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The study of singularly perturbed initial value problems for nonlinear systems of ordinary differential equations parallels the analysis underlying the development of numerical algorithms for obtaining solutions to systems of stiff differential equations. This paper seeks to emphasize the advantages of combining these two substantial research efforts. It develops insight and intuition based on a sequence of solvable model problems, and it relates a variety of literature scattered throughout asymptotic and numerical analyses, stability and control theory and specific topics in applied mathematical modeling.

The asymptotic solution to the linear state regulator problem with cheap control is obtained for the situation where the limiting solution is a singular arc of first order and the initial impulse is that of a delta function. The presentation simplifies previous studies and allows generalization to further cases.

The asymptotic solution of a singularly perturbed linear state regulator problem with quadratic cost has been considered by Yackel and Wilde, O’Malley, and Kung, among others. Hamilton–Jacobi theory converts the problem to a singularly perturbed two-point boundary value problem. This problem can be solved directly or by solving a terminal value problem for the Riccati gains and, then, an initial value problem for the states. For a $2 \times 2$ time-invariant system, it is shown that both methods require the same hypotheses.

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.

This paper obtains asymptotic expansions for initial value problems of the form \[ \begin{gathered} (1)\qquad \dot x(t) = f(t,x(t), x(t - \mu ),\dot x(t - \mu )),\quad t \geqq 0, \hfill \\ (2)\qquad x(t) = \phi (t), \quad - \mu \leqq t \leqq 0 , \hfill \\ \end{gathered} \] as the positive delay parameter $\mu $ tends to zero. The critical hypotheses are: (i) that the reduced problem ((1)–(2) with $\mu = 0$) has a unique solution $X_0 (t)$, and (ii) that $| {f_u (t,x,y,u)} | < a < 1$ everywhere. Then $x(t)$ will converge to $X_0 (t)$ as $\mu \to 0$ and boundary layer behavior will occur in higher order approximations near $t = 0$.