## 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.

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.

Original language | English |
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Pages (from-to) | 134-139 |

Journal | IEEE Control Systems Letters |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- Optimal control
- discounted occupation measures
- moments
- sum-of-squares
- infinite linear programming