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Fundamental solution and asymptotic stability of linear delay differential equations. (English) Zbl 1099.34067

Given the linear delay system \[ \dot x_k(t) = -\sum_{\ell=0}^m \sum_{j=1}^n a_{kj}^{(\ell)} x_j\big(t-\tau_{kj}^{(\ell)}\big)\;, \quad k=1,\dots,n,\quad t\geq 0\;,\tag{1} \] with \(a_{kj}^{(0)}\), \(a_{kj}^{(\ell)}\in\mathbb R\), \(\tau_{kj}^{(0)}=0\), \(\tau_{kj}^{(\ell)}\geq 0\) for \(k,j=1,\dots,n\); \(\ell=1,\dots,m\). By considering the associated scalar linear delay system \[ \dot y_k(t) = -\sum_{\ell=0}^m a_{kk}^{(\ell)} y_k \big(t-\sigma_k^{(\ell)}\big)\;,\quad k=1,\dots,n,\quad t\geq 0\;,\tag{2} \] sufficient conditions for the asymptotic stability of system (1) are obtained. It turns out that it depends on the behavior of the fundamental solution of (2).

MSC:

34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
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