Optimal control problems with no turning back

In this paper we consider the maximization of a payoff functional subject to a differential equality contraint over the class of monotonically increasing functions with values in 0, 1]. We will show that an optimal control exists, derive the system of inequalities (similar to a quasi-variational inequality) that the value function satisfies and derive various properties of the value function sufficient to characterize it. Furthermore, we derive a perturbation result using the theory of Lipschitz controls. Finally, we also consider the case when the control functions are of bounded total variation and relate the problems considered herein to the impulse control problem of Bensoussan-Lions.