Stability Criteria for First Order Delay Differential Equation with m Commensurate Delays
by
Baruch Cahlon
Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4401
Coauthors: Darrell Schmidt

The aim of this paper is to study the asymptotic stability of the zero solution of the delay differential equation

y'(t)+ n å j=0  ajy(t-jt)=0

(1)

where t > 0, a_j, j=0, ..., m are constants. In our previous paper [1], we considered equation (1) with one delay and complex coefficients. There are many studies on first order equations with more than one delay, see [1, 2, 3, 4, 5] with applications in population dynamics, see [6, 7, 8, 9]. In these studies necessary conditions or sufficient conditions were derived. For some commensurate cases see [6, 7, 8]. In this paper, we obtain explicit stability criteria for certain values of the coefficients in (1) and establish a robust algorithmic test for asymptotic stability of the zero solution. The authors developed stability criteria of both form of higher order equations with one delay, see [10] and [11]. An algebraic approach was developed in [12] where the authors study equation (1) over Rings.

Date received: February 22, 2007