Quasi-regular asymptotic behavior of the solution of a singularly perturbed Cauchy problem for linear systems of differential matrix equations

The author constructs an asymptotic expansion in powers of epsilon of the solution on a finite interval of t of the singularly perturbed matrix Cauchy problem epsilon dot{Z}=A(t)Z+ZB(t),quad Z(0,epsilon)=Z_0, where A(t) and B(t) are real square matrices depending smoothly on t. It is assumed that the eigenvalues of the matrices are simple and have non-positive real parts. A norm estimate of the error term is given. An expansion is also given for the solution of the non-homogeneous Cauchy problem.