Diagonalisation method in the asymptotical theory for solutions of systems of ODE

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.