  The question about asymptotical behaviour of solutions for the system $\dot
x=A_\nu x+f$ for big values of the parameter $\nu\in\frak A$ is considered. An
approach to the reduction of a large class of problems to easily solvable
problem with diagonal coefficient matrix is described. This approach is based
on the approximation of summable matrix function $A_\nu$ by specially
constructed smooth matrix functions.
