On the stability of hybrid difference-differential systems
Hybrid difference-differential systems are considered to include some class of neutral differential-difference equations after a suitable transformation. It is shown that the Cauchy problem for a class of hybrid systems has a unique solution. For systems with constant coefficients, \[ x_1'(t) = A_{11}x_1(t) + A_{12}x_2(t),\quad x_2(t) = A_{21}x_1(t) + A_{22}(t-h), \] it is proven that the system is asymptotically stable if and only if \(A_{22}\) is a Schur matrix and all characteristic roots of the system have negative real parts. Further, when the systen degenerates to a scalar equation, i.e. the matrices \(A_{ij}\) become numbers \(a_{ij}\), the above condition for asymptotic stability can be realised in terms of these coefficients.
scientific article; zbMATH DE number 5611080
scientific article