Attainability of the minimal exponents in the class of infinitesimal perturbations (Q5960226)

For given \(n\in \mathbb{N}\), the set \(\mathcal M_n\) of systems of the type \[ \dot x = A(t)x,\qquad x\in \mathbb{R}^n,\tag{1} \] is considered, where \(A: \mathbb{R}^+\to \text{End} \mathbb{R}^n\) is bounded and piece-wise continuous on \(\mathbb{R}^+ = [0;\infty)\). The Lyapunov exponents \(\lambda_1\leq\dots\leq\lambda_n\) to system (1) are assumed to be functionals on the set \(\mathcal M_n\). It is proved that the lower mobility boundary of one of the intermediate Lyapunov exponents to system (1) is attainable in the class of perturbations tending to zero at infinity.