Linear nonautonomous systems of differential equations with a quadratic integral (Q2659259)

The author considers the linear nonautonomous system of differential equations \[\dot x=A(t)x,\quad x\in \mathbb{R}^n,\tag{1}\] admitting the quadratic integral \[F(x,t)=(B(t)x,x)/2\tag{2}\] where the bracket (,) stands for the inner product on \(\mathbb{R}^n\). It is proved that a linear system (1) is stable iff a first integral (2) is a positive definite quadratic form.