It is shown that the Fedorov procedure for finding $L$-optimal designs always converges to an $L$-optimal design, even when the limiting design is singular. Two lemmas are given first. An example illustrates the case of singular limiting design.
"A Convergence Theorem in $L$-Optimal Design Theory." Ann. Statist. 5 (4) 790 - 794, July, 1977. https://doi.org/10.1214/aos/1176343902