On quadratic Liapunov functionals for linear difference equations
By means of a quadratic Lyapunov functional defined in an appropriate \(L_2\) space, the author studies some stability properties of vector linear difference equations of the form \[ x(t)= \sum^k_{j=1} A_j x(t- r_j), \qquad 0< r_1< r_2< \dots< r_k< \infty, \] where \(A_j\), \(j=1, 2, \dots, k\) are given \(N\times N\) real matrices. In particular, the author gives a criterion which determines stability with respect to the delays \(r_j\). For related results see the papers of \textit{J. Hurt} [SIAM J. Numer. Anal. 4, 582-596 (1967; Zbl 0264.65076)] and \textit{W. R. Melvin} [J. Math. Anal. Appl. 48, 749-763 (1974; Zbl 0311.39002)].
scientific article; zbMATH DE number 894234
scientific article