  We consider a class of exit time stochastic control problems for diffusion
processes with discounted criterion, where the controller can utilize a given
amount of resource, called "fuel". In contrast to the vast majority of existing
literature, concerning the "finite fuel" problems, it is assumed that the
intensity of fuel consumption is bounded. We characterize the value function of
the optimization problem as the unique continuous viscosity solution of the
Dirichlet boundary value problem for the correspondent Hamilton-Jacobi-Bellman
(HJB) equation. Our assumptions concern the HJB equations, related to the
problems with infinite fuel and without fuel. Also, we present computer
experiments, for the problems of optimal regulation and optimal tracking of a
simple stochastic system with the stable or unstable equilibrium point.
