Continuity of the Value Function for Deterministic Optimal Impulse Control with Terminal State Constraint

TL;DR

This paper investigates the continuity of the value function in deterministic optimal impulse control problems with terminal state constraints, introducing conditions for continuity and deriving a related Hamilton-Jacobi-Bellman QVI.

Contribution

It presents an intrinsic condition ensuring the value function's continuity and derives a QVI for the problem using Bellman dynamic programming.

Findings

Identifies conditions for value function continuity

Derives a Hamilton-Jacobi-Bellman type QVI

Discusses the uniqueness of the viscosity solution

Abstract

Deterministic optimal impulse control problem with terminal state constraint is considered. Due to the appearance of the terminal state constraint, the value function might be discontinuous in general. The main contribution of this paper is the introduction of an intrinsic condition under which the value function is continuous. Then by a Bellman dynamic programming method, the corresponding Hamilton-Jacobi-Bellman type quasi-variational inequality (QVI, for short) is derived for which the value function is a viscosity solution. The issue of whether the value function is characterized as the unique viscosity solution to this QVI is carefully addressed and the answer is left open challengingly.