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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.

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, [

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, [

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.

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

We now consider a functional

satisfying

The initial condition of (

All the linear operators

There is an interesting set of references on the application of Hyers-Ulam method to stability of differential equations (c.f. [

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.

Assume that

(i) The error norm is in between the current solution and the nominal one on

(ii) Assume that property (i) holds by

replacing the constraint

replacing

replacing “1” in the numerator of (

The nominal and current unique solutions of (

Theorem

It turns out that all fundamental matrices of form (

Note that if

Direct sufficient conditions for the fulfilment of Theorem

The following properties hold.

(i) Theorem

(ii) Theorem

A particular stability result of the perturbed system under that of the nominal one follows.

Consider the perturbed differential system

for sufficiently large

Then, the nominal and perturbed solutions are bounded for all time and

It turns out that if

Note that a close result to Corollary

A close result to Corollary

A further close result to Corollary

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

If

Assume that the nominal differential system is globally exponentially stable and

The author declares that he has no conflict of interests.

The author is very grateful to the Spanish Government for its support through Grant DPI2012-30651 and to the Basque Government for its support through Grants IT378-10 and SAIOTEK S-PE13UN039. He is also grateful to the University of Basque Country for its support through Grant UFI 2011/07. Finally, he is also thankful to the reviewers for their interesting suggestions.