On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems

TL;DR

This paper explores a relaxed linear programming framework for infinite horizon optimal control problems, establishing equivalence with traditional formulations and employing moment-sum-of-squares relaxations for approximation and control design.

Contribution

It introduces a weaker set of assumptions for the LP formulation and applies polynomial data techniques to approximate solutions and synthesize controllers.

Findings

Proved equivalence of LP and control problem under mild assumptions.

Demonstrated the use of moment relaxations for approximating the value function.

Provided an illustrative example of the approach.

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.