A procedure for determination of the exponential stability of certain differential-difference equations

Certain continuity properties of the spectra of linear autonomous differential-difference equations which depend on a parameter are developed. These results are used to obtain a practical criterion for determination of the exponential stability of these systems. Introduction. In this paper a problem arising from the study of linear differential difference equations is considered. Suppose we are given an ^-dimensional family of differential difference equations of the type d_ dt x(t) S Bjx(t — ahj) = £ Ajx(t ~ ahj), a > 0, (0.1) 7=1 where 0 = h0 < h, < h2 < • • • < hm are fixed constants. We ask the questions: "In what manner do the stability properties of (0.1) depend on a change in a? In particular, if (0.1) is asymptotically stable for a = 0, how large can we take a and still preserve this property?" In the case where Bj = 0 for all j, i.e. the retarded case, the answer to this last question is quite easily given. In this case the roots of the characteristic equation (see e.g. [5]), although infinite in number, possess a continuity property which allows one to determine or at least approximate the size of the largest q for which a e [0, q) guarantees that the corresponding system (0.1) is asymptotically stable. If Bj ^ 0 for some j, the above questions become more difficult. The procedure used in this paper to study these questions may be outlined as follows. For each a in [0, °o) we define a(a) = sup Re{s: ^ is in the point spectrum of (0.1)}. Then for a > 0 we can prove that a (a) is continuous. In the retarded case a(a) is also continuous at a = 0. In the general case a(a) is continuous at a = 0 for a particular class of systems which contains the uniformly exponentially stable systems. Thus classes of systems (0.1) for which c(a) is continuous on [0, °°) provide us with a method of determining whether for a given a„ the system is uniformly asymptotically stable. For if <r(0) 0 and q?o [0, q\ where q is the smallest number which satisfies (t(cv) = 0, then a(a0) < 0. But from the general theory of retarded and neutral functional equations (see e.g. [6] and [7] we know that <r(a0) < guarantees that for the value a0 (0.1) is uniformly exponentially stable. The idea of studying stability though examination of the manner in which the point spectrum of (0.1) varies as a varies is similar to the method of D-partitions used to study the stability of retarded linear systems. A good outline of this method can be found in [5, Received September 8. 1977.