2) is said to be positive if for any
and
, the solution
for
.
Also, we follow the standard definitions for stabilities of the zero solution.
Definition 2.2.
The zero solution of (1.1) is said to be
-
(i)
uniformly stable if for any
there exists a
such that if
and
is an initial function on
with
then
for all
;
-
(ii)
uniformly asymptotically stable if it is uniformly stable, and if there exists a
such that, for any
there exists an
with the property that, if
and
is an initial function on
with
then
for all
.
Definition 2.3.
The zero solution of (1.2) is said to be
-
(i)
uniformly stable if for any
there exists a
such that if
and
is an initial function on
with
then
for all
;
-
(ii)
uniformly asymptotically stable if it is uniformly stable, and if there exists a
such that, for any
there exists an
with the property that, if
and
is an initial function on
with
then
for all
.
Here and subsequently, denotes the Z-transform of
; that is,
, which is defined for
under our assumption
. Then,
is called the characteristic operator associated with (1.1) (or (1.2)). In [6, 7], under some restrictive conditions on
, we discussed the uniform asymptotic stability of the zero solution of (1.2) in connection with the summability of the fundamental solution
, as well as the invertibility of the characteristic operator
; see also [9–12] for the case that
is finite dimensional. Moreover, we have shown in [4] the equivalence among these three properties without such restrictive conditions; more precisely, we have established the following.
Theorem 2.4 (see [4, Theorem 1]).
Let , and assume that
are all compact. Then the following statements are equivalent.
-
(i)
The zero solution of (1.1) is uniformly asymptotically stable.
-
(ii)
The zero solution of (1.2) is uniformly asymptotically stable.
-
(iii)
.
-
(iv)
For any
such that
, the operator
is invertible in
.
3. Stability for Positive Volterra Difference Equations
In this section, we will prove that the uniform asymptotic stability of the zero solution of positive Volterra difference equations (1.1) and (1.2) is, in fact, characterized by the invertibility of the operator for
. To this end we need some observations on the spectral radius of the
-transform of the convolution kernel
.
First of all, we show the relation between the positivity of Volterra difference equations and that of the sequence of bounded linear operators .
Proposition 3.1.
Equation (1.1) is positive if and only if all are positive.
Proof.
Suppose that all are positive. Then from (2.11) each element
(
of the fundamental solution is also positive; so that by virtue of (2.12),
for
, where
and
are arbitrary. Hence, (1.1) is positive. Conversely, suppose (1.1) to be positive. Then we have in particular
for
, which implies that
. Let
and for
,
be any function such that
except
. Then it follows from (2.12) that

which implies that for
. Thus,
for
since
is arbitrary.
By using (2.13) one can verify the following proposition quite similarly.
Proposition 3.2.
Equation (1.2) is positive if and only if all are positive.
In what follows, we assume that and each
is compact. For any closed operator
on
we denote by
,
, and
the spectrum, the point spectrum, and the resolvent set of
, respectively. Also denote by
the interior of the unit disk
of the complex plane. Then for the uniform asymptotic stability of the zero solution of (1.1) we have the following criterion.
Theorem 3.3.
Suppose that (1.1) is positive. If , the zero solution of (1.1) is uniformly asymptotically stable.
Proof.
In view of Theorem 2.4 it is sufficient to show that is invertible for
with
. Suppose by contradiction that
is not invertible for some
with
. Then
and hence
since
and
is compact. Let
,
be an eigenvector of the operator
for the eigenvalue
. Then,
. We generally get
for any
. Notice that the spectral radius of
is less than
by the assumption. Therefore, it follows from the well known Gelfand's formula (see, e.g., [13, Theorem 10.13]) for the spectral radius of bounded linear operators that
, which implies
. On the other hand, since
,
are positive by Proposition 3.1, we get

by (2.7); and generally,

for any . Therefore,

and it follows from the lattice norm property that

This is a contradiction, because we must get by (3.5).
The converse of Theorem 3.3 also holds. To see this we need another proposition. Let be the spectral radius of
for
.
Proposition 3.4.
Suppose that are all positive. Then,
is nonincreasing and continuous as a function on
Proof.
Let . Then,

Observe that the resolvent of
(
is given by
whenever
. Then, we deduce from (3.6) that

Note also that under our assumption is a positive operator for
since
is closed in
. In particular,
is also positive and hence for
,

by (2.7). Therefore so that

Now, let us assume that holds for some
and
with
. Since
is positive, it follows from [5, Chapter 5, Proposition 4.1] that
. Observe that if
and
, then
is invertible in
with
. Since
, the above observation leads to the fact that
as
; consequently, we get
as
. On the other hand it follows from
for
that
and the function
is continuous on
. Hence, we get
, which is a contradiction. Consequently,
for
.
We next show the left continuity of on
. If
is not left continuous at some
, we have

Since , we have
. Notice that
is an open set. Hence, it follows that if
is small enough,
that is,
for such an
. On the other hand, by virtue of [5, Chapter 5, Proposition 4.1] again, the positivity of
yields
; this is a contradiction.
is right continuous as well in
. Indeed, if it is not so, there exists a
such that

In view of the positivity of we have
. Also, since
is compact,
is an isolated point of its spectrum
; in particular, there exists an
such that
. By the continuity of
there corresponds an
such that
is invertible in
for
. Moreover, by the fact that
for
, one can see

from which readily follows because of the positivity of
. Therefore, passing to the limit
, we deduce that
. Let
be any number such that
. By the same reasoning, we see that
. Since

we get

Then, for , it follows from (2.7) that

hence

for any with
. This is a contradiction, because
as
. The proof is now completed.
Theorem 3.5.
Suppose that (1.1) is positive. If the zero solution of (1.1) is uniformly asymptotically stable, then .
Proof.
Set . To prove the theorem it is sufficient to show
. Assume that
. Since by Proposition 3.4  
is continuous, there exists
such that
, that is,
. It follows from the positivity of
, together with its compactness, that
belongs to
; hence, there exists an
such that
, or equivalently

Setting , we have

so that is a solution of (1.2). By virtue of Theorem 2.4 and our assumption, (1.2) is uniformly asymptotically stable and therefore
as
, which is impossible because
for all
. Thus we must have
.
Combining the results above with Theorem 2.4, we have, for positive Volterra difference equations, the equivalence among the uniform asymptotic stability of the zero solution of (1.1) and (1.2), the summability of the fundamental solution and the invertibility of the operator outside the unit disk.
Theorem 3.6.
Let the assumptions in Theorem 2.4 hold. If, in addition, ac.
-
(i)
The zero solution of (1.1) is uniformly asymptotically stable.
-
(ii)
The zero solution of (1.2) is uniformly asymptotically stable.
-
(iii)
-
(iv)
The operator
is invertible in
for
.
Before concluding this section, we will give an example to which our Theorem 3.6 is applicable. In [6, 7], following the idea in [14], we have shown that Volterra difference equations on a Banach space are naturally derived from abstract differential equations on
with piecewise continuous delays of type

where denotes the Gaussian symbol and
is the inifinitesimal generator of a strongly continuous semigroup
, of bounded linear operators on
, and
,
are bounded linear operators on
such that

Recall that a function with
is called a (mild) solution of (3.19) on
, if
is continuous on
, and satisfies the relation

In case of for some
, the relation above yields that

Letting , we get the Volterra difference equation

where are bounded linear operators on
defined by

for Conversely, if
satisfies (3.23) with
, then the function
extended to nonintegers
by the relation

is a (mild) solution of (3.19). Thus, abstract differential equations of type (3.19) lead to Volterra difference equations on .
Now suppose that the semigroup is compact. Then,
is continuous in
with respect to the operator norm ([15]) and also
, defined by the relation (3.24) are compact operators on
(see [6, Proposition 1]). Moreover, it follows from (3.20) that
. Moreover in the restricted case where
, are given by
for some
with
, we know by [7, Proposition 1] that the spectrum of the characteristic operator
of (3.23) is given by the formula:

Hence, in the restricted case, Theorem 2.4 implies that the zero solution of (3.23) is uniformly asymptotically stable if and only if

for all and
.
We further assume that is a complex Banach lattice, the compact semigroup
on
is positive, and that
are all nonnegative. Then the sequence
defined by (3.24) meets the assumptions in Theorem 3.6. Noticing that
, we know by Theorem 3.6 in the further restricted case that the zero solution of (3.23) is uniformly asymptotically stable if and only if

for all .
4. Robust Stability and Some Stability Radii of Positive Volterra Difference Equations
Let (1.1) be uniformly asymptotically stable, that is, the zero solution of (1.1) is uniformly asymptotically stable, and consider a perturbed difference equation of the form

where are given operators corresponding to the structure of perturbations and
is an unknown (disturbance) parameter. Here
and
are also assumed to be complex Banach lattices. Our objective in this section is to determine various stability radii of (1.1) provided that
are all positive; for this topic in case that the space
is finite dimensional, see, for example, [16] and the references therein. By the stability radius of (1.1) we mean the supremum of positive numbers
such that the uniform asymptotic stability of the perturbed (4.1) persists whenever the size of the perturbation
, measured by the
-norm
, is less than
(for precise definitions see the paragraph preceding Theorem 4.3).
Here and hereafter we also assume that for any perturbation ,
,
, are all compact, although this assumption is not necessary in the case that at least one of the operators
and
is compact. In what follows, we define
by convention.
Theorem 4.1.
Let ,
, be positive. Suppose that (1.1) is uniformly asymptotically stable and
,
are both positive. Then the perturbed (4.1) is still uniformly asymptotically stable if

We need the following lemma to prove the theorem.
Lemma 4.2.
Under the same assumptions as in Theorem 4.1 we have

Proof.
Since (1.1) is uniformly asymptotically stable, , the spectral radius of
, is less than
by Theorem 3.5. In particular, it follows that
is convergent and coincides with
. Let
,
, be given. As in the proof of Theorem 3.3, one can see
for
and
; hence
, so that
. Therefore,

The positivity of and
then implies

for , and we thus obtain

which completes the proof.
Proof.
Assume that perturbed (4.1) is not uniformly asymptotically stable for some satisfying
, that is,

Then, by Theorem 2.4 there exists a with
such that
is not invertible. So
since
is compact. Hence, there corresponds an
with
satisfying
. By virtue of the uniform asymptotic stability of (1.1) we know that
is invertible; and therefore we get

so that and
. In view of Lemma 4.2,

which gives , a contradiction to (4.7). The proof is completed.
Let be the set of all compact operators mapping
into
. We introduce three classes of perturbations defined as
,
and
. Then the complex, real and positive stability radius of(1.1) under perturbations is defined, respectively, by

where the convention is used. By definition, it is easy to see that
. On the other hand Theorem 4.1 yields the estimate

provided that the assumptions of the theorem are satisfied. In fact, these three radii coincide, that is, we have the following.
Theorem 4.3.
Let ,
, be positive. Suppose that (1.1) is uniformly asymptotically stable and
,
are both positive. Then

Proof.
By the fact , combined with (4.11), it is sufficient to prove that

We may consider the case since otherwise the theorem is trivial. Suppose by contradiction that (4.13) does not hold. Then there is an
,
, such that

By the same reasoning as in the proof of Lemma 4.2, one can see that and hence
. Also by (2.10) one may choose
with
such that

Now let . Since
, there exists a positive
,
, satisfying
(see [17, Proposition 1.5.7]). Consider a map
defined by

Then, it is easy to see that and
. Hence,

By setting , we have

Notice that because
, and that
, or equivalently
. Define a perturbation
by

Then . Moreover,

and in particular , which means, by Theorem 3.6 (or Theorem 2.4), that the perturbed (4.1) with
is not uniformly asymptotically stable. Therefore,

which contradicts (4.14). Consequently we must have (4.13), and this completes the proof.
References
Berman A, Plemmons RJ: Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematic. Academic Press, New York, NY, USA; 1979:xviii+316.
Luenberger DG: Introduction to Dynamical Systems: Theory, Models and Applications. John Wiley & Sons, New York, NY, USA; 1979.
Commault C, Marchand N (Eds): Positive Systems, Lecture Notes in Control and Information Sciences. Volume 341. Springer, Berlin, Germany; 2006.
Murakami S, Nagabuchi Y: Stability properties and asymptotic almost periodicity for linear Volterra difference equations in a Banach space. Japanese Journal of Mathematics 2005, 31(2):193-223.
Schaefer HH: Banach Lattices and Positive Operators, Die Grundlehren der Mathematischen Wissenschaften. Volume 215. Springer, New York, NY, USA; 1974:xi+376.
Furumochi T, Murakami S, Nagabuchi Y: Volterra difference equations on a Banach space and abstract differential equations with piecewise continuous delays. Japanese Journal of Mathematics 2004, 30(2):387-412.
Furumochi T, Murakami S, Nagabuchi Y: A generalization of Wiener's lemma and its application to Volterra difference equations on a Banach space. Journal of Difference Equations and Applications 2004, 10(13–15):1201-1214.
Furumochi T, Murakami S, Nagabuchi Y: Stabilities in Volterra difference equations on a Banach space. In Differences and Differential Equations, Fields Institute Communications. Volume 42. American Mathematical Society, Providence, RI, USA; 2004:159-175.
Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Application, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.
Elaydi SN: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 2nd edition. Springer, New York, NY, USA; 1999:xviii+427.
Elaydi S, Murakami S: Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type. Journal of Difference Equations and Applications 1996, 2(4):401-410. 10.1080/10236199608808074
Elaydi S, Murakami S: Uniform asymptotic stability in linear Volterra difference equations. Journal of Difference Equations and Applications 1998, 3(3-4):203-218.
Rudin W: Functional Analysis, International Series in Pure and Applied Mathematics. 2nd edition. McGraw-Hill, New York, NY, USA; 1991:xviii+424.
Wiener J: Generalized Solutions of Functional-Differential Equations. World Scientific, Singapore; 1993:xiv+410.
Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.
Ngoc PHA, Naito T, Shin JS, Murakami S: Stability and robust stability of positive linear Volterra difference equations. International Journal of Robust and Nonlinear Control. In press
Meyer-Nieberg P: Banach Lattices, Universitext. Springer, Berlin, Germany; 1991:xvi+395.
Acknowledgment
The first author is partly supported by the Grant-in-Aid for Scientific Research (C), no. 19540203, Japan Society for the Promotion of Science.
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Murakami, S., Nagabuchi, Y. Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices. Adv Differ Equ 2008, 598964 (2008). https://doi.org/10.1155/2008/598964
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DOI: https://doi.org/10.1155/2008/598964