Further remarks on Markus-Yamabe instability for time-varying delay differential equations

摘要

We study the stability of time-varying delay differential equations of type x ( t )= Ax ( t - τ ( t )), when the delay τ ( t ) takes values in an interval [0, τ m ], for some τ m 0, and A is a n × n real matrix. The fundamental question that we consider is the following: is the system exponentially stable when every individual (constant-delay) system x ( t ) = Ax ( t - τ ), for τ ε [0, τ m ], is exponentially stable? This is nothing else than the so-called Markus-Yamabe instability. The answer to this question is ‘no’ for one-dimensional systems, as illustrated in the literature. The situation is more complicated for n -dimensional systems, n ≥ 2, and the previous question remains open for a general matrix A and a general τ m as above. Nevertheless in this paper we show that the answer is still ‘no’ for particular classes of A and τ m .