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A stability analysis for a class of differential-delay equations having time-varying delay. (English) Zbl 0735.34063
Delay differential equations and dynamical systems, Proc. Conf., Claremont/CA (USA) 1990, Lect. Notes Math. 1475, 225-242 (1991).
[For the entire collection see Zbl 0727.00007.]
Consider the system \(x'(t)=A_ 0x(t)+A_ 1x(t-h)\); let \(H_ s=\{h\in[0,\infty)\), \(\det(sI-A_ 0-A_ 1\exp(-sh))\neq 0\) for all \(s\in\mathbb{C}\) with \(\text{Re} s\geq 0\}\). Assume that the function \(h:[0,\infty)\to H_ s\) is bounded and continuous and moreover there exist a compact subset \(D\in H_ s\) and \(\hat t\geq 0\) such that \(h(t)\in D\) for \(t\geq\hat t\) and \(h\) is differentiable for \(t>\hat t\). By using a suitable Lyapunov functional, it is proved that there exist \(\mu_ 1,\mu_ 2,-1<\mu _ 1<0<\mu_ 2\), such that if \(h\) is as above and \(\mu_ 1<h'(t)<\mu_ 2\) for \(t\geq\hat t\), \(\int^ \infty_{\hat t}| h'(t)| dt<\infty\), then all solutions of the system \(x'(t)=A_ 0x(t)+A_ 1x(t-h(t))\) are bounded and tend to zero as \(t\to\infty\).

34K20 Stability theory of functional-differential equations