Linear Programming Approach to Optimal Control Problems with Unbounded State Constraint

This paper is devoted to a study of infinite horizon optimal control problems with time discounting and time averaging criteria in continuous time. It is known that these problems are related to certain infinite-dimensional linear programming problems, but to facilitate the analysis of these LP problems, it is usually assumed that all admissible trajectories remain in a compact set. In the recent paper (Shvartsman in Discrete Contin Dyn Syst Series B 29(1):110–123, 2024), a problem without the latter assumption was considered, and Alexandroff compactification was used to carry out the analysis. In this paper, we carry over and further extend the compactification approach to problems in continuous time and show applications of the obtained results to estimating Abel and Cesàro limits of the optimal value functions.