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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

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
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