Stability Criteria for Certain Third Order Delay Differential Equations
by
Baruch Cahlon and Darrell Schmidt
Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309-4401

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

y'''(t) = p1 y''(t)+p2y''(t-\tau)+q1 y'(t)+q2y'(t-\tau)+r1y(t)+r2y (t-\tau) \tag1

where \tau > 0 is a constant and p₁, p₂, q₁, q₂, r₁ and r₂ are constants. In our previous paper we consider equation (1) with q₂=0 and r₁=0 which was a mechanical robotics model with damping and delay. There is not much study on the delay differential equations of the third order, and in particular, there is no study of practical stability criteria of the zero solution of (1). One can transform (1) into a system of a first order. The study on systems does not yield practical stability criteria. It is clear that with six independent parameters one cannot expect to get region of stability. Our goal is to derive algorithmic type stability criteria for certain coefficients.

In this paper we obtain practical (either easily checked or algorithmically type checked) stability criteria for the zero solution of (1) for certain coefficients of (1).

Date received: March 3, 2003