Uniqueness of classical solutions to the exploratory HJB equation

Establish uniqueness, under appropriate conditions, among arbitrary classical solutions of the exploratory Hamilton–Jacobi–Bellman equation on the collision-free configuration space, including a suitable growth condition and boundary data on the polar collision set.

Background

The paper studies the discounted entropy-regularized exploratory control problem for the semi-discrete quadratic Wasserstein energy. It proves that the associated value function is a classical interior solution of the exploratory HJB equation on the collision-free configuration space and identifies the optimal state-dependent temperature feedback.

The collision set is polar for the controlled diffusions in dimensions d≥2, so the HJB analysis imposes no boundary condition there. The authors explicitly restrict their uniqueness result: they establish identification of the value-function solution but do not establish uniqueness among all classical solutions without additional growth conditions and data on the collision set. Determining whether an appropriate comparison or uniqueness theorem holds is therefore left unresolved.