Optimality conditions for $\mathbb {L}^p$ problems with reflected dynamics

We look into optimality considerations for control problems consisting in minimizing, for $p\in (1,\infty ]$, the $\mathbb {L}^{p}$ norms of a Borel measurable cost function, in finite time, and over all trajectories associated with a controlled dynamics which is reflected in a compact prox-regular set. This is done via the linear programming formulations of the problem inspired by the considerations in [Goreac et al., Journal of Differential Equations 290 (2021), 78–115]. The optimality conditions are given in terms of support of the optimal measures and the conditions echo the maximizers of Hamiltonians.