Convergence of singular perturbations in singular linear systems

Abstract

A convergence criterion for singular perturbations in linear systems is established. The criterion is useful for regular and singular systems. It is assumed that the matrices of the system depend analytically on ε. It is proved that if the real parts of the divergent eigenvalues tend to −∞, the solution of the system converges to the solution for ε = 0 uniformly in compact subsets of ]0, +∞[. On the other hand, if there is a divergent eigenvalue whose real part does not tend to −∞, then there are initial values such that the solution does not converge at some points. The criterion is valid both for the homogeneous equation and for the nonhomogeneous equation if functions are sufficiently smooth. The use of distributions allows the analysis with inconsistent initial conditions.