On a property of the reducibility exponent of linear differential systems

We establish that the reducibility exponent (Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system

$$dot x = A(t)x, x in mathbb{R}^n , t geqslant 0$$

, with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1_A+Q) with an arbitrary piecewise continuous perturbation Q satisfying the condition (overline {lim } _{t to + infty } t^{ - 1} ln left| {Q(t)} right| leqslant - sigma ) is reducible to the original system (1 _A ) by some Lyapunov transformation.