On upward mobility of the highest exponent of a linear differential system under perturbations of the coefficients from the simplest classes with degeneracies

We obtain some upper bound for the highest exponent of a linear differential system with perturbation matrix $Q$, satisfying the condition $\|Q (t)\|\le N_Q\beta(t)$, $t\ge0$, where $N_Q>0$ is some constant depending on $Q$, and $\beta$ is an arbitrary fixed nonnegative piecewise continuous bounded function that is infinitesimal in the mean on the positive semiaxis.