Stabilization of a system with a constant and a linear delay

Abstract

INTRODUCTION Systems of differential equations with a constant delay have been investigated quite comprehensively [1, 2]. The asymptotic properties of stationary systems with a linear delay (that is, a delay given by a linear function of the argument) have been much less studied. However, such systems are used in the theory of radioactive decay [1] as well as in the problem on the stability of vibrations of the pantograph of a moving locomotive when passing a contact system mast [3, 4]. For some velocities of the locomotive, the pantograph undergoes unstable vibrations (one observes the so-called “pantograph detachment” from the contact wire when passing an elastic mast [5]); consequently, such systems should be stabilized. This problem was considered in [4]. Systems with a constant and a linear delay are a further generalization of such systems. Sufficient conditions for the asymptotic stability (as well as instability) were obtained in [6]. In the present paper, we suggest a stabilization method for such systems on the basis of sufficient conditions for the asymptotic stability earlier obtained in [6, 7].