Abstract
In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented.
Highlights
Perturbed differential systems, which can serve as adequate and convenient for analysis mathematical models of real-life multi-time-scale dynamical systems, are studied extensively in the literature
Since a singularly perturbed system depends on a small parameter ε > 0, its characteristic equation depends on this parameter
Using this feature of the characteristic equation, the structure of the set of its roots, valid for all sufficiently small ε, can be studied. Such a study can be carried out using a decomposition of the original singularly perturbed system into two much simpler ε-free subsystems, the slow and fast ones
Summary
Perturbed differential systems, which can serve as adequate and convenient for analysis mathematical models of real-life multi-time-scale dynamical systems, are studied extensively in the literature (see e.g., [1,2,3,4,5] and references therein). Using this feature of the characteristic equation, the structure of the set of its roots, valid for all sufficiently small ε, can be studied Such a study can be carried out using a decomposition of the original singularly perturbed system into two much simpler ε-free subsystems, the slow and fast ones. To the best of our knowledge, the exact slow-fast decomposition of singularly perturbed time-delay systems was developed only for the systems which are not of the neutral type Such a result was proposed in [9] where a singularly perturbed linear autonomous system with small delays both, point-wise and distributed, in the fast state variable was analyzed. A further extension of the exact slow-fast decomposition method was proposed in [6,10] where singularly perturbed linear autonomous systems with point-wise and distributed small delays in both, slow and fast, state variables were studied. For the asymptotic slow-fast decomposition of singularly perturbed time-delay differential systems of both, non neutral and neutral, types one can see e.g., [1,3,12,13,14,15,16,17,18,19,20] and references therein
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
