Instability in the critical case of a pair of pure imaginary roots for a class of systems with aftereffect (Q2714639)

The system of integro-differential equations NEWLINE\[NEWLINE \frac{dx}{dt} = Ax +\int_0^t K(t-s)x(s) ds + F(x,\widetilde{y},t),\quad x\in \mathbb{R}^n,\quad \widetilde{y}\in \mathbb{R}^m,\tag{1} NEWLINE\]NEWLINE is considered, where \(A\) is a constant \(n\times n\)-matrix, \( K(t)\in C(\mathbb{R}_+,\mathbb{R}^{n\times n}) \) and \( \|K(t)\|\leq C \exp(-\beta t)\), \( C\), \(\beta=\text{const}>0\). It is assumed that the characteristic equation corresponding to (1) has a couple of pure imaginary roots. Instability conditions are established for the solutions to system (1). The sign of the Lyapunov constant is determined in the problem on rotating motion of a solid with visco-elastic supports.