# Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.


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## Body

## 1. Introduction

The background literature on Hyers-Ulam-Rassias analysis is abundant and many different problems have been solved with it under the basis that there is a perturbation of a nominal equation and that a norm upper-bounding function of the error is obtained, [1–13]. A variety of results in this field have been obtained, in particular, for perturbations of additive and subadditive functions [5–7]. Special attention to the asymptotic properties of the Cauchy equation is paid in [5]. On the other hand, some inequalities related to the exponential function are proposed and investigated in [1]. Closed problems related to the slopes and mean values of exponential functions are discussed in [2]. Also, extensions to functions of several variables and to the study of approximate homomorphisms are discussed and solved in [3, 4], which are nowadays classical studies in the field of Hyers-Ulam-Rassias stability. A discussion with several results of asymptotic aspects is given in [5] close to the asymptotic derivability which is a very important issue in nonlinear analysis. The cubic function is studied in [8] from the Hyers-Ulam stability point of view while its relations to the related stability quadratic functional functions, symmetric biadditive functions are also commented. Different kinds of perturbed differential equations of first order are investigated in [11–13] in the light of the Hyers-Ulam-Rassias stability analysis.On the other hand, it is well known that time-delay dynamic systems are a very relevant field of research in dynamic systems and functional differential equations because of their intrinsic theoretical interest since the required formalism lies in that of functional differential equations, then infinite dimensional, and since there are a wide range of applicability issues in modelling aspects of physical systems, like queuing systems, teleoperated systems, war and peace and biological models and transportation systems, also finite impulse response filtering, and so forth. Another important useful application is the inclusion of delays in the description of epidemic models so as to obtain richer information about the disease propagation and to take it into account in the design of vaccination laws. See, for instance, [14–23] and references therein. The stability and the derivation of approximate solutions of some kinds of functional equations have been also investigated in [24, 25] and references therein, under close analysis methods.This paper is concerned with the study of the solutions of perturbed time-delay differential systems and their comparison and asymptotic properties of convergence to those of the corresponding unperturbed ones. The differential systems involve a combined fashion linear dynamics of point delays, finitely distributed time-delays, and infinitely distributed Volterra-type delays as well as perturbations involving nonlinear dynamics depending on further delays, in general, and which can be unknown with just slight “a priori” knowledge on an upper-bounding function on the supremum of the trajectory solution norm. External nonnecessarily identical forcing terms can be also present in both the nominal and the current differential functional equations. The number of delays of the perturbed equation and that of its nominal versions might be distinct and the matrices describing the linear delayed and delay-free dynamics of both differential systems might be also distinct. There are two problems focused on in the paper; namely, firstly the paper focuses on the asymptotic convergence to zero of the error between both nominal and current solutions irrespective of the stability properties of the nominal differential system, if any, and, secondly such a problem is revisited together with the stability or asymptotic stability of both the nominal and the perturbed functional differential systems.Notation. Consider
(1)R0+∶=R+∪0,R+∶=z∈R:z>0,R0-∶=R-∪0,R-∶=z∈R:z<0,C0+∶=z∈C:Rez≥0,C+∶=z∈C:Rez>0,C0-∶=z∈C:Rez≤0,C-∶=z∈C:Rez<0,Z0+∶=Z+∪0,Z+∶=z∈Z:z>0,
whereR,C, andZare the sets of real, complex, and integer numbers, respectively. The complex imaginary unity is i=-1. A finite subset of j consecutive positive integers starting with 1 is denoted by j-∶=1,2,…,j. The set R-hθ:=-θ,0∪R0+ will be used to define the solution of functional differential equations on R0+ including its initial condition on -θ,0.C
e
R
0
+
∶
=
C
R
0
+
,
C
n is the Banach space of continuous functions from R0+ into Cn endowed with the supremum norm⁡xα=sup⁡0≤τ<∞xτα.C
R
-
θ
∶
=
C
-
θ
,
∞
,
C
n is the Banach space of continuous functions from -θ,∞ into Cn endowed with the supremum norm⁡ϕα=sup⁡-θ≤τ<∞ϕτα; ∀ϕ∈Ce-θ∶=C-θ,0,Cn (defined below) is an initial condition, for some given vector norm ·α.C
e
-
θ
∶
=
{
ϕ
=
ϕ
1
+
ϕ
2
:
ϕ
1
∈
C
-
θ
,
ϕ
2
∈
B
0
-
θ
}, ϕ0=x0, with C-θ∶=C0-θ,0,X, that is, the set of continuous mappings from -h,0 into the Banach space X with norm ϕ-α∶=ϕα=sup⁡ϕ(t)α:-θ≤t≤0; · denotes the Euclidean norm of vectors in Cn and matrices in Cn×n, and B0-θ∶=ϕ:-θ,0→X is the set of real bounded vector functions on X endowed with the supremum norm having support of zero measure.x
t denotes the solution string within t-θ,t pointwise defined by the solution xt.It is said that the delays associated with Volterra-type dynamics are infinitely distributed because the contribution of the delayed dynamics is made under an integral over0,∞ as t→∞; that is, xt-τ-hi′ acts on the dynamics of xt from τ=0 to τ=t for finite t and as t→∞.Dom
⁡
H is the definition domain of the operator H.


## 2. Perturbed and Nominal Differential System

We now consider a functionalnth order differential system with point and, in general, both infinite-type Volterra-type and finite-distributed delays in a more general context that is the approaches of [16, 21–23], since it includes the contributions of both structured and unstructured delayed dynamics with point and finite- and infinite-distributed delays, as well as the presence of nonlinear dynamics. Such a differential equation obeys the widely general structure:
(2)x˙t-Lxt-gt=0,
where
(3)Lxt≡L0xt+ft,xt-θb,(4)L0xt≡A0xt+∑i=1m0Aitxt-hi+∑i=0m0′∫0tdαiτAαiτxt-τ-hi′+∑i=m0′+1m0+m0′′∫t-hi′tdαit-τAαiτxτ-θ0+g0t(5)ft,xt-θb=f0t,xt-θa+f1t,xt-θ3;gt=g0t+g~t,(6)f1t,xt-θ3=∑i=1mA~itxt-hi+∑i=0m′∫0tdαiτA~αiτxt-τ-hi′+∑i=m′+1m′+m′′∫t-hi′tdαit-τA~αiτxτ-θ1,xt≡ϕt; t∈-θ,0 for some given initial condition vector function ϕ∈Ce-θ, where θ0*=max⁡max⁡1≤i≤m0hi,max⁡1≤i≤m0′hi′,max⁡m0′+1≤i≤m0′+m0′′hi′+θ0 and θ*=max⁡θ0*,θa are, respectively, the maximum delays of the unperturbed (or perfectly modelled nominal system, being associated with the operators L0xt) and the nominal system subject to unmodeled and perhaps nonlinear nonstructured dynamics, of maximum delay θa, which includes the contribution to the dynamics of the possibly nonlinear function f0t,xt-θa while the maximum delay of the current system is
(7)θ=max⁡max⁡1≤i≤mhi,max⁡1≤i≤m′+m′′hi′,max⁡m′+1≤i≤m′+m′′hi′+θ1,θb,
where θb=max⁡θa,θ3 is the maximum delay of its unmodeled dynamics with
(8)θ3=max⁡max⁡1≤i≤m0hi,max⁡1≤i≤m0′hi′,max⁡m0′+1≤i≤m0′+m0′′hi′+θ1.
The nonnegative real constants θ0 and θ1, if they are not zero in (4) and (6), modulate the finitely distributed delays with the functions that configure their contributions under the integral symbols. The objective is the comparison of the solution of the current dynamic functional equation (2), subject to (3)–(6), to that of its nominal version x˙t-L0xt-f0t,xt-θa-g0t=0. The following, rather weak, hypotheses are made.(1)
L:CeR-θ→Cn is a bounded linear functional defined by the right hand side of (2).(2)
hk and hl′ and hj′+θ0,1 (k=1,2,…,m; l=0,1,…,m′, j=m′+1,…,m′+m′′) are nonnegative real point delays, infinite-time distributed Volterra-type delays (i.e., the first m′ distributed delays), and finite time-interval distributed delays with h0=h0′=0 such that m≥m0, m′≥m0′, and m′′≥m0′′, where the 0 subscripts stand for the nominal equation. The finitely distributed nominal and current delays have an increasing factor θ0≥0 and θ1≥0 for formulation generality purposes.(3)
g0,g:R0+→Cn is piecewise continuous, f0:R0+×CR-θ→Cn describes a perturbed linear dynamics, and
(9)xtCR-θ∶=x:-θ,τ⟶X,τ≤t0,τ>t,satisfyingxt=ϕt, ∀t∈-θ,0, is a string of the solution of (2).The following, rather nonrestrictive in practice, hypotheses are made.(H.1)
The initial condition of (2) is ϕ∈Ce-θ. Roughly speaking, ϕ∈B0-θ if and only if it is almost everywhere zero except at isolated discontinuity points within -θ,0 where it is bounded. Thus, ϕ∈Ce-θ if and only if it is almost everywhere continuous in -θ,0 except possibly on a set of zero measure of bounded discontinuities. Ce-θ is also endowed with the supremum norm since ϕ=ϕ1+ϕ2, some ϕ1∈C-θ, ϕ2∈B0-θ for each ϕ∈Ce-θ. In the following, the supremum norms on L(X) are also denoted by ·.(H.2)
All the linear operatorsAk,A~k(0≤k≤m), Aαk,A~αk(0≤k≤m′+m′′), with the abbreviated notation Aα0=Aα, are in L(X)∶=L(X,X), the set of linear operators on X, of dual X*, which are closed and densely defined with respective domain and range D(Aαi) and R(Aαi)⊂X (i=0,1,…,m′). The functions αi∈C0(0,∞,C)∩BVloc⁡(C+)(i=0,1,…,m′) and αi∈C0(-h,0,C) (i=0,1,…,m′+m′′) are everywhere differentiable with possibly bounded discontinuities on subsets of zero measure of their definition domains with ∫0∞eνtdαi(t)<∞ for some nonnegative real constant ν(i=0,1,…,m′). If αi· is a matrix function αi:0,∞×X*→LX,X* then it is in C0(0,∞,Cn×n)∩BVloc⁡(C+n×n) with ∫0∞eνtdαi(t)<∞ and its entries are everywhere time differentiable with possibly bounded discontinuities within a subset of zero measure of their definition domains.


## 3. Main Results

There is an interesting set of references on the application of Hyers-Ulam method to stability of differential equations (c.f. [11–13]). The first and third ones are first-order equations, respectively, linear and nonlinear while the second one [12] is of linear time-varying type. Such differential equations in those references are delay-free, so that they are of a nonfunctional type. In the following, we develop a related formal stability analysis of functional differential equations with internal delays under the forms (2)–(6) and satisfying hypotheses (H.1)-(H.2). Note that the studied equations have several types of time-varying linear delayed dynamics (as, e.g., point delays, finitely distributed delays, and Volterra-type infinitely distributed delays). Note also that, furthermore, nonautonomous nonlinear dynamics can be considered in (3) under the generic structure of (4)-(5), that is, involving if suited any of the various types of delays plus nonlinear unmodeled terms under the functional f0 which can be unstructured and unmeasurable. In that case, an upper-bounding function of generic structure, the supremum of the norm of the state, will be assumed to be known in the formal subsequent developments. This general structure of the functional equation of dynamics and its stability study under the Hyers-Ulam-Rassias formalism are the main contribution of the paper. It turns out while it is a well-known feature that Hyers-Ulam-Rassias method of analysis in differential equations relies basically on comparing the perturbed differential equation with the unperturbed (or nominal) one. In [11–13], the analysis is performed based on the error in between both differential equations while, in the current paper, it is based on the direct analysis of the error between both of them. Other important characteristics of the proposed formalism are that it is based on a dynamic system description rather than on simple differential equations and that the fundamental matrix can be defined based on different comparison systems which can be delay-free or it can contain a number of delays. The formal treatment proceeds in such a way that the delayed dynamics which is not accounted for, if any, to define the fundamental matrix is considered as a contribution to the forcing part of the differential system. The analysis is neither based in a formalism stated on the Hilbert space framework, as in [15], nor in Lyapunov stability theory, including the use of matrix equalities and inequalities as in [17, 18, 21], but on the analysis of the solution error in between the nominal and the current differential systems for all time. In particular, note that the approach used in [16] is not directly applicable in this context since the differential system at hand is time-varying and the Laplace transform methods cannot be used.The main results of this section consist of a main theorem and three corollaries related to sufficiency-type conditions for guaranteeing the theorem and particular cases as well as several remarks related to their applications to further potential particular cases. The basic main result follows below.Theorem 1.
Assume thatg~≡0 and f0:R0+×Cn→Cn is subadditive, θ≥θ*, and f0t,xt-θb≤K00sup⁡0≤δ≤hxt-θb-δ+K01 for some K00,K01∈R0+; ∀t∈R0+. Let μtn=μ1,μ2,…,μm0tn be any m0-tuple defined from a piecewise constant m0-binary vector function μ:tn,tn+1×Z0+→μ1tn,…,μm0tn for any combination of values of the set of binary variables μitn∈0,1; ∀i∈m-0=1,2,…,m0 defining a fundamental matrix of the nominal unforced differential system x˙t=L0xt of the form
(10)Ψμtnt,tn=eA0t-tn×I+∑i=1m0μitncccccccccc×∫tnteA0tn+τAiτΨμtnτ-hiUτ-hidτ;dddddddddddddddddddddddddddddddddddd∀t∈tn,tn+1;∀n∈Z0+ with initial conditions with Ψμtnt,t=I; ∀t∈R0 and Ψμtnt,tn=0 for t<tn, where I is the n-identity matrix and Ut is the Heaviside function, which satisfies the differential system:
(11)Ψ˙μtnt,tn=A0Ψμtnt,tn+∑i=1m0μitnAitUt-hiΨμtnt-hi,tn;ddddddddddddddd∀t∈tn,tn+1,∀n∈Z0+.
Then, the following properties hold.
(i) The error norm is in between the current solution and the nominal one ontn,∞; ∀n∈Z0+ is upper-bounded by a prescribed positive norm bound E if et0≤E; the fundamental matrix defined from a binary vector function μ:tn,tn+1×Z0+→μ1tn,…,μm0tn fulfills for a matrix induced vector norm Ψμtntn,t≤ψμtntn,t<1; ∀t∈tn,tn+1, ∀n∈Z0+, such that the upper-bounding function ψμ··,·:Z0+×R0+→R0+ satisfies ψμtntn,tn=1; ∀n∈Z0+, and the subsequent constraint holds for some sufficiently small real constant T*∈R+:
(12)t-tn≤ef22+fτ,eτ-θb∑i=1m01-μitnAiτ-11-ψμtntn,t×max⁡tn≤τ≤t∑i=1m01-μitnAiτψμtntn,τdddddddffdd×∑i=1m01-μitnAiτdddddddffdsssss+∑i=0m0′∫0τdαiσAαiσsssssssssssfssssss+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσdddddddfdfsssfss+fτ,eτ-θb∑i=1m01-μitnAiτ-1;dddddddddddddd∀t∈tn,tn+1,∀n∈Z0+
for some strictly increasing sequence t0,t1,… satisfying
(13)T*≤Tn∶=tn+1-tn≤min⁡h*tn,max⁡argrn∈R+:rn=zn;ssssssssssssssssssssssddddddsssssssssss∀n∈Z0+,
where
(14)zn≤+fτ,eτ-θb∑i=1m01-μitnAiτ-11-ψμtntn,tn+zn×∑i=1m01-μitnAiτψμtntn,tn+znffffdff×max⁡tn≤τ≤tn+1∑i=1m01-μitnAiτdddddddddddddddddd+∑i=0m0′∫0τdαiσAαiσddddddddddddddddddd+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσddddddddddddddddddd+fτ,eτ-θb∑i=1m01-μitnAiτ-1;dddddddddddddddddddddddddddddddd∀n∈Z0+(15)h*tn∶=min⁡min⁡1≤i≤m01-μitnhi,min⁡m0<i≤mhi,h1′;dddddddddddddddddddddddddiidddddddddddd∀n∈Z0+;∀ϕ∈Ce-θ∶=C-θ,t0,Cn with θ=max⁡max⁡1≤i≤mhi,max⁡1≤i≤m′+m′′hi′,max⁡m′+1≤i≤m′+m′′hi′+θ1,θb≥θ*, with xt0=φt0 for any given t0∈R0+. If sup⁡0≤t≤t0et≤E for any given absolutely continuous vector function of initial conditions φ:-θ,0→Cn then sup⁡0≤t<∞et≤E.
(ii) Assume that property (i) holds by(a)
replacing the constraintΨμtntn,t≤ψμtntn,t<1 with Ψμtntn,t≤ψμtntn,t<Kn; ∀t∈tn+γn,tn+1 for some real sequence γn with γn≥γ>0; ∀n∈Z0+;(b)
replacingE by a nonnegative real sequence En satisfying En+1=KnEn for each n∈Z+ and some decreasing sequence of nonnegative real numbers Kn such that lim⁡sup⁡n→∞Kn=K for some real constant K∈0,1;(c)
replacing “1” in the numerator of (12) and (14) by Kn; ∀n≥n0∈Z0+ for some given ε<1-K∈R+ and n0=n0ε∈Z0+ such that Kn≤K+ε<1.Then, et→0 exponentially as t→∞.Proof.
The nominal and current unique solutions of (2)–(4) [16, 17] are, respectively, given by
(16)x0t=Ψμtnt,t0x0t0+∫t0tΨμtnt,τddddd×∑i=m0′+1m0+m0′′∫τ-hi′τdαi∑i=1m01-μiAiτx0τ-hisssddsdsss+∑i=0m0′∫0τdαiσAαiσx0τ-σ-hi′ssssddddsss+∑i=m0′+1m0+m0′′∫τ-hi′τdαi(τ-σ)Aαiσx0σ-θ0dτ+∫t0tΨμt,τgτdτ;∀t≥t0∈R0+xt=Ψμtnt,t0xt0+∫t0tΨμtnt,τddddd×∑i=1m01-μiAiτxτ-hiddddddddd+∑i=0m0′∫0τdαiσAαiσxτ-σ-hi′ddddddddd+∑i=m0′+1m0+m0′′∫τ-hi′τdαi(τ-σ)Aαiσxσ-θddddddddd+fτ,xτ-θb∑i=1m01-μiAiτxτ-hidτ+∫t0tΨμtnt,τgτdτ;∀t≥t0∈R0+
and, since f1:R0+×Cn→Cn is structurally additive in its second argument, then also subadditive, and f0:R0+×Cn→Cn is subadditive in its second argument; then f1:R0+×Cn→Cn is subadditive in its second argument so that h:R0+×Cn→Cn is also subadditive in its second argument. Thus,
(17)et∶=xt-x0t=Ψμtnt,tnetn+∫tntΨμtntn,τdddd×∑i=1m01-μiAiτeτ-hidddsddd+∑i=0m0′∫0τdαiσAαiσeτ-σ-hi′ddddsdd+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσeσ-θdddsddd+fτ,eτ-θb∑i=1m01-μiAiτeτ-hidτ;∀t∈tn,tn+1 for any strictly increasing sequence t0,t1,…,tn,… and any absolutely continuous initial condition φt on t0-h,t0 with xt0=φt0 for any given t0∈R0+ where μtn=μ1tn,μ2tn,…,μm0tn is any m0-tuple defined for any combination of values of the set of binary variables μi∈0,1; ∀i∈m-0=1,2,…,m0 with
(18)ft,xt-θb=f0t,xt-θadddddddddd+∑i=1mA~itxt-hidddddddddd+∑i=0m′∫0tdαiσA~αiσxt-σ-hi′dddddddddd+∑i=m′+1m′+m′′∫t-hi′tdαit-σA~αiσxσ-θ1dddddddddddddddddddddddddddddd∀t≥t0∈R0+;et∶=xt-x0t=Ψμtnt,tnetn+∫tntΨμtntn,τddddd×∑i=1m01-μinAiτeτ-hissssssssssss+∑i=0m0′∫0τdαiσAαiσeτ-σ-hi′ssssssssssss+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσeσ-θssssssssssss+fτ,xτ-θb∑i=1m01-μinAiτeτ-hidτ.
Assume for any n∈Z0+ that sup⁡t0≤t≤tnet≤E,Ψμtntn,t≤ψμtntn,t<1 (note that Ψμtntn,tn=I; ∀n∈Z0+; ∀t∈tn,tn+1, ∀n∈Z0+). Note also that Tn=tn+1-tn; ∀n∈Z0+, and define
(19)htn=max⁡max⁡1≤i≤m01-μitnhi,max⁡1≤i≤m0′hi′,ffffffmax⁡m0′+1≤i≤m0′+m0′′hi′+θ0,θ.
Note that (12) holds with a bounded denominator for 0≤τ≤Tn≤h*tn; ∀n∈Z0+ since f0t,xt-θa≤K00sup⁡0≤δ≤hxt-θa-δ+K01 then ft,xt-θb≤K0sup⁡0≤δ≤hxt-θb-δ+K1; ∀t∈R0+ from (5)-(6) note the fact that linearity of f1 grows non faster than linearity with respect to sup⁡0≤δ≤hxt-θb-δ; ∀t∈R0+. As a result, if sup⁡t0≤t≤tnet≤E, then(20a)e˙t-Let=L0et+ft,xt-θb=L-L0x0t,(20b)et-tn≤Ψμtntn,tetn+t-tnsup⁡tn-htn≤τ≤t-h*tneτ×max⁡tn≤τ≤t∑i=1m01-μitnAiτΨμtntn,tssssssdssss×∑i=1m01-μitnAiτsssssssssssssss+∑i=0m0′∫0τdαiσAαiσsssssssssssssss+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσsssssssssssssss+fτ,xτ-θb∑i=1m01-μitnAiτ≤Eψμtnt,tn+t-tnE×max⁡tn≤τ≤t∑i=1m01-μitnAiτ+∑i=0m0′∫0τdαiσAαiσψμtnτ,tnddddddddd×∑i=1m01-μitnAiτssssssssssddssss+∑i=0m0′∫0τdαiσAαiσssssssssddssssss+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσssssssssddssssss+fτ,xτ-θb∑i=1m01-μitnAiτ+∑i=0m0′∫0τdαiσAαiσ≤E;∀t∈tn,tn+1;∀n∈Z0+.Thus, if sup⁡t0≤t≤tn+1-hn*et≤E and Ψμtntn,t<1 for t∈tn,tn+1, ∀n∈Z0+ then, for any given n∈Z0+, sup⁡t0≤t≤tnet≤E<+∞⇒sup⁡t0≤t≤tn+1et≤E; if constraint (12) holds, subject to (13)–(15) (note that the second min-max part of (13) comes from (12) for τ=tn+1, and then tn+1-hn*≤tn, ∀n∈Z0+, and that the upper-bound of zn is continuous and is zero if zn=0; ∀n∈Z0+), the matrix function Ψμtntn,t from Z0+×R×R to Cn×n is everywhere continuous in Z0+×R×R even if μtn+≠μtn (i.e., if the matrices of the dynamics used to define the fundamental matrix change at t=tn for some n∈Z0+). Then, property (i) has been proved by complete induction. The proof of property (ii) is direct from property (i) together with sup⁡tn-1≤t≤tnet≤En; ∀n∈Z+ and En→0 at exponential rate as n→∞ since Ψμtntn,t≤ψμtntn,t<Kn, lim⁡sup⁡n→∞Kn=K<1.Theorem1 involves the assumption that θ≥θ*; that is, the total delay involved in the current system is not less than that of the nominal one subject to unmodeled dynamics. The above assumption is made for presentation clarity. Its removal is not difficult by replacing θ→θ-=max⁡θ,θ* in the proof which would be the maximum delay appearing in the error in between the solutions xt and x0t if any of the two situations θ≥θ* or θ<θ* holds. Note that the definition of appropriate terms in the function f1, θ1, equal to or distinct from θ0, or the integers defining maximum point and distributed numbers of delays in both differential equations, that is, m·,m·′,m·′′ versus its nominal values, might allow the cancellation of some of the delayed dynamics contributions in the nominal system if suited. As a result of the case, θ<θ* can be also easily considered in the formulation.Remark 2.
It turns out that all fundamental matrices of form (10), satisfying (11), for any values of μ in the set 0,1×0,1⋯m0︸⋯0,1 are useful to construct the solution since the dynamics of the point delays which do not contribute to the homogeneous part of the solution are transferred to the forced solutions through contributing coefficients of the form 1-μi; ∀i∈m-0=1,2,…,m0. Note also that the definition of the fundamental matrix dynamics (10) as a solution of (11) allows dealing with the stability conditions under more general condition than the approaches used in [14–20]. See also Remark 3 and Corollaries 4 and 5.Remark 3.
Note that ifψμ··,·:Z0+×R0+→R0+ is nonincreasing in tn+ηn,tn+1 and strictly decreasing in tn,tn+ηn, ∀n∈Z0+ for some positive real sequence ηn, with ηn≤T*≤Tn, then the right-hand side of (12) is positive for all n∈Z0+ since ψμtntn,tn=1. Thus, Theorem 1(i) is applicable if its fundamental matrix has a norm Ψμtntn,t<1 being nonincreasing in tn+ηn,tn+1 and strictly decreasing in tn,tn+ηn; ∀n∈Z0+. Theorem 1(ii) is applicable if Ψμtntn,t<Kn; ∀t∈tn,tn+1; ∀n∈Z0+ and it is strictly decreasing in time subintervals tn,tn+ηn; ∀n∈Z0+.Direct sufficient conditions for the fulfilment of Theorem1 are given in the subsequent result by using the upper-bounding function of the constraint for the potentially unknown nonlinear nonautonomous unmodeled dynamics contribution f0t,xt-θa≤K0sup⁡0≤δ≤hxt-θa-δ+K1.Corollary 4.
The following properties hold.
(i) Theorem1(i) holds if the upper-bound of (12) is replaced with the following one:
(21)1-K0-ψμtntn,t×max⁡tn≤τ≤t∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσψμtntn,τssssssssdssss×∫τ-hi′τdαiτ-σAαiσK1+∑i=1m01-μitnAiτsdddddsddssss+∑i=0m0′∫0τdαiσAαiσdddddsddddd+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσ-1;ddddddddddsdddddddddddddd∀t∈tn,tn+1,∀n∈Z0+
provided that ψμtntn,t<1-K0 with K0, K1∈R+ being such that ft,xt-θb≤K0sup⁡0≤δ≤hxt-θb-δ+K1; ∀t∈R0+.
(ii) Theorem1(ii) holds if the upper-bound of (12) is replaced with
(22)K-K0-ε-ψμtntn,t×max⁡tn≤τ≤t∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσψμtntn,τdddddddd×K1+∑i=1m01-μitnAiτdddddssdsddd+∑i=0m0′∫0τdαiσAαiσdddddsssdddd+∑i=m0′+1m0+m0′′∫τ-hi′τdαiτ-σAαiσ-1;dddddddddddddddddddd∀t∈tn,tn+1,∀n∈Z0+
provided that ψμtntn,t<K+ε-K0 with K=lim⁡sup⁡n→∞Kn<1; ∀t∈tn+γn,tn+1 with γn≥γ>0; ∀n≥n0=n0ε∈Z0+ for any given ε<1-K∈R+.A particular stability result of the perturbed system under that of the nominal one follows.Corollary 5.
Consider the perturbed differential systemLxt≡L0xt+ft,xt-θ with linear nominal version subject to a single point delay h1>0 of dynamics L0xt≡∑i=01Aixt-hi, with h0=0 and A0 being a stability matrix of stability abscissa -ρ0<0, under any initial vector function satisfying hypothesis H.1 with ft,xt-θb=f0t,xt-θa+f1t,xt-θ3 with ft,xt-θb≤K0sup⁡0≤δ≤hxt-θb-δ+K1 for some K0,K1∈R0+; ∀t∈R0+, and f0t,xt-θ0 being subadditive. Assume that(1)
μ2A1<ρ0 (or if μ2A12<ρ0), where μ2A1 is the 2-matrix measure of A1;(2)
ψ1tn,t≤Ke-ρ0-δ0t-tn; ∀t∈tn,tn+1, ∀n∈Ζ0+ for any arbitrary δ0∈R+, K=KΨ01+A12/ρ0 and KΨ0∈R+ being such that eA0t2≤KΨ0e-ρ0t; ∀t∈R0+;(3)
0≤K0<K1-e-ρ0γ for some γ∈R+;(4)
Tn is subject to (13) so that (14) takes the form
(23)zn≤K-K0-ψ1tn,tK1max⁡tn≤τ≤tψ1tn,τ;∀t∈tn+γn,tn+1withγn≥γ>0for sufficiently largen0∈Z0+ and (15) becomes h*tn=h1; ∀n∈Z0+.
Then, the nominal and perturbed solutions are bounded for all time andx0t→0, xt→0, and et→0 exponentially fast as t→∞ so that the nominal and the perturbed differential systems are globally exponentially Lyapunov stable. The condition that et is bounded for all time and et→0 exponentially fast as t→∞ still holds if condition 1 (global asymptotic stability of the nominal differential system) is removed.Proof.
It turns out that ifA0 is a stability matrix of stability abscissa -ρ0<0 and μ2A1<ρ (or if μ2A12<ρ0 since μ2A1≤A12) then the solution of L0xt≡∑i=01Aixt-hi is bounded for any initial solution subject to hypothesis (H.1). Furthermore, x0t→0 as t→∞ exponentially for the fundamental matrix function for μ1t≡1:
(24)Ψ1t,tn=eA0t-tn×I+∫tnteA0tn+τA1Ψ1τ-h1Uτ-h1dτ;ssssssdddssssssssgdggssssss∀t∈tn,tn+1;∀n∈Z0+
being the unique solution of
(25)Ψ˙1t,tn=A0Ψ1t,tn+A1tUt-h1Ψ1t-h1,tn;ssdddsssss∀t∈tn,tn+1,∀n∈Z0+
with Ψ1t,t=I; ∀t∈R0 and Ψ1t,tn=0 for t<tn; ∀n∈Z0+ [16, 21–23] with Ψ1t,tn2≤ψ1tn,t; ∀t∈tn,tn+1, ∀n∈Z0+. Furthermore,(a)
eA0t2≤KΨ0e-ρ0t for some KΨ0∈R+; ∀t∈R0+ since A0 is a stability matrix, and(b)
(26)Ψ1t,t′2≤eA0t-t′2×1+∫t′teA0t′+τA1Ψ1τ-h1Uτ-h1dτ2≤KΨ0e-ρ0t-t′1+KΨ0A12ρ0≤KΨ0+K~e-ρ0-δ0t-t′≤KΨ0e-ρ0t-t′+K~e-ρ0-δ0t-t′for all t≥t′∈R0+ and any arbitrary δ0∈R+ if K~≥KΨ0A12/ρ0. Thus,
(27)Ψ1t,tn2≤ψ1tn,t≤Ke-ρ0-δ0t-tn≤K-K0=KΨ01+A12ρ0-K0;∀t∈tn+γn,tn+1, ∀n∈Z0+ for γn≥γ>0 and sufficiently large γ∈R+. On the other hand, from Corollary 4 for μ1t≡1, then leading to (23), one gets that et is bounded for all t∈R0+ for any initial solution of the perturbed differential system subject to hypothesis (H.1) and et→0 as t→∞ exponentially fast. As a result, xt2 is bounded for all t∈R0+ and xt→0 as t→∞ exponentially fast.Remark 6.
Note that a close result to Corollary5 may be formulated for the special case when f1t,xt-θ=A~1txt-h1 by replacing K0→K00, K1→K01 in (23) with K00≤K0 and K01≤K1 being such that f0t,xt-θa≤K00sup⁡0≤δ≤hxt-θa-δ+K01.Remark 7.
A close result to Corollary5 may be formulated if A0+A1 is a stability matrix of stability abscissa -ρ<0 by replacing ρ0→ρ in conditions 1–3 and K1→2K1 in condition 4. In the case that f1t,xt-θ=A~1txt-h1 (see Remark 6) a further close result is obtained by replacing K1max⁡tn≤τ≤tψ1tn,τ→2max⁡tn≤τ≤tψ1tn,τ+A~1τ2 in condition 4 of Corollary 5 since A1xt-h1-xt2≤2A12xt2 with xt being the string of solution in t-h1,t. The associated proofs are direct from Corollary 5 by rewriting the nominal differential system as L0xt≡A0+A1xt+A1xt-h1-xt.Remark 8.
A further close result to Corollary5 can be obtained by considering the nominal differential system to be L0xt≡A0xt so that A1xt-h1 defines a perturbed delayed dynamics. In this case, we consider the fundamental matrix to be Ψ0t,tn=eA0t-tn with Ψ0tn,t2≤ψ0t,tn≤Kψ0e-ρ0t; ∀t∈tn,tn+1, ∀n∈Z0+. Conditions 1–3 are restated with the replacement ψ1tn,τ→ψ0tn,τ for t∈tn,tn+1, ∀n∈Z0+ while constraint (23) of condition 4 is changed to
(28)zn≤K-K0-ψ0tn,tmax⁡tn≤τ≤tψ0tn,τK1+A1+A~1τ2;dddddddddddddd∀t∈tn,tn+1,∀n∈Z0+.
Then, the nominal and perturbed solutions are bounded for all time and x0t→0, xt→0, and et→0 exponentially fast as t→∞ so that the nominal and the perturbed differential systems are globally exponentially Lyapunov stable under conditions 1–3 and condition 4 modified with (28). The condition that et2 is bounded for all time and et→0 exponentially fast as t→∞ still holds if condition 1 (global asymptotic stability of the nominal differential system) is removed. Variants of this result under the considerations of Remarks 6 and 7 are direct.Note that the above results imply that both the nominal system and the current perturbed one have trajectory solutions which converge asymptotically to zero under any initial conditions. It is easy to see thatx=0 is a globally asymptotically stable equilibrium point and also a fixed point of the state-trajectory solution, under the various conditions of Corollary 5 as well as their variants in Remarks 3–8; that is, it is a globally stable attractor for all the trajectory solutions. The relevance of fixed point theory in stability of perturbed differential systems has been also emphasized in some background literature. See, for instance, [10, 20, 24, 26–28] and references therein. On the other hand, Hyers-Ulam stability has been also invoked in difference-type linear and nonlinear equations as, for instance, in [25] and several background references therein. Note that difference equations are sometimes got from the discretization of continuous-time system either via the use of numerical tools or by the use of physical sampling and hold devices and that the stability of such discretized systems can be, in general, either studied independently of that of their continuous-time counterparts, via “ad hoc” discrete analysis methods, or based with the stability properties of the continuous-time version with extra conditions on the sequence of sampling instants (see, e.g., [22]).Ifg~=g-g0 is nonzero, then the extension of all the above results is direct for the cases that it is either bounded on R0+, if the fundamental matrix is absolutely integrable, or square-integrable, if the fundamental matrix is absolutely integrable on R0+. Note that if the nominal differential system is exponentially stable, then a fundamental matrix is of exponential negative order, and then both absolutely integrable and square-integrable exist. Then one has the following result.Corollary 9.
Assume that the nominal differential system is globally exponentially stable andg~:R0+→Rn is either bounded or integrable or square-integrable. Then, a fundamental matrix Ψμtntn,t exists such that Theorem 1, Corollaries 4 and 5, and their extensions of Remarks 3–8 still hold for some E>∫tntΨμt,τg~τdτ if the needed denominator of (21) to (23) is, in each case, corrected with the additive term ∫tntΨμt,τg~τdτ or with any of its upper-bounds:
(29)max⁡tn≤τ≤tψμt,τ∫tntg~τdτ;max⁡tn≤τ≤tgτ∫tntΨμt,τdτ;∫tntgτdτ21/2∫tntΨμt,τdτ21/2.Outline of Proof. Note that (20a) is modified as follows:
(30)e˙t-Let=∫tntΨμt,τg~τdτ≤L-L0x0t+Λtn,t
for some nonnegative real function Λ:Z0+×R0+×R0+→R0+ such that Λtn,t is of some of the forms of (29) leading to Λtn,t→0 and L-L0x0t+Λtn,t→0 for t∈tn,tn+1 as n→∞; the nominal differential system is globally exponentially stable and g~:R0+→Rn is either bounded or integrable or square-integrable.



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