Singular Perturbations of Two-Point Boundary Value Problems Arising in Optimal Control

This paper considers a two-point boundary value problem which arises from an application of the Pontryagin maximal principle to some underlying optimal control problem. The system depends singularly upon a small parameter, $\varepsilon $. It is assumed that there exists a continuous solution of the system when $\varepsilon = 0$, known as the reduced solution. Conditions are given under which there exists an “outer solution”, and “left and right boundary-layer solutions” whose sum constitutes a solution of the system which degenerates uniformly on compact sets to the reduced solution. The principal tool used in the proof is a Banach space implicit function theorem.