We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson–Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.
"A control problem with fuel constraint and Dawson–Watanabe superprocesses." Ann. Appl. Probab. 23 (6) 2472 - 2499, December 2013. https://doi.org/10.1214/12-AAP908