Stability Criteria for Certain Third Order Delay Differential Equations
by
Baruch Cahlon
Oakland University, Dept. Of Math., Rochester MI. 48309, USA
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) = p1 y''(t)+p2y''(t-\tau)+q1 y'(t)+q2y'(t-\tau)+r1y(t)+r2y (t-\tau) \tag 1

where \tau > 0, p₁, p₂, q₁, q₂, r₁ and r₂ are constants. In a previous paper, we considered equation (1) with q₂=0 and r₁=0 which arose from a robotic with damping and delay. Equations of the type (1) appear in many applications, for instance, in stabilization of inverted pendulums. This problem is intersting either in biology, to explain the self-balancing of human body, or in robotics, to construct biper robots. One such example is balancing of a stick. In this example the delay \tau is the human reflexes and linearization of the mathematical model leads to the type of equation (1) . Equations of the form of (1) can be used as test equations for numerical methods. The authors are not aware of a comprehensive study of this important equation. There are no practical stability criteria of the zero solution of (1). It is clear that with six independent parameters of (1) one cannot expect to get a region of stability. Our goal is to derive algorithmic type stability criteria for certain coefficients.

It is practically impossible to give all the possible values of the coefficients of (1) in one paper, however, the method of this paper can provide the way to derive the stability criteria for the other cases of the coefficients of equation (1) which we will not cover in this paper.

Note that with \tau = 0 the zero solution of (1) is asymptotically stable if and only if

P=p1+p2 < 0,     Q=q1+q2 < 0,   R=r1+r2 < 0,     \tag 2a

and

PQ > -R.     \tag 2b

We will demonstrate some cases when stability occurs with \tau > 0 and conditions (2a-2b) are not valid. In other words we expose some rare cases where the delay can stabilize equation (1).

Date received: January 22, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calu-63.