We will discuss the characterization of the value function for some
control problems with an integral constraint. Usually, some controllability conditions
are imposed to guarantee the continuity of the value function and its characterization
as unique constrained-viscosity solution to a HJB equation. Here, we will mainly
focus on the case where no controllability assumption is satisfied. In this
case the value function is discontinuous but its epigraph can still be described
by introducing an auxiliary control problem without state constraints.
The theoretical results will be commented on a specific problem of optimal exercise of
swing contracts in energy markets.