Title: On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems,
Abstract: We revisit the linear programming approach to deterministic, continuous time,
infinite horizon discounted optimal control problems. In the first part, we
relax the original problem to an infinite-dimensional linear program over a
measure space and prove equivalence of the two formulations under mild
assumptions, significantly weaker than those found in the literature until now.
The proof is based on duality theory and mollification techniques for
constructing approximate smooth subsolutions to the associated
Hamilton-Jacobi-Bellman equation. In the second part, we assume polynomial data
and use Lasserre's hierarchy of primal-dual moment-sum-of-squares semidefinite
relaxations to approximate the value function and design an approximate optimal
feedback controller. We conclude with an illustrative example.