On the computation of singular controls
Joseph Flaherty, Robert E. O’Malley · IEEE Transactions on Automatic Control · 1977
We consider singular optimal control problems consisting of a state equation\dot{x}=Ax+bufor vectorsxand scalarsuand a cost functionalJ = \frac{1}{2} \int\min{0}\max{T}(x'Qx+\epsilon^{2}u^{2})dtto be minimized for|u|\leq mand\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.