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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 $$t\to\infty$$.

##### MSC:
 34K20 Stability theory of functional-differential equations
##### Keywords:
delay equations; stability; Lyapunov functional