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Stability of linear neutral delay-differential systems. (English) Zbl 0669.34074
Suppose that A, B and C are real $$n\times n$$ matrices and $$\mu (A)=\lim_{h\to 0+}h^{-1}[\| I-hA\| -1].$$ The paper deals with the system $$(i)\quad \dot x(t)=Ax(t)+Bx(t-\tau)+C\dot x(t-\tau),$$ where $$\tau >0$$. Suppose that $$\| C\| <1$$ and $$\mu (A)+(\| B\| +\| A\| \| C\|)(1-\| C\|)^{-1}<0.$$ Then there exist $$M\geq 1$$, $$\alpha >0$$ such that $$\| x(t,\Phi)\| +\| \dot x(t,\Phi)\| \leq 2M\cdot \| \Phi \| \cdot e^{-\alpha t},$$ $$t\in [0,+\infty)$$, where x($$\cdot,\Phi)$$ is a solution of (i) with $$x(t)=\Phi(t),$$ $$\dot x(t)={\dot \Phi}(t)$$ on [-$$\tau$$,0]. A method of differential inequalities is used.
Reviewer: Z.Kamont

##### MSC:
 34K20 Stability theory of functional-differential equations 34A40 Differential inequalities involving functions of a single real variable 34D05 Asymptotic properties of solutions to ordinary differential equations
##### Keywords:
method of differential inequalities
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##### References:
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