On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems
This work provides a theoretical foundation for using linear programming in optimal control, but the results are incremental, extending existing theory with weaker assumptions.
The paper revisits the linear programming approach for deterministic infinite horizon discounted optimal control, proving equivalence between the original problem and an infinite-dimensional linear program under weaker assumptions than previously known. It also applies Lasserre's hierarchy to approximate the value function and design an approximate optimal feedback controller for polynomial data.
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.