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

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear timevarying 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.


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][2][3][4][5][6][7][8][9][10][11][12][13].A variety of results in this field have been obtained, in particular, for perturbations of additive and subadditive functions [5][6][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][12][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][15][16][17][18][19][20][21][22][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.
where R, C, and Z are the sets of real, complex, and integer numbers, respectively.The complex imaginary unity is i = √ −1.A finite subset of  consecutive positive integers starting with 1 is denoted by  := {1, 2, . . ., }.The set R −ℎ  := [−, 0) ∪ R 0+ will be used to define the solution of functional differential equations on R 0+ including its initial condition on 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 over [0, ∞) as  → ∞; that is, ( −  − ℎ   ) acts on the dynamics of () from  = 0 to  =  for finite  and as  → ∞.
Dom() is the definition domain of the operator .

Perturbed and Nominal Differential System
We now consider a functional th 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][22][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: where (,  −  ) =  0 (,  −  ) +  1 (,  − 3 ) ;  () =  0 () + g () , () ≡ ();  ∈ [−,0] for some given initial condition vector function  ∈   (−), where  * 0 = max(max 1≤≤ 0 (ℎ  ), max 1≤≤  0 (ℎ   ), max   0 +1≤≤  0 +  0 (ℎ   )+ 0 ) and  * = max ( * 0 ,   ) are, respectively, the maximum delays of the unperturbed (or perfectly modelled nominal system, being associated with the operators  0   ) and the nominal system subject to unmodeled and perhaps nonlinear nonstructured dynamics, of maximum delay   , which includes the contribution to the dynamics of the possibly nonlinear function  0 (,  −  ) while the maximum delay of the current system is where   = max(  ,  3 ) is the maximum delay of its unmodeled dynamics with The nonnegative real constants  0 and  1 , 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 ẋ () −  0   −  0 (,  −  )− 0 () = 0.The following, rather weak, hypotheses are made.
The following, rather nonrestrictive in practice, hypotheses are made. (H. ) are everywhere differentiable with possibly bounded discontinuities on subsets of zero measure of their definition domains with and its entries are everywhere time differentiable with possibly bounded discontinuities within a subset of zero measure of their definition domains.

Main Results
There is an interesting set of references on the application of Hyers-Ulam method to stability of differential equations (c.f.[11][12][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  0 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 wellknown 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][12][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 involves the assumption that  ≥  * ; 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  →  = max(,  * ) in the proof which would be the maximum delay appearing in the error in between the solutions () and  0 () if any of the two situations  ≥  * or  <  * holds.Note that the definition of appropriate terms in the function  1 ,  1 , equal to or distinct from  0 , or the integers defining maximum point and distributed numbers of delays in both differential equations, that is,  (⋅) ,   (⋅) ,   (⋅) 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,  <  * can be also easily considered in the formulation.
Direct sufficient conditions for the fulfilment of Theorem 1 are given in the subsequent result by using the upper-bounding function of the constraint for the potentially unknown nonlinear nonautonomous unmodeled dynamics contribution ‖ 0 (,  −  )‖ ≤  0 sup 0≤≤ℎ ‖ −  − ‖ +  1 .

Corollary 4. The following properties hold.
(i) Theorem 1(i) holds if the upper-bound of (12) is replaced with the following one: A particular stability result of the perturbed system under that of the nominal one follows.
Then, the nominal and perturbed solutions are bounded for all time and  0 () → 0, () → 0, and () → 0 exponentially fast as  → ∞ so that the nominal and the perturbed differential systems are globally exponentially Lyapunov stable.The condition that ‖()‖ is bounded for all time and () → 0 exponentially fast as  → ∞ still holds if condition 1 (global asymptotic stability of the nominal differential system) is removed.
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 that  = 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][27][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]).
If g =  −  0 is nonzero, then the extension of all the above results is direct for the cases that it is either bounded on R 0+ , if the fundamental matrix is absolutely integrable, or square-integrable, if the fundamental matrix is absolutely integrable on R 0+ .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 and g : R 0+ → R for some nonnegative real function Λ : (Z 0+ × R 0+ ) × R 0+ → R 0+ such that Λ(  , ) is of some of the forms of (29) leading to Λ(  ,) → 0 and ‖( −  0 ) 0 ‖ + Λ(  , ) → 0 for  ∈ [  ,  +1 ) as  → ∞; the nominal differential system is globally exponentially stable and g : R 0+ → R  is either bounded or integrable or square-integrable.