A control problem with fuel constraint and Dawson-Watanabe superprocesses

TL;DR

This paper addresses control problems with fuel constraints using Dawson-Watanabe superprocesses, linking probabilistic methods to PDE solutions and providing bounds on functional blow-up behavior.

Contribution

It introduces a novel approach to solve control problems with fuel constraints via log-Laplace transforms of superprocess functionals, connecting stochastic processes with PDEs.

Findings

Solution method using log-Laplace transforms of superprocess functionals

Connection between superprocess solutions and quasilinear PDEs with singular terminal conditions

Development of bounds on blow-up behavior of functionals

Abstract

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.