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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
34K20 Stability theory of functional-differential equations
34D20 Stability of solutions to ordinary differential equations