Singular Perturbation of Autonomous Linear Systems

Abstract

Let $X_\varepsilon (t) = \exp (({{A + B} /\varepsilon })t)$ where A, B are $n \times n$ matrices. It is shown that $X_\varepsilon (t)$ converges pointwise for $t > 0$ as $\varepsilon \to 0^ + $ if and only if Index $B \leqq 1$ and the nonzero eigenvalues of B have negative real part. An explicit representation of the limit of $X_\varepsilon (t)$ is given. These results are applied to the singularly perturbed system $\dot x = A_1 (\varepsilon )x + A_2 (\varepsilon )y$, $\varepsilon \dot y = B_1 (\varepsilon )x + B_2 (\varepsilon )y$. This paper differs from earlier work both in the derivation of necessary and sufficient conditions and in the explicit forms for the limits.