A Singular Singularly-Perturbed Linear Boundary Value Problem

We consider the asymptotic solution of boundary value problems for the vector system \[\begin{gathered} \dot x = A(t,\varepsilon )x + B(t,\varepsilon )y + C(t,\varepsilon ), \hfill \\ \varepsilon \dot y = E(t,\varepsilon )x + F(t,\varepsilon )y + G(t,\varepsilon ) \hfill \\ \end{gathered} \] as $\varepsilon \to 0$ under the assumption that the matrix $F(t,0)$ is singular. A full set of asymptotic solutions is constructed assuming that $F(t,0)$ can be block-diagonalized, the reduced problem is consistent, and a new stability condition holds. Boundary value problems are then solvable if an appropriate “boundary” matrix is nonsingular for $\varepsilon \ne 0$. Such problems arise in optimal control theory, among other applications.