On the computation of singular controls

We consider singular optimal control problems consisting of a state equation
\dot{x}=Ax+bu
for vectors
x
and scalars
u
and a cost functional
J = \frac{1}{2} \int\min{0}\max{T}(x'Qx+\epsilon^{2}u^{2})dt
to be minimized for
|u|\leq m
and
\epsilon=0
. By considering the problem as
\epsilon \rightarrow 0
, singular perturbation concepts can be used to compute solutions consisting of bang-bang controls followed by singular arcs. The procedure further develops a numerical technique proposed by Jacobson, Gershwin, and Lele [18], as well as additional analytic methods developed by other authors.