  We present a methodology for bounding the error term of an asymptotic
solution to a singularly perturbed optimal control (SPOC) problem whose exact
solution is known to be computationally intractable. In previous works, reduced
or computationally tractable problems that are no longer dependent on the
singular perturbation parameter $\epsilon$, where $\epsilon$ represents a
small, non-negative number, have provided asymptotic error bounds of the form
$O(\epsilon)$. Specifically, the optimal solution $\bar{V}$ of the reduced
problem has been shown to be asymptotically equivalent in $\epsilon$ to the
optimal solution $V(\epsilon)$ of the singularly perturbed problem in the sense
that $|V(\epsilon)-\bar{V}| =O(\epsilon)$ as $\epsilon \rightarrow 0$. In this
paper, we improve on this result by incorporating a duality theory into the
SPOC problem and derive an upper bound $\chi_u(\epsilon)$ and lower bound
$\chi_l(\epsilon)$ of $V(\epsilon)$ that hold for arbitrary $\epsilon$ and,
furthermore, satisfy the inequality $|\chi_u(\epsilon)-\chi_l(\epsilon)| \leq C
\epsilon$ for small $\epsilon$, with the constant $C$ determined. We carry out
numerical experiments to illustrate the computational savings obtained for the
upper and lower bound. In particular, we generate a set of 50 random SPOC
problems of a specific form and show that for $\epsilon$ smaller than
$10^{-2}$, it becomes faster, on average, to solve for the bounds rather than
the SPOC problem and for $\epsilon=10^{-5}$, the computational time for the
upper and lower bounds is approximately 20 times faster, on average, than that
of the SPOC problem.
