Linear conic optimization for nonlinear optimal control

TL;DR

This paper introduces a linear conic framework for nonlinear optimal control problems, enabling the use of convex optimization techniques to approximate solutions of complex control tasks.

Contribution

It develops a novel infinite-dimensional linear conic formulation for nonlinear optimal control, linking primal occupation measures with dual value function bounds.

Findings

Establishes approximation results connecting original problems with conic relaxations.

Demonstrates relevance of semidefinite programming relaxations for numerical solutions.

Provides simple examples illustrating the theoretical framework.

Abstract

Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem. Various approximation results relating the original optimal control problem and its linear conic formulations are developed. As illustrated by a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite programming relaxations used to approximate numerically the solutions of the infinite-dimensional linear conic problems.