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We solve a class of control problems with fuel constraint by means of thelog-Laplace transforms of $J$-functionals of Dawson-Watanabe superprocesses.This solution is related to the superprocess solution of quasilinear parabolicPDEs with singular terminal condition. For the probabilistic verificationproof, we develop sharp bounds on the blow-up behavior of log-Laplacefunctionals of $J$-functionals, which might be of independent interest.

A control problem with fuel constraint and Dawson-Watanabe superprocesses