We revisit the linear programming approach to deterministic, continuous time,infinite horizon discounted optimal control problems. In the first part, werelax the original problem to an infinite-dimensional linear program over ameasure space and prove equivalence of the two formulations under mildassumptions, significantly weaker than those found in the literature until now.The proof is based on duality theory and mollification techniques forconstructing approximate smooth subsolutions to the associatedHamilton-Jacobi-Bellman equation. In the second part, we assume polynomial dataand use Lasserre's hierarchy of primal-dual moment-sum-of-squares semidefiniterelaxations to approximate the value function and design an approximate optimalfeedback controller. We conclude with an illustrative example.
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