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Criteria for asymptotic stability of linear time-delay systems. (English) Zbl 0557.93058
The paper deals with stability analysis of the time-delay system (1) $$\dot x=Ax(t)+Bx(t$$-$$\tau)$$, where $$A,B\in R^{n\times n}$$, $$x(t)\in R^ n$$, $$\tau >0$$. Let $$\mu$$ (X) be the matrix measure for $$X\in C^{n\times n}$$ derived from some matrix norm $$\| X\|$$ and let $$\lambda_ i(X)$$ be the eigenvalues of X. The main result of the article is the following theorem: The system (1) is asymptotically stable if there holds $\mu (A)+\max_{y\in \Delta}\mu (Be^{-\tau yj})<0\quad for\quad \max_{y\in \Delta}\mu (Be^{-\tau yj})\geq -1/\tau$ $1+\tau \max_{y\in \Delta}\mu (Be^{-\tau yj})e^{1-\tau \mu (A)}<0\quad for\quad \max_{y\in \Delta}\mu (Be^{-\tau yj})<-1/\tau.$ Here $$\Delta$$ denotes the range of the values taken by the solution y of $$y=Im$$ $$\lambda_ i(A+Be^{-\tau yj}.e^{-Re\quad \tau s})$$, Re $$s\geq 0$$, $$s\in C$$, $$\tau >0$$, $$j^ 2=-1$$.
Reviewer: G.Leonov

##### MSC:
 93D20 Asymptotic stability in control theory 34K20 Stability theory of functional-differential equations 93C05 Linear systems in control theory 93C99 Model systems in control theory
##### Keywords:
stability analysis; time-delay system
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