Infinite frequency solutions to discontinuous control systems with variable delay

Abstract

In this article, we study the frequency of zeros in the solution tou′(t)=f(t, u(t−τ(t))), whereτ is a delay that depends on time, andf is a discontinuous function. First we show examples of solutions that have infinite frequency for autonomous systems with variable delay, and for non-autonomous systems with constant delay. Then we prove that infinite frequency solutions cannot come from finite frequency data. Also we prove that under certain conditions on the delay, the zero solution is the only solution that has infinitely many zeros in each interval of a fixed length.