Regular finite fuel stochastic control problems with exit time

TL;DR

This paper studies finite fuel stochastic control problems with bounded fuel consumption, characterizing the value function via viscosity solutions of HJB equations and providing computer experiments for optimal regulation and tracking.

Contribution

It introduces a novel bounded fuel consumption model for exit time stochastic control problems and characterizes the value function as a unique viscosity solution of the associated HJB equation.

Findings

Value function characterized as unique viscosity solution

Bounded fuel consumption impacts optimal control strategies

Computer experiments demonstrate practical applications

Abstract

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.