Singular Perturbation of Autonomous Linear Systems

Let X_ε (t) = exp ((A + B /ε )t) where A, B are n × n matrices. It is shown that X_ε (t) converges pointwise for $t > 0$ as ε → 0^ + if and only if Index B 1 and the nonzero eigenvalues of B have negative real part. An explicit representation of the limit of X_ε (t) is given. These results are applied to the singularly perturbed system x = A₁ (ε )x + A₂ (ε )y, ε y = B₁ (ε )x + B₂ (ε )y. This paper differs from earlier work both in the derivation of necessary and sufficient conditions and in the explicit forms for the limits.