doi:10.1155/2010/869608
Research Article
Stability of Difference Equations and Applications
to Robustness Problems
Bogdan Sasu
Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis¸oara, C. Coposu Blvd. No. 4, 300223 Timis¸oara, Romania
Correspondence should be addressed to Bogdan Sasu,[email protected]
Received 3 November 2009; Accepted 23 February 2010
Academic Editor: Alberto Cabada
Copyrightq2010 Bogdan Sasu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to obtain new necessary and sufficient conditions for the uniform exponential stability of variational difference equations with applications to robustness problems. We prove characterizations for exponential stability of variational difference equations using translation invariant sequence spaces and emphasize the importance of each hypothesis. We introduce a new concept of stability radiusrstabA;B, C for a variational system of difference
equationsAwith respect to a perturbation structureB, Cand deduce a very general estimate for the lower bound ofrstabA;B, C. All the results are obtained without any restriction concerning
the coefficients, being applicable for any system of variational difference equations.
1. Introduction
In the last decades an increasing interest was focused on the asymptotic properties of the most general class of evolution equations—the variational systems and a number of open questions were answered, increasing the applicability area not only to partial differential equations but also to systems arising from the linearization of nonlinear equationssee1–10 and the references therein. In this context a special attention was devoted to the general case of variational systems of difference equations of the form:
xθn1 Aσθ, nxθn, ∀θ, n∈Θ×N, A
no measurability or continuity conditions are needed, this system often models the families of equations proceeding from the linearization of nonlinear equations and also extends the nonautonomous case in infinite-dimensional spacessee1,2,4–10and the references therein.
In recent years a notable progress was made in the study of the qualitative properties of various classes of difference equations see4–8, 11–32. The input-output methods or the so-called “theorems of Perron type” have proved to be important tools in the study of the asymptotic behavior of difference equations like stabilitysee27,29,30, expansiveness see6, dichotomysee1,5,8,15,17,19,20,32, and trichotomy as wellsee7,18,20. A distinct method for the study of exponential stability relies on the convergence of some associated series and this was used in28for difference equations and in4for variational difference equations. The exponential stability of difference equations with several variable delays and variable coefficients was studied in16, where the authors obtained interesting conditions for global exponential stability using new computational formulas with respect to the coefficients. The uniform asymptotic stability of positive Volterra difference equations was recently studied in 25, the authors proving the equivalence between the uniform asymptotic stability of the zero solution, the summability of the fundamental solution, and the invertibility of an associated operator outside the unit disk. A very efficient method in the study of the stability of difference equations is represented by the so-called “freezing technique,” which was used in 22 for the study of absolute stability of discrete-time systems with delay and also in23in order to deduce explicit conditions for global feedback exponential stabilizability of discrete-time control systems with multiple state delays. In the study of the asymptotic behavior of discrete-time systems, there is an increasing interest in finding methods arising from control theorysee5–8,21–27,29–32. This is motivated by the fact that besides their large applicability area, the control-type techniques can be also applied to the analysis of the robustness of diverse properties in the presence of perturbations see5,25,26,30,32,33. In this context it is natural to extend the study to the variational case. Thus, two main questions arise: which is the most general framework for the study of the stability of variational difference equations using control type methods and how one may apply the new techniques in order to determine the behavior of the initial system in the presence of perturbations. In what follows, our attention will focus in order to provide complete answers to these open questions.
In this paper we propose a new study concerning the stability of variational difference equations and the robustness of this property. We associate with the system A a family of control systems SA {Sθ}θ∈Θ and we attack the subject from the perspective of
the solvability of SA between two Banach sequence spaces invariant under translations.
We split the class of Banach sequence spaces which are invariant under translations into two central subclasses and deduce necessary and sufficient conditions for uniform and exponential stability with respect to the solvability of the control system SA when the
Next, we apply the stability results and we propose a new approach for stability robustness of variational difference equations. We study the stable behavior of the system A in the presence of a general perturbation structure B, C with B ∈ ∞Θ,LU, X,
C ∈ ∞Θ,LX, Y, by means of the stability radius rstabA;B, C see Definition 4.3. Our target is to obtain a lower bound for the stability radius of variational systems of difference equations as well as to determine the largest class of Banach sequence spaces within the robustness properties hold. With this purpose we associate with the system A an input-output control system SA {Sθ}θ∈Θ and consider WN the general
class of all Banach sequence spaces W with the property that if there is M > 0 such that |s · χ{0,...,n}|W ≤ M, for all n ∈ N, then s ∈ W and |s|W ≤ M. For every
Banach sequence space W ∈ WN, we introduce the index λWA;B, C : supθ∈ΘΛθW,
where {Λθ
W}θ∈Θ is the family of input-output operators associated with the system SA
and we obtain a lower bound for rstabA;B, C in terms of λWA;B, C. Thus, we point
out an interesting connection between the family of input-output operators and the size of the smallest perturbation in the presence of which the perturbed system loses its exponential stability. The variational case requires a special analysis and the methods are substantially more complicated compared to those used in the nonautonomous case see 30, 33. We note that the study is done without any restriction or assumption on the coefficients, the obtained results being applicable for any system of variational difference equations.
2. Banach Sequence Spaces
In this section, for the sake of clarity, we will recall some basic definitions and properties of Banach sequences spaces. LetZdenote the set of the integers, letNdenote the set of all non negative integers, letRdenote the set of all real numbers, and letSN,Rbe the linear space of all sequences s : N → R. Let N∗ N\ {0}. For every setA ⊂ Nwe denote by
χA the characteristic function of the setA. For everys ∈ SN,Rwe consider the sequence
s:N → Rdefined bys0 0 andsn sn−1, for alln∈N∗.
Definition 2.1. A linear spaceB⊂ SN,Ris called anormed sequence spaceif there is a mapping | · |B:B → Rsuch that
i|s|B 0 if and only ifs 0;
ii|αs|B |α||s|B, for allα, s∈R×B;
iii|sγ|B≤ |s|B|γ|B, for alls, γ ∈B;
ivifs, γ ∈ SN,Rhave the property that|sj| ≤ |γj|, for allj ∈Nandγ ∈B, then
s∈Band|s|B≤ |γ|B.
If, moreover,B,| · |Bis complete, thenBis calledBanach sequence space.
Definition 2.2. A Banach sequence spaceB,| · |B is calledinvariant under translationsif for
everys∈Bthe sequencesbelongs to Band|s|B |s|B.
Example 2.3Orlicz sequence spaces. Letϕ :R → 0,∞be a nondecreasing left continuous function which is not identically 0 or∞on0,∞. TheYoung functionassociated with ϕis
Yϕt
Remark 2.5. IfB∈ QN, then the following properties hold: ifor everyA⊂N,χA ∈B;
sn → uinB. According toLemma 2.7we deduce thatu s, and sos∈B. This implies that
Another interesting property of the classVNis the following.
Lemma 2.10. Let ϕN,R be an Orlicz space. Then either ϕN,R ∈ VN or ϕN,R
Notation. We denote by WNthe class of all Banach sequence spaces B ∈ QN with the property that ifs∈ SN,Rand there isM >0 such that|s·χ{0,...,n}|B≤M, for alln∈N, then
s∈Band|s|B≤M.
Example 2.12. Any Orlicz sequence space Oϕ belongs to the class WN. Consequently,
Remark 2.13. The classWNwill play a fundamental role for a new study of the robustness of uniform exponential stability presented inSection 4.
Notation. LetX, · be a real or complex Banach space. For every Banach sequence space
B∈ QNwe denote byBN, Xthe space of all sequencess:N → Xwith the property that the mappingNs : N → R, Nsm smbelongs toB.BN, Xis a Banach space with
respect to the normsBN,X: |Ns|B.
3. Stability of Variational Difference Equations
LetX be a real or complex Banach space and letLXbe the Banach algebra of all bounded linear operators onX. Throughout this paper the norm on X will be denoted by · . For everyT ∈ LX, the norm ofTis defined by
T: inf{M >0 :Tx ≤Mx, ∀x∈X} sup
x≤1Tx. 3.1
We denote byΔN, Xthe linear space of all sequencess:N → Xwith the property that the set{j ∈N:sj/0}is finite and byIdthe identity operator onX.
LetΘ, dbe a metric space and letE X×Θ. Letσ:Θ×Z → Θbe a discrete flow on Θ, that is,σθ,0 θandσθ, mn σσθ, m, n, for allθ, m, n∈Θ×Z2.
Let{Aθ}θ∈Θ ⊂ LX. We consider the variational system of variational difference equations A. We note that in the particular case when Θ N and σθ, n θn we obtain the case of difference equations. There are several distinct directions of generalizing the case of difference equations. One of the most interesting methods is to consider them in the general framework of dynamic equations on time scales see 35, 36, having a wide potential for applications in the study of population dynamics. Another method is to consider them as particular cases of variational difference equations, which often proceed from the linearization of nonlinear equationssee2,10and the references therein. It is also interesting to note that the exponential stability of a variational equation is equivalent with the exponential stability of the variational difference equation associated with it. Therefore, concerning the stability of variational equations it is recommended to study the discrete-time case, because no measurability or continuity conditions are required.
The discrete cocycle associated with the systemAis
Φ:Θ×N−→ LX, Φθ, n
⎧ ⎨ ⎩
Aσθ, n−1· · ·Aθ, n∈N∗,
Id, n 0.
3.2
Remark 3.1. The discrete cocycle satisfiesΦθ, mn Φσθ, n, mΦθ, n, for allθ, m, n∈
Definition 3.2. The systemAis said to be
iuniformly stableif there isK >0 such thatΦθ, n ≤K, for allθ, n∈Θ×N; iiuniformly exponentially stableif there areK, ν >0 such thatΦθ, n ≤Ke−νn, for all
θ, n∈Θ×N.
We associate with the systemAthe input-output systemSA {Sθ}θ∈Θ, where for everyθ∈Θ,
xθn1 Aσθ, nxθn sn, n∈N,
xθ0 0.
Sθ
Remark 3.3. For everyθ, s∈Θ×ΔN, X, the solution of2.1has the form:
xθ,sn n
k 1
Φσθ, k, n−ksk−1, ∀n∈N∗. 3.3
Definition 3.4. LetU, V be two Banach sequence spaces withU, V ∈ QN. The systemSAis
said to beUN, X, VN, X-stableif the following assertions hold:
ifor everys∈ΔN, Xandθ∈Θthe solutionxθ,sbelongs to VN, X;
iithere isL >0 such thatxθ,sVN,X≤LsUN,X, for allθ, s∈Θ×ΔN, X.
UN, Xis called the input spaceandVN, Xis called the output space.
Our first result provides a sufficient condition for uniform stability and is given by the following.
Theorem 3.5. LetU, V ∈ QN. If the systemSAisUN, X, VN, X-stable, then the system
Ais uniformly stable.
Proof. LetL >0 be given byDefinition 3.4. Letθ∈Θand letx∈X. We consider the sequence
s:N−→X, sn χ{0}nx. 3.4
Then,s∈ΔN, XandsUN,X FU1x. We observe thatxθ,sn Φσθ,1, n−1x, for
alln∈N∗. Letp∈N. Then from
Φ
σθ,1, pxχ{p1}k≤ xθ,sk, ∀k∈N, 3.5
we obtain that
Φ
SettingK LFU1/FV1it follows that
Φσθ,1, nx ≤Kx, ∀n∈N. 3.7
Taking into account thatKdoes not depend onθorx, from3.7we have that
Φσθ,1, n ≤K, ∀n∈N, ∀θ∈Θ. 3.8
Letθ∈Θ. By applying relation3.8forθ σθ,−1, we deduce thatΦθ, n ≤K, for all
n∈Nand thus the proof is complete.
Corollary 3.6. The systemA is uniformly stable if and only if the system SAis ∞N, X,
1N, X-stable.
Proof. Necessity is immediate via Definition 3.4 and sufficiency follows fromTheorem 3.5.
The first main result of this section is the following.
Theorem 3.7. LetU, V ∈ QNbe such that eitherU∈ UNorV ∈ VN. If the systemSAis
UN, X, VN, X-stable, then the systemAis uniformly exponentially stable.
Proof. LetL > 0 be given byDefinition 3.4. FromTheorem 3.5it follows that there isM >0 such that
Φθ, nx ≤Mx, ∀θ, n∈Θ×N, ∀x∈X. 3.9
Case 1. If U ∈ UN then, according toRemark 2.5ivwe deduce that there is a sequence
δ:N → Rwithδ∈U\1N,R. Leth∈N∗be such that
h
j 0
δj≥ eLM|δ|U
FV1 . 3.10
Letθ∈Θand letx∈X. We consider the sequence
s:N−→X, sn χ{0,...,h}nδnΦθ, n1x. 3.11
We have thats∈ ΔN, Xand using3.9we obtain thatsn ≤ Mxδn, for alln∈N, which implies that
sUN,X≤Mx|δ|U. 3.12
According to our hypothesis we deduce that
We observe that
xθ,sh1
⎛
⎝h
j 0
δj
⎞
⎠Φθ, h1x. 3.14
Then from
xθ,sh1χ{h1}n≤ xθ,sn, ∀n∈N 3.15
and using relations3.13and3.14we obtain that
⎛
⎝h
j 0
δj
⎞
⎠Φθ, h1xFV1≤ xθ,s
VN,X≤LMx|δ|U. 3.16
From relations3.10and3.16it follows thatΦθ, h1x ≤1/ex. Settingp h1 and taking into account thatpdoes not depend onθorxwe have that
Φ
θ, px≤
1
e
x, ∀θ∈Θ, ∀x∈X. 3.17
Letν 1/pand letK Me. Letθ∈Θand letn∈N. Then there arek∈Nandj ∈ {0, . . . , p−1} such thatn kpj. Using relations3.9and3.17we have that
Φθ, n ≤MΦθ, kp≤Me−k≤Ke−νn. 3.18
Case 2. IfV ∈ VNthen there isp∈N∗such that
FV
p≥eLM2FU1. 3.19
Letθ∈Θand letx∈X. We consider the sequence
u:N−→X, un χ{0}nΦθ,1x. 3.20
Thenu∈ΔN, Xand according to relation3.9we have that
uUN,X≤MFU1x. 3.21
It is easy to see thatxθ,un Φθ, nx, for alln∈ N∗. Then, using relation3.9we deduce
that
Φ
which implies that
The second main result of this section is the following.
Theorem 3.8. LetU, V ∈ QNbe such that eitherU∈ UNorV ∈ VN. The following assertions
Proof. iThis follows fromTheorem 3.7.
In what follows, we prove that the result obtained inTheorem 3.8is the most general in this topic. Precisely, we will show that ifU /∈ UNandV /∈ VN, then theUN, X, VN, X -stability of the systemSAdoes not assure the uniform exponential stability of the system
A.
Example 3.9. LetΘ Rand letσ :Θ×Z → Θ,σθ, m θm. Thenσis a discrete flow on Θ. We consider the function
ϕ:R−→0,∞, ϕt
⎧ ⎨ ⎩
t1, t≥0,
et, t <0. 3.28
It is easy to see thatϕis nondecreasing with limt→ ∞ϕt ∞. LetXbe a Banach space. For everyθ∈Θ, let
Aθ:X −→X, Aθx ϕθ
ϕθ1x. 3.29
The discrete cocycle associated with the systemAis
Φθ, nx ϕθ
ϕθnx, ∀x∈X, ∀θ, n∈Θ×N. 3.30
We associate with the systemAthen input-output systemSA {Sθ}θ∈Θ. Using3.30it follows that
xθ,sn n
k 1
ϕθk
ϕθnsk−1, ∀n∈N
∗, ∀θ, s∈Θ×ΔN, X. 3.31
Let U, V ∈ QN with U /∈ UN and V /∈ VN. Then, fromRemark 2.5 we have thatU 1N,Rand fromRemark 2.9we obtain thatc
0N,R⊂V. Then, there isL >0 such that
|w|V ≤Lw∞, ∀w∈c0N,R. 3.32
We prove that the system SA isUN, X, VN, X-stable. Indeed, letθ ∈ Θand let s ∈
ΔN, X. Then, there ish∈N∗such thatsj 0, forj ≥h. From3.31we have that
xθ,sn
1
ϕθn
h1
k 1
Sinceϕt → ∞ast → ∞, we obtain thatxθ,sn → 0 asn → ∞. This shows thatxθ,s ∈
c0N, X, soxθ,s∈VN, X. Moreover, from
xθ,sn ≤ n
k 1
sk−1 ≤ s1N,X, ∀n∈N, 3.34
we have that
xθ,sc0N,X≤ s1N,X. 3.35
From relations3.32and3.35it follows that
xθ,sVN,X≤Ls1N,X. 3.36
Taking into account that L does not depend on θ or s it follows that the system SA is
1N, X, VN, X-stable. But, for all that, it is easy to verify that there are notK, ν >0 such thatΦθ, n ≤Ke−νn, for allθ, n∈Θ×N, so the systemAis not uniformly exponentially
stable.
Corollary 3.10. Letp, q∈1,∞withp, q/ 1,∞. The following assertions hold:
iif the system SA is pN, X, qN, X-stable, then the system A is uniformly
exponentially stable;
iiifp ≤ q, then the systemAis uniformly exponentially stable if and only if the system
SAispN, X, qN, X-stable.
Corollary 3.11. LetW ∈ QN. The systemAis uniformly exponentially stable if and only if the
systemSAisWN, X, WN, X-stable.
Proof. This follows fromTheorem 3.8observing that ifW /∈ UN, thenW 1N,R, and so
W∈ VN.
Definition 3.12. LetU, V ∈ QN. The systemSAis said to be completelyUN, X, VN, X
-stableif the following properties hold:
ifor everys∈UN, Xand everyθ∈Θthe solutionxθ,sbelongs to VN, X;
iithere isL >0 such thatxθ,sVN,X≤LsUN,X, for allθ, s∈Θ×UN, X.
Theorem 3.13. LetU, V ∈ QNwithU∈ UNorV ∈ VN. The following assertions hold:
iif the system SA is completely UN, X, VN, X-stable, then the system A is
uniformly exponentially stable;
iiifU⊂ V, then the systemAis uniformly exponentially stable if and only if the system
SAis completelyUN, X, VN, X-stable.
Corollary 3.14. Letp, q∈1,∞withp, q/ 1,∞. The following assertions hold:
iif the system SA is completely pN, X, qN, X-stable, then the system A is
uniformly exponentially stable;
iiifp ≤ q, then the systemAis uniformly exponentially stable if and only if the system
SAis completelypN, X, qN, X-stable.
Corollary 3.15. LetW ∈ QN. The systemAis uniformly exponentially stable if and only if the
systemSAis completelyWN, X, WN, X-stable.
Remark 3.16. Let W ∈ QN. If the system A is uniformly exponentially stable, then according toCorollary 3.15, for everyθ∈Θ, the operator
Γθ:WN, X−→WN, X, Γθs xθ,s 3.37
is correctly defined and this is a bounded linear operator. Moreover, ifL > 0 is given by Definition 3.12, then we have that supθ∈ΘΓθ ≤L.
Remark 3.17. From the above results it follows that in the study of exponential stability of variational difference equations using input-output techniques one may work with Banach sequence spaces which are invariant under translations, such that either the input space contains at least a sequence whose series is divergentor the output space has unbounded fundamental function. Moreover, according toCorollary 3.11we deduce that when the input space and the output space coincide, then there is no other requirement on the underlying sequence spaces and it is sufficient to consider any Banach sequence space which is invariant under translations and contains at least a characteristic function of a singleton.
4. Applications to Robustness of Exponential Stability
In this section, by applying our main results we will study the persistence of the exponential stability in the presence of variational structured perturbations.
Notations. If Y, Z are two Banach spaces, we denote by LY, Z the Banach space of all bounded linear operatorsT : Y → Z. If Y Z, we denote LY, Y LY. If Θ, d is a metric space, we consider∞Θ,LY, Z : {R : Θ → LY, Z | supθ∈ΘRθ < ∞}, which is a Banach space with respect to the norm
|R|: sup
LetΘ, dbe a metric space and letσ : Θ×Z → Θbe a discrete flow on Θ. LetX
be a Banach space and let{Aθ}θ∈Θ ⊂ LX. We consider the linear system of variational difference equationsA. For everyD∈∞Θ,LXwe consider the perturbed system
xθn1 Aσθ, n Dσθ, nxθn, θ, n∈Θ×N. AD
Remark 4.1. IfΦAis the discrete cocycle associated to the systemAandΦADis the discrete
cocycle associated to the system4.2, then the following perturbation formulas are satisfied:
ΦADθ, n ΦAθ, n n
k 1
ΦAσθ, k, n−kDσθ, k−1ΦADθ, k−1,
ΦADθ, n ΦAθ, n n
k 1
ΦADσθ, k, n−kDσθ, k−1ΦAθ, k−1
4.2
for allθ, n∈Θ×N∗.
In what follows we suppose that the systemA is uniformly exponentially stable. The main question is how large may be the norm of the perturbationD∈∞Θ,LXsuch that the perturbed system4.2remains uniformly exponentially stable. With this purpose we introduce in the following.
Definition 4.2. The number
rstabA: sup{r >0 : ∀D∈∞Θ,LX
with|D|< r ⇒ADis uniformly exponentially stable 4.3
is called the stability radiusof the systemA.
We note that in the existent literature there is not an explicit computational formula for the stability radius of systems, only in some sporadic special cases. For linear retarded systems of differential equations onRn, some interesting formulas were obtained by Ngoc
and Son in26, Theorems 3.5 and 3.10, but the estimates are still complicated. For the case of positive linear retarded systems which are Hurwitz stable, the authors succeeded to deduce a formula for the stability radius corresponding to multiaffine perturbations see Theorem 4.7 and the following example. In 25 Murakami and Nagabuchi obtained an explicit formula for the stability radius of uniformly asymptotically stable positive Volterra difference equations on Banach latticessee25, Theorem 4.3. Generally, in order to analyze the persistence of the exponential stability in the presence of perturbations, it is interesting to find a lower bound for the stability radius of systems see 3, 25,26, 30, 33 because in this manner we estimate the possible size of the disturbance operator under which the additivelyperturbed system remains exponentially stable.
In what follows, using the results obtained in the previous section we will estimate a lower bound for the stability radius of the systemA. We will provide a detailed study, when the systemAis subject to a very general perturbation structure.
Definition 4.3. The number
rstabA;B, C: sup{r >0 :∀P ∈∞Θ,LY, U
with|P|< r ⇒ABP Cis uniformly exponentially stable 4.4
is calledthe stability radiusof the systemAsubject to the perturbation structureB, C.
Remark 4.4. In particular, ifU Y XandBIθ CIθ Id, for allθ∈Θ, then
rstabA;BI, CI rstabA. 4.5
Up to now, there is not an explicit formula for the computation of the stability radius of variational systems. Based on our stability results, we will deduce a lower bound for
rstabA;B, C. With this purpose we associate with the systemAthe input-output control systemSA {Sθ}θ∈Θ, where for everyθ∈Θ,
xθn1 Aσθ, nxθn Bσθ, nun, n∈N
xθ0 0
yθn Cσθ, nxθn, n∈N
Sθ
Let W ∈ WN. Since the system A is uniformly exponentially stable, according to Corollary 3.15there is L > 0 such that for everyθ, s ∈ Θ×WN, Xthe corresponding solution
xθ,sn n
k 1
ΦAσθ, k, n−ksk−1, n∈N∗ 4.6
has the property
xθ,sWN,X ≤LsWN,X. 4.7
Letθ∈Θand letu∈WN, U. Then, the sequence
su:N−→X, sun Bσθ, nun 4.8
has the property
This implies thatsu ∈WN, X. According to4.7we have thatxθ,su ∈ WN, Xand from
4.7and4.9it follows that
xθ,suWN,X≤L|B| uWN,U. 4.10
Observing that the solutionyθ,uof the system2.2has the property
yθ,un Cσθ, nxθ,sun, ∀n∈N 4.11
and since
yθ,un≤ |C| xθ,sun, ∀n∈N, 4.12
we deduce thatyθ,u∈WN, Yand from4.10and4.12we have that
yθ,u
WN,Y≤λuWN,U, 4.13
whereλ |C|L|B|. Taking into account thatλdoes not depend onθoruit follows that
yθ,u
WN,Y≤λuWN,U, ∀θ, u∈Θ×WN, U. 4.14
In this context, for everyθ∈Θ, it makes sense to consider the operator
Λθ
W :WN, U−→WN, Y, ΛθWu yθ,u. 4.15
We have thatΛθW is a correctly defined and bounded linear operator. Moreover, from4.14 we have that
λWA;B, C: sup θ∈Θ
Λθ
W<∞. 4.16
Remark 4.5. The family{Λθ
W}θ∈Θis called the family of input-output operatorsassociated to the systemSA.
Remark 4.6. For everyθ, u∈Θ×WN, Uwe have that
yθ,u
WN,Y≤λWA;B, CuWN,U. 4.17
In what follows we suppose that there isδ > 0 such thatCθx ≥ δx, for all θ, x∈Θ×X.
Theorem 4.7. IfP ∈∞Θ,BY, Uhas the property that
|P|< 1
λWA;B, C,
then for everyθ, x∈Θ×X, the sequence
γθ,x:N−→X, γθ,xn ΦABP Cθ, nx 4.19
belongs toWN, X. Moreover, there isM >0such that
γθ,x
WN,X ≤Mx, ∀θ, x∈Θ×X. 4.20
Proof. LetK, ν >0 be such that
ΦAθ, n ≤Ke−νn, ∀θ, n∈Θ×N. 4.21
LetP ∈∞Θ,BY, Ube such that|P|λWA;B, C <1. We setα: |P|λWA;B, C. Let
θ, x∈Θ×X. For everyn∈N∗we denote by
γn :N−→X, γnk χ{0,...,n}kγθ,xk,
qn:N−→Y, qnk Cσθ, kγnk,
un:N−→U, unk Pσθ, kqnk.
4.22
Letn∈N∗. We have thatun∈ΔN, U, soun∈WN, U. Since
unk ≤ Pσθ, kqnk≤ |P|qnk, ∀k∈N, 4.23
we deduce thatunWN,U≤ |P| qnWN,Y. Fromun∈WN, U, usingRemark 4.6we have
that
yθ,u
nWN,Y≤λWA;B, CunWN,U≤λWA;B, C|P|qnWN,Y αqnWN,Y. 4.24
FromRemark 4.1we obtain that
qnk Cσθ, kΦAθ, kxyθ,unk, ∀k∈ {1, . . . , n}, 4.25
which via4.21implies that
qnk≤ |C|Ke−νkxy
θ,unk, ∀k∈N
∗. 4.26
Sinceqn0 Cθx ≤ |C|x, using4.26we deduce that
qnk≤ |C|Kxeνk yθ,u
whereeν : N → R,eνk e−νk. Sinceeν ∈ 1N,R, we have thateν ∈ W. Then, setting
m |C|K|eν|W from4.27and4.24it follows that
qn
WN,Y≤mxyθ,unWN,Y≤mxαqnWN,Y, 4.28
which implies that
qn
WN,Y≤
m
1−αx. 4.29
According to our hypothesis we have thatδγnk ≤ qnk, for allk∈N, soδγnWN,X ≤
qnWN,Y. SettingM m/δ−αδ, from4.29we deduce thatγnWN,X≤Mx. SinceM
does not depend onn,θ, orx, we have that
γθ,xχ{0,...,n}
WN,X≤Mx, ∀n∈N, ∀θ, x∈Θ×X. 4.30
Taking into account thatW ∈ WNfrom4.30it follows thatγθ,x ∈WN, X, for allθ, x∈
Θ×Xandγθ,xWN,X≤Mx, for allθ, x∈Θ×X.
The main result of this section is the following.
Theorem 4.8. The following estimate holds:
rstabA;B, C≥ 1
λWA;B, C. 4.31
Proof. SinceAis uniformly exponentially stable, according toCorollary 3.11we have that the associated systemSA {Sθ}θ∈Θ see p. 7 isWN, X, WN, X-stable. So there is
LA >0 such that
xθ,sWN,X≤LAsWN,X, ∀θ, s∈Θ×ΔN, X. 4.32
Let P ∈ ∞Θ,BY, U with |P|λWA;B, C < 1. Denote by α : |P|λWA;B, C. Let
M >0 be given byTheorem 4.7.
We consider the systemSABP C {SPθ}θ∈Θ, where
zθn1 Aσθ, n BP Cσθ, nzθn sn, n∈N,
zθ0 0
SPθ
Lets∈ΔN, Xand letθ∈Θ. Sinces∈ΔN, X, there ish≥2 such thatsj 0, for allj≥h. Then we observe that
zθ,sn h
k 1
ΦABP Cσθ, k, n−ksk−1
ΦABP Cσθ, h, n−h h
k 1
ΦABP Cσθ, k, h−ksk−1
ΦABP Cσθ, h, n−hzθ,sh, ∀n≥h.
4.33
According toTheorem 4.7the sequence
γ:N−→X, γn ΦABP Cσθ, h, nzθ,sh 4.34
belongs toWN, Xand
γ
WN,X≤Mzθ,sh. 4.35
SinceWis invariant under translations, we have that
γh:N−→X, γhn
⎧ ⎨ ⎩
γn−h, n≥h,
0, n∈ {0, . . . , h−1} 4.36
belongs toWN, Xand
γh
WN,X γWN,X. 4.37
Then, from
zθ,sn ≤χ{0,...,h−1}nzθ,snγhn, ∀n∈N, 4.38
we deduce thatzθ,s ∈ WN, X. In what follows, we prove that the second condition from
From4.43we obtain that
Cσθ, nzθ,sn ≤ |C|
1−αxθ,sn. 4.44
SinceCσθ, nzθ,sn ≥δzθ,sn, using4.44it follows that
zθ,sn ≤ |C|
δ1−αxθ,sn, ∀n∈ {2, . . . , h}. 4.45
We note thatzθ,s0 xθ,s0 0 andzθ,s1 xθ,s1 s0. Then, settingq max{|C|/δ−
αδ,1}from relation4.45we have that
zθ,sn ≤qxθ,sn, ∀n∈ {0, . . . , h}. 4.46
Then, from relations4.38and4.46we deduce that
zθ,sn ≤qxθ,snγhn, ∀n∈N. 4.47
This implies that
zθ,sWN,X≤qxθ,sWN,XγhWN,X. 4.48
From relations4.35,4.37, and4.46we obtain that
γh
WN,X≤Mzθ,sh ≤Mqxθ,sh. 4.49
From
xθ,shχ{h}n≤ xθ,sn, ∀n∈N 4.50
using the translation invariance ofWwe have that
xθ,shFW1≤ xθ,sWN,X. 4.51
Then from relations4.48–4.51it follows that
zθ,sWN,X≤q
1 M
FW1
xθ,sWN,X. 4.52
Finally, settingL qLA1 M/FW1, from relations4.32and4.52we deduce that
Taking into account thatLdoes not depend onθorsit follows that the systemSABP Cis
WN, X, WN, X-stable. By applyingTheorem 3.8we deduce that the systemABP C
is uniformly exponentially stable and the proof is complete.
Remark 4.9. FromTheorem 4.8we deduce that the property of uniform exponential stability of a system of variational difference equations is preserved in the presence of structured perturbations, provided that the norm of the perturbation factor is less than 1/λWA;B, C
and this estimate holds for any sequence spaceWin the general classWN. The study points out an interesting connection between the family of the input-output operators associated with the control system SA and the size of the “largest” perturbation in the presence of
which the perturbed system still remains uniformly exponentially stable.
The central result of this section is the following.
Theorem 4.10. The following estimate holds:
rstabA;B, C≥ sup W∈WN
1
λWA;B, C. 4.54
As a consequence, we deduce the following.
Corollary 4.11. SettingλpA;B, C λpN,RA;B, Cone has that
rstabA;B, C≥ sup p∈1,∞
1
λpA;B, C. 4.55
Acknowledgments
The author would like to thank the referees for carefully reading the manuscript and for helpful comments, which led to the improvement of the paper. The work was supported by CNCSIS – UEFISCSU, project number PN II – IDEI 1081/2008 and project number PN II – IDEI 1080/2008.
References
1 S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,”Journal of Differential Equations, vol. 120, no. 2, pp. 429–477, 1995. 2 C. Foias¸, G. R. Sell, and R. Temam, “Inertial manifolds for nonlinear evolutionary equations,”Journal
of Differential Equations, vol. 73, no. 2, pp. 309–353, 1988.
3 A. L. Sasu and B. Sasu, “A lower bound for the stability radius of time-varying systems,”Proceedings of the American Mathematical Society, vol. 132, no. 12, pp. 3653–3659, 2004.
4 A. L. Sasu, “New criteria for exponential stability of variational difference equations,” Applied Mathematics Letters, vol. 19, no. 10, pp. 1090–1094, 2006.
5 B. Sasu and A. L. Sasu, “Input-output conditions for the asymptotic behavior of linear skew-product flows and applications,”Communications on Pure and Applied Analysis, vol. 5, no. 3, pp. 551–569, 2006. 6 B. Sasu, “New criteria for exponential expansiveness of variational difference equations,”Journal of
Mathematical Analysis and Applications, vol. 327, no. 1, pp. 287–297, 2007.
8 B. Sasu, “On dichotomous behavior of variational difference equations and applications,”Discrete Dynamics in Nature and Society, vol. 2009, Article ID 140369, 16 pages, 2009.
9 B. Sasu, “On exponential dichotomy of variational difference equations,”Discrete Dynamics in Nature and Society, vol. 2009, Article ID 324273, 18 pages, 2009.
10 G. R. Sell and Y. You,Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002.
11 R. P. Agarwal,Difference Equations and Inequalities, vol. 228 ofMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
12 R. P. Agarwal and M. Pituk, “Asymptotic expansions for higher-order scalar difference equations,”
Advances in Difference Equations, vol. 2007, Article ID 67492, 12 pages, 2007.
13 R. P. Agarwal and D. O’Regan,Ordinary and Partial Differential Equations. With Special Functions, Fourier Series, and Boundary Value Problems, Universitext, Springer, New York, NY, USA, 2009.
14 N. Apreutesei and V. Volpert, “Solvability conditions for infinite systems of difference equations,”
Journal of Difference Equations and Applications, vol. 15, no. 7, pp. 659–678, 2009.
15 L. Berezansky and E. Braverman, “On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations,”Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 511–530, 2005.
16 L. Berezansky and E. Braverman, “Exponential stability of difference equations with several delays: recursive approach,”Advances in Difference Equations, vol. 2009, Article ID 104310, 13 pages, 2009. 17 C. Cuevas and C. Vidal, “Discrete dichotomies and asymptotic behavior for abstract retarded
functional difference equations in phase space,”Journal of Difference Equations and Applications, vol. 8, no. 7, pp. 603–640, 2002.
18 C. Cuevas and C. Vidal, “Weighted exponential trichotomy of linear difference equations,”Dynamics of Continuous, Discrete & Impulsive Systems, vol. 15, no. 3, pp. 353–379, 2008.
19 F. Cardoso and C. Cuevas, “Exponential dichotomy and boundedness for retarded functional difference equations,”Journal of Difference Equations and Applications, vol. 15, no. 3, pp. 261–290, 2009. 20 S. Elaydi and K. Janglajew, “Dichotomy and trichotomy of difference equations,”Journal of Difference
Equations and Applications, vol. 3, no. 5-6, pp. 417–448, 1998.
21 S. Elaydi,An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
22 R. Medina, “Absolute stability of discrete-time systems with delay,”Advances in Difference Equations, vol. 2008, Article ID 396504, 14 pages, 2008.
23 R. Medina, “Stabilization of discrete-time control systems with multiple state delays,”Advances in Difference Equations, vol. 2009, Article ID 240707, 13 pages, 2009.
24 R. Medina and M. Pituk, “Asymptotic behavior of a linear difference equation with continuous time,”
Periodica Mathematica Hungarica, vol. 56, no. 1, pp. 97–104, 2008.
25 S. Murakami and Y. Nagabuchi, “Uniform asymptotic stability and robust stability for positive linear Volterra difference equations in Banach lattices,”Advances in Difference Equations, vol. 2008, Article ID 598964, 15 pages, 2008.
26 P. H. A. Ngoc and N. K. Son, “Stability radii of positive linear functional differential equations under multi-perturbations,”SIAM Journal on Control and Optimization, vol. 43, no. 6, pp. 2278–2295, 2005. 27 M. Pituk, “A criterion for the exponential stability of linear difference equations,”Applied Mathematics
Letters, vol. 17, no. 7, pp. 779–783, 2004.
28 K. M. Przyłuski and S. Rolewicz, “On stability of linear time-varying infinite-dimensional discrete-time systems,”Systems & Control Letters, vol. 4, no. 5, pp. 307–315, 1984.
29 K. M. Przyłuski, “Remarks on the stability of linear infinite-dimensional discrete-time systems,”
Journal of Differential Equations, vol. 72, no. 2, pp. 189–200, 1988.
30 B. Sasu and A. L. Sasu, “Stability and stabilizability for linear systems of difference equations,”Journal of Difference Equations and Applications, vol. 10, no. 12, pp. 1085–1105, 2004.
31 A. L. Sasu, “Stabilizability and controllability for systems of difference equations,”Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 821–826, 2006.
32 A. L. Sasu, “Exponential dichotomy and dichotomy radius for difference equations,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 906–920, 2008.
33 D. Hinrichsen, A. Ilchmann, and A. J. Pritchard, “Robustness of stability of time-varying linear systems,”Journal of Differential Equations, vol. 82, no. 2, pp. 219–250, 1989.
35 M. Bohner and A. Peterson,Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2001. 36 S. Hilger,Ein Makettenkalk ¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Universit¨at W ¨urzburg,