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On the investigation of stability of linear systems of large dimension with lag. (Russian) Zbl 0588.34051
Let the large dimension differential-difference system $$\dot x($$t)$$=Ax(t)+Bx(t-\tau)$$, $$x(t)\in {\mathbb{R}}^ N$$, $$\tau =const>0$$, be decomposed as $(1)\quad \dot x_ i(t)=A_ ix_ i(t)+B_ ix_ i(t- \tau)+\sum_{j\neq i}(C_{ij}x_ j(t)+D_{ij}x_ j(t-\tau)),\quad i=1,...,n,$ where $$x_ i(t)\in {\mathbb{R}}^{N_ i}$$, $$N_ 1+...+N_ n=N$$. Under the natural assumption that the systems $$\dot Z_ i(t)=A_ iz_ i(t)+B_ iz_ i(t-\tau)$$ are asymptotically stable the authors give conditions for asymptotic stability of (1) in terms of $$\| C_{ij}\|$$, $$\| D_{ij}\|$$ and $$\tau$$ using the technique of Lyapunov functions.
Reviewer: M.M.Konstantinov
##### MSC:
 34K20 Stability theory of functional-differential equations 34D20 Stability of solutions to ordinary differential equations