On the properties of Bogdanov perturbations of linear differential systems (Q2461871)

The author considers linear differential equations in \(\mathbb{R}^n\) of the form \[ \dot x= A(t)x,\quad x\in\mathbb{R}^n,\quad t\geq 0\tag{1} \] and \[ \dot y= A(t)y+ Q(t)y,\quad y\in\mathbb{R}^n,\quad t\geq 0,\tag{2} \] where \(Q(t)\) is a Bogdanov perturbation, i.e. the integral \[ R(t)= -\int^\infty_0 Q(s)\,ds \] exists and is finite for each \(t\geq 0\). Sufficient conditions for the asymptotically equivalence between (1) and some system of the form (2) with piecewise continuous perturbation \(\widetilde Q\) on \([0,\infty)\) are derived.